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UBC Theses and Dissertations

Visco-plastically lubricated multi-layer flows with application to transport in pipelines Sarmadi, Parisa 2019

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Visco-Plastically Lubricated Multi-Layer Flows withApplication to Transport in PipelinesbyParisa SarmadiB.Sc., University of Tehran, 2012M.Sc., University of Tehran, 2014A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSTDOCTORALSTUDIES(Mechanical Engineering)The University of British Columbia(Vancouver)December 2019c© Parisa Sarmadi, 2019The following individuals certify that they have read, and recommend to the Fac-ulty of Graduate and Postdoctoral Studies for acceptance, the thesis entitled:Visco-Plastically Lubricated Multi-Layer Flows with Application to Trans-port in Pipelinessubmitted by Parisa Sarmadi in partial fulfillment of the requirements for the de-gree of Doctor of Philosophy in Mechanical Engineering.Examining Committee:Ian A. Frigaard, Mechanical Engineering, MathematicsSupervisorSarah Hormozi, Mechanical Engineering, Ohio UniversityCo-SupervisorDana Grecov, Mechanical EngineeringSupervisory Committee MemberGregory A. Lawrence, Civil EngineeringUniversity ExaminerNeil Balmforth, MathematicsUniversity ExaminerGijs Ooms, TU DelftExternal ExaminerAdditional Supervisory Committee Members:Gwynn Elfring, Mechanical EngineeringSupervisory Committee MemberSavvas G. Hatzikiriakos, Chemical and Biological EngineeringSupervisory Committee MemberiiAbstractThe thesis presents a novel triple-layer core-annular flow method in which we pur-posefully position an unyielded skin of a visco-plastic fluid between the core andthe lubricating fluid to eliminate the possibility of interfacial instabilities. Specifi-cally, the skin layer is shaped which allows for lubrication force to develop as thecore rises under the action of buoyancy forces. The motivation originally stemsfrom lubricated transport of heavy viscous oils. The objective is to reduce thefrictional pressure gradient while avoiding interfacial instabilities.For this aim, first, we study this methodology for a steady periodic length ofestablished flow, to establish the feasibility for the pipelining application. Second,we address the equally important issue of how in practice to develop a triple-layerflow with a sculpted visco-plastic skin, all within a concentric manifold by con-trol of the flow rates of the individual fluids. The axisymmetric simulation estab-lishes that these flows may be stably established in a controlled way. We developa long-wavelength analysis of the extensional flow to predict the minimal yieldstress required to maintain the skin rigid. Third, we extend the feasibility of themethod to large pipes and higher flow rates by considering the effects of inertiaand turbulence in the lubricating layer. We show that the method can generateenough lubrication force for wide range of parameters if the proper wave shapeis imposed on the unyielded skin. Then, three-dimensional computations are per-formed to capture the buoyant motion of the core to reach its equilibrium position.The study shows that development lengths (times) for the core to attain equilibriumare relatively long, meaning extensive computation. We also present a simplifiedanalytical model using the lubrication approximation and equations of motion forthe lubricant and skin layers, to quickly estimate motion to the balanced configura-iiition for a given shape and initial conditions. Finally, we show an explicit advantageof the proposed method in producing stable core-annular flows in regimes whereconventional core-annular flows are unsuitable. In summary, we establish the po-tential of this new method for the stable and efficient transport of highly viscousfluids along pipelines.ivLay SummaryMulti-layer flows have broad applications in industry, such as heavy oil pipelining,co-extrusion, and many coating flows. This work is motivated mainly by a lubri-cated pipeline flow method. The oil industry has shifted toward heavier oils over20 years ago as light oil is progressively consumed. One of the important techni-cal challenges concerns a cost-effective transportation. In the lubricated pipelineflow, a low viscous fluid is used to lubricate the pipeline via a core-annular flow.This method suffers from interfacial instabilities which means that the oil and wa-ter may mix, making it harder to separate downstream. A layer of yield stress fluidis purposefully positioned between the oil and lubricant to stabilize the flow. Theidea is to maintain yield stress layer completely unyielded. In this thesis, we inves-tigate this triple-layer core-annular flow theoretically and computationally to studyits feasibility and gain insights into how it can be used industrially.vPrefaceThe research presented in the current thesis is conducted by the author, ParisaSarmadi, under the supervision of Prof. Ian A. Frigaard and co-supervision ofProf. Sarah Hormozi. The following papers are published or submitted for pub-lication:• P. Sarmadi, S. Hormozi, and I. A. Frigaard. Triple-layer configuration for stablehigh-speed lubricated pipeline transport. Phys. Rev. Fluids, 2:044302, 2017[157].This paper was co-authored with I. A. Frigaard and S. Hormozi. I was thelead investigator, responsible for major areas of concept formation, data col-lection and analysis. I. A. Frigaard and S. Hormozi supervised the researchand contributed to manuscript edits.• P. Sarmadi, S. Hormozi, and I. A. Frigaard. Flow development and inter-face sculpting in stable lubricated pipeline transport. J. Non-Newton. FluidMech., 261:6080, 2018 [158].This paper was co-authored with I. A. Frigaard and S. Hormozi. I was thelead investigator, responsible for major areas of concept formation, imple-mentation of the code, running, data compilation, and analysis. I. A. Frigaardand S. Hormozi supervised the research and contributed to manuscript edits.• P. Sarmadi and I. A. Frigaard. Stable core-annular horizontal flows in inacces-sible domains via a triple-layer configuration. Chem. Eng. Sci.:X, 3:100028,2019 [155].This paper was co-authored with I. A. Frigaard. I was the lead investigator,viresponsible for major areas of concept formation, data collection and anal-ysis. I. A. Frigaard supervised the research and contributed to manuscriptedits.• P. Sarmadi and I. A. Frigaard. Inertial effects in triple-layer core-annular pipelineflow. Phys. Fluids., 31 (10):103102, 2019 [155].This paper was co-authored with I. A. Frigaard. I was the lead investigator,responsible for major areas of concept formation, data collection and anal-ysis. I. A. Frigaard supervised the research and contributed to manuscriptedits.• P. Sarmadi, O. Mierka, S. Turek, S. Hormozi, and I. A. Frigaard. Three di-mensional simulation of flow development of triple-layer lubricated pipelinetransport. J. Non-Newton. Fluid Mech., 274:104201, 2019 [159].This paper was co-authored with O. Mierka, S. Turek, S. Hormozi, and I. A.Frigaard. I was the lead investigator, responsible for major areas of con-cept formation, implementation of the code, running, data compilation, andanalysis. O. Mierka and S. Turek developed the FEM-CFD package, super-vised the implementation of the code, and contributed to manuscript edits.I. A. Frigaard and S. Hormozi supervised the research and contributed tomanuscript edits.viiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xivNomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiiiList of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxviAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxviiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxix1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Industrial motivation . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 Lubricated pipeline flow . . . . . . . . . . . . . . . . . . 21.2 Limitations of lubricated pipeline flow method . . . . . . . . . . . 41.2.1 Objectives of the thesis . . . . . . . . . . . . . . . . . . . 51.3 Yield stress fluid . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4 Multiphase flows . . . . . . . . . . . . . . . . . . . . . . . . . . 8viii1.4.1 Multi-fluid flows . . . . . . . . . . . . . . . . . . . . . . 91.5 Multi-layer viscous shear flow instability . . . . . . . . . . . . . . 101.5.1 Visco-plastic fluids and shear flow stability . . . . . . . . 111.5.2 Visco-plastically lubricated flow (VPL) . . . . . . . . . . 121.6 Underlying analytical methods used in the thesis . . . . . . . . . . 131.6.1 Lubrication problems . . . . . . . . . . . . . . . . . . . . 131.6.2 Integral method . . . . . . . . . . . . . . . . . . . . . . . 131.7 Overview of computational methods to resolve fluid interfaces . . 141.7.1 Volume of fluid method (VOF) . . . . . . . . . . . . . . . 151.7.2 Level-set method . . . . . . . . . . . . . . . . . . . . . . 151.7.3 Phase-field method . . . . . . . . . . . . . . . . . . . . . 161.7.4 Interface tracking . . . . . . . . . . . . . . . . . . . . . . 161.8 Computational methods relevant to visco-plastic fluids . . . . . . 171.8.1 Regularization method . . . . . . . . . . . . . . . . . . . 171.8.2 Variational principles and numerical methods . . . . . . . 181.9 The triple-layer flow concept . . . . . . . . . . . . . . . . . . . . 211.9.1 Core levitation . . . . . . . . . . . . . . . . . . . . . . . 221.9.2 Visco-plastic sculpting . . . . . . . . . . . . . . . . . . . 221.10 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 Triple-Layer Visco-Plastically Lubricated Pipe Flow . . . . . . . . . 262.1 Flow description . . . . . . . . . . . . . . . . . . . . . . . . . . 282.1.1 Concentric core-annular flow . . . . . . . . . . . . . . . . 302.2 Eccentric core-annular flow . . . . . . . . . . . . . . . . . . . . . 332.2.1 Lubricating layer . . . . . . . . . . . . . . . . . . . . . . 362.2.2 Solution method . . . . . . . . . . . . . . . . . . . . . . 392.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402.3.1 Example flow . . . . . . . . . . . . . . . . . . . . . . . . 422.3.2 Parametric variations . . . . . . . . . . . . . . . . . . . . 432.3.3 Balance position . . . . . . . . . . . . . . . . . . . . . . 502.3.4 Estimating the minimal yield stress in the skin . . . . . . 542.4 Discussion and conclusion . . . . . . . . . . . . . . . . . . . . . 56ix3 Flow Development and Interface Sculpting in Stable Lubricated PipelineTransport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593.1 Modelling flow development . . . . . . . . . . . . . . . . . . . . 603.1.1 Skin sculpting . . . . . . . . . . . . . . . . . . . . . . . 633.1.2 Computational method . . . . . . . . . . . . . . . . . . . 683.2 Computational results . . . . . . . . . . . . . . . . . . . . . . . . 723.2.1 Rigid skin . . . . . . . . . . . . . . . . . . . . . . . . . . 723.2.2 Shorter wavelengths . . . . . . . . . . . . . . . . . . . . 783.2.3 Yielding of the skin . . . . . . . . . . . . . . . . . . . . . 803.2.4 Varying m and Re . . . . . . . . . . . . . . . . . . . . . . 833.2.5 Manifold design . . . . . . . . . . . . . . . . . . . . . . 843.3 Extensional model . . . . . . . . . . . . . . . . . . . . . . . . . 863.3.1 Scaling the equations . . . . . . . . . . . . . . . . . . . . 863.3.2 Solution method and interface evolution . . . . . . . . . . 913.3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 963.4 Discussion and conclusions . . . . . . . . . . . . . . . . . . . . . 1024 Inertial Effects in Triple-Layer Core-Annular Pipeline Flow . . . . 1084.1 Flow description . . . . . . . . . . . . . . . . . . . . . . . . . . 1114.1.1 Lubricating layer . . . . . . . . . . . . . . . . . . . . . . 1124.2 Inertial-laminar flow . . . . . . . . . . . . . . . . . . . . . . . . 1154.2.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 1174.3 Turbulent flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1214.3.1 Inertial-turbulent flow . . . . . . . . . . . . . . . . . . . 1264.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 1274.4 Redesigning the waveform . . . . . . . . . . . . . . . . . . . . . 1314.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1355 3D Simulation of Triple-Layer Flow Development . . . . . . . . . . 1385.1 Modelling flow development . . . . . . . . . . . . . . . . . . . . 1395.1.1 Computational method . . . . . . . . . . . . . . . . . . . 1425.1.2 Validation and benchmarking . . . . . . . . . . . . . . . 1445.2 Computational results . . . . . . . . . . . . . . . . . . . . . . . . 147x5.2.1 Outflow boundary condition . . . . . . . . . . . . . . . . 1495.2.2 Development of eccentric flows . . . . . . . . . . . . . . 1525.2.3 Parametric study . . . . . . . . . . . . . . . . . . . . . . 1555.2.4 Yielding of the skin . . . . . . . . . . . . . . . . . . . . . 1585.3 Core rise model . . . . . . . . . . . . . . . . . . . . . . . . . . . 1615.3.1 Governing equations . . . . . . . . . . . . . . . . . . . . 1615.3.2 Lubrication approximation and reduced model . . . . . . 1625.3.3 Method of solution . . . . . . . . . . . . . . . . . . . . . 1645.3.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 1655.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1716 Stable Triple-Layer Flow in Inaccessible Domains . . . . . . . . . . 1736.1 Flow description . . . . . . . . . . . . . . . . . . . . . . . . . . 1776.1.1 Solution method . . . . . . . . . . . . . . . . . . . . . . 1806.1.2 Estimating a minimal yield stress . . . . . . . . . . . . . 1806.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1816.2.1 Geometric features and oil hold up ratio . . . . . . . . . . 1836.2.2 Feasibility study . . . . . . . . . . . . . . . . . . . . . . 1856.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1937 Summary and Future Research Direction . . . . . . . . . . . . . . . 1957.1 Contributions of the thesis . . . . . . . . . . . . . . . . . . . . . 1957.1.1 Contributions of the individual chapters . . . . . . . . . . 1957.1.2 Synopsis of contributions . . . . . . . . . . . . . . . . . . 2017.2 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2017.2.1 Rheological characterization . . . . . . . . . . . . . . . . 2027.2.2 Computational limitations . . . . . . . . . . . . . . . . . 2027.2.3 Industrial applicability . . . . . . . . . . . . . . . . . . . 2037.3 Future directions . . . . . . . . . . . . . . . . . . . . . . . . . . 2037.3.1 Experimetal investigation . . . . . . . . . . . . . . . . . . 2037.3.2 Optimization problem . . . . . . . . . . . . . . . . . . . 2047.3.3 Visco-plastically lubricated flows of visco-elstic fluids . . 2047.3.4 Turbulent drag reduction in the lubrication layer . . . . . 206xiBibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208A Some Complementary Information About Software . . . . . . . . . 225A.1 PELICANS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225A.1.1 PLIC advection scheme . . . . . . . . . . . . . . . . . . 226A.2 FEATFLOW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227A.2.1 MPSC . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228B Supporting Materials . . . . . . . . . . . . . . . . . . . . . . . . . . 230B.1 Method of solution used in Chapter 4 . . . . . . . . . . . . . . . . 230B.2 Coefficients of modified Reynolds equation for turbulent flow usedin Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231B.3 Required minimal yield stress . . . . . . . . . . . . . . . . . . . 232xiiList of TablesTable 2.1 Dimensional parameter ranges found in heavy crude oil pipelin-ing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43Table 2.2 Ranges of key dimensionless numbers expected in typical pipelin-ing operations. . . . . . . . . . . . . . . . . . . . . . . . . . . 44Table 4.1 Lubrication layer flow types based on Re[3], Re[3]δ , and δ/ f .Here Re[3]c is a nominal critical value for transition to turbulence. 114Table 5.1 Studied meshes . . . . . . . . . . . . . . . . . . . . . . . . . 145Table 6.1 Flow properties for Fig. 6.1 from [10] and Fig. 6.2 from [161].Note that Jˆo = J1 and Jˆw = J2 on Fig. 6.1. . . . . . . . . . . . . 182Table 6.2 Key dimensionless variables for [10] and [161]. . . . . . . . . 182Table 6.3 Sample cases used for this study. Symbols presented in the tableare shown on Fig. 6.1 and 6.2. . . . . . . . . . . . . . . . . . . 183xiiiList of FiguresFigure 1.1 Cartoons of the flow types in horizontal flow when the oil islighter with permission from [83]. . . . . . . . . . . . . . . . 3Figure 1.2 Cartoons of a flow map in horizontal flow when the oil islighter with permission from [83]. Here, Uo and Uw are theoil and water superficial velocities, respectively. . . . . . . . . 4Figure 1.3 Schematic of the triple-layer core-annular flow within an in-flow manifold and its conceptual zones. . . . . . . . . . . . . 6Figure 1.4 Examples of visco-plastic fluids. . . . . . . . . . . . . . . . . 7Figure 1.5 Example of interface shapes sculpted near the inflow pipe andthen frozen into a shaped solid interface advected downstream,(for parameters and description, see Fig. 15a in [72]. . . . . . 24Figure 2.1 (a) Conceptual zones of triple-layer flow. (b) Flow develop-ment within the inflow manifold. . . . . . . . . . . . . . . . . 27Figure 2.2 Cross section of the pipe with triple-layer configuration. . . . 29Figure 2.3 Schematic of the outer radius variation of skin layer with zˆ,characterized via the mean outer radius rˆ2,0 and streamwisevariation ∆rˆ2Φ(zˆ). . . . . . . . . . . . . . . . . . . . . . . . 30Figure 2.4 Scaled pressure gradient (Gc) for representative m/δ and r1,r2,0 = 0.95. . . . . . . . . . . . . . . . . . . . . . . . . . . . 33Figure 2.5 Intensity of flow rate of lubricating fluid (Q3/(1−r22,0)=Wp/2)for representative m/δ and r1, (with m = 0.0001). . . . . . . . 34Figure 2.6 Variations in the minimal B from (2.8), for representative r1and r2,0, (with m = 0.0001). . . . . . . . . . . . . . . . . . . 34xivFigure 2.7 Axial pressure gradient for the current study (solid line) andfrom Ooms et al. [123]. . . . . . . . . . . . . . . . . . . . . . 40Figure 2.8 Periodic pressure variation along z for the current study (solidline) and from Ooms et al. [125]. . . . . . . . . . . . . . . . . 41Figure 2.9 Sawtooth outer skin profile Φ(z). . . . . . . . . . . . . . . . . 41Figure 2.10 Variations with z at fixed y of (a) lubricant layer thicknessh(y,z) and (b) lubrication pressure Pl(y,z). . . . . . . . . . . . 42Figure 2.11 (a) Layer-averaged azimuthal velocity and (b) layer-averagedaxial velocity, for the same parameters as in Fig. 2.10. . . . . 43Figure 2.12 Normalized pressure gradient (G/Go) for different a and l′. . . 45Figure 2.13 Flow rate of lubricating fluid (Q3) for different a and l′. Insetshows the total flow rate and partial flow rate due to Wp oflubricating layer when l′ = 0.2. . . . . . . . . . . . . . . . . 45Figure 2.14 Computed net lubrication force able to balance Fl for differenta and l′. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46Figure 2.15 Normalized pressure gradient (G/Go) for different a and λ . . 47Figure 2.16 Computed net lubrication force able to balance Fl for differenta and λ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48Figure 2.17 Normalized pressure gradient (G/Go) for different m/δ and r1. 48Figure 2.18 Flow rate of lubricating fluid (Q3) for different m/δ and r1. . . 49Figure 2.19 Computed net lubrication force able to balance Fl for differentm/δ and r1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 49Figure 2.20 Normalized pressure gradient (G/Go) for different r1 and e. . 50Figure 2.21 Computed net lubrication force able to balance Fl for differentr1 and e. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51Figure 2.22 Variation of the eccentricity (e) required to balance Fl , for dif-ferent a and Fl . . . . . . . . . . . . . . . . . . . . . . . . . . 52Figure 2.23 Variation of the minimal layer thickness hmin = 1− e− a forthe balance eccentricity. . . . . . . . . . . . . . . . . . . . . 52Figure 2.24 Normalized pressure gradient (G/Go) in the balance state, fordifferent a and Fl . . . . . . . . . . . . . . . . . . . . . . . . . 53Figure 2.25 Flow rate of lubricating fluid (Q3) in balance state, for differenta and Fl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53xvFigure 2.26 Shear stress at r = r2. . . . . . . . . . . . . . . . . . . . . . . 54Figure 2.27 Variation of |Pl|/(x2− x1). . . . . . . . . . . . . . . . . . . . 55Figure 2.28 Computed minimal dimensionless yield stress Bmin needed tokeep the skin layer completely unyielded in balance state, fordifferent a and Fl . . . . . . . . . . . . . . . . . . . . . . . . . 56Figure 3.1 Schematic of the triple-layer CAF within an inflow manifold,together with supposed velocity development. . . . . . . . . . 61Figure 3.2 Flow rates (Q2,Q3) for different (r1,r2), for m = 0.005: solidshaded line is Q2; broken shaded line is Q3. The black brokenline shows the value of r2 where Q2 is maximal. . . . . . . . . 65Figure 3.3 (a) Desired interface positions along the pipe: r2(z) = solidblack line (here (a,m)= (0.15,0.005)); r1 = broken black line.(b) Q2 (solid black line) and Q3 (broken black line) needed toshape sawtooth skin-lubrication interface of (a). . . . . . . . . 66Figure 3.4 Plug velocity variation of example 1 in Fig. 3.3. . . . . . . . . 66Figure 3.5 (a) Desired interface positions along the pipe: r2(z) = solidblack line (here (a,m)= (0.15,0.005)); r1 = broken black line.(b) Q2 (solid black line) and Q3 (broken black line) needed toshape the dinosaur skin-lubrication interface of (a). . . . . . . 67Figure 3.6 Flow rates for different r2, where (r1,m) = (0.78,0.005). . . 67Figure 3.7 (a) Schematic of the dimensionless triple-layer flow geome-try in an axisymmetric pipe; (b) detail example computationalmesh close to the manifolds. . . . . . . . . . . . . . . . . . . 69Figure 3.8 Effect of mesh size on: (a) L2 norm of difference between exitvelocity w(r,L) and analytical velocity W (r); (b) exit concen-tration profile of skin fluid c[2]; (c) exit velocity profile; (d)detail of the exit velocity in fluid 1 close to the interface. . . . 71Figure 3.9 Concentration colormap for different times. . . . . . . . . . . 73Figure 3.10 Speed colormap for different times. . . . . . . . . . . . . . . 74Figure 3.11 Stress components at: a) t = 1 and b) t = 2. In each panel,from top to bottom we show: the position of the skin layer,then τzz, τrz, τθθ , τrr. . . . . . . . . . . . . . . . . . . . . . . 75xviFigure 3.12 Averaged normal stresses in skin layer τ¯s,i: variation with z fordifferent times,  : τ¯s,r,  : τ¯s,z, F : τ¯s,θ . The narrow panelat the top of each figure shows the skin-lubrication interfaceposition r2(z). . . . . . . . . . . . . . . . . . . . . . . . . . . 76Figure 3.13 Entry average extensional stress (τ¯zz|z=1) variation in time. . . 77Figure 3.14 Axial velocity development in the inflow manifold at t = 4. . . 78Figure 3.15 Results of imposing (a) the sawtooth shape with period T = 2and (b) the dinosaur skin shape with period T = 2: solid line =intended shape and (–) simulation result, after 2 time periods. 79Figure 3.16 Concentration colormap for different times. . . . . . . . . . . 79Figure 3.17 Speed colormap for different t. . . . . . . . . . . . . . . . . . 80Figure 3.18 Averaged normal stresses in skin layer τ¯s,i: variation with zfor different times. The narrow panel at the top of each figureshows the skin-lubrication interface position r2(z). . . . . . . 81Figure 3.19 Concentration colormap for t = 1, 2, 3, 4, 5, respectively. . . 82Figure 3.20 τzz colormap for t = 1, 2, 3, 4, 5, respectively. Grey regionindicates the plug. . . . . . . . . . . . . . . . . . . . . . . . 82Figure 3.21 Spatiotemporal variations in the yielded and unyielded parts ofthe skin. Grey region is plug and the white region is yieldedskin layer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83Figure 3.22 Variation of (a) minimal required Bingham number Bcr to keepthe skin completely unyielded (b) pressure drop along the pipewith viscosity ratio m. . . . . . . . . . . . . . . . . . . . . . 84Figure 3.23 Effect of Re on minimal required Bingham number Bcr to keepthe skin completely unyielded (b) average of stress magnitude(|τ¯|=∣∣∣∫ 10 c[2]rτdr/∫ 10 c[2]rdr∣∣∣, τ = (1/2Στ2i j)1/2) at t = T andRe = 1, 10, 50, 100, 200. . . . . . . . . . . . . . . . . . . . 85Figure 3.24 Concentration colormap at times t = 2, 6; where ξ1 < ξ2. . . . 86Figure 3.25 Speed colormap at t = 2, 6, respectively; where ξ1 < ξ2. . . . 86Figure 3.26 (a) Skin and lubricating flow rates required to generate saw-tooth wave in the skin-lubrication interface when Q1 = 1. (b)Computed values of inflow extensional stress (τ¯zz(0, t)) for dif-ferent B˜. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97xviiFigure 3.27 Spatiotemporal variations in the yielded and unyielded parts ofthe skin for different B˜ number. . . . . . . . . . . . . . . . . 98Figure 3.28 Effect of B˜ on the solution at t = T/2, plotted against z˜ ; (a)skin-lubrication interface r2 and the inset shows the r2 pro-file close to the maximum of the interface, (b) core-skin inter-face r1, (c) skin velocity Wp, and (d) skin-averaged extensionalstress τ¯zz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99Figure 3.29 Temporal change of the solution. Plotted against z˜ when skinyields and B˜ = 1; (a) skin-lubrication interface r2, (b) core-skin interface r1, (c) skin velocity Wp, and (d) skin-averagedextensional stress τ¯zz where the red lines mark |τ¯zz|= B˜ = 1. . 100Figure 3.30 Effect of skin-lubrication interface shape on the solution. Toppanel shows the imposed shape, the middle panel indicates thespatiotemporal variations in the yielded and unyielded partsof the skin (grey region and white region show the plug andyielded skin, respectively.), and the bottom panel shows the in-flow extensional stress τ¯zz(0, t) for yielded and unyielded skin.(a) Dinosaur shape with B˜= 0.65 and B˜cr = 0.776 and (b) sinewave with B˜ = 1.65 and B˜cr = 1.86. . . . . . . . . . . . . . . 102Figure 3.31 Evolution of interfaces in time. . . . . . . . . . . . . . . . . . 103Figure 3.32 (a) Inlet flow rates versus time; (b) Plug velocity; (c) Meanpressure gradient; (d) Lubrication pressure variation in z˜ fordifferent times. . . . . . . . . . . . . . . . . . . . . . . . . . 104Figure 4.1 (a) Shaped skin profile in 3D; (b) cross section of the pipe withtriple-layer configuration, illustrating the coordinates. . . . . . 110Figure 4.2 Lubrication layer thickness variation in z and y. . . . . . . . . 118Figure 4.3 Lubrication pressure colormap for different Re[3]δ . . . . . . . 119Figure 4.4 Layer-averaged axial velocity ( ¯¯w) colormap in z and y for dif-ferent Re[3]δ . . . . . . . . . . . . . . . . . . . . . . . . . . . 120Figure 4.5 Lubrication layer axial velocity profile (w(r,y,z)) variation alongthe pipe for different Re[3]δ . . . . . . . . . . . . . . . . . . . 122xviiiFigure 4.6 Generated lubrication force variation with Re[3]δ , (a) l′ = 0.25and (b) l′ = 0.75 for e = 0.1, 0.2, 0.3; where (a,λ ) = (0.4,1). 123Figure 4.7 Lubrication force Fl variation with a and l′, (a) inertialess lam-inar flow and (b) inertialess turbulent flow with Re[3] = 5000;where (e,λ ) = (0.3,1). . . . . . . . . . . . . . . . . . . . . 127Figure 4.8 Generated lubrication force variation with Re[3] for l′= 0.25, 0.75(a) δ/ f = 1 and (b) δ/ f = 10. . . . . . . . . . . . . . . . . . 128Figure 4.9 Generated lubrication force variation with wave amplitude (a)and wave breaking point (l′) for Re[3] = 20000, 300000 anddifferent δ/ f . . . . . . . . . . . . . . . . . . . . . . . . . . . 129Figure 4.10 Lubrication pressure colormap for different Re[3]. . . . . . . . 130Figure 4.11 Lubrication layer averaged axial velocity profile (w¯(r,y,z)) vari-ation along the pipe for different Re[3]. . . . . . . . . . . . . . 131Figure 4.12 Lubrication layer thickness (h) variation with y and z. . . . . . 133Figure 4.13 (a) Lubrication pressure variation along the pipe at different yand (b) lubrication layer velocity profile at y = 0, 1 at different z.134Figure 4.14 Generated lubrication force (Fl) versus eccentricity e for dif-ferent wave breaking point l′ = 0.25, 0.5, 0.75. . . . . . . . . 135Figure 4.15 Generated lubrication force variation with wave shape (a & l′)for inertial-laminar and inertial-turbulent flow. . . . . . . . . . 136Figure 5.1 Schematic of the dimensionless triple-layer flow geometry. . . 140Figure 5.2 (a) Fully developed velocity profile (W (r,L)), (b) L2 norm ofdifference between exit velocity W (r,L) and analytic velocityW (r), (c) phase property Ψ(r,L), and (d) strain rate for differ-ent mesh sizes. . . . . . . . . . . . . . . . . . . . . . . . . . 146Figure 5.3 (a) Fully developed velocity profile (W (r,L)) for different reg-ularization parameters and analytic solution. (b) Closer look atthe core flow velocity profile. . . . . . . . . . . . . . . . . . . 147Figure 5.4 Strain rate colormaps in logarithmic scale log10(γ˙) and stresscolormaps (τ). . . . . . . . . . . . . . . . . . . . . . . . . . 148xixFigure 5.5 L2 errors for fully developed triple-layer flow computed viaregularisation. We use the solution of ε = 0.00001 as the ref-erence solution and compute the error of other values. . . . . 149Figure 5.6 (a) Phase maps; solid black line shows the border of skin layerusing axisymmetric model computed by PELICANS. (b) Ve-locity colormaps. . . . . . . . . . . . . . . . . . . . . . . . . 150Figure 5.7 Effects of outflow boundary condition on the flow: (a) phasemaps, (b) velocity colormaps, and (c) stress colormaps in dif-ferent directions. . . . . . . . . . . . . . . . . . . . . . . . . 151Figure 5.8 Phase maps at t = 0, 2T, 5T, 10T, 15T and the black lineshows the border of unyielded skin/plug. . . . . . . . . . . . . 152Figure 5.9 Velocity colormap and mesh deformation at different t shownfor a single wavelength. . . . . . . . . . . . . . . . . . . . . 154Figure 5.10 (a) Eccentricity variations in time at z = 10, 20, 30, 40, 50.(b) Detailed of eccentricity variation in time at z = 20. . . . . 155Figure 5.11 (a) Fully developed shaped skin and the lubricant velocity col-ormap, (b) lubrication pressure variation in z at different az-imuthal positions, and (c) azimuthal velocity colormap of lu-brication layer shown over a single wavelength at t = 20T . . . 156Figure 5.12 Variation of eccentricity with t at z = 25 for different densitydifferences. . . . . . . . . . . . . . . . . . . . . . . . . . . . 156Figure 5.13 (a) Variation of eccentricity in z at t = 10T, 15T, 20T, 21T forvery long domain (L = 60), the red solid line shows the pointwhere the eccentricity reaches 90% of its balanced value. (b)Eccentricity variation in t at z = 20 for L = 30, L = 60. . . . . 157Figure 5.14 Variation of eccentricity in t at z = 20 for different wave am-plitude. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158Figure 5.15 (a) Variation of eccentricity in t at z = 20 for different coreReynolds number Re = 1, 0.1, 0.01 and ∆φ = 0.03. (b) Varia-tion of eccentricity in t at z= 20 for different density differences.159Figure 5.16 Strain rate colormap in logarithmic scale log10(γ˙) on yieldedshaped skin at t = 4T . . . . . . . . . . . . . . . . . . . . . . 160xxFigure 5.17 Schematic of the periodic triple-layer eccentric core-annularflow a) cross section and b) side view. . . . . . . . . . . . . . 160Figure 5.18 (a) Calculated eccentricity with core rise model, (b) generatedlubrication force variation in time using core rise model, (c) ec-centricity variation in time using simulations, and (d) variationof lubrication force in time. . . . . . . . . . . . . . . . . . . . 166Figure 5.19 (a) Variation of eccentricity in time calculated by core risemodel for various density differences. (b) Computed balancedeccentricity by 3D computations (e3D) versus calculated ec-centricity by core rise model (eCRM). . . . . . . . . . . . . . . 167Figure 5.20 Effect of initial condition on the solution calculated by corerise model. (a) Variation of eccentricity in time and (b) varia-tion of planar angel (Θ) in time. . . . . . . . . . . . . . . . . 169Figure 5.21 Effect of initial condition on the solution using 3D computa-tions. (a) Variation of eccentricity in time and (b) variation ofplanar angel (Θ) in time. . . . . . . . . . . . . . . . . . . . . 170Figure 6.1 Horizontal flow map reproduced from [10], showing observedflow regimes for oil and water superficial velocities and param-eters as in Table 6.1. . . . . . . . . . . . . . . . . . . . . . . 174Figure 6.2 Horizontal flow map reproduced from [161], showing observedflow regimes for oil and water superficial velocities and param-eters as in Table 6.1. . . . . . . . . . . . . . . . . . . . . . . 175Figure 6.3 a) Shaped skin profile in 3D; b) cross section of the pipe withtriple-layer configuration, illustrating the coordinates. . . . . . 176Figure 6.4 Sawtooth outer skin profile Φ(z). . . . . . . . . . . . . . . . . 183Figure 6.5 Variations of core and mean skin radii (r1,r2,0) with relativelubricant input ratio (εl,R) for the different lubricant-skin andoil superficial velocities of cases I-III. . . . . . . . . . . . . . 184Figure 6.6 (a) Core radius (r1) and (b) mean skin radius (r2,0) variationswith relative lubricant input ratio (εl,R) and oil input ratio (εo). 185xxiFigure 6.7 (a) G0/G, (b) τˆy,min, and (c) balanced eccentricity variationswith relative lubricant input ratio, for different εo where theflow parameters are as in [10]. . . . . . . . . . . . . . . . . . 186Figure 6.8 (a) G0/G, (b) τˆy,min, and (c) balanced eccentricity variationswith relative lubricant input ratio for different εo where theflow parameters are as in [161]. . . . . . . . . . . . . . . . . 188Figure 6.9 (a) G0/G, (b) τˆy,min, and (c) balanced eccentricity variationswith relative lubricant input ratio for different wavelengths wherethe fluids and pipe properties are as in [10]. . . . . . . . . . . 189Figure 6.10 (a) G0/G, (b) τˆy,min, and (c) balanced eccentricity variationswith relative lubricant input ratio for different wave amplitude(a) where the fluids and pipe properties are similar to [10]. . . 190Figure 6.11 Feasible triple-layer flows for the flow parameters of [161],showing variations with Jˆo and Jˆls. (a) r1. (b) r2,0. (c) εl,R,d .(d) G0/G. (e) τˆy,min. (f) Balanced core eccentricity position. . 192Figure 6.12 Feasible triple-layer flows for the flow parameters of [10], show-ing variations with Jˆo and Jˆls. (a) εl,R,d . (b) G0/G. (c) τˆy,min.(d) Balanced core eccentricity position. . . . . . . . . . . . . 194Figure A.1 The interfaces of three-layer flow tracked by PLIC and MUSCLmethods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227Figure A.2 Color function C variation across the domain of fully devel-oped flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227xxiiNomenclatureΩ: fluid domainµˆ: fluid viscosityρˆ: fluid densityτˆy: yield stresszˆ: axial coordinaterˆ: radial coordinateθ : azimuthal coordinaterˆ1: inner radiusrˆ2: outer radiusrˆ3: pipe radius measured from core centerrˆ2,0: mean outer radius∆rˆ: wave amplitudeRˆ: pipe radiusΦ(zˆ): wave shape functiondˆ: eccentricityLˆ: axial length scale (wave length/pipe length)ˆ˙γ: strain rateτˆ: stressQˆ1: core flow rateQˆ2: skin flow rateQˆ3: lubricant flow rateWˆ0: velocity scale (core superficial velocity)Vˆ : volumePˆ: pressurexxiiiPˆ∗G: linear pressure scalePˆ∗l : lubrication pressure scalea: dimensionless wave amplitudel′: wave breaking point lengthz: dimensionless axial coordinatex: scaled radial coordinatey: scaled azimuthal coordinatee: dimensionless eccentricityG: dimensionless constant pressure gradientλ : scaled wave lengthm: viscosity ratioB: Bingham numberWp: dimensionless plug velocityδ : aspect ratio of the thin lubricating layer to circumferential length scaleh: dimensionless lubricating layer thicknessφ : density ratioFl: scaled buoyancy forceRe: Reynolds numberRe[3]: lubricant effective Reynolds numberc: color functionξ1: inner manifold lengthξ2: outer manifold lengthT : time periodTb: breaking timeε: aspect ratio in Chapter 3ε: regularization parameter in Chapter 5tˆ: timet: dimensionless timetˆv: viscous time scaletˆa: advective time scaleB˜: Bingham number scaled with lubrication pressure scaleBcr: critical Bingham numberRecr: transition Reynolds numberxxivf : friction factorκ: von Ka´rma´n constantA: damping constantβ : pressure gradient parameteruw: wall shear velocityΨ: phase fieldεo: oil input ratioJˆo: oil superficial velocityJˆls: lubricant-skin superficial velocityεl,R: relative lubricant input ratioHo: oil hold-up ratioxxvList of AbbreviationsALE arbitrary Lagrangian-EulerianCAF core-annular flowsC-P Couette-PoiseuilleHB Herschel-Bulkley1D one-dimensionalPLIC Piecewise Linear Interface Calculation3D three-dimensional2D two-dimensional2CAF two-layer core-annular flowVPL visco-plastically lubricated flowsVOF Volume of FluidWCAF wavy core-annular flowxxviAcknowledgmentsThis research has been carried out at the University of British Columbia, supportedfinancially by Natural Sciences and Engineering Research Council of Canada. Ireally appreciate this support. I acknowledge MITACS Globalink program for pro-viding me with funding to visit TU Dortmund University. Also awards and schol-arships coming from the Department of Mechanical Engineering of the Universityof British Columbia are really appreciated.First I would like to thank my PhD supervisor, Prof. Ian Frigaard for all of hissupport during this research. Without his support, knowledge, patience, and senseof humor, I would not be able to finish this work. Ian, I have been extremely luckyto have a supervisor who cares so much about his students.I am deeply grateful to Prof. Sarah Hormozi for her supervision, guidance, andadvice. She has always provided me friendly help.I gratefully acknowledge Prof. Stefan Turek and Dr. Otto Mierka for their su-pervisions and guidance for numerical part of project. Three dimensional simula-tions would not be possible without their help and support.Moreover, my sincere thank goes to my advisory committee members, Prof. Sav-vas Hatzikiriakos, Prof. Dana Grecov, and Prof. Gwynn Elfring. Their guidancehelped me in all the time of research and writing of this thesis.Completion of this thesis, along a journey that started many years ago, wouldnot be possible without endless love and support of my parents, Farkhondeh andAbbas, and my brothers, Pooyan and Payman. I am thankful to all of them.Most importantly, my heart-felt thanks to my love and true friend. Vahid youhave always supported me and been by my side throughout this journey. I amforever thankful for having you in my life.xxviiI thank my dearest friends, Zamzam and Haleh who have always supportedme and encouraged me throughout this endeavor. I also would like to express mygratitude to all my labmates in Complex Fluids Lab at UBC for their support andmaking my grad life more enjoyable and fun.Last but certainly not least, I would like to thank my new family for their loveand support.xxviiiDedicationTo my parentsfor their unconditional love,encouragement, andendless supportChapter 1IntroductionMulti-layer flow attracts much attention because of its broad industrial applica-tions. Some of the examples are:• Lubricated transport processes, where a lubricating fluid lies in a layer be-tween the wall of a duct and the transported fluid.• Co-extrusion processes, where a product is made of more than one layersimultaneously.• Film coating processes, where a layer is applied to a fluid substrate.All these processes suffer from interfacial instabilities, specifically the instabilitiesrise because of viscosity mismatch. Consequently, stability analysis of multi-layerflows and providing methods to eliminate the interfacial instabilities are of specialinterest practically. In this thesis, we focus on one such method that uses visco-plastic fluid to freeze the interfaces and stabilize the flow. We study the problem inpipe geometry which has mostly application in lubricated pipeline flow and a briefintroduction is presented on this application.1.1 Industrial motivationThe demand for heavy oil has been largely increasing over the past 20 years, dueto both more abundant reserves and shortages of light conventional oil [152, 164].Over the years, this has led researchers to investigate methods and technologies1to improve upstream heavy oil production [109, 150, 152]. High viscosity, highpour points, and low API gravity (high density) are the characteristics of heavy oil.Some of the shared common characteristics of heavy crude oil and natural bitumenrelevant to pipelining that distinguish them from medium and light crude oils areas follows [107]: (i) density of 935kg/m3 or higher and (ii) absolute viscosity inthe range of 0.1−1000Pa.One of the important technical challenges concerns cost-effective transporta-tion, due to its significantly increased viscosity. The management of frictionalpressure losses and ensuring continued flow (flow assurance) have become im-portant areas of pipeline engineering: both operationally and in design. Severalmethods are in use, or under development, to enhance the heavy crude oil trans-portation, such as emulsion formation, droplet suspension, dilution, use of dragreducing agents, and heating [44]. Unfortunately, most of these methods are ex-pensive and have negative impacts on the environment. This has led to a revival ofinterest in core-annular flows (CAF), originally proposed 50 years ago as a methodof reducing friction [21].1.1.1 Lubricated pipeline flowIn CAF flow, the heavy crude oil phase is in the center of the pipe and low viscousfluid (usually water) flows near the wall surface. Conceptually, the lower viscosityof the lubricating fluid reduces friction.For more than 50 years, there has been an ongoing interest in research onliquid-liquid two-layer core-annular flow (2CAF); see e.g. [16, 17, 82, 83, 91, 92,120, 143–145]. Significant viscosity and density differences between two fluidlayers result in interfacial instabilities at even modest flow rates, which can com-promise transportation effectiveness, see e.g. [5, 6, 73, 74], and/or can result in dis-persed phases that are harder to separate downstream. Joseph, Bannwart, Sotgia,Trallero, and many others have studied oil-water flow configurations for variousflow rates, oils, pipe diameters etc. The aim of many of these studies is to pro-duce maps that delineate where flows of different identifiable flow regimes may befound, in terms of the flow control parameters; see e.g. [2, 4, 9, 10, 25, 54, 123,133, 142, 162, 171].2Figure 1.1: Cartoons of the flow types in horizontal flow when the oil islighter with permission from [83]. (a) Oil in water dispersion, (b) bub-bles, (c) slugs, (d) churns, (e) stratified flow, (f) corkscrew core flow, (g)flying core flow, (h) flying core flow, and (i) water in oil dispersion.Flow regimeMulti-phase flows of liquid-liquid and gas-liquid are important in the oil and gas in-dustry and many studies have been performed on the flow regime characterization,see e.g. [82]. The flow regimes of horizontal two-phase oil-water flow are shownschematically in Figs. 1.1 & 1.2 for different input data, oil superficial velocity andwater superficial velocity [83].Dispersion of oil in water is always observed if there is a large amount of waterand the water velocity is much higher than the oil velocity. Oil bubbles in waterarise from capillary instabilities in the presence of shear. If more oil is added, thedroplets grow in size, limited by the pipe radius, and oil slugs appear in the waterphase. These slugs can merge with even more oil added, and this leads to stratifiedand core-annular flow. For very large amount of oil, the oil fraction becomes closeto one, and the flow regime changes to a continuous oil phase with water droplets.We can see that core-annular flows appear for very limited combinations of oil andwater flow rates and it is highly unstable.3Figure 1.2: Cartoons of a flow map in horizontal flow when the oil is lighterwith permission from [83]. Here, Uo and Uw are the oil and water su-perficial velocities, respectively.1.2 Limitations of lubricated pipeline flow methodSeveral studies have investigated CAF configurations and the stability of suchflows, see e.g. [5, 73, 82, 83, 130, 162]. CAF configurations are found only forrelatively high oil superficial velocities. However, this configuration is only sta-ble for high oil input ratios. Many of the flows have bubbles or dispersed phasesand we see that large regions of the parameter space are simply inaccessible to thecore-annular configuration in practice.The stability analysis of such flows has attracted much attention, e.g. [136]showed the stability of perfect CAF without density mismatched can only be achievedfor small range of parameters. A capillary instability can disturb the interfaceat low oil superficial velocities and by increasing the oil superficial velocity, thecapillary effects are stabilized and CAF configurations can be observed. How-ever, perfect CAF configurations are unstable for higher velocity and lead to wavycore-annular flow (WCAF) configurations due to interfacial stresses. Some stud-ies, e.g. [161], show a significant operating window where CAF configurations areachievable. However, this configuration is bounded by other unfavorable regimes.4On the other hand, perfect CAF configurations (no-wavy interface) are hardly everachievable and can only exist for density matched fluids, e.g. WCAF configura-tions have been found in experimental observations [5]. In practice, transitions be-tween the regimes, unfavourable pressure drop reduction and various other physicalconstraints limit the operational range for core-annular flows to a relatively smallparameter domain. This small functionality range makes this method challeng-ing industrially, i.e. the operating conditions (pump power, pipe geometry, fluidproperties, etc.) must be compatible with the stability of the configurations andinstability of CAF configurations in low oil superficial velocity makes the stoppingand restarting the flow problematic.Density mismatched layers impose another difficulty on CAF method, i.e. buoy-ancy force can move the core radially and if no counterbalancing force is applied,the core will eventually touch the pipe wall. In the case of WCAF, it has beenshown that waves are a necessary component to generate sufficient lubricationforce capable of counterbalancing the buoyancy force, see e.g. [5, 121–123, 125].In [120, 121, 123, 124], the authors investigate the levitation mechanism theoreti-cally and assume the core viscosity is infinitely large leading to negligible interfacedeformation. As a simple representation of a shape variation, they used series ofsawtooth wave shape as input. In reality, the sawtooth wave are unstable due tofinite oil viscosity, see [5], and the lubrication pressure is highest where the gapbetween core and pipe wall is smallest. Therefore, the wave must become steepenwhere the annular is smallest, which reduces the lubrication force. Some literatureshows that viscous lubrication is insufficient and inertia is required to balance thebuoyancy of the core, see [6, 124].1.2.1 Objectives of the thesisThis thesis concerns the establishment of practical multi-layer pipe flow with theaid of a visco-plastic fluid, in a core-annular flow configuration. A triple-layerstructure is considered in which a visco-plastic (yield stress) fluid is inserted be-tween transported and lubricating fluid, acting as a skin layer. This is combinedwith the usual low viscosity fluid along the pipe wall, to lubricate transport, andheavy viscous oil is the core fluid. The yield stress is used to eliminate the pos-5Inflow Manifold 1) Crude Oil2) Skin3) LubricantTransient length Steady pipelineQ^1Q^2Q^3Figure 1.3: Schematic of the triple-layer core-annular flow within an inflowmanifold and its conceptual zones.sibility of interfacial instability growth, by remaining rigid. A schematic of thetriple-layer core-annular flow within an inflow manifold is shown in Fig. 1.3, andthis is the system that we study in depth. Throughout the thesis, the main objectivesinclude the following.• Establishing the feasibility of the triple-layer visco-plastically lubricated pipeflow for some practical classes of pipelining flows.• Developing a practical and robust method to shape the interfaces (skin sculpt-ing) via control over flow rates to generate lubrication force.• Exploring aspects of flow development for various fluid properties and ge-ometries.• Investigating the effect of turbulence and inertia on pressure drop reductionand lubrication force.• Identifying limitations of the triple-layer core-annular flow in pipeline trans-portation flows.1.3 Yield stress fluidThe term “yield-stress fluid” is used to categorize a range of non-Newtonian fluidswhich behave as a rigid solid if the imposed stress on them is below a threshold,which is called the yield stress (τˆy). The fluid deforms as a (nonlinearly) viscousfluid if the shear stress in the fluid exceeds the yield stress [33]. There are many ex-amples of these fluids in everyday life: gels, creams, toothpaste, chocolate, yogurt,6Figure 1.4: Examples of visco-plastic fluids. (a) Hair gel, (b) melted choco-late, (c) toothpaste, (d) lava, (e) drilling mud, and (f) avalanche.soft butter, lava, mucus, cement slurries, drilling mud, mine tailings (see Fig. 1.4)[23, 41, 169]. Materials which exhibit yield stress behaviour have very complexrheological behaviour so, the description of such materials is a most difficult taskin rheology. The constitutive model needs to be simple enough to be used theoret-ically and computationally and yet predict actual properties of such materials. Thesimplest and most common rheology model is attributed to Bingham [13], whichcorresponds (in 1D) to adding a yield stress to the Newtonian viscosity:{τˆ =(µˆ+ τˆy| ˆ˙γ|)ˆ˙γi j if τˆ > τˆyˆ˙γ = 0 if τˆ ≤ τˆy,(1.1)whereˆ˙γi j =∂ uˆi∂ xˆ j+∂ uˆ j∂ xˆi,7ˆ˙γ =[123∑i, j=1[ ˆ˙γi j]2]1/2τˆ =[123∑i, j=1[τˆi j]2]1/2, (1.2)here ˆ˙γ , uˆ, and τˆ are the shear rate, velocity, and stress. In this model, yield stressmaterials are rigid when they are unyielded and they have linear flow curve whenthey yield. The Bingham model has two physical parameters, µˆ and τˆy, represent-ing the plastic viscosity and yield stress, respectively.Later, Herschel and Bulkley [58] added a power-law dependency to the Bing-ham model to explicitly fit shear-thinning behaviour in rheological measurementsof yield stress materials:{τˆi j =(Kˆ ˆ˙γn−1+ τˆy| ˆ˙γ|)ˆ˙γi j if τˆ > τˆyˆ˙γ = 0 if τˆ ≤ τˆy,(1.3)where Kˆ and n are the consistency parameter and power law index. Since manyyield stress materials exhibit shear-thinning behaviour, the Herschel-Bulkley (HB)model is able to represent the rheological behaviour of more yield stress materialsthan the Bingham model.1.4 Multiphase flowsMultiphase flows can be defined as systems of different fluid phases or fluid andsolid phases. Multiphase flows have broad applications in everyday life. In indus-try, there are countless examples, such as power plants, cavitation pumps, oil andgas industries, pipelines, plastic industries, sprays, fluidized beds, distillation andmany others. In nature, we can mention sandstorms, sediment transport, geysers,volcanic eruptions, acquifiers, clouds, rain, and blood flow. These flows can beclassified by two general topologies: (i) dispersed flow and (ii) separated flow. Thedispersed flow is a flow which contains finite particles, drops or bubbles (the dis-persed phase) distributed in a connected volume of the continuous phase. Two ormore continuous streams of different fluids separated by interfaces can be definedas a separated flow [19, 137].Investigating multiphase flows is very complicated not only because of theinteraction among many entities/phases, such as bubbles, drops, or particles im-8mersed in the fluid, but also because of the very physics of the phenomena, e.g. theadvancing of a solid-liquid-gas contact line, the transition between different gas-liquid flow regimes, disturbed interfaces, turbulence, and many others. There arethree approaches to study these complex flows: experimental studies, theoreticalmodeling, and numerical simulations. Full-scale laboratory models are possible forsome applications. But in many cases, only a scaled model (a prototype) is feasiblefor experiment, and in a restricted parameter range, i.e. full dimensional similarityis not possible. This situation then requires a reliable theoretical or computationalmodel to extrapolate the results.Several theoretical models have been proposed. However, these models havesome limitations specially in intermediate Reynolds number flows and difficultiesto be applied, such as boundary conditions. The trajectory model and the two-fluidmodel [18, 79, 177–179] are two well-known models to study dispersed flows,such as granular flows and bubbly flows. Free streamline theory [14, 186, 187] isan example to model separated flows. Consequently, computational methods are animportant tool to study multiphase flows. Computations can solve actual practicalproblems in some cases, e.g. Direct Numerical Simulation, or they study crucialphysical aspects, such as effects of gravity or surface tension. In this thesis weare interested in immiscible multi-fluid flows and we provide brief introduction oninvestigating this class of flows computationally later (§1.7).1.4.1 Multi-fluid flowsThe flow of immiscible fluids can be found in several configurations: layers, slugs,rollers, sheets, bubbles, drops, and dynamic emulsions and foams with very wideapplications in industry, such as fingering in oil recovery, lubricated pipeline flow,air entertainment in porous media, formation of emulsions, co-extrusion of plasticsand other composite polymeric materials, compound jets, coating, fiber industry,and etc. [82].In these flows, fluids are separated by interfaces and the flows are driven byprescribed forces of the usual type [51]. One common way to investigate theseproblems is using a “one-fluid” formulation. In this method, fluids are identifiedby a marker function that is advected by the flow. Several methods have been9developed for that purpose, such as the Volume of Fluid (VOF) method, the level-set method, and the phase-field method. Later in the chapter, we briefly discussvarious computational method to resolve the interface (§1.7).1.5 Multi-layer viscous shear flow instabilityThe stability of two superposed fluids of different viscosities was initially studiedby Yih [190]. He found the plane Couette and Poiseuille flow can be unstable evenat small Re number. More work has been done on stability of two-layer Couette,Poiseuille, Couette-Poiseuille (C-P) flow, see e.g. [60, 65–67, 82, 189]. Govindara-jan and Sahu [52] extensively review the literature and Joseph et al. [83] include thestudies of core-annular flows. The physical view of the instabilities of two-layerviscous stratified flow have been identified in [22, 61, 93]. They have shown thatshear instabilities arise from the jump in viscosity at the interface. One interest-ing feature which has application in lubricated core-annular flow is that when thethinner layer is less viscous, the flow can be stable in long-wave limit; see e.g. [3]which provides the stability diagram in the plane of the viscosity and thickness ra-tio between the fluid layers in the absence of surface tension and gravity effects. In[3], there is a discontinuity (jump) of viscosity at the interface and Ern et al. [35]studied the inetrfacial instability of the continuous viscosity stratification. Sahuand Govindarajan [146, 147] also show that the continuous viscosity stratificationcan alter the absolute instability of the flow. In general, large surface tension isrequired to have a linearly stable immiscible iso-dense flow while the lubricatingfluid is less viscous. Surface tension stabilizes short wavelength interfacial modesand the the viscosity ratio tends to stabilize long wavelength instabilities [110].Apart from Newtonian fluids, linear stability of multi-layer flows of other in-elastic non-Newtonian fluids have been studied, e.g. [87, 163, 182, 183] considerpower law fluids and in [132], Carreau-Yasuda and Bingham-like fluids are stud-ied. None of these studies consider the stability analysis of yield stress fluids andthey lead to results of that are qualitatively similiar to Newtonian fluids, i.e. lin-ear interfacial instabilities arise at small-to-moderate Re numbers. The physicalexplanations presented in [22, 61] can be extended to purely viscous generalizedNewtonian fluids [110] as the dominant feature at the fluid-fluid interface is a dis-10continuity in a finite constant viscosity between two fluids and not the nonlinearityof the fluid.1.5.1 Visco-plastic fluids and shear flow stabilityInitially, a linear stability analysis of plane Poiseuille flow of a Bingham fluid sub-jected to two-dimensional disturbances was performed by Frigaard et al. [40]. In[40], the linear stability equations (eignenvalue problem) was solved with modi-fied boundary conditions at the yield surface to be consistent with Newtonian fluid.Later, Frigaard [37] studied the super-stability of a multi-layer plane Poiseuilleflow of two Bingham fluids in which the fluid next to the channel wall has higheryield stress than the fluid in the centre of the channel, using the same boundaryconditions as in [40]. Although [37, 40] show enhanced stability compared to theanalogous Newtonian flows, if the physically correct boundary conditions are used,both flows are stable to all Re. This was shown by Kabouya and Nouar [84] foran annular flow and the methods have been generalised to plane channel flows ine.g. [111]. The three-dimensional linear stability analysis of a plane channel flowwas studied by Frigaard and Nouar in [38] and they derived the eigenvalue bounds.Nouar et al. also questioned the treatment of odd and even perturbation usedby [40] and they implemented the correct conditions at the yield surface and foundno unstable modes. In [105], the key feature of the linear stability analysis is usedin which the plug region remains intact. They considered the asymptotic limit oflinear stability with small yield stress leading to a rigid linearly stable sheet in thecentre of a plane channel.The nonlinear stability analysis of Bingham fluid Poiseuille flows in pipes andplane channels based on the energy method was performed in [117]. This studyshowed that the increasing yield stress increases the critical Re number for transi-tion.In [110], it was shown that core-annular pipe flow can be nonlinearly stable atsignificant Reynolds numbers and produce stability bounds. Hormozi et al. ex-tended results of [110] by considering the core-annular flow of an Oldroyd-B fluidlubricated by a Bingham fluid. They showed the exponential decay of an energyfunctional for sufficiently small Reynolds number and Weissenberg number, i.e. the11core-annular flow base state is stable and the nearby states are asymptotically sta-ble in a case of changing domain. Transient growth phenomena has been studiedin [118].Later in [106], weakly nonlinear stability analysis of visco-plastic fluid wasperformed for Rayleigh-Be´nard-Poiseuille flow and they showed that the range ofthe validity of the amplitude equation was limited. [106] is the only work whichconsider this type of analysis as the nonlinearity of yield stress fluids is not simplyin the inertial terms, but also in the shear stress and in the existence of unyieldedplug regions, which leads to a complicated problem.On the other hand, stability analysis of yield stress fluids using regulariza-tion models has been studied in [39]. Frigaard and Nouar [39] studied the planePoiseuille flow of the Bingham fluid which is linearly stable flow. They showedthat the regularized models can predict the spurious modes as the boundary con-ditions for regularized models are different. These spurious eigenvalues dependon the regularization parameters and can give rise to unstable modes. [20] stud-ied the linear stability of the circular Couette flow of visco-plastic fluid for bothBingham and regularized model. The authors showed that the critical conditionsare not affected by regularization, although the regularized model is fundamentallydifferent from a true plane Bingham Poiseuille flow. In [104], a linear stabilityanalysis of a Rayleigh-Be´nard-Poiseuille flow is performed for yield stress fluidsusing the Bingham and regularized models. They showed that different regulariza-tion models predict different instability results. However, the results can convergeto the exact Bingham results for small enough regularization parameter.1.5.2 Visco-plastically lubricated flow (VPL)The idea of the triple-layer structure originates from visco-plastically lubricatedflows (VPL), as studied by Moyers-Gonzalez et al. [110] and Hormozi et al. [70].In these flows the yield stress is used to eliminate the possibility of linear interfacialinstability growth, by remaining rigid. Frigaard [37] studied the linear stability ofa multi-layer plane Poiseuille flow of two Bingham fluids, exposing the role of theyield stress in freezing the interface. Moyers-Gonzalez et al. [110] extended theanalysis to visco-plastically lubricated viscous core-annular pipe flows and derived12non-linear stability bounds. Huen et al. [75] demonstrated experimentally thatthese flows can be stably established. The range of applications was extended byHormozi and co-authors [68, 70, 71]. When the interface is formed by two yieldedfluids, the stability is lost and interfaces are vulnerable to the usual range of linearinstabilities, as explored by Sahu et al. [148, 149].1.6 Underlying analytical methods used in the thesisIn the thesis, we investigate the lubrication force, generated by eccentric core po-sitioning, using analytical methods from tribology, as briefly explained in §1.6.1.Later in the thesis, we include the effect of inertia and turbulence by using inertialand turbulent tribology which is an integral method, see §1.6.2.1.6.1 Lubrication problemsIn Chapter 2 & Chapter 5, we study the positioning of the core by using meth-ods from tribology. These methods are derived for the “thin-gap” limit, in whichthe distance between boundaries is small comapred with the lateral gap width. Inthis case, using classical lubrication scaling results in an approximate analyticalsolution, the famous Reynolds equation from which we can calculate the pressuredistribution in the gap; see [141]. The combination of relative motion of the bound-aries and a thin but nonparallel gap can generate very high pressures [166]. Thehigh generated pressure tends to keep the surfaces from coming into contact evenwhen they are subjected to a considerable normal load [96]. This is the basis forlubrication between moving surfaces in many mechanical systems, such as sliderbearings and journal bearings.1.6.2 Integral methodIn Chapter 4, we study the effect of inertia on the generated lubrication force. Thin-film lubrication assumes that inertial forces are negligible in comparison to viscousforces. However, it is possible that the inertial forces are significant, specially inthe case of a low viscous lubricant with a high velocity. Therefore, the nonlinearinertial terms should be considered in hydrodynamics of a thin-film flow.An average velocity profile method can be used to approximate the effect of13inertia on thin-film flows. This method is derived based on the negligible effectof inertia on the velocity profile, see e.g. [24, 27, 88]. The method is based on themomentum equation of von Ka´rma´n, which is obtained by integrating the boundarylayer equation across the layer [85]. The advantage of this method is that if a certaindefinite form is assumed for velocity profile, an ordinary differential equation isderived from momentum equation. However, the method fails to predict accuratelythe separation point in adverse pressure gradients and compute stability analysis.1.7 Overview of computational methods to resolve fluidinterfacesInterface capturing and interface tracking are two mesh-based methods to resolvethe interfaces. Eulerian description with fixed mesh is implemented in interfacecapturing method to advect the marker (color) function. Mainly, two color func-tions are used: the volume of fluid method (VOF) [46, 62, 134, 140], or a signeddistance function in the level-set method [165]. Briefly, the marker function C isintroduced such that C = 1 for one fluid and C = 0 for other fluid. Once the fluidvelocity is known, the marker function is advected by the fluid and integrating (1.4)leads to an updated C.∂C∂ t+u.∇C = 0. (1.4)Where u is the fluid velocity. In case of miscible flows, (1.4) changes to:∂C∂ t+u.∇C =1Pe∇2C. (1.5)where Pe = RˆUˆ0/Dˆm is the Pe´clet number with Dˆm molecular diffusivity. In lu-bricated pipeline flow, typically we have Pe ∼ 106− 1010, which leads to negli-gible diffusive effects in (1.5). Alternatively, the capillary number is defined asCa = µˆWˆ/σˆ , where in pipeline flow, µˆ is the viscosity of the heavy oil and σˆ isthe interfacial tension of heavy oil-water. Heavy oil pipeline flows lead to high Caand negligible capillary effects, typically we have Ca∼ 104−106. In this thesis, weconsider the immiscible limit of miscible fluids (Pe→ ∞), or equivalently an im-miscible flow at infinite capillary number, i.e. surface tension effects are negligible14due to high radius of pipes typical in pipeline applications (e.g. Rˆ∼ 10cm-1m).The properties at each point in the domain can be calculated by using themarker function and interpolating. As an example if the property of each fluidis constant, the property ( f ) of each cell is:fi = f1C+ f2(1−C). (1.6)here, f1 is the property of fluid where C = 1 and f2 is the property where C = 0.Note, this type of interpolation is not valid for all properties.In interface tracking methods, the interface is described by a set of particles orinterconnected mesh points, which are either part of a secondary mesh advectedrelative to a fixed primary mesh (background mesh) or are a subset of the primarymesh.1.7.1 Volume of fluid method (VOF)In the VOF method, the color function represents the volume of one phase at eachcomputational point. In this method, the interface is reconstructed based on colorfunction and the method is able to conserve the respective mass of phases exactly,however, the interface representation suffers from low accuracy. the VOF methodis one of the most widely used interface capturing method and lots of improvementhas been made on its implementations [57, 160, 188, 191]. The simplest types ofVOF method are the simple line interface calculation [62, 116]. In this method,the interface is reconstructed with a piecewise constant function aligned to oneof the coordinates resulting low accuracy. The more accurate methods are knownas the Piecewise Linear Interface Calculation (PLIC) [42, 99, 131]. The slope ofthe interface in the cell is determined by the gradient of the volume of fractiondistribution in PLIC method.1.7.2 Level-set methodThe level-set method [126, 165] is the main alternative to the VOF method to di-rectly advect the marker function. Similar to the VOF method, the level-set methodcan capture interfaces which undergo a topological transition, such as breaking ordrops coalescence. Unlike the VOF method, level-set method can accurately com-15pute the interfacial normal and curvature, but it does not conserve the mass verywell. In addition, this method needs frequent reinitialisation to keep the level-set asigned distance function. Some researchers have proposed the combination of theVOF and level-set methods to use the advantages of both, see e.g. [8].1.7.3 Phase-field methodThe phase-field method introduces the phase function to identify the different flu-ids. The phase function is updated by a nonlinear advection-diffusion equationknown as the Cahn-Hilliard equation. The equation includes both diffusion termswhich smear the interface and anti-diffusive terms to prevent the interface frombecoming too thick. The key to the modification is the introduction of a properlyselected free-energy function, ensuring that the thickness of the interface remainsof the same order as the grid spacing. This method has been widely implemented inthe simulation of solidification, e.g. [90] as well as for fluid dynamic simulations,e.g. see [80, 81].1.7.4 Interface trackingAs mentioned before, interface capturing is able to capture the complicated inter-face motion due to topological changes. However, the interfaces are not resolved bythe mesh and the accuracy of interface capturing can be affected. Unlike interfacecapturing method, interface tracking method resolves the interfaces by mesh defor-mation and this method is able to make the interface a true discontinuity. Therefore,interface tracking methods lead to more accurate representation of the interface andcurvature calculation, but topological changes such as breaking and merging of in-terfaces have to be included manually. The arbitrary Lagrangian-Eulerian (ALE)methods are the common ways to track the interface. These methods allow theinterface mesh to be moved in the Lagrangian way while the interior meshes maybe adjusted in an arbitrary manner, with the goal of preserving well shaped meshelements during the simulation [11, 34]. The ALE method has been widely usedfor simulations of fluid-structure interactions [7, 184].161.8 Computational methods relevant to visco-plasticfluidsThere are few problems in simple geometries for which there are classical solutionsfor Bingham and HB models, e.g. Poiseuille flow. As mentioned before in §1.3,numerical and analytical investigation of yield stress models are difficult due tothe singularity of these models. There are two general approaches to solve thisproblem: (i) methods solving the exact Bingham model, and (ii) methods removingthe singularity by modifying the effective viscosity model. Here, we review thesestrategies.1.8.1 Regularization methodThis method eliminates the singularity of the effective viscosity in Bingham/HBmodel by adding a small parameter, ε , to the constitutive equation µˆe,ε = f (µˆ, τˆy,ε)such thatlimˆ˙γ→0µˆe,ε < ∞ and limε→0µˆe,ε = µˆe.This method was initially proposed by Glowinski et al. [50], who also analyzedthe error of these methods. Several regularized viscosity models can be definedwhich satisfy above presented conditions. Some more commonly used regulariza-tion models are listed below:Bercovier and Engelman model from (1980); see [12]:µˆe,ε = µˆ+τˆy√ˆ˙γ2+ ε2. (1.7)Bi-viscosity models from (1984); see [43, 98]:{µˆe,ε = µˆε if ˆ˙γ <ετˆyµˆµˆe,ε = µˆ+(1−ε)τˆyˆ˙γ ifˆ˙γ ≥ ετˆyµˆ .(1.8)Papanastasiou model from (1987); see [127]:µˆe,ε = µˆ+τˆyˆ˙γ(1− exp(−ˆ˙γε)). (1.9)17In any regularization method, unyielded fluid is replaced with a very high vis-cosity fluid, i.e. there is no true unyielded regions with zero strain rate in the flow.Convergence of results of a regularized scheme is verified by evaluating the nu-merical solution at decreasing values of ε to ensure that a desired precision isaccomplished. Then, the unyielded region can be defined as τˆ ≤ τˆy. However,the regularization parameter (ε) may affect the yield surfaces. This is the mainregularization defect.For smaller ε the problem becomes progressively stiffer as we approach the sin-gularity ε → 0 and one might intuitively expect that the regularized model shouldconverge to the exact Bingham model by letting ε → 0. Frigaard and Nouar [39]showed that this intuition is only valid for the velocity field and not the stress field,i.e. the predicted yield surfaces by regularization method may not converge to theexact Bingham model. They also illustrated that using Papanastasiou model (1.9)results in faster convergence in comparison with other schemes where the stressis far from the yield limit. However, the regularization method converges poorlywhen the stress is close to the stress limit for most of the domain.Stability analysis of yield stress fluid can be also affected by regularized mod-els. Small disturbances may only disturb the boundaries of the plug asymptoticallyand may not break the plug completely. As the yielded and unyielded regionsare not decoupled in linear stability analysis when using regularized models, thespurious eigenvalues can be introduced to the stability problem [39]. Despite thementioned disadvantages, using regularization method can be helpful as it is easyto implement specially in commercial software, its convergence is faster, and thevelocity field is guaranteed to converge to the exact Bingham model. However, ifthe accurate resolution of the stress field is required, multiplier based method is abetter choice, see e.g. [49, 110]. We review such methods in §1.8.2.1.8.2 Variational principles and numerical methodsThe weak formulation of visco-plastic flow problems has received much attentionafter the work of Prager [135]. In [135], two variational principles were statedfor the Stokes flow of Bingham fluid: velocity minimization and stress maximiza-tion. These principles can be generalized to the Herschel-Bulkley and many other18rheological models of generic type [76]:{τˆ =(φ( ˆ˙γ)+ τˆy| ˆ˙γ|)ˆ˙γ if τˆ > τˆyˆ˙γ = 0 if τˆ ≤ τˆy.(1.10)where, φ( ˆ˙γ) represents the effective plastic viscosity of the fluid. Lets assumethe Stokes flow and the flow is governed by momentum equations, continuity, andboundary conditions:∇.σˆ + ρˆ fˆ = 0, in Ω, (1.11a)∇.uˆ = 0, in Ω, (1.11b)σˆ .n = tˆ, on St , (1.11c)uˆ = uˆs, on Sv. (1.11d)Here, σˆ , ρˆ, ρˆ fˆ, and uˆ are Cauchy stress tensor, fluid density, body force, andvelocity vector, respectively and σˆ =−pˆI+ τˆ .The velocity potential function (Φ( ˆ˙γ)) is defined as,{Φ= 0, if ˆ˙γ = 0dΦd ˆ˙γ = τˆ, ifˆ˙γ > 0.(1.12)Let vˆ be a kinematically admissible velocity field for the problem (1.11). Thenthe unique solution uˆ minimizes the following functional J(vˆ):J(vˆ) =∫ΩΦdVˆ −∫Ωρˆ fˆ.vˆdVˆ −∫Sttˆ.vˆdSˆ. (1.13)Similarly, if we define stress potential function (Ψ( ˆ˙γ)) as:{Ψ= 0, if τˆ ≤ τˆydΨdτˆ =ˆ˙γ, if τˆ > τˆy.(1.14)If σˆ ′ is a statically admissible stress field and uˆ the unique velocity field solution forthe problem (1.11).Then the stress solution (σˆ ) maximizes the following functional19(K(σˆ ′)):K(σˆ ′) =−∫ΩΨdVˆ +∫Svσˆ ′.n.uˆsdSˆ. (1.15)Proof of these two principles can be found in [76]. It also showed that bothfunctional J and K will take same limiting value when evaluated with the actualfield solution.K(σˆ ′)≤ K(σˆ) = J(uˆ)≤ J(vˆ). (1.16)The two potential functions (first term on the RHS of (1.13) and (1.15)) can beexpanded for Bingham fluid as:∫ΩΦdVˆ =12µˆ∫Ωˆ˙γ(vˆ) : ˆ˙γ(vˆ)dVˆ + τˆy∫Ω|| ˆ˙γ(vˆ)||dVˆ , (1.17)and, ∫ΩΨdVˆ =− 18µˆ∫Ω(τˆ ′− τˆy)2+dVˆ , (1.18)where (τˆ ′− τˆy)+ indicates the positive part of (τˆ ′− τˆy).Augmented Lagrangian methodAugmented Lagrangian method is a robust numerical method which can solve theexact yield stress models. This method is founded on the variational principlesmentioned above. Glowinski and his coworkers developed the numerical algo-rithm, see [36, 48–50]. They showed that the solution uˆ of the problem (1.11) canminimize (1.13). To minimize (1.13), the conventional gradient-type algorithmcannot be used as the relation is non-differentiable, due to the second term in RHSof 1.17. The functional J(vˆ) is convex and it can be optimized by replacing thefunctional J(vˆ) with a Lagrangian functional which introduces a Lagrange multi-plier to the problem, which becomes a saddle point problem. Then, the Lagrangianfunction is augmented with an additional penalty term to stabilize the computa-tions. Fortin and Gelowinski proposed set of augmented Lagrangian methods in[36], have been called ALG1-ALG4 later. ALG2 has attracted more attentions inengineering applications and specifically in solving yield stress fluid flows.Lets consider Bingham model and the problem (1.11) for a Bingham fluid to20briefly discuss ALG2. The Lagrangian functional is as follow:L(uˆ, γˆ, Tˆ) =12µˆ∫Ωγˆ : γˆdVˆ + τˆy∫Ω||γˆ||dVˆ −∫Ωρˆ fˆ.uˆdVˆ−∫Sttˆ.uˆdSˆ+∫Ω(γˆ− ˆ˙γ(uˆ)) : TˆdVˆ+a2µˆ∫Ω(γˆ− ˆ˙γ(uˆ)) : (γˆ− ˆ˙γ(uˆ))dVˆ . (1.19)where, Tˆ is the Lagrange multiplier and a is the augmentation parameter. Solvingsaddle point problem L is the same as minimizing J, see [36, 48]. ALG2 followsthree main steps for optimization process.Algorithm 1 ALG21: Procedure2: n← 03: γˆ0, Tˆ0← 0 (or any initial guess)4: loop (Uzawa algorithm):5: if error < convergence then close.6: find uˆn+1 and pˆn+1 satisfying−aµˆ∆uˆn+1 =−∇pˆn+1+∇.(Tˆn−aµˆ γˆn)+ ρˆ fˆ in Ω,∇.uˆn+1 = 0 in Ω,with given BC.7: γˆn+1←{0, if ||Σˆ|| ≤ τˆy,(1− τˆy||Σˆ||) ||Σˆ||µˆ(1+a) , if ||Σˆ||> τˆy.where Σˆ= Tˆ+aµˆ ˆ˙γ(uˆn+1)8: Tˆn+1← Tˆn+aµˆ ( ˆ˙γ(uˆn+1)− γˆn+1)9: n← n+110: goto loopThis numerical method can manage the exact Bingham model and the detailedalgorithm can be found in [36]. To achieve a fine resolution of yield surface, adap-tive meshing can be employed, see e.g. [154].1.9 The triple-layer flow conceptIn this thesis we study a novel triple-layer lubricated pipeline flow with a visco-plastic fluid as one of the layers (see Fig.1.3). Here, we review the related literature21that the novel method is originated from beside VPL, explained in §1.5.2.1.9.1 Core levitationOne of the important features of the triple-layer method is that a modest densitydifference, such as that typical in pipelining, can be accommodated. Core-annularflows are often observed to adopt wavy/corrugated interfaces e.g. [161]. Althoughthese initially attracted interest as a hydrodynamic instability, e.g. bamboo insta-bilities, perhaps the most relevant mechanical feature is that the interface shapegenerates a lift force via hydrodynamic lubrication. Indeed without any skewingof the wave shape and an eccentrically positioned core, the flow cannot support adensity difference between the fluids; see e.g. [5, 121, 123]. Generation of lubrica-tion pressures requires that the interface profile varies in the streamwise direction,whereas the eccentricity of the core focuses the differential pressure above/belowthe core. The differential pressure can be used to balance the density mismatchbetween the fluids (the oil usually being lighter). More recently, Ooms and co-workers [122, 125] have studied eccentric core-annular flow of a very-viscous core,i.e. a solid, analytically and semi-analytically using hydrodynamic lubrication the-ory.1.9.2 Visco-plastic sculptingIn this thesis, the unyielded skin layer needs to be shaped to generate sufficientlubrication force counterbalancing the density differences. Concentric manifoldshave been used to establish multi-layer flows for many years. Specific to flowsin which a yield stress fluid is used [75] demonstrated that stable flows of givendesign radii can be routinely established. Also in [75], calibration studies in whichan inflow tube was pinched and released, established that a single flow disturbancecould be frozen into the interface. Experimentally, it is found that a short develop-ment distance from the manifold is needed before the stresses relax sufficiently forthe interface to become unyielded and a parallel VPL is established. The develop-ment distance was studied further computationally in [70], over ranges of Reynoldsnumber, Bingham number and viscosity ratio. In [69] it was observed experimen-tally (and accidentally) that not only singular events (such as in [75]) could be22frozen into the unyielded interface, but also unstable interfacial waves that grow inthe development region can be captured.Two principal avenues are available to sculpt a given shape: (i) controlled vari-ation in the flow rates of the lubricant (and/or skin); (ii) mechanical variation of theaperture. The first of these is the subject of our this thesis. In the limit of a slowlyvarying interface shape, the mapping from flow rate to shape is direct. The sculpt-ing process can also be influenced by fluid rheology, which may control interfacialdeformation before the skin relaxes to a solid shape.In [69] it was observed experimentally that unstable interfacial waves wouldfreeze into the unyielded interface as VPL developed. Hormozi et al. [72] usedcomputational and experimental methods to show that variations in the individualflow rates of the 2 fluid streams allowed some degree of control over the interfaceshape (i.e. wavelength and amplitude). Figure 1.5 shows an example flow in whicha diamond-shaped interface is sculpted into the flow near the inflow pipe and thenfreezes into a shaped solid interface which is advected downstream. In relatedwork, Maleki et al. [102] showed how liquid droplets could be encapsulated in aflowing stream of unyielded fluid.1.10 Thesis outlineIn Chapter 2, we present the underlying triple-layer core-annular method studiedthroughout the thesis, with application in lubricated pipeline flow. We purposefullyposition an unyielded skin of a visco-plastic fluid between the oil and the lubricat-ing fluid. The objective is to reduce the frictional pressure gradient while avoidinginterfacial instability. We study this methodology in both concentric and eccen-tric configurations and show its feasibility for a wide range of geometric and flowparameters found in oil pipelining. The eccentric configuration benefits the trans-port process via generating lift forces to balance the density differences among thelayers. We use classical lubrication theory to estimate the leading order pressuredistribution in the lubricating layer and calculate the net force on the skin. We ex-plore the effects of skin shape, viscosity ratio, and geometry on the pressure drop,the flow rates of skin and lubricant fluids, and the net force on the skin. In addition,we estimate the yield stress required in order that the skin remains unyielded and23Figure 1.5: Example of interface shapes sculpted near the inflow pipe andthen frozen into a shaped solid interface advected downstream, (for pa-rameters and description, see Fig. 15a in [72].ensure interfacial stability.In Chapter 3, we address the important issue of how in practice to developa triple-layer flow with a desired sculpted/shaped visco-plastic skin, all within aconcentric inflow manifold. First, we use a simple one-dimensional (1D) modelto control layer thickness via flow rates of the individual fluids. This is used togive the input flow rates for an axisymmetric triple-layer computational simulation.This uses a finite element discretization with the augmented Lagrangian method torepresent the yield surface behavior accurately and a PLIC method to track the in-terface motion. The shaped interface induces extensional stresses in the skin layer,which eventually leads to yielding of the skin, as the axisymmetric computationsillustrate. We study this directly by developing a long-wavelength/quasi-steadyanalysis of the extensional flow. This allows us to predict the minimal yield stressrequired to maintain the skin rigid, for a given shape, all while maintaining a con-stant flow rate of the transported oil.In Chapter 4, we extend the feasibility of the method to large pipes and higher24flow rates by considering the effects of inertia and turbulence in the lubricationlayer. We show that the method can generate enough lubrication force to balancethe buoyancy force for wide range of density differences and pipe sizes if the propershape is sculpted into the unyielded skin layer.In Chapter 5, we present three-dimensional (3D) triple-layer computations whichcapture the buoyant motion of the core to reach its equilibrium position. The 3Dcomputations are performed using a finite element method and adaptively alignedmeshes to track dynamically the interfaces, benchmarked against axisymmetriccomputations from Chapter 3. The study shows that these flows may stably becomeestablished with control over interface shape, but development lengths (times) forthe core to attain equilibrium are relatively long, meaning extensive 3D computa-tion. We also present a simplified analytical model using lubrication approximationand equations of motion for the lubricant and rigid skin layers. This model allowsus to quickly estimate motion to the balanced configuration for a given shape andinitial conditions.In Chapter 6, we show an explicit advantage of the proposed method: namelythat we are able to produce stable core-annular flows in regimes where conven-tional core-annular flows are unsuitable, i.e. either dispersed phases or instabilityresult. Essentially the method can give stable flows for a very wide range of fluidinput ratio, although not all will produce the desired reduction in frictional pressurelosses.Chapter 7 of the thesis contains a summary of the results of the thesis andrecommendations for future work.25Chapter 2Triple-Layer Visco-PlasticallyLubricated Pipe Flow1In this chapter we present a novel method of crude oil transportation via core-annular flow, and establish its feasibility for some practical classes of pipeliningflows. A triple-layer structure is considered in which a visco-plastic fluid is insertedbetween transported and lubricating fluid, acting as a skin layer. This is combinedwith the usual low viscosity Newtonian fluid along the pipe wall, to lubricate trans-port, and heavy viscous oil is the core fluid. This underlying triple-layer flow is thesame as we study throughout the thesis.The idea of the triple-layer structure originates from VPL, as studied by Moyers-Gonzalez et al. [110] and Hormozi et al. [70]. In these flows the yield stress is usedto eliminate the possibility of interfacial instability growth, by remaining rigid.Frigaard [37] studied the linear stability of a multi-layer plane Poiseuille flow oftwo Bingham fluids, exposing the role of the yield stress in freezing the interface.Moyers-Gonzalez et al. [110] extended the analysis to visco-plastically lubricatedviscous core-annular pipe flows and derived non-linear stability bounds. Huenet al. [75] demonstrated experimentally that these flows can be stably established.The range of applications was extended by Hormozi and co-authors [68, 70, 71].In fact the VPL concept would be very suitable for stabilizing a core-annular flowon its own, but is not effective at reducing the pressure drop. If one could use1A version of this chapter has been published [157].26Core fluidSkinLubricantInflow ManifoldTransient length Steady pipeline(a)(b) rw ˆˆ  rw ˆˆ  rw ˆˆrˆzˆ1Qˆ2Qˆ tQ ˆˆ3 tˆ3Core-skin development Shaping zoneFigure 2.1: (a) Conceptual zones of triple-layer flow. (b) Flow developmentwithin the inflow manifold.a yield stress fluid with a high yield stress and low plastic viscosity, this wouldbe ideal. However, unfortunately the molecular structure that gives rise to a yieldstress tends to mean that the viscosity is not low in such materials. The triple-layer,using a low viscous lubricant, circumvents this barrier.The second component contributing to the triple-layer structure comes from ourunderstanding of CAF. Ooms, Bai and others initiated the study of how shaped ec-centrically positioned core-annular interfaces generate differential lubrication pres-sures around the core; see e.g. [5, 121–123, 125]. Varying interface profile in thestreamwise direction generates the lubrication pressure which is different above/-below the core due to eccentricity.This chapter combines these two component ideas. We proceed in a number ofsteps.I. We consider that the lubricating layer to be relatively thin, so that even forsignificant Reynolds numbers inertial terms are negligible at leading order.We then use classical lubrication theory to estimate the leading order pres-sure distribution in the lubricating fluid layer.II. By integrating over the lubricating fluid layer we calculate the net force on27the skin and core fluids.III. By adjusting the shape of the skin we may make the net force positive or neg-ative. By adjusting the eccentricity of the layer we may increase or decreasethe net force.IV. By balancing with the buoyancy force acting on the skin and core fluids, weestablish the equilibrium position of the transported core.V. Finally, on calculating the stresses within the lubrication layer we can esti-mate the yield stress required to keep the skin rigid.This leaves one final piece in the puzzle: namely how can we form shapes inthe visco-plastic skin in such a way that they remain rigid when formed, what flu-ids should be used for the skin and can this be done continuously, as the fluids arepumped and pipelined. This part of the process concept is discussed in Chapter 3.We envisage the process sequentially in 3 parts (Fig. 2.1a): (a) a concentric inflowmanifold in which the multi-layer flow is constructed; (b) a transient length, onleaving the manifold, within which the core fluid floats to its steady eccentric po-sition; (c) steady fully developed flow along a pipeline. This chapter is directed atthe mechanics of (c).This chapter starts by introducing the flow setup and notation (§2.1) and thenmoves onto a concentric flow configuration that allows semi-analytical solution.Next, the eccentric case is examined for various variables, following through thesteps (I) - (V) outlined above. The chapter closes with a discussion of the feasibilityof the proposed method for lubricated transport.2.1 Flow descriptionIn this chapter, we study core-annular configurations that allow the generation oflift via hydrodynamic lubrication, while at the same time resisting interfacial defor-mation via the introduction of an unyielded skin layer. Consider therefore a sectionof the pipe, periodic in the streamwise zˆ-direction and assumed horizontal for sim-plicity. The entire flow domain is denoted Ω and the 3 individual fluid domains byΩ1, Ω2, and Ω3. Fluid 1 denotes the core fluid (viscous Newtonian heavy oil), with28Figure 2.2: Cross section of the pipe with triple-layer configuration.viscosity µˆ [1] and density ρˆ [1]. The skin layer is fluid 2, modelled simply as a Bing-ham fluid, with µˆ [2], τˆ [2]y , and ρˆ [2] denoting its viscosity, yield stress, and density,respectively. Fluid 3 is the lubrication layer (assumed to be a low viscosity Newto-nian fluid) with viscosity µˆ [3], and density ρˆ [3]. Figure 2.2 indicates schematicallythe positions of the 3 fluids, within a cross-section of the pipe at fixed zˆ. Through-out the thesis we denote dimensional quantities with a ·ˆ symbol and dimensionlessquantities without.The outer radius of the skin may vary with zˆ: rˆ = rˆ2 = rˆ2,0 +∆rˆ2Φ(zˆ), but theinner radius (rˆ = rˆ1) is uniform; see Fig. 2.3. The skin fluid is assumed to have asufficiently high yield stress that it remains rigid (unyielded). Thus, fluids 1 and3 remain separated. In general, we might assume that the yield stress required ismoderately large (estimated later) and the pipe diameter is > 0.1m, so that anysurface tension effects are negligible in comparison to the other stresses.The governing equations for the flow are the Navier-Stokes equations, in eachfluid domain. The traction and velocity vectors are continuous across each interface(neglecting surface tension, as argued above). The flow is periodic in zˆ and no-slip29Figure 2.3: Schematic of the outer radius variation of skin layer with zˆ,characterized via the mean outer radius rˆ2,0 and streamwise variation∆rˆ2Φ(zˆ).conditions are satisfied at the pipe wall. Constitutive equations for the 3 fluids are:τˆi j [k] = µˆ [k] ˆ˙γi j, k = 1,3, (2.1)τˆ [2]i j =[µˆ [2]+τˆ [2]y∣∣ ˆ˙γ∣∣]ˆ˙γi j⇐⇒ τˆ [2] > τˆ [2]y , (2.2)ˆ˙γ = 0 ⇐⇒ τˆ [2] ≤ τˆ [2]y , (2.3)whereˆ˙γi j =∂ uˆi∂ xˆ j+∂ uˆ j∂ xˆi,ˆ˙γ =[123∑i, j=1[ ˆ˙γi j]2]1/2, τˆ [2] =[123∑i, j=1[τˆ [2]i j ]2]1/2. (2.4)2.1.1 Concentric core-annular flowWe start with a brief analysis of fully developed steady concentric flows, with nobuoyancy and a uniform skin layer. In order to scale the equations, we focus ontransport of the core fluid, which is assumed to have flow rate Qˆ1. We use Qˆ1 todefine the velocity scale: Wˆ0 = Qˆ1/piRˆ2, which we see is the velocity of the heavyoil, if transported alone in the pipe. We scale all lengths with Rˆ. The stress scale is30µˆ [3]Wˆ0/(Rˆ− rˆ2,0), used for both the deviatoric stresses and pressure, representingthe shear stress in the lubrication layer.(r,z) =(rˆ, zˆ)Rˆ, u =uˆWˆ0,−Gc = ∂ p∂ z =∂ pˆ∂ zˆRˆ(Rˆ− rˆ2,0)µˆ [3]Wˆ0, τzr =τˆzr(Rˆ− rˆ2,0)µˆ [3]Wˆ0.This leads to a problem governed by 2 dimensionless radii, r1 & r2,0, and 2further dimensionless groupsm =µˆ [3]µˆ [1], B =τˆ [2]y (Rˆ− rˆ2,0)µˆ [3]Wˆ0.Here m is the viscosity ratio and B is Bingham number. Provided that the skinlayer remains unyielded, the second viscosity ratio µˆ [2]/µˆ [1] plays no role in theflow. Steady unidirectional flow is governed by the z-momentum equation.−Gc = 1r∂∂ r[rτ [k]zr ], (2.5)where k = 1, 2, 3 represent the three fluid layers. The shear stresses and velocitiesare constant at each interface; Gc is constant representing the modified axial pres-sure gradient for the concentric flow. In the main case of interest, when the skinlayer is completely unyielded the axial velocity is:W (r) =Wp[1+mr21− r21− r22,0]0≤ r < r1,Wp r1 ≤ r < r2,0,Wp1− r21− r22,0r2,0 ≤ r < 1,(2.6)whereWp =Gc(1+ r2,0)4,is the plug velocity. The pressure gradient Gc is found by ensuring a unit flow rate31through Ω1 (due to the chosen scaling):Gc =8pimδr41 +2r21pi(1+ r2,0). (2.7)Where δ =(Rˆ− rˆ2,0)/piRˆ, which is the aspect ratio of the thin lubricant layerthickness to the circumferential length-scale. Finally, by calculating the shearstresses we will estimate the yield stress required in order to have an unyieldedskin layer:B >Gcr2,02=4pir2,0mδr41 +2r21pi(1+ r2,0). (2.8)In other words, for a given geometry this type of flow becomes feasible for asufficiently large yield stress. Note also that Gc ∼O(1), indicating that the viscousstress in the lubrication layer is the relevant scale, and since typically m/δ  1,the constraint on B is not severe, i.e.τˆ [2]y ∼ µˆ[3]Wˆ0Rˆ(1− r2,0).Assuming (2.8) to hold, the viability of the lubrication process depends on theconsumption of skin and lubricant fluids, plus whether or not the frictional pressurehas been reduced.The scaled flow rates (Q2,Q3) are:Q2 = 2∫ r2,0r1W (r)r dr = [r22,0− r21]Wp, (2.9)Q3 = 2∫ 1r2,0W (r)r dr =[1− r22,02]Wp. (2.10)Note that due to the scaling, we have Q1 = 1. Figure 2.4 plots Gc for representativem/δ and r1. Over the range plotted, the pressure gradient is significantly lessthan the pressure gradient Go required for the heavy oil to flow alone in the pipe,(Go = 4pi/mδ ). Also, for small m/δ , we observe that Gc is quite independent fromm/δ , as expected from (2.7).The flow rate of the skin layer is given by (2.9) and provided that m 1 and3210−3 10−2 10−1 1000.70.750.80.85m/δr 1  2.5 3 3.5 4Figure 2.4: Scaled pressure gradient (Gc) for representative m/δ and r1,r2,0 = 0.95.both skin and lubricant layers are thin, we can expect that Wp ≈ 1. Thus, Q2 scalesprimarily with [r22,0− r21] in any lubrication regime that is effective. Fig. 2.5 plotsthe intensity of the flow rate of the lubrication layer (Q3/(1− r22,0) =Wp/2). Weobserve that we are able to achieve relatively small flow rates of both lubricant andskin fluids by varying r2,0 and r1. For typical heavy crude viscosity ratios (m 1)the flow rate change with m is negligible both for skin and lubricant. Finally, weshow variations in the criterion (2.8) for different r1 and r2,0; see Fig. 2.6. We seethat the minimal B increases as r2,0 approaches the wall, as is expected.To summarize, this simple 1D model suggests that there are parameter regimesin which the triple-layer flows can provide a viable transportation method, depend-ing of course on fluid costs and on the difficult question of establishing the triple-layer flow, in a development stage somewhere upstream. However, the concentricsolution is not feasible as a core-annular flow as no lift force is generated to balancethe density differences.2.2 Eccentric core-annular flowNote that the concentric solution of the previous section has pressure gradient onlyin the axial direction. In the typical case where the oil density differs from that of33Figure 2.5: Intensity of flow rate of lubricating fluid (Q3/(1− r22,0) =Wp/2)for representative m/δ and r1, (with m = 0.0001).0.75 0.8 0.85 0.9 0.950.70.750.80.850.9r2,0r1  1.2 1.4 1.6 1.8 2Figure 2.6: Variations in the minimal B from (2.8), for representative r1 andr2,0, (with m = 0.0001).34the lubricant and skin fluid, there is a net transverse buoyancy force, so that theunderlying configuration is unlikely to be concentric. A uniform eccentric annularflow also generates only axial pressure gradients and, although different from theconcentric considered above, cannot support density differences. As recognised in[125], it is necessary to have both eccentricity and axial variation in the lubricationlayer in order to generate transverse lift forces via viscous lubrication. As with anytransport process there is an underlying constant pressure drop along the pipeline.Superimposed on this is a periodic (in z) variation in the pressure, which is gov-erned by the local thickness of the lubricant. In this way, differential lubricationpressures are generated that are able to counter the transverse buoyancy force.Anticipating the above scenario, we use a classical lubrication scaling of theequations. As shown in Fig. 2.2, we position our cylindrical coordinates at the cen-tre of Ω1, which has uniform radius rˆ1. The skin layer outer radius is rˆ2(zˆ), whichhas mean position rˆ2,0 and axial variation as in Fig. 2.3. The pipe wall is denotedrˆ = rˆ3(θ), with variation due to the eccentricity. We assume the mean thickness ofthe outer lubricant layer is thin, relative to circumferential and axial length-scales,piRˆ & lˆ respectively. In other words, δ = (Rˆ− rˆ2,0)/(piRˆ) 1 and assume thatλ = lˆ/(piRˆ)∼ O(1). Below we calculate the leading order in δ shear stresses andpressure in the lubricant layer. However, in typical scenarios considered m 1also (even m. δ ), and the parameter m/δ occurs in the leading order expressions.Therefore, the practical limit we consider here is the distinguished limit: δ → 0,with m/δ = finite.Our solution below is parameterized by 3 dimensionless scalars: the mean fric-tional pressure gradient along the pipe (G), the plug velocity (Wp) and the ec-centricity (e) of the core. These 3 scalars are determined by satisfying 3 integralconstraints. Firstly, the flow rate of fluid 1 has been specified in the formulationadopted:Qˆ1 = Wˆppi rˆ21 +pi rˆ418µˆ [1]Gˆ. (2.11)35Secondly, the pressure drop along the pipe is balanced by the wall shear stresses:0 =∫ 2pi0∫ rˆ30rˆ([−Pˆ+ τˆzz]zˆ=lˆ− [−Pˆ+ τˆzz]zˆ=0)drˆdθ+∫ 2pi0∫ lˆ0rˆ3τˆzr|rˆ=rˆ3 dθdzˆ. (2.12)Thirdly, the static pressure and viscous shear forces acting on the skin in the verticaldirection balance the weight of liquid.0 =(ρˆ [3]− ρˆ [2])gˆVˆ [2]+(ρˆ [3]− ρˆ [1])gˆVˆ [1]−∫ 2pi0∫ lˆ0rˆ2(Pˆ|rˆ=rˆ2 cosθ + τˆrθ |rˆ=rˆ2 sinθ)dθdzˆ, (2.13)where Vˆ [1] and Vˆ [2] are the volume of core and skin fluids, respectively.The flow rate and pressure drop constraints vary linearly with the frictionalpressure gradient and the plug velocity, which are easily incorporated into the so-lution, whereas (2.13) varies non-linearly with e. To begin our analysis we fix theeccentricity e and compute the frictional pressure gradient and plug velocity, us-ing the flow rate and pressure drop conditions. Later in the paper we include thevertical force balance (2.13) to determine e.2.2.1 Lubricating layerWe assume symmetry about a central vertical plane through the pipe, define z= zˆ/lˆ,define scaled azimuthal coordinate y = θ/pi with y ∈ [0,1], extending from the topto bottom of the lubricant annulus, and define x:rˆ = rˆ2,0+piRˆδx.Velocity components in axial and azimuthal directions are scaled with Wˆ0 and thatin the radial direction with δWˆ0. We break the pressure into 3 parts: a constantaxial pressure gradient, a periodic lubrication pressure (coming from the variationin layer thickness), and a hydrostatic pressure component:Pˆ =−Pˆ∗GGz+ Pˆ∗l Pl(x,y,z)+piδ ρˆ [3]gˆRˆxcos(piy). (2.14)36The lubrication pressure scale (Pˆ∗l ) is chosen to balance the leading order shearstress gradients:Pˆ∗l =piRˆµˆ [3]Wˆ0(Rˆ− rˆ2,0)2. (2.15)The axial pressure gradient scale (Pˆ∗G) is similar to that is used in the concentricsolution (§2.1.1):Pˆ∗G =piλ µˆ [3]Wˆ0(Rˆ− rˆ2,0). (2.16)Note we use Wˆ0 in the pressure scale Pˆ∗ instead of the dimensional plug speedWˆp because these are similar for small m, and since Wˆ0 is readily accessible. Theleading order shear stresses scale with δ Pˆ∗l .With the above scaling the leading order momentum equations are:0 =∂Pl∂x, (2.17a)0 =−∂Pl∂y+∂ 2v∂x2, (2.17b)0 =− 1λ∂Pl∂ z+∂ 2w∂x2, (2.17c)0 =∂u∂x+∂v∂y+1λ∂w∂ z. (2.17d)We can see that Pl is only function of (y,z). For the present study we assumethat the skin layer moves only axially, with speed Wp to be determined. Boundaryconditions for the lubricant are: (u,v,w) = (0,0,0) at x = x3(y) and (u,v,w) =(0,0,Wp) at x = x2(z). The functions x3(y) and x2(z) are derived from the wall andouter skin positions, to leading order in δ . The thickness of the lubricant layer isdenoted h(y,z) = x3(y)− x2(z), which is given to leading order by:h(y,z) = 1− ecospiy−aΦ(z)+O(δ ), (2.18)where the eccentricity is e = dˆ/(Rˆ− rˆ2,0) and the amplitude a = ∆rˆ2/piRˆδ . dˆis the radial distance between the pipe and core centres; see Fig. 2.2. Note thatthe dimensionless axial skin thickness variation Φ(z) has zero mean and maximalamplitude 1.37We treat the system (2.17a) - (2.17d) and boundary conditions in the usualway, eliminating the averaged (y,z)-velocity components with a stream functionand then deriving the following Reynolds equation for the pressure:∂∂y[h3∂Pl∂y]+1λ 2∂∂ z[h3∂Pl∂ z]=−6Wpλ∂h∂ z. (2.19)As we see, apart from depending on the lubrication layer geometry h, (2.19) isdriven linearly by Wp which is determined from the flow rate constraint and hori-zontal force balance. The velocity in the core fluid is easily found as a superposi-tion of a parabolic Poiseuille profile on the plug velocity. Integrating across Ω1 thedimensionless form of (2.11) is:1 =Wpr21 +mδr418piG. (2.20)For the horizontal force balance, note that only the constant pressure gradientcontributes to the pressure drop term (the other terms being periodic in z). Thenon-dimensional version of (2.12) is:0 =G+2∫ 10∫ 10r3∂w∂x(x3,y,z) dydz, (2.21)where r3 = r2,0+piδx3.Finally, to determine the vertical displacement or eccentricity e, we scale thevertical force balance. The constant axial pressure gradient makes no contribution.To leading order in δ the balance is as follows:∫ 10∫ 10r2Pl(y,z)cospiy dydz−Fl(1−ρ[1− r21r22,0])= 0. (2.22)In the above equation,Fl =(ρˆ [3]− ρˆ [1])gˆRˆ2Pˆ∗lr22,0 = δ(ρˆ [3]− ρˆ [1])gˆpiRˆ2r22,02piRˆ[µˆ [3]Wˆ0(Rˆ− rˆ2,0)] ,38which we easily identify as the ratio of buoyancy forces to lubrication pressureforces (the viscous scale amplified by δ−1). The parameter ρ arises if there is adensity difference between skin and core fluids:ρ =ρˆ [1]− ρˆ [2]ρˆ [3]− ρˆ [1] .For simplicity we shall assume that ρ = 0 in this study, so that Fl must balancewith the lubrication force generated through Pl .2.2.2 Solution methodNote that the Reynolds equation describes flow in a symmetric geometry h(y,z)and we thus seek a symmetric solution by imposing symmetry conditions at y = 0and y = 1, together with periodicity in z. As the lubricant is Newtonian (2.19) islinear and is symmetric due to h(y,z). Consequently we may find the solution asPl =WpP[0]l , driven by the skin/plug velocity Wp. For fixed geometry the solutionP[0]l corresponds to the pressure fields generated at unit plug velocity. To find P[0]lwe have used a second order central finite difference approximation to discretize(2.19) and then solve the linear algebraic system with Matlab.Substitution of the velocity into (2.21) leads to:G = 2∫ 10∫ 10[Wph− h2λ(∂Pl∂ z)]dydz, (2.23)(r3 ∼ 1+O(δ )), and G is eliminated by:G =(1−Wpr21) 8pir41δm. (2.24)Now Wp can be found by solving (2.23) and (2.24) simultaneously:Wp =1r211+mδr214pi∫ 10∫ 10[1h− h2λ∂P[0]l∂ z]dydz. (2.25)390 0.1 0.2 0.3 0.4 0.5 0.600.511.522.533.54eGFigure 2.7: Axial pressure gradient for the current study (solid line) and fromOoms et al. [123].We verify the solution by comparison with that of Ooms et al. [123], who con-sider a 2-layer eccentric core annular flow with wavy profile in the axial directionand infinite relative viscosity between core and lubricant. Figure 2.7 shows thecomparison of the axial pressure gradient. The results match well, with the verysmall discrepancy attributable to numerical error. In [123] the core viscosity isinfinite so it has rigid body motion and the core wall is wavy. A second compar-ison is shown in Fig. 2.8, this time comparing the pressure variations between thecurrent model and results of [125]. The small differences are probably because ofdifferent solution methods: Ooms et al. use an asymptotic approximation to solvethe Reynolds equation.2.3 ResultsFor all the results below we assume a sawtooth profile for the skin shape functionΦ(z), as illustrated in Fig. 2.9. The parameter l′ is referred to as the break point ofthe wave. This configuration is chosen as it is relatively simple and the asymmetryin z is necessary to generate lubrication pressures that can support the buoyancyforces of the core and skin fluids; see [125].400 0.2 0.4 0.6 0.8 1−1−0.500.511.52zPl(0,z)Figure 2.8: Periodic pressure variation along z for the current study (solidline) and from Ooms et al. [125].Figure 2.9: Sawtooth outer skin profile Φ(z).41(a) (b)Figure 2.10: Variations with z at fixed y of (a) lubricant layer thickness h(y,z)and (b) lubrication pressure Pl(y,z).2.3.1 Example flowThe flows are parameterized by 6 dimensionless parameters: (r1,e,m/δ , l′,λ ,a)if e is specified and Fl calculated. Alternatively Fl may be specified and e cal-culated. We discuss typical parameters in more detail and explore the paramet-ric variations below in §2.3.2. As an illustration of an example flow we solvewith (r1,e,m/δ , l′,λ ,a) = (0.87,0.6,0.1,0.7,1,0.3), assuming a sufficiently largeBingham number to prevent yielding of the skin. Here e is set relatively large, toemphasize eccentricity effects.The lubrication layer thickness and associated periodic pressure are indicatedin Fig. 2.10, at different azimuthal positions y. As expected, it can be seen that forthe narrower gaps (near the top of the pipe) the amplitude of the pressure variationincreases, generating significant pressures within the lubricant. In this case wehave l′ = 0.7, which results in a net upwards lubrication force. Note that in generalwe would have l′ < 0.5 to generate a net downwards (positive) force.Figure 2.11 shows the variation of the layer-averaged velocity in the lubricationlayer. Here we have unwrapped the annulus into the (y,z)-plane. We observe asmall secondary azimuthal flow relative to the main axial flow, i.e. fluid is squeezedin/out of the narrower parts of the annulus near the top of the pipe (y = 0).42(a) (b)Figure 2.11: (a) Layer-averaged azimuthal velocity and (b) layer-averagedaxial velocity, for the same parameters as in Fig. 2.10.Variable Symbol RangeCrude oil viscosity µˆ [1] 0.1−100Pa.sCrude oil density ρˆ [1] 770−980kg/m3Pipe radius Rˆ 0.05−0.61mCrude oil velocity Wˆ0 0.5−5m/sTable 2.1: Dimensional parameter ranges found in heavy crude oil pipelining.2.3.2 Parametric variationsTo develop intuition about the effect of geometric parameters and fluid proper-ties on the main process variables (G,Q2,Q3,Fl), we now systematically explorevariations over representative ranges of variables. Dimensional parameters rele-vant to these flows are extracted from the literature and summarized in Table 2.1;e.g. [1, 15, 64, 86, 103, 108, 114, 152] and many others. Based on Table 2.1, wecan find the ranges over which the non-dimensional numbers typically vary (seeTable 2.2).In terms of our study and the proposed method, the variables r1 and r2,0 shouldremain close to 1 to minimize consumption of lubricant and skin fluids. This is es-sentially an economic constraint: typical lubricant fractions used vary in the range5−20%. Although δ  1, typically also m < δ and to some extent one could vary43Dimensionless number Symbol RangeAspect ratio δ 0.01−0.03Viscosity ratio m 0.00001−0.01Density ratio ρˆ [3]/ρˆ [1] 1.02−1.3Buoyancy to lubrication force ratio Fl 0.014−2Table 2.2: Ranges of key dimensionless numbers expected in typical pipelin-ing operations.m/δ < 1 in designing the flow. The waveform Φ(z), should be regarded as beingdesignable, i.e. amplitude a, break point l′, and wavelength λ . However, we mightexpect λ = O(1) and note that a+ e < 1 to avoid contact. The eccentricity e willeventually be determined below to balance the lubrication force.Varying skin profileHere we fix a moderate eccentricity and explore the effects of wave shape on theflow, for (r1,e,m/δ ,λ ) = (0.87,0.3,0.1,1). The wave shape Φ(z) is governed byits amplitude (a), the break point of the wave (l′), and its wave length (λ ). Figure2.12 shows the normalized pressure gradient required, G/Go. It can be seen thatthe effect of l′ is negligible, whereas increasing a decreases the pressure gradientsignificantly. This arises because the pressure gradient is determined dispropor-tionately by the narrowest parts of the lubrication layer.The required flow rate of lubricating fluid (Q3) is presented in Fig. 2.13. Aswith the pressure gradient (Fig. 2.12), the variation of the flow rate does not changesignificantly with l′. It seems that Q3 increases as a increases. This may seemat odds with the pressure gradient variation in Fig. 2.12, which decreases. Notehowever, that the flow rate of lubricating layer includes two components: a pressuredriven flow rate and plug velocity flow rate, say Q3,Wp . The inset of Fig. 2.13 showsQ3 and Q3,Wp . The latter is the dominant contribution to Q3 and increases witha: the pressure driven component decreases (see the difference between the twocurves in the inset). Variations in Q2 are not shown, but are negligible.The effect of l′ thus far seems insignificant, but this only considers (G,Q2,Q3).440 0.1 0.2 0.3 0.4 0.5 0.60.20.40.60.8al′  0.01 0.015 0.02Figure 2.12: Normalized pressure gradient (G/Go) for different a and l′;fixed parameters (r1,e,m/δ ,λ ) = (0.87,0.3,0.1,1).0.1 0.2 0.3 0.4 0.5 0.60.20.40.60.8al′  0.06 0.068 0.0750.3 0.4 0.5 0.60.050.060.070.08a  Q3Q3,WpFigure 2.13: Flow rate of lubricating fluid (Q3) for different a and l′; fixedparameters (r1,e,m/δ ,λ ) = (0.87,0.3,0.1,1). Inset shows the totalflow rate and partial flow rate due to Wp of lubricating layer whenl′ = 0.2.450 0.1 0.2 0.3 0.4 0.5 0.60.20.40.60.8al′  −0.3 −0.2 −0.1 0 0.1 0.2 0.3Figure 2.14: Computed net lubrication force able to balance Fl for different aand l′: fixed parameters (r1,e,m/δ ,λ ) = (0.87,0.3,0.1,1). The solidblack line is where no density difference can be supported (Fl = 0).If instead we compute the leading order net lubrication force from (2.22), i.e.∫ 10∫ 10Pl(y,z)cospiy dydz,this gives the value of the dimensionless Fl that can be supported. As we recall,Fl represents the dimensionless ratio of buoyancy force to the scaling for the netlubrication force. Figure 2.14 plots the variation of the net lubrication force with l′and a. The main point to note is that the net lubrication force changes sign along acritical curve, say ac(l′). In a balanced system, satisfying (2.22), this figure showsFl . Thus for a significant positive Fl , in this example we need l′ < 0.45 and asufficiently large a. In this example, we have fixed the eccentricity at a modeste = 0.3, and the supported Fl is modest. As we have remarked in the previoussubsection, Fl < 0 are generated for l′ > 0.5.We now explore the effects of the wavelength (λ ). The normalized pressuregradient (G/Go) is shown in Fig. 2.15. Although there is a modest variation witha, λ has an insignificant effect on the pressure drop. Similarly, the effect of λ onthe flow rates Q2 and Q3 is found to be negligible. Figure 2.16 shows the effectof λ on the computed net lubrication force. Although there is a variation, it is not460.1 0.2 0.3 0.4 0.5 0.612345aλ  0.01 0.015 0.02Figure 2.15: Normalized pressure gradient (G/Go) for different a and λ ;fixed parameters (r1,e,m/δ , l′) = (0.87,0.3,0.1,0.2).significant.In conclusion, we fix (λ , l′) = (1,0.2) for the remainder of the study: varyingλ has little effects and l′ in this range ensures a buoyancy balance for positive e.The amplitude a remains as a control parameter, but is also limited by e.Varying viscosity ratio and geometryWe now study variations in (r1,m/δ ). We first fix (e, l′,λ ,a) = (0.3,0.2,1,0.5)and vary (r1,m/δ ) within admissible ranges. Figure 2.17 shows the variation ofthe normalized pressure gradient, G/Go. As expected this increases with m/δ , andalso decreases with r1. To aid visualization, we have superimposed the contourG/Go = 0.1, and see that even for m/δ ≈ 0.7 we have a pressure gradient less than10% of the crude oil pressure gradient (and indeed normally we would expect tobe far below that). Note however, that this is with relatively thin lubricating layer.Regarding the flow rates, the scaled skin fluid flow rate is Wp(r22,0− r21). Asdiscussed previously, variations in Wp are small for small m/δ . Figure 2.18 plotsthe flow rate of the lubricant (Q3), which is practically invariant with small m/δ ,but decreases with r1 for larger m/δ / 1. Thus, to control acceptably G/Go weadjust m/δ , either via m or r2,0 (δ ). The flow rates (consumption) of skin and470.1 0.2 0.3 0.4 0.5 0.612345aλ  0 0.05 0.1 0.15 0.2 0.25 0.3Figure 2.16: Computed net lubrication force able to balance Fl for differenta and λ ; fixed parameters (r1,e,m/δ , l′) = (0.87,0.3,0.1,0.2). Thesolid black line is where no density difference can be supported (Fl =0).10−2 10−1 1000.750.80.850.9m/δr 1  0.05 0.1 0.15 0.2Figure 2.17: Normalized pressure gradient (G/Go) for different m/δ and r1;fixed parameters (e, l′,λ ,a) = (0.3,0.2,1,0.5). The solid white con-tour marks G/Go = 0.1.4810−2 10−1 1000.750.80.850.9m/δr 1  0.08 0.09 0.1Figure 2.18: Flow rate of lubricating fluid (Q3) for different m/δ and r1; fixedparameters (e, l′,λ ,a) = (0.3,0.2,1,0.5).10−2 10−1 1000.750.80.850.9m/δr 1  0.06 0.065 0.07 0.075 0.08 0.085Figure 2.19: Computed net lubrication force able to balance Fl for differentm/δ and r1; fixed parameters (e, l′,λ ,a) = (0.3,0.2,1,0.5).lubricant fluids may then be partially controlled by varying r1.The computed net lubrication force that is able to balance Fl is shown for thesame parameters in Fig. 2.19. It can be seen, m/δ has a relatively small effect onthe lubrication force, which varies mostly with r1. Having said this the variationshere are relatively modest due to the small eccentricity e.490.75 0.8 0.85 0.90.10.20.30.4r1e  0.016 0.018 0.02 0.022Figure 2.20: Normalized pressure gradient (G/Go) for different r1 and e;fixed parameters (m/δ , l′,λ ,a) = (0.1,0.2,1,0.5).The final geometric parameter is the eccentricity e. The flow rates are notsignificantly affected by e, so we focus on the pressure gradient and lubricationforce. Figure 2.20 shows the scaled pressure gradient for different eccentricity andcore radius, where (m/δ , l′,λ ,a) = (0.1,0.2,1,0.5). We can see, pressure gradientmostly varies with r1, except when e+a ≈ 1, where we see a significant decreasewith e. Thus, in total we see that varying e does not have any detrimental effectson the pressure gradient.The net lubrication force which is able to balance density difference is shownin Fig. 2.21 for different r1 and e. Here by contrast we see that the net lubricationforce varies strongly with e and weakly with r1. As might be expected, moreeccentric flow configurations are able to support larger density differences with thecore fluid. The mechanism is straightforward: the lubricant layer at the top of thepipe thins and contributes larger pressures to the net lubrication force.2.3.3 Balance positionIn general, for fixed (r1,m/δ , l′,λ ,a) as we increase the eccentricity the net lu-brication force increases; see Fig. 2.21 for example. Eventually, according to thevalue of a, the skin contacts the wall, which is to be avoided. However, before thiseccentricity is attained a certain range of Fl can be supported.500.75 0.8 0.85 0.90.10.20.30.4r1e  0 0.05 0.1 0.15 0.2 0.25 0.3Figure 2.21: Computed net lubrication force able to balance Fl for differentr1 and e; fixed parameters (m/δ , l′,λ ,a) = (0.1,0.2,1,0.5). Solidwhite line shows Fl = 0.1.For a fixed specified Fl we iteratively solve (2.22) for e. For fixed (r1,m/δ , l′,λ ,a),equation (2.22) appears to be a monotonically increasing function of e, with a sin-gle root that represents the balance eccentricity. This is straightforwardly foundusing a bisection method. Figure 2.22 shows the eccentricity needed to generatebuoyancy forces for different values of Fl . It should be noted that in these figuresat each Fl , e is adjusting with respect to a to ensure a sufficiently narrow gap onthe top of the pipe, i.e. the narrowest gap is h(y,z) = hmin = 1− e− a, and it is inthe vicinity of the narrowest gap that we generate the largest lubrication pressures.However, there is in fact a significant variation in hmin as (a,Fl) vary, as illustratedin Fig. 2.23, i.e. the combination of a and e results in the distribution of h(y,z)about hmin. As we see the largest hmin appear to arise in the range a≈ 0.55−0.65,for these parameters and as we expected, for smaller Fl , hmin is larger.The relative pressure gradient is presented in Fig. 2.24 for different a and Fl .The larger values of a give rise to smaller equilibrium e and reduced frictionalpressures. However, in all cases (even though m/δ = 0.1), the relative pressuregradient is relatively small.The flow rate of the skin fluid is largely unaffected by the balance position ofthe core fluid (i.e. finding e). As expected, it varies with wave amplitude and as510.1 0.2 0.3 0.4 0.5 0.6 0.70.050.10.150.20.250.3aFl  0.2 0.4 0.6 0.8Figure 2.22: Variation of the eccentricity (e) required to balance Fl , for differ-ent a and Fl . Fixed parameters are (m/δ , l′,λ ,r1) = (0.1,0.2,1,0.87).Solid white line is a guide to the eye showing e = 0.3.0.1 0.2 0.3 0.4 0.5 0.6 0.70.050.10.150.20.250.3aFl  0.05 0.1 0.15 0.2 0.25Figure 2.23: Variation of the minimal layer thickness hmin = 1− e− a forthe balance eccentricity. Fixed parameters are (m/δ , l′,λ ,r1) =(0.1,0.2,1,0.87).520.1 0.2 0.3 0.4 0.5 0.6 0.70.050.10.150.20.250.3aFl  0.01 0.02 0.03 0.04Figure 2.24: Normalized pressure gradient (G/Go) in the balance state,for different a and Fl . Fixed parameters (m/δ , l′,λ ,r1) =(0.1,0.2,1,0.87).0.1 0.2 0.3 0.4 0.5 0.6 0.70.050.10.150.20.250.3aFl  0.07 0.08 0.09 0.1Figure 2.25: Flow rate of lubricating fluid (Q3) in balance state, for differenta and Fl . Fixed parameters (m/δ , l′,λ ,r1) = (0.1,0.2,1,0.87).the skin layer gets narrower, flow rate of skin fluid decreases. The flow rate of thelubricant is shown in Fig. 2.25. The required flow rate increases when the waveamplitude a increases.53Figure 2.26: Shear stress at r = r2, for (m/δ , l′,λ ,a,r1,e) =(0.1,0.2,1,0.5,0.87,0.3136).2.3.4 Estimating the minimal yield stress in the skinThe final important design parameter is the yield stress required to keep the skinlayer completely unyielded. In practice the stresses generated in the lubricationlayer are quite localised. As illustration, for (m/δ , l′,λ ) = (0.1,0.2,1) we fix(a,r1,Fl)= (0.5,0.87,0.1) and compute the equilibrium eccentricity e= 0.3135858.For this solution we plot the shear stress at the interface with the skin layer (r = r2)in Fig. 2.26. The shear stress is continuous at the interface and therefore this repre-sents one measure of the deviatoric stresses within the skin layer. We can see thatthis is not extreme.Secondly, if we consider the x-momentum equation in the skin layer:∂τ [2]xy∂y+1λ∂τ [2]xz∂ z=∂P∂x+O(δ 2), (2.26)we see that the shear stresses in the skin also adjust to accommodate normal stressgradients. The axial pressure gradient −Gz transmits across the skin, driving the54Figure 2.27: Variation of |Pl|/(x2 − x1), for (m/δ , l′,λ ,a,r1,e) =(0.1,0.2,1,0.5,0.87,0.3136).flow. The lubrication pressure however vanish on the inner radius r = r1. There-fore, we estimate:∂P∂x≈ Plx2− x1 ,for the right-hand side of (2.26). We then integrate the shear stressed throughoutthe skin, in (y,z) to give a rough estimate for τ [2]:τ [2] ≈ |Pl|x2− x1 .The normal stresses are typically one order larger than the shear stresses in lubri-cation problems and here we again amplify by insisting that the normal stressesvanish at r = r1. Thus, the above estimate is generally much larger than that basedon the shear stresses alone. Figure 2.27 shows the variation in |Pl|/(x2−x1) for thesame parameters as in Fig. 2.26. The maximum normal stress is in the narrowestpart of lubricating layer, but this is modulated slightly by having larger x2−x1. Thestresses are clearly much larger than in Fig. 2.26.550.1 0.2 0.3 0.4 0.5 0.6 0.70.050.10.150.20.250.3aFl  5 10 15 20 25 30Figure 2.28: Computed minimal dimensionless yield stress Bmin needed tokeep the skin layer completely unyielded in balance state, for differenta and Fl; fixed parameters (m/δ , l′,λ ,r1) = (0.1,0.2,1,0.87). Thesolid white contour marks Bmin = pi .Now we simply consider as an estimate of the shear stress the maximum of|Pl|/(x2− x1), i.e. we require the minimal yield stress to satisfyB > Bmin = max(y,z)( |Pl|x2− x1). (2.27)We believe that this should give a reasonable estimate of the maximal skin shearstresses, which we note are generally indeterminate for a yield stress fluid. Tosummarize, for fixed (m/δ , l′,λ ,a,r1,Fl) we first compute the equilibrium valueof e, and then the minimal Bmin from (2.27). Figure 2.28 plots Bmin for the sameparameters as in the previous section. It can be seen that Bmin is O(1) and adoptsmoderate values within a wide range of (a,Fl).2.4 Discussion and conclusionWe have developed a lubrication model of a 3-layer lubrication flow aimed atheavy-oil transport. We have then systematically explored the parametric varia-tions of the solutions, with the aim of establishing the feasibility of this methodol-ogy.56An attractive feature of our results is that many dimensionless parameter varia-tions do not appear to cause significant variation in the critical parameters (G,Q2,Q3)that would be considered as process costs. For example, (λ , l′) have very limitedeffect, m/δ affects primarily G. The flow rates of skin and lubricant are onlymarginally affected by any parameters other than r1, a and r2,0, but anyway re-main within ranges that are economic compared to current practices in lubricatedpipelining.The feasibility therefore rests with the range of Fl that is supported. Althoughwe may find Fl ≈ 2, these relate to larger pipelines and larger density differences.A great many situations are covered by Fl / 0.3 as explored above. Note also thatsmaller Fl are straightforwardly accommodated with smaller e. Nevertheless, weconsider our methodology to be targeted at modest diameter pipelines.To recover the meaning of Fl if we simply insert dimensional parameters intothe definition of Fl:Fl =piδ 22m(ρˆ [3]− ρˆ [1])gˆrˆ22,0µˆ [1]Wˆ0.For example, if 150kg/m3 density difference needs to be supported for a geo-metrical configuration, (r1,m/δ , l′,λ ) = (0.87,0.1,0.2,1), we find Fl ≈ 0.1 for(m, Rˆ,Wˆ0, µˆ [1]) = (0.005,0.1m,1m/s,10Pa.s). For these same parameters the solidwhite contour in Fig. 2.28 corresponds to a yield stress of τˆ [2]y ≈ 10Pa. Similarcalculations lead us to the conclusion that, for sensible λ , δ , and typical m, densitydifferences and mean velocities, we would be able to use this method in pipelinesof moderate radii, e.g. 10−20cm.Yield stress values in the range 1− 102Pa are easily realised in polymer gels,even at relatively low concentrations, and/or in various emulsions. Therefore, itwould appear that design of a suitable skin system is feasible, e.g. using an emul-sion of the transported oil & water. For extreme viscosities of oil, using viscosifiedlubricants (i.e. relative to water) can still provide a reasonable reduction G/Gowhile allowing for a designed variation of m/δ . Decreasing the density of thelubricant is less practical if water-based fluids are used.This estimates of required yield stress developed in this chapter are quite crude.In Chapter 3 we will develop estimates based on an axisymmetric computationalmodel and also on a reduced model that targets the flow, in Chapter 6 we explore in57more detail where these flows might produce a good reduction (G/G0), in regimeswhere two-layer CAF cannot be used.58Chapter 3Flow Development and InterfaceSculpting in Stable LubricatedPipeline Transport1In Chapter 2 we studied the steady periodic eccentric triple-layer flow, establishingthat a shaped rigid skin could generate sufficient lift force from viscous lubricationto balance buoyancy and estimated the yield stresses required in the skin for this tohappen. In this chapter, we focus on how the triple-layer flows might be developedin practice.We use both computational and analytical methods. We consider a concentricflow, within an inflow manifold designed to establish the triple-layer flow, and justdownstream of the manifold. The concentric flow is assumed, partly for practicalease and partly because the downstream eccentric configurations of Chapter 2 willanyway arise naturally as the flow develops, i.e. it is assumed that any initiallyconcentric buoyant core will rise slowly in the pipe until the buoyancy force of thecore fluid is balanced by the lubrication forces. The flow development as the corerises is studied in Chapter 5.In this chapter we therefore focus on the idea of controlled sculpting of a de-sired shape of interface into a visco-plastic (yield stress) fluid, which will itself hold1A version of this chapter has been published [158].59the interface shape after a (short) development length. The scenario considered isthat of a throughput Qˆ1 of heavy oil along a pipeline of radius rˆ1. To reduce pres-sure the flow passes through a concentric inflow manifold, within which the skinfluid and lubricant fluid are added, and continues to flow down a pipeline of radiusRˆ > rˆ1 in the multi-layer configuration at significantly reduced pressure drop.An outline of this chapter is as follows. In §3.1 we introduce a simple skinsculpting model, with control of the interface positions and skin shape via flowrate control. This relies on using the axisymmetric fully developed Poiseuille flowand is expected to be valid in the long-wavelength limit; see §3.1.1. This is usedthroughout the chapter, as inputs for both an axisymmetric two-dimensional (2D)computation and later for a simplified extensional flow model. Section 3.2 presentsresults from the axisymmetric 2D model. This shows that these flows can be estab-lished with a rigid skin and with reasonable control of the interface to approximatea designed shape. We study the effects of various flow parameters, explore de-tails of the stress fields generated and explore what happens when the skin yields.Yielding occurs when extensional stresses within the skin are generated from thenormal stresses transmitted from the lubricating layer. In §3.3 we develop a sim-plified semi-analytical model that is targeted at resolving the extensional stresses.This uses a long-wavelength approximation, retaining leading order shear and ex-tensional stresses, neglecting inertial terms. This provides a quick method for es-timating the interface behaviour and computing the minimal yield stresses neededto maintain the skin rigid; see §3.3.3. The chapter concludes with a brief summary.3.1 Modelling flow developmentOur aim in this chapter is to establish the feasibility of developing steady periodictriple-layer flows with a shaped interface of the type illustrated in Fig. 3.1. Thedevelopment flow is assumed to occur in an inflow manifold, with a concentricarrangement, through which the three fluids may be pumped at controlled flowrates. Fluid 1 denotes the core fluid (a viscous Newtonian heavy oil), with viscosityµˆ [1] and density ρˆ [1]. The skin layer is fluid 2, modeled simply as a Bingham fluid,with µˆ [2], τˆ [2]y , and ρˆ [2] denoting its viscosity, yield stress, and density, respectively.Fluid 3 is the lubrication layer (assumed to be a low viscosity Newtonian fluid)60Lubricant (3)Skin (2)Core fluid (1)Shaping zone(a) (b)(1)(2)(3)Figure 3.1: Schematic of the triple-layer CAF within an inflow manifold, to-gether with supposed velocity development.with viscosity µˆ [3], and density ρˆ [3].Two approximations are made throughout the chapter for simplicity. First, weignore density differences between the fluids. The inflow manifold is likely tobe horizontally oriented and buoyancy certainly has an effect on the steady estab-lished downstream flow [157], resulting in eccentricity of the core. However, itis assumed that this equilibrium eccentric configuration is achieved slowly down-stream. This allows us to consider a simplified axisymmetric flow, in which ourfocus is on shaping the interface and establishing the triple-layer structure. Sec-ondly, interfacial tension will be ignored. The key idea of the flow is use the yieldstress of fluid 2 to separate the other two fluids, with the implication that this is thedominant stress. Additionally in terms of application area, pipelining radii tend tobe in the range 5−50cm so that bulk capillary effects are minimal.The governing equations for the flow are the Navier-Stokes equations in eachfluid domain. The traction and velocity vectors are continuous across each interface(neglecting interfacial tension, as argued above). Constitutive equations for thethree fluids are:τˆi j [k] = µˆ [k] ˆ˙γi j, k = 1,3, (3.1)τˆ [2]i j =[µˆ [2]+τˆ [2]y∣∣ ˆ˙γ∣∣]ˆ˙γi j⇐⇒ τˆ [2] > τˆ [2]y , (3.2)ˆ˙γ = 0 ⇐⇒ τˆ [2] ≤ τˆ [2]y , (3.3)whereˆ˙γi j =∂ uˆi∂ xˆ j+∂ uˆ j∂ xˆi,61ˆ˙γ =[123∑i, j=1[ ˆ˙γi j]2]1/2, τˆ [2] =[123∑i, j=1[τˆ [2]i j ]2]1/2. (3.4)In order to scale the equations, we focus on transport of the core fluid, whichis assumed to have flow rate Qˆ1. We use Qˆ1 to define the velocity scale: Wˆ0 =Qˆ1/piRˆ2, which is the mean velocity of the heavy oil, if transported alone in thepipe. We scale all lengths with Rˆ. The stress scale is µˆ [3]Wˆ0/Rˆ, used for boththe deviatoric stresses and pressure, representing the shear stress in the lubricationlayer.(r,z) =(rˆ, zˆ)Rˆ, t =tˆWˆ0Rˆ, u =uˆWˆ0, p = pˆRˆµˆ [3]Wˆ0, τ = τˆRˆµˆ [3]Wˆ0.This leads to a problem governed by 2 dimensionless radii, r1 & r2,0, where r2,0 isthe mean value of r2(z), and 4 further dimensionless groupsm =µˆ [3]µˆ [1], m2 =µˆ [2]µˆ [3], B =τˆ [2]y Rˆµˆ [3]Wˆ0, Re =ρˆRˆWˆ0µˆ [3].Here m 1 is the viscosity ratio of lubricant to transported fluid. The Binghamnumber B, is a measure of yield stress to viscous stress in the lubricant and Re is aReynolds number, also based on the lubricant. The parameter m2 plays little roleas for the most part the skin layer is unyielded.The dimensionless model considered in this chapter is as follows:∇ ·u = 0, (3.5)Re[∂u∂ t+(u ·∇)u]=−∇p+∇ · τ. (3.6)To treat this system analytically we assume that the three fluids are separated bytwo interfaces, that evolve following a kinematic condition. In dimensionless form,62the constitutive laws within each fluid areτ [1]i j =1mγ˙i j, (3.7)τ [3]i j = γ˙i j, (3.8) τ[2]i j =[m2+B|γ˙|]γ˙i j⇐⇒ τ [2] > B,γ˙ = 0 ⇐⇒ τ [2] ≤ B. (3.9)To solve (3.5) & (3.6) computationally, different options are available for dealingwith the different fluids. Here we adopt a volume-of-fluids approach, which isequivalent to advecting the volume fractions of the three fluids. Physically, thismay be interpreted as either the large Capillary number limit of an immiscibleflow, or the large Pe´clet number limit of a miscible flow. Thus, we track the fluidsvia:∂c[k]∂ t+u ·∇c[k] = 0, k = 1,2. (3.10)where c[k] represents the volume fraction of fluid k and c[3]= 1−c[1]−c[2]. The con-centrations are used to define τ in terms of the individual component fluid stressesτ [k], here by using a simple linear interpolation.Equations (3.5), (3.6) & (3.10) are solved over the domain 0 ≤ r ≤ 1 and 0 ≤z≤ L = Lˆ/Rˆ for various parameters to gain insight into the development flow; see§3.2. No-slip conditions are imposed at the walls, symmetry conditions on thez-axis and outflow conditions at the exit to the flow domain. The computationaldomain length L is chosen to be sufficiently longer than the interior walls of theinlet manifold and so that the flow dynamics downstream of the manifold appearindependent of the computational length. Hence an stress free outflow condition isimposed at z = L, see e.g. [32, 138, 151]. This leaves the inflow conditions, whichplay a critical role in controlling the shape of the skin.3.1.1 Skin sculptingThe axisymmetric flow described in the previous section has a steady fully de-veloped Poiseuille flow solution, as with most multi-layer flows. We first use thissolution to derive the relationship between the interface positions and the flow rates63in each fluid. We then invert this relationship: using the flow rates to control the(steady fully developed) interface positions. This inverse relationship is used tomap between a desired interface shape (the sculpted skin) and the required flowrates. Of course, whether the method works in practice depends on stability of theresulting flow and on the variations being sufficiently slow, in time and space, aswill be apparent later in the chapter.We look for an axisymmetric Poiseuille flow solution, driven by constant pres-sure gradient: G = − ∂ p∂ z > 0. The shear stresses are given by τ[k]rz =−Gr2 wherek = 1,2,3 and we assume that B is sufficiently large that the skin layer is unyieldedin this 1D solution. Thus, the skin moves with a constant plug speed Wp. ThePoiseuille flow solution is:W (r) =G4[(1− r22)+m(r21− r2)], 0≤ r < r1,G4(1− r22), r1 ≤ r < r2,G4(1− r2), r2 ≤ r < 1,(3.11)Dimensionless flow rates are found by integrating W (r) over each layer. We findthatQ1 =Gr214[(1− r22)+mr212], (3.12)and due to our scaling Q1 = 1, which leads to:G =4/r211− r22 + m2 r21, (3.13)andQ2 =G4(1− r22)(r22− r21) =1r21(1− r22)(r22− r21)1− r22 + m2 r21, (3.14)Q3 =G8(1− r22)2 =12r21(1− r22)21− r22 + m2 r21. (3.15)In terms of the sculpting problem, (3.14) & (3.15) are simply a mapping fromthe interface positions (r1,r2) 7→ (Q2,Q3). For a given lubricant m 1 is a fixed640.7 0.8 0.9 10.70.750.80.850.90.95100.10.20.30.40.50.6Figure 3.2: Flow rates (Q2,Q3) for different (r1,r2), for m = 0.005: solidshaded line is Q2; broken shaded line is Q3. The black broken lineshows the value of r2 where Q2 is maximal.process parameter. The transported fluid 1 is very viscous and often pipelined athigh flow rates. Hence practically speaking one might not want to disturb this flow,i.e. by augmenting a throughput of fluid 1 by the two outer layers. Therefore, letsassume that r1 is fixed, e.g. the radius of the inner pipe upstream of the inflowmanifold. Typical variations in (Q2,Q3) with (r1,r2) are sketched in Fig. 3.2 form = 0.005. We see that Q2 vanishes along r1 = r2, and Q3 vanishes as r2→ 1.Below the dotted line (indicating maximal Q2) and above r1 = r2 in Fig. 3.2,there is a large (control) region of feasible flow for sculpting. In this region wesee that the shaded contours of (Q2,Q3) are monotone and independent, meaningthat the mapping between (Q2,Q3) and (r1,r2) is 1-to-1 and invertible. At fixed r1,each r2 in this control region gives a unique relationship between Q2 and Q3. Al-ternatively, selecting Q2 and Q3 to satisfy this functional relationship produces thedesired r2. This mapping is effectively: (Q2,Q3)(t) 7→ (r1,r2(t)), holding r1 fixed.To produce a desired waveform that is sculpted into the skin-lubricant interface,we must also map from t 7→ z. The axial position of the skin/interface is simplyz =∫ tWp dt =∫ t 1r2111+ mr212(1−r22)dt. (3.16)65(a) (b)Figure 3.3: (a) Desired interface positions along the pipe: r2(z) = solid blackline (here (a,m) = (0.15,0.005)); r1 = broken black line. (b) Q2 (solidblack line) and Q3 (broken black line) needed to shape sawtooth skin-lubrication interface of (a).0 5 101.6341.6351.6361.637Figure 3.4: Plug velocity variation of example 1 in Fig. 3.3.For small m and constant r1, the plug velocity is almost constant and this mappingis nearly linear.In practice the above computations are easily resolved and we illustrate themapping with a couple of examples. For these, we describe our desired r2(z) as thesum of a mean value and a perturbation: r2 = r2,0+a(1− r2,0)Φ(z). Here, Φ(z) isthe shape of interface, which has mean value equal to zero and maximum absolutevalue of 1; a < 1 measures the perturbation amplitude. Example 1 is a sawtoothprofile (as explored previously in Chapter 2); Fig. 3.3 shows the desired interfacepositions and computed required flow rates. The plug velocity variation for thisexample is shown in Fig. 3.4.As a second example, to illustrate the robustness of this procedure we calcu-late the flow rates necessary to produce the dinosaur profile illustrated in the inset66(a) (b)Figure 3.5: (a) Desired interface positions along the pipe: r2(z) = solid blackline (here (a,m) = (0.15,0.005)); r1 = broken black line. (b) Q2 (solidblack line) and Q3 (broken black line) needed to shape the dinosaurskin-lubrication interface of (a).0.8 0.85 0.9 0.95 100.20.40.6Figure 3.6: Flow rates for different r2, where (r1,m) = (0.78,0.005). : Q2,  : Q3, F : Q2 +Q3. The black broken line shows the valueof r2 where Q2 is maximal.of Fig. 3.5a. Again the calculation of the input flow rates presents no difficulty;see Fig. 3.5b. We observe that the sculpted profile is essentially mimicked in thevariation of Q2, as the plug velocity does not vary much and the skin flow rate issimply area × plug velocity. To make transparent the ease of this process, Fig. 3.6shows the two flow rates plotted against r2 for (r1,m) = (0.78,0.005) (as in theseexamples): Q2 increases and Q3 decreases, both approximately linearly over thisrange.67Finally, we can make a first estimate of the minimum yield stress requiredto keep the skin layer completely unyielded, from Bmin > max(Gr2/2). We canexpress G in terms of the total flow rate:G =8(1+Q2+Q3)(1− r42)+mr41, (3.17)and have: r22 = 1+m2 r21− 4r21G . After some algebra:G =1r21−11r21− (1+Q2+Q3)2r21(1− r21), (3.18)and bounding r2 < 1:Bmin =G2=1r21−11r21− (1+Q2+Q3)1r21(1− r21), (3.19)which has the advantage of expressing Bmin in terms of r1 and the total flow rate:quantities that may be more amenable to measurement. As an example, for (r1,r2,m,µˆ [1], Rˆ,Wˆ0) = (0.8,0.9,0.005,10Pa.s,10cm,1m/s), Bmin = 16.07 which resultsτˆ [2]y = 8.03Pa. In practice, this estimate of Bmin is quite conservative. The varyingskin amplitude produces extensional stresses (that we estimate later) and furtherdownstream the eccentrically positioned steady flow produces even larger lubrica-tion shear stresses that must be bounded; see Chapter 2.3.1.2 Computational methodOur main computational results are obtained by solving the model equations (3.5),(3.6) & (3.10) numerically with the boundary conditions described. At time t,according to the desired interface profile (r1(t),r2(t))we compute the relevant flowrates corresponding to the Poiseuille flow described in §3.1.1, as this is the shapewe want downstream of the inflow manifold. According to the three flow rates(1,Q2,Q3) we impose a uniform inflow velocity at z = 0 for each fluid stream. Asseen in Fig. 3.7, within the manifold the three inlet streams are separated with shortO(1) and thin (δ = 0.02) walls, which allow the individual streams to become fully68𝑟𝑧𝑟2𝑟11𝐿𝑢 = 𝜏𝑟𝑧 = 0𝑢 = 𝑤 = 0𝑄1 = 1𝑄2 𝑡𝑄3 𝑡 OutflowCore fluidSkinLubricant(b)(a)𝜉1𝜉2Figure 3.7: (a) Schematic of the dimensionless triple-layer flow geometry inan axisymmetric pipe; (b) detail example computational mesh close tothe manifolds.developed before merging. Equally the inlet walls have different lengths which aredenoted by ξ1 and ξ2, so that the interface between fluids 2 & 3 forms before theinterface between fluids 1 & 2 (ξ1 > ξ2). The reverse manifolds, i.e. inner manifoldshorter than the outer one (ξ1 < ξ2), is examined and the results are presented laterin §3.2.5. Initial conditions are set to be a motionless uniform concentric flow, withc[k] = 1 only in each fluid k layer (c[k] = 0 elsewhere). Inflow conditions for eachfluid stream are also c[k] = 1 at inlet k and c[k] = 0 for the other inlets.The model equations are discretized using a mixed finite-element/finite volumemethod. The weak form of Navier-Stokes equations are solved with the Galerkinfinite-element method using iteration. The computations are carried out on a struc-tured rectangular mesh, with linear elements (Q1− iso−non−con f ) for the veloc-ity and constant elements (Q0) for the pressure discretization. The divergence-freecondition is enforced by an augmented Lagrangian technique [48]. And, with thischoice of elements, the inf-sup condition is satisfied. The skin layer contains a yieldstress fluid which presents some computational complications; see e.g. [39, 153].Here we use the augmented Lagrangian method [36] to handle the yield stress be-haviour.The augmented Lagrangian method introduces additional relaxed variables,to avoid the singular viscosity arising at yield surfaces and stress indeterminacywithin plug regions. The additional relaxed strain rate and the Lagrange multi-69plier (deviatoric stress) fields are also discretized using constant elements (Q0).At each time step, the velocity and pressure, Lagrange multiplier and strain rateare updated sequentially using an Uzawa algorithm and the entire system is iter-ated until converged. More details of the algorithm can be found in [69, 70, 185],which are identical except for the different choice of elements. The calculationis implemented in C++ as an application of PELICANS. A brief introduction toPELICANS is provided in appendix (§A.1).The concentration equations (3.10) use a finite volume scheme with the Piece-wise Linear Interface Calculation (PLIC) volume-of-fluid method [139] to trackthe motion of interfaces (A brief explanation of PLIC method can be found in ap-pendix §A.1.1). This method reconstructs interfaces by segmenting each cell, andthen advects the interfaces using a Lagrangian method. We use 250 elements acrossthe pipe and 20 elements per unit radius along the pipe. The radial discretization isvery fine but allows the triple-layer structure to be resolved.For the results presented below we set the length of computational domain toL= 15 and mostly have modest Re. We have also experimented with L= 10−30 indifferent calculations performed. The main point here is that the flows consideredhave large yield stress (B) and m 1, with the consequence that fluid layers 1& 2 have large (effective) viscosity and the development lengths are consequentlyshort. The downstream flows are then relatively insensitive to larger L. Apartfrom the large effective viscosity, note that once the unyielded skin forms, fluids 1and 3 are separated and have development lengths that scale with a representativereduced width, i.e. 1− r2,0 in fluid 3. Later in the chapter we shall develop a long-wavelength extensional flow model that explicitly assumes (1− r2,0)/L 1.As an example of the convergence behaviour of the discretized system, we usethe Poiseuille solutions in §3.1.1 to define an analytical solution W (r). We in-tegrate the model equations from initial conditions until steady, imposing only aconstant flow rate. We compare the numerical solution at z = L with the analyti-cal solution. For parameters (r1,r2,0,m,m2,Re,L) = (0.5,0.8,0.005,200,10,15),Fig. 3.8a shows the difference between the analytical solution and the computedsolution at the exit, for decreasing mesh sizes and fairly typical parameters. Theconcentration c[2] at the exit is shown in Fig. 3.8b, in which we see that the jump inconcentration is smeared over a few cells and mesh refinement reduces the thick-70(a) (b)(c) (d)c[2]Figure 3.8: Effect of mesh size on: (a) L2 norm of difference between exit ve-locity w(r,L) and analytical velocity W (r); (b) exit concentration profileof skin fluid c[2]; (c) exit velocity profile; (d) detail of the exit velocity influid 1 close to the interface. Flow parameters are (r1,r2,0,m,m2,Re,L)= (0.5,0.8,0.005,200,10,15). ∆r is the radial mesh size.ness of this diffuse layer. The computed exit velocity is shown in Fig. 3.8c, againfor the different radial mesh sizes. The fluid 1 & 2 velocities appear plug-like atthis scale, but in fact only fluid 2 is unyielded. Figure 3.8d shows the fluid 1 veloc-ity and part of the fluid 2 layer velocity; the former has a parabolic profile (barelyvisible in Fig. 3.8c due to the small value of m).713.2 Computational resultsWe now present a number of computational examples that establish the feasibilityof sculpting the triple-layer flow via the method explained. For most of these wechoose B large enough to maintain the skin layer completely unyielded downstreamof the inflow manifold, although we also look at what happens when the skin yields.For the computed examples we have thicker skin and lubrication layers than mightbe used in practice, so that we can better visualize the variations in sculpted skinshape. The main dimensionless parameters are Re, m, B. The flow rates of skinlayer and lubricating layer are then controlled to shape the skin-lubricating layerinterface.3.2.1 Rigid skinHere we fix parameters as (r2,0,a,r1,B,m,Re,T,Tb,L) = (0.75,0.6,0.4,500,0.001,10,2,0.5,15) and vary inflow rates to produce a sawtooth wave on fluid 2 & 3 in-terface. Here T is the period of the sawtooth wave and Tb < T is the time atwhich sawtooth attains its maximum value of r2(t), according to the sculpted in-flow rates. Figure 3.9 shows the interface development after imposing periodicflow rates (Q2,Q3). Initially, the pipe is filled with three fluids and parallel inter-faces in equilibrium positions. The sawtooth wave shape develops in time near theend of the inflow manifold and we can see it remains sculpted into the interface asit propagates along the pipe.The velocity magnitude (speed) colormap is shown in Fig.3.10. Unyieldedplug regions in the skin are indicated by a grey region on the speed colormap.Because of the high yield stress, B, the plug forms in the fluid 2 layer soon afterthe inflow manifold and does not break during the flow rate oscillations. Note thata key reason for using the augmented Lagrangian method in our computation wasto properly represent the unyielded regions of the flow. We observe a thin lineof yielded material positioned downstream of the outer manifold and extendinginto the skin plug. This appears to be related to downstream decay of the stresssingularity at the corners of the skin-lubrication manifold.Figure 3.11 shows each component of the deviatoric stress as colormaps at t = 1and t = 2. It can be seen that within the skin layer the shear stress is dominated by72ξ1ξ2Figure 3.9: Concentration colormap for t =1, 1.5, 2, 2.5, 3, 3.5, 4, 4.5, 5, 5.5, 6, 6.5, respectively; where(r2,0,a,r1,B,m,Re,T,Tb,L) = (0.75,0.6,0.4,500,0.001,10,2,0.5,15).Dark blue = fluid 1; light blue/green = fluid 2; yellow = fluid 3.73  Figure 3.10: Speed colormap for t = 1, 1.5, 2, 2.5, 3, 3.5, 4, 4.5, 5, 5.5, 6, 6.5,respectively; where (r2,0,a,r1,B,m,Re,T,Tb,L) = (0.75,0.6,0.4,500,0.001,10,2,0.5,15). Grey region shows unyielded fluid 2 (plug).the normal stresses. The normal stresses are otherwise largely insignificant awayfrom the inflow manifold in fluids 1 & 3, which are primarily shear flows. In-deed, the shear stresses are of comparable size across the pipe. The largest normalstresses appear to be generated within the inflow manifold and just after, decay-ing slightly with distance downstream. The three normal stress components are ofcomparable size and oscillate both spatially and temporally, i.e. between extensionand compression.To understand these variations we first simplify by averaging the normal stresses74  (a)ξ1ξ2(b)ξ1ξ2(a)(b)Figure 3.11: Stress components at: a) t = 1 and b) t = 2 for(r2,0,a,r1,B,m,Re,T,Tb,L) = (0.75,0.6,0.4,500,0.001,10,2,0.5,15).In each panel, from top to bottom we show: the position of the skinlayer, then τzz, τrz, τθθ , τrr. The colorbar scale is the same for allstresses.750.812 4 6 8 10 12 14-20002000.812 4 6 8 10 12 14-20002000.812 4 6 8 10 12 14-20002000.812 4 6 8 10 12 14-20002000.812 4 6 8 10 12 14-20002000.812 4 6 8 10 12 14-2000200Figure 3.12: Averaged normal stresses in skin layer τ¯s,i: vari-ation with z for different times (T denotes one pe-riod of wave),  : τ¯s,r,  : τ¯s,z, F : τ¯s,θ ; where(r2,0,a,r1,B,m,Re,T,Tb,L) = (0.75,0.6,0.4,500,0.001,10,2,0.5,15).The narrow panel at the top of each figure shows the skin-lubricationinterface position r2(z).((τs,1,τs,2,τs,3) = (τrr,τθθ ,τzz)) across the skin layer:τ¯s,i =∫ 10rc[2]τs,i dr/∫ 10rc[2] dr. (3.20)Although this definition has only physical meaning as a stress for τzz, as it is aver-aging over its acting surface, it gives a valid definition for an averaged-skin valuefor τrr and τθθ . Variation in the averaged normal stresses in the skin with z atdifferent times is shown in Fig. 3.12. The narrow panel at the top of each figureshows the skin-lubrication interface position (r2(z)). To some extent τ¯rr appears torespond to the changes in r2 and mostly τ¯zz has an opposite sign. We also see thatall the normal stress components decay in magnitude with z, although this may bea consequence of the outflow condition at z = L. Note that within the unyieldedplug the computed stresses are indeterminate, i.e. those computed are admissiblebut not necessarily unique.760 2 4 6 8-50050Figure 3.13: Entry average extensional stress(τ¯zz|z=1) variation in time; where(r2,0,a,r1,B,m,Re,T,Tb,L) = (0.75,0.6,0.4,500,0.001,10,2,0.5,15).Vertical dashed line indicates the time period T and  symbols indi-cate when the inflow rates are designed to give maximal r2.The influence of the oscillation on the normal stresses is less ambiguous closeto the inflow. Figure 3.13 shows variation of the average extensional stress nearthe entry (z = 1). We can see that time-periodic behavior in τ¯zz|z=1 is beginningto emerge after the first few time periods. The dash line shows the time period(T = 2) and the square symbols indicate when the inflow rates are designed to givemaximal r2 (i.e. t = Tb, modulo T ). This appears to coincide with the maximum inextensional stresses, as might be intuitive.Figure 3.14 shows the velocity development in the entry part of pipe. Becauseof the high Bingham number B, the visco-plastic fluid flows as a plug in the middleannular region, developing rapidly. The skin-lubrication manifold is shorter andwe see this interface develops soon after exiting. At this particular time, we seethat the skin layer spreads rapidly outwards: in the radial direction the velocityis plug-like, but is varying in the streamwise direction. After exiting the secondmanifold (core-skin), we see a fairly rapid development of the core-skin interface.Later in §3.2.5 we explore other possible manifolds.Figure 3.15 shows the output skin wave shape of the simulation after two timeperiods, compared against the input shape. This improves slightly with t as the771  0.80.60.40.20  Figure 3.14: Axial velocity development inthe inflow manifold at t = 4; where(r2,0,a,r1,B,m,Re,T,Tb,L) = (0.75,0.6,0.4,500,0.001,10,2,0.5,15).For zk = 0, 0.25, 0.35, 0.5, 0.75, 1, 2, 5, 6, 8, 14.solution becomes progressively periodic, but there remains a discrepancy wherethere are rapid changes in inflow rates. During these time periods, transients inthe development region are largest and allow some relaxation from the intendedshape. In Fig. 3.15b, we show the results of varying the inflow rates to get theprevious dinosaur shaped interface (again at t = 2T = 4). The agreement betweensimulation output and interface shape input is surprisingly good for this complexshape.3.2.2 Shorter wavelengthsAs we decrease the wavelength of the inflow oscillation, we expect to move fur-ther from the long-wavelength/slowly varying scenario where the control on inputflow rates is expected to predict the desired output interface shape. We might alsoexpect that streamwise gradients in the various stresses become more significant.Figure 3.16 shows the result of halving the period (T = 1) on the previous compu-tations. After the initial start-up transients have exited the domain we see that theinterfaces adopt established positions for which the sawtooth profile is evident andhas amplitude similar to that in Fig. 3.9 earlier.The velocity variations are illustrated in Fig. 3.17 and appear comparable to78(a) (b)Figure 3.15: Results of imposing (a) the sawtooth shape with period T = 2and (b) the dinosaur skin shape with period T = 2: solid line = intendedshape and (–) simulation result, after 2 time periods; parameters are(r2,0,a,r1,B,m,Re,L) = (0.75,0.6,0.4,500,0.001,10,15).ξ1ξ2Figure 3.16: Concentration colormap for t =0.3, 0.5, 1, 1.3, 1.5, 2, 3, respectively; where(r2,0,a,r1,B,m,Re,T,Tb,L) = (0.75,0.6,0.4,500,0.001,10,1,0.3,15)79Figure 3.17: Speed colormap for t = 0.3, 0.5, 1, 1.3, 1.5, 2, 3, respectively;where (r2,0,a,r1,B,m,Re,T,Tb,L) = (0.75,0.6,0.4,500,0.001,10,1,0.3,15).The shaded grey region indicates unyielded plugs within the skinlayer.those for the longer wavelength case Fig. 3.10, including the size of developmentregion before the plug forms in the skin layer.Figure 3.18 shows the oscillations in the skin-averaged normal stresses as thesimulation progresses; compare with Fig. 3.12. As before, the three componentshave comparable magnitude. The streamwise variation is increased, due to theshorter wavelength, but the overall sizes of stress variation are similar to that forthe longer wavelength case. The overall impression that we have is that the de-velopment of these flows, formation of the unyielded shaped skin, size of stressesgenerated etc., is not particularly sensitive to the period T , at least over some rangeof wavelengths that are significantly larger than the interface amplitudes generated.3.2.3 Yielding of the skinHaving computed flows with an unyielded and sculpted skin, we are able to esti-mate the maximal stresses and then re-compute with a reduced yield stress B toobserve what happens when the skin yields. Here we present one example, similar800.812 4 6 8 10 12 14-20002000.812 4 6 8 10 12 14-20002000.812 4 6 8 10 12 14-20002000.812 4 6 8 10 12 14-20002000.812 4 6 8 10 12 14-20002000.812 4 6 8 10 12 14-2000200Figure 3.18: Averaged normal stresses in skin layer τ¯s,i: vari-ation with z for different times (T denotes one pe-riod of wave),  : τ¯s,r,  : τ¯s,z, F : τ¯s,θ ; where(r2,0,a,r1,B,m,Re,T,Tb,L) = (0.75,0.6,0.4,500,0.001,10,1,0.3,15).The narrow panel at the top of each figure shows the skin-lubricationinterface position r2(z).to those above but with B reduced from 500 to 150. The concentration colormapis shown in Fig. 3.19, which can be compared with Fig. 3.9. Although similar, itappears that the skin-lubrication interface profile is less angular than in Fig. 3.9and that there are slight bulges in the core-skin interface.Figure 3.20 plots τzz and unyielded regions in the skin layer. We see that theskin yields for large positive τzz, which occurs just after the maximal skin layer,i.e. in these regions the skin is stretched axially. This axial stretching as the fluidyields also accounts for the less angular sawtooth shapes, comparing Fig. 3.19 withFig. 3.9. Where the skin yields is also where we observe the non-uniformities inthe inner interface.Figure 3.21 shows the yielded and unyielded regions of the skin layer plottedas a spatiotemporal plot, over one time period. To construct this plot we have post-processed our 2D results. A position (z, t) is only shaded as unyielded if the entirethickness of the skin layer is unyielded. We have seen in Fig. 3.20 that the skin81ξ2 ξ1Figure 3.19: Concentration colormap for t = 1, 2, 3, 4, 5, respec-tively; where (r2,0,a,r1,B,m,m2,Re,T,Tb,L) = (0.75,0.6,0.4,150,0.001,1000,10,2,0.5,15).Figure 3.20: τzz colormap for t = 1, 2, 3, 4, 5, respectively; where(r2,0,a,r1,B,m,m2,Re,T,Tb,L) = (0.75,0.6,0.4,150,0.001,1000,10,2,0.5,15). Grey region indicates the plug.82Figure 3.21: Spatiotemporal variations in the yielded and unyielded parts ofthe skin; where (r2,0,a,r1,B,m,m2,Re,T,Tb,L) = (0.75,0.6,0.4,150,0.001,1000,10,2,0.5,15). Grey region is plug and the white region isyielded skin layer.yields when we have large positive τzz. From Fig. 3.21 we see that the length of theyielded skin region remains nearly the same at each time and because of the smallvariations in skin velocity, the unyielded regions are advected with same velocityalong the pipe (straight line in Fig. 3.21).3.2.4 Varying m and ReWe have computed a large number of flows, centered around the base set of param-eters (r2,0,a,r1,T,Tb,L) = (0.75,0.6,0.4,2,0.5,15), considered in our examplesabove and with the same sawtooth pattern skin-lubrication interface. Many of theseare qualitatively similar to those already illustrated. A critical design parameter inthese flows is the critical value of B, say Bcr, below which the skin yields down-stream of the manifold, i.e. this determines a critical yield stress for the skin.Figure 3.22a shows variations in Bcr with the viscosity ratio m. Figure 3.22bshows the corresponding variations in axial pressure gradient. Both increase sig-nificantly with m, as expected due to the increased stress gradient generated in thelubrication layer (which translate to pressures, normal and extensional stresses inthe skin). Evidently, the idea of drag reduction in these layered flows limits our83(a) (b)Figure 3.22: Variation of (a) minimal required Bingham num-ber Bcr to keep the skin completely unyielded (b) pres-sure drop along the pipe with viscosity ratio m; where(r2,0,a,r1,B,Re,T,Tb,L) = (0.75,0.6,0.4,1000,10,2,0.5,15).interest to m 1.Reynolds number effects on Bcr are much more significant; see Figure 3.23.As Re increases the position of maximal stress in the skin also shifts downstreamas well as the value increases. This suggests that the increases in Bcr is linked tochanges in the development length. This is an area that needs further study, in orderthat these flows can be achieved at significant Re.3.2.5 Manifold designAll simulations performed above have used the same inflow manifold design, inwhich the skin-lubrication manifold is shorter than that of the core-skin (ξ1 > ξ2).The walls of our manifolds have finite thickness and we have seen some singu-lar behavior in the stresses, associated with the corner of these walls. Evidently,that aspect of the geometry could be improved in any practical design, but for thequadrilateral elements used in our computational code, these simple geometries areeasy to implement.In general, for a given design of skin-lubrication interface shape it seems ad-visable to position the manifold walls approximately at r1 and r2,0. For the flowsconsidered we have selected the core-skin manifold radius in this way, which wouldanyway be practical as the inner radius is kept constant in the procedure of §3.1.1.84(a) (b)Figure 3.23: Effect of Re on minimal required Bingham num-ber Bcr to keep the skin completely unyielded (b) av-erage of stress magnitude (|τ¯| =∣∣∣∫ 10 c[2]rτdr/∫ 10 c[2]rdr∣∣∣,τ =(1/2Στ2i j)1/2) at t = T and Re = 1, 10, 50, 100, 200; where(r2,0,a,r1,B,m,T,Tb,L) = (0.75,0.6,0.4,5000,0.001,2,0.5,15).For the outer manifold radius we have instead taken a radius less than r2,0, so thatin general the skin layer expands. It is unclear if varying this manifold radius mayhave consequences for the flow stability. In a broader view of this process, it islikely that r1 might anyway be fixed by the upstream radius of the pipeline de-livering the core fluid to the manifold. On the other hand, according to the fluidproperties, one might experiment with different sculpted shapes and mean flowrates for skin and lubricant. Thus, having the outer manifold wall matching r2,0 isperhaps impractical.We have explored the question of which manifold wall should be longer, find-ing that the configuration used tends to produce more stable flows. As an illustra-tion, Figs. 3.24 & 3.25 show concentrations and speed colormaps for the same baseset of parameters as previously, but with the short and long manifold walls reversed(ξ1 < ξ2). We observe that the skin layer now initially expands inwards towardsthe core fluid, while there is still a skin-lubrication solid wall. Further downstreamthe core and skin bulge outwards. It should be mentioned, there is a thin layerof lubricating fluid with thickness of a cell in the bulged skin area, otherwise thecomputations will stop. The unyielded skin does not form along the length of the85ξ1ξ2Figure 3.24: Concentration colormap at times t = 2, 6; where(r2,0,a,r1,B,m,m2,Re,T,Tb,L) = (0.75,0.6,0.4,500,0.001,1000,2,0.5,15)and ξ1 < ξ2.Figure 3.25: Speed colormap at t = 2, 6, respectively; where(r2,0,a,r1,B,m,Re,T,Tb,L) = (0.75,0.6,0.4,500,0.001,2,0.5,15)and ξ1 < ξ2. The grey region indicates the plug.computational domain.3.3 Extensional modelAs observed in the previous section, the main contribution to the deviatoric stresswithin the skin layer comes from the normal stresses, i.e. these dictate the mini-mal yield stress required to maintain the skin rigid. Therefore, here we exploit theaspect ratio of the flow to derive a simplified model that will allow us to approx-imate the extensional stresses within the skin. To derive the model we return todimensional variables to discuss the various scales.3.3.1 Scaling the equationsWe derive the model under the assumption of small aspect ratio. We suppose thatthe desired sculpted interface is to be periodic in zˆ, with period Lˆ. Variations inthe downstream flow of the skin and lubricant occur due to variations in interfaceshape, hence layer thickness, which have amplitude Rˆ(1− r2,0). Therefore, an86appropriate aspect ratio ε to consider is:ε =Rˆ(1− r2,0)Lˆ. (3.21)We scale zˆ with Lˆ: z˜= zˆ/Lˆ. In the lubricant and skin layers we scale radial distancesonto x:rˆ = rˆ2,0+(Rˆ− rˆ2,0)x ⇒ x = r− r2,01− r2,0 .In terms of the x-coordinate the skin occupies x1≤ x≤ x2, with x1 =(r1−r2,0)/(1−r2,0)< 0 and x2 = aΦ(z˜). The lubricant occupies x2 ≤ x ≤ 1. The core fluid layeris not assumed thin: hence r = rˆ/Rˆ for fluid 1.As before, the axial velocities are scaled with Wˆ0, which we recall was definedfrom the fixed steady imposed flow of fluid 1. Flow rates at the inflow are definedby the procedure in §3.1.1. On scaling with piRˆ2Wˆ0 the individual flow rates at theinflow in fluids 1-3 are 1, Q2(t) and Q3(t), respectively. The dimensionless timehere is t = Wˆ0tˆ/Lˆ.Radial velocities in each layer are scaled with εWˆ0. In the skin and lubricantlayers this follows from the differential scaling of (rˆ, zˆ). In the core fluid, wesuppose that the axial velocity w[1] depends on (z˜, t) only via the interface positionsr1 and r2. Implicitly then∂w[1]∂ z˜=∂w[1]∂ r1∂ r1∂ z˜+∂w[1]∂ r2∂ r2∂ z˜= O(ε) ⇒∣∣∣∣∣∂u[1]∂ r∣∣∣∣∣= O(ε).Thus, on integrating the continuity equation radially we find u[1] ∼ O(ε) in thecore.We now clarify the previous notion of long-wavelength approximation, in whichnonlinear inertial terms and imposed accelerations are neglected. This is validwhen the viscous terms dominate the momentum balance. There are two potentialviscous timescales:tˆv,1 =ρˆ [1]Rˆ2µˆ [1], tˆv,3 =ρˆ [3](Rˆ− rˆ2,0)2µˆ [3],i.e. assuming that the skin layer (which is also thin) responds much faster than87the lubricant layer. The advective timescale is tˆa = Lˆ/Wˆ0, which we note is alsorepresentative of the period of the interface oscillation. The long-wavelength as-sumption is that ε  1, and that:max{tˆv,1, tˆv,3} tˆa. (3.22)We see that in terms of the dimensionless groups used earlier, (3.22) implies:mReλpi 1, and εRe(1− r2,0) 1. (3.23)Here λ = Lˆ/piRˆ ∼ O(1) is the ratio of axial to circumferential length-scales, useddownstream for the steady established flow in Chapter 2. Typically m < ε for usualheavy oils and λ ∼ OS(1) so that (3.23) amounts to the usual lubrication/thin-filmapproximation for neglecting inertial terms.We now consider the different scales for the stress present in each layer. Firstly,note that pressure losses are generated by shear in the lubrication layer, leading tothe following pressure scale:Pˆ∗l =µˆ [3]Wˆ0LˆRˆ2(1− r2,0)2, (3.24)which balances with the leading order shear stress gradients:|τˆ [3]rz | ∼ µˆ [3] Wˆ0Rˆ(1− r2,0)= εPˆ∗l . (3.25)Regardless of the state of the skin, the traction vector is continuous between twomedia, meaning normal and tangential components are transmitted across the in-terfaces. Thus, the same pressure scale is used in each fluid layer and we expect(3.25) to give the shear stress scale in the skin and core fluids. For the normaldeviatoric stress components τˆ [k]rr ∼ τˆ [k]zz , in the core and lubricant layers, which areNewtonian, the velocity scaling implies τˆ [k]rr ∼ τˆ [k]zz ∼ ε|τˆ [3]rz | ∼ ε2Pˆ∗l . Within theskin layer, as we are interested in large yield stresses where the skin is unyieldedor just yielded, we expect an extensional flow. The pressure in the Newtonian lay-ers is balance by the total normal stress, i.e. part of the pressure generated in the88lubrication layer may be absorbed by τˆ [2]rr , i.e. τˆ[2]rr ∼ τˆ [2]zz ∼ Pˆ∗l .Lubricant layerWe now construct the solution, from the pipe wall inwards. The leading ordermomentum equations are:−∂ p[3]∂x= 0, (3.26a)−∂ p[3]∂ z˜+∂τ [3]xz∂x= 0. (3.26b)The next order viscous stresses are O(ε2) and the neglected inertial terms areO(εRe(1− r2,0)) in the z-momentum equation. The pressure is constant acrossthe lubricating layer. The shear stress and the velocity are found using conditionsw[3] =Wp at the skin and w[3] = 0 at the wall.τ [3]xz =2x−1− x22∂ p[3]∂ z˜+Wpx2−1 , (3.27)w[3] =(x−1)(x− x2)2∂ p[3]∂ z˜+Wpx−1x2−1 , (3.28)Here Wp denotes the plug velocity, which we note it is also the interfacial velocityto leading order, (see below).Skin layerWith the assumed stress scales, the leading order momentum equations are:−∂ p[2]∂x+∂τ [2]xx∂x= 0, (3.29a)−∂ p[2]∂ z˜+∂τ [2]zz∂ z˜+∂τ [2]xz∂x= 0. (3.29b)Normal and tangential stresses continuity at the x2 are:− p[3](x+2 ) = τ [2]xx (x−2 )− p[2](x−2 ), (3.30)89τ [3]xz (x+2 )+∂x2∂ z˜p[3](x+2 ) = τ[2]xz (x−2 )−∂x2∂ z˜[τ [2]zz (x−2 )− p[2](x−2 )],Noting that at leading order, τ [2]zz +τ[2]xx = 0, in the skin, the above is combined with(3.30) to give:τ [3]xz (x+2 ) = τ[2]xz (x−2 )−2∂x2∂ z˜τ [2]zz (x−2 ). (3.31)From the x-momentum equation and normal stress conditions, we see in particularthat p[2]− τ [2]xx is constant across the skin, transmitting the lubricant layer pressure.As the leading order deviatoric stresses are normal stresses, these will deter-mine the leading order strain rates. Having shear stresses one order smaller in εimplies the same for the shear component of the strain rate. This is compatibleonly with an axial velocity of formw[2](x, z˜, t) =Wp(z˜, t)+ ε2w˜[2](x, z˜, t). (3.32)Thus, at leading order the plug and interface velocities are identical. Although werefer to Wp(z˜, t) as a plug velocity, extensional strain rates may still occur at leadingorder. These are related to τ [2]zz by the leading order constitutive relation:∂Wp∂ z˜=12m2sgn(τ [2]zz )max(|τ [2]zz |− B˜,0). (3.33)Where B˜ = τˆ [2]y /Pˆ∗l . Note that the different Bingham number (compared to thecomputational model) comes from the scaling of the extensional model. The twoBingham numbers are related by: B˜/B = piλε2.Core fluidAt the core-skin interface, x = x1 (r = r1) we have:−p[1](x−1 ) = τ [2]xx (x+1 )− p[2](x+1 ) (3.34)τ [1]xz (x−1 )+∂x1∂ z˜p[1](x−1 ) = τ[2]xz (x+1 )−∂x1∂ z˜[τ [2]zz (x+1 )− p[2](x+1 )] ⇒1m∂w[1]∂ r= τ [1]rz (r−1 ) = τ[1]xz (x−1 ) = τ[2]xz (x+1 )−2∂x1∂ z˜τ [2]zz (x+1 ). (3.35)90Shear stress in the core fluid is only generated at the interface, confirming thatthe choice of scaling (τˆ [1]rz ∼ εPˆ∗l ) is correct. The core axial velocity will have aconstant component Wp and a radially varying component due to the shear above.The shear component is found from the leading order momentum equations:∂ p[1]∂ r= 0, (3.36a)∂ p[1]∂ z˜= εpiλ1r∂∂ r(rτ [1]rz ) =εpiλm1r∂∂ r(r∂w[1]∂ r). (3.36b)We find:1m∂w[1]∂ r= τ [1]rz =1εpiλr2∂ p[1]∂ z˜, ⇒ w[1] = m4εpiλ∂ p[1]∂ z˜(r2− r21)+Wp. (3.37)3.3.2 Solution method and interface evolutionWe first consider solving for the normal stresses in the skin and the plug velocity.Using the normal stress continuity conditions we define P(z˜, t) as:P(z˜, t) = p[1](z˜, t) = p[3](z˜, t) = p[2](z˜, t)− τ [2]xx (z˜, t). (3.38)Substituting P for p[2] into the z-momentum equation in the skin, we integrateacross the skin, and substitute from the tangential stress conditions:2∫ x2x1∂∂ z˜τ [2]zz dx = (x2− x1)∂P∂ z˜ + τ[2]xz (x+1 )− τ [2]xz (x−2 ),= (x2− x1)∂P∂ z˜ + τ[1]rz (r−1 )− τ [3]xz (x+2 )+2∂x1∂ z˜τ [2]zz (x+1 )−2∂x2∂ z˜τ [2]zz (x−2 ),Substituting the shear stress expressions from core and skin leads to the followingequation for development of the mean extensional stresses along the skin:2∂∂ z˜[∫ x2x1τ [2]zz dx]= (x2− x1)∂P∂ z˜ + τ[1]rz (r−1 )− τ [3]xz (x+2 )=[r12εpiλ+1+ x2−2x12]∂P∂ z˜+Wp1− x2 . (3.39)91The right-hand side contains a linear combination of ∂P∂ z˜ and Wp. The pressuregradient can be eliminated by considering the flow rate along the pipe, as we nowexplain.The oil flow rate and the interface positions at the inflow (z˜ = 0) are the designparameters of the flow, assumed to be designed via the procedure of §3.1.1, i.e. atz˜ = 0, the flow rates are 1, Q2(t) and Q3(t), in layers 1-3 respectively; Wp(0, t),r1(0, t) & r2(0, t) are each specified. Downstream, the flow rates in the individ-ual layers may vary, but with the total flow rate fixed from the inflow conditions(Q0(t) = 1+Q2(t)+Q3(t)):3∑k=1Q[k](z˜, t) = Q0(t), ⇒ ∂∂ z˜3∑k=1Q[k](z˜, t) = 0, (3.40)whereQ[1] = 2∫ r10rw[1] dr, Q[2] = 2∫ r2r1rw[2] dr, Q[3] = 2∫ 1r2rw[3] dr.Condition (3.40) arises from the incompressibility of the fluids and requires thecontinuity of both components of velocity at the interfaces. We integrate withrespect to r to find each of the three flow rates:Q[1] = Wpr21−mr418εpiλ∂P∂ z˜, (3.41)Q[2] = Wp(r22− r21), (3.42)Q[3] = Wp(1+2r2)(1− r2)3− (1− r2)3(1+ r2)12(1− r2,0)2∂P∂ z˜. (3.43)Having computed Wp(z˜, t) and the pressure gradient, the flow rates are specifiedabove we may evolve the interface positions (in time) by integrating forward:∂∂ t[r21]+∂∂ z˜Q[1] = 0, (3.44)∂∂ t[r22]−∂∂ z˜Q[3] = 0. (3.45)to solve the above hyperbolic system, we use the total variation diminishing (TVD)92finite difference scheme of [31].Computation of Wp(z˜, t) and the pressure gradient (hence the flow rates above)varies according to whether the skin partially yields or not. We know that forsufficiently large yield stress the skin does not yield (as we have seen in our com-putations), and consider this special situation first.B˜ = ∞, the rigid skinWhen rigid, the skin velocity is given by the imposed Wp(0, t). The flow rate isfixed in fluid 1, and therefore:Q[2](z˜, t)+Q[3](z˜, t) = Q0(t)−1, (3.46)which is used with (3.42) & (3.43) to determine the pressure gradient in the skinlayer and Q[2] & Q[3] at each r1(z˜, t) and r2(z˜, t). The rigid skin means that (3.41)decouples from (3.42) & (3.43). The lubrication pressure gradient generated influid 3 is not transmitted through the skin to the core fluid layer, i.e. physically, thecore and lubricant layer behave as independent streams separated by a solid skin.Note that the flow problem here is similar to that we have solved downstream inthe established periodic flow; see Chapter 2.Partially yielded skinIn this case, we assume that normal stresses transmit through the skin, as in (3.38).On summing and rearranging (3.41) - (3.43), we have:∂P∂ z˜=−Q0(t)−Wp(r22 +(1+2r2)(1−r2)3)(1−r2)3(1+r2)12(1−r2,0)2 +mr418εpiλ. (3.47)93We eliminate the pressure gradient in (3.39), so that the mean extensional stressesin the skin evolve downstream according to:2∂∂ z˜[(x2− x1)τ¯zz] = Wp 11− x2 +(r12εpiλ+1+ x2−2x12)(r22 +(1+2r2)(1− r2)3)(1− r2)3(1+ r2)12(1− r2,0)2 +mr418εpiλ−Q0(t)(r12εpiλ+1+ x2−2x12)(1− r2)3(1+ r2)12(1− r2,0)2 +mr418εpiλ. (3.48)The plug velocity Wp evolves in z˜ according to (3.33), in which we may replaceτ [2]zz by its skin-averaged value since Wp is independent of x:∂Wp∂ z˜=12m2sgn(τ¯zz)max(|τ¯zz|− B˜,0). (3.49)Mathematically, using the skin-averaged stress τ¯zz provides a closed system of 2differential equations. However, we should examine its physical consistency. First,note that when |τ¯zz| > B˜ and the skin is yielded, then the form of axial velocity inthe skin guarantees that variations in τ [2]zz across the skin are of lower order, i.e. τ¯zz =τ [2]zz + o(ε). On the other hand, when |τ¯zz| ≤ B˜ and the skin is unyielded, then wemay have variations in τ [2]zz across the skin, although P= p[2]+τ[2]zz is independent ofx. In this case, as the skin is unyielded there is no streamwise variation in Wp. Thestress τ [2]zz is indeterminate in this case, and all that we determine (by integrating(3.48)), is the averaged value:τ¯zz ≡ 1(x2− x1)∫ x2x1τ [2]zz dx.For a given interface configuration r1(z˜, t) & r2(z˜, t) (equivalently x1(z˜, t) &x2(z˜, t)) the mean extensional stress and plug velocity are obtained by integratingthe coupled system of differential equations (3.48) & (3.49), downstream in z˜. In-flow conditions for Wp come from the sculpted inflow conditions at z˜ = 0. Theother initial (inflow) condition for τ¯zz(0, t), is unknown in general and must be94determined.Periodicity and the inlet stress: Note that Wp(0, t) is the input plug velocity,determined from the target interface profile. As the intended interface profile isperiodic (in space), the control input Wp(0, t) is also periodic in time: Wp(0, t) =Wp(0, t +T ). We suppose that τ¯zz(0, t) is such that it allows the intended periodicsolution, which will be periodic in both time and space. We can construct Wp(z˜, t)as follows:Wp(z˜, t) =Wp(0, t)+∫ z˜0∂Wp∂ z˜(s, t) ds, (3.50)and on integrating in t over one period will find the length of fluid that has passedthrough z˜:L(z˜) =∫ T0Wp(z˜, t) dt.Evidently, if the solution is periodic in both z˜ and t, then L(z˜) must be independentof z˜ and represents the spatial wavelength of the solution. Furthermore, as we havescaled axial lengths using spatial wavelength, we must have L(z˜) = 1. In particular,we can evaluate at z˜ = 0: ∫ T0Wp(0, t) dt = L = 1. (3.51)This constraint determines the time period T , which we see can be computed aheadof time from the control input. Since the periodicity is in both time and space wesee that:Wp(0, t) =Wp(0, t+T ) =Wp(1, t). (3.52)We are now in a position to calculate τ¯zz(0, t), which we do iteratively. Forgiven τ¯zz(0, t)we integrate forward the system (3.48) & (3.49) to find Wp(1, t; τ¯zz(0, t))and then adjust τ¯zz(0, t) to satisfy (3.52). Using the above procedure we computeWp(z˜, t) and τ¯zz(z˜, t) at each timestep. We evaluate the pressure gradients and flowrates from the system (3.41) - (3.43) and integrate forward (3.44) & (3.45) in time.The condition (3.52) appears to determine a unique value of τ¯zz(0, t). Intu-itively this may be because we expect Wp(1, t; τ¯zz(0, t)) to increase with τ¯zz(0, t)(although we have not proven this). Note too that since the extensional stresses arecoupled to the plug velocity, we can also assume that the extensional stresses are95periodic in z˜ and t, i.e. there is always same amount of compression and extensionat each spatial wavelength or in one time period. This periodicity is found in thecomputed solutions.3.3.3 ResultsWe now present some example results where the skin yields partially. We impose asawtooth interface profile in the skin-lubrication interface, via control of the inputflow rates; see Fig. 3.26a. The computed inflow stresses τ¯zz(0, t), required to satisfy(3.52), are shown in Fig. 3.26b for a range of increasing B˜.Figure 3.27 shows spatiotemporal variations in the yielded and unyielded partsof the skin for the sawtooth profile of Fig. 3.26a. Four different values of B˜ arecomputed over two time periods (here T = 0.64). We can see, as B˜ increases, theunyielded regions expand and for big enough B˜, the whole region will be unyielded(for this example, the minimal B˜ = 1.665). Although we have yielded bands ofskin at the lower B˜, the variation in plug velocity (slope of the bands in Fig. 3.27)is minimal.Figure 3.27 can also be compared to our earlier simulation results (see Fig. 3.21).Since the plug velocity variation along the pipe is very small, the unyielded regionsare advected with almost same velocity (Wp). The order of the required minimalBingham number to maintain the skin rigid is comparable in these two methods,however, the values are not identical. This because of both different boundary con-ditions and a different geometric domain. For example, Bcr calculated from theaxisymmetric simulations is ≈ 200 which equivalent to B˜cr = 1.89 in our exten-sional flow model, on taking r2,0 = 0.75, L = 15. However, in the axisymmetricsimulations we solving an initial boundary value problem with a specific inflowmanifold geometry. We are essentially estimating the maximal stresses and theseare often found in the development region just after exiting the manifold. Yield-ing of the skin must happen here in any axisymmetric computation whereas in theextensional model we have no inflow manifold to consider, only inflow parame-ters. The extensional model provides instead an estimate to the critical yield stressneeded downstream of the inflow manifold.Figure 3.28 provides a more detailed look at the effects of B˜ on the solution96  (b)(a)Figure 3.26: (a) Skin and lubricating flow rates required to generate saw-tooth wave in the skin-lubrication interface when Q1 = 1. (b) Com-puted values of inflow extensional stress (τ¯zz(0, t)) for different B˜, B˜ =∞, 1.65, 1.5, 1, 0.7. Other parameters are (r2,0,r1,0,a,λ ,m/ε,m2) =(0.9,0.8,0.5,20,0.01,100).97(a) (b)(d)(c)Figure 3.27: Spatiotemporal variations in the yielded and unyielded partsof the skin for different B˜ number, B˜ = 0.7, 1, 1.5, 1.65; where(r2,0,r1,0,a,λ ,m/ε,m2) = (0.9,0.8,0.5,20,0.01,100). Grey regionis plug and white region is yielded skin layer.at t = T/2. In Fig. 3.28a, the skin-lubrication interface (r2) is shown for differentB˜. The inset shows the r2 profile close to the maximum of the interface, wherewe see a small variation form the angular profile (converging for large B˜). Theinterface is rounded and tilted upstream. This deviation is small due to the largeskin viscosity ratio (m2) and small relative viscosity (m). Figure 3.28b presents thecore-skin interface. Although it is desired that this interface remains flat, whichis the case when the skin layer is completely unyielded, we can see that there isa small localised change in r1. This arises from the variation in pressure gradient98(a) (b)(d)(c)Figure 3.28: Effect of B˜ on the solution at t = T/2, plotted against z˜ ;(a) skin-lubrication interface r2 and the inset shows the r2 profileclose to the maximum of the interface, (b) core-skin interface r1, (c)skin velocity Wp, and (d) skin-averaged extensional stress τ¯zz (leg-ends are the same as Fig. 3.28d), B˜ = ∞, 1.65, 1.5, 1, 0.7; where(r2,0,r1,0,a,λ ,m/ε,m2) = (0.9,0.8,0.5,20,0.01,100).along the pipe, i.e. pressure can be transmitted to the core fluid from the lubricatinglayer through normal stresses in the yielded skin layer. We see a correspondingchange in Wp with B˜ (slight yielding before/after the peak in the interface); seeFig. 3.28c. The extensional stress distribution in the skin is shown in Fig. 3.28d,where we can see both the periodic nature and that the stresses converge at large B˜,allowing us to estimate a critical B˜.The interfaces evolution is explored in Fig. 3.29a & 3.29b for B˜ = 1. We cansee that the core-skin interface changes along the pipe because of pressure gradient99(a) (b)(d)(c)Figure 3.29: Temporal change of the solution. Plotted against z˜when skin yields and B˜ = 1; (a) skin-lubrication interfacer2, (b) core-skin interface r1, (c) skin velocity Wp, and (d)skin-averaged extensional stress τ¯zz where the red lines mark|τ¯zz| = B˜ = 1 (legends are the same as Fig. 3.29d); where(r2,0,r1,0,a,λ ,m/ε,m2) = (0.9,0.8,0.5,20,0.01,100).variation as mentioned before. Both components of the core flow rate (Poiseuilleand Wp parts) contribute to the growth in disturbance of r1 as time evolves. InFig. 3.29c, the skin velocity profile is shown: periodic in both time and z˜. Theextensional stress in the skin (τzz) is presented in Fig. 3.29d where the red linesmark |τ¯zz|= B˜ = 1, beyond which the skin is yielded. We can see that the imposedperiodic skin velocity results in a periodic extensional stress. This is also used toestimate the minimal Bingham number required for the skin to remain completelyunyielded (for this example B˜cr = 1.665).100Other shapesThe extensional model can be applied to an arbitrary interface shape. Figure 3.30shows the results of imposing the previous dinosaur profile and a sine wave profile.We have kept the parameters similar for both simulations, (r2,0,r1,0,a,λ ,m/ε,m2)= (0.9,0.8,0.5,20,0.01,100), and have reduced B˜ below B˜cr to allow both flowsto yield marginally. The interface profiles evidently represent the target profilesreasonably well. The minimal Bingham numbers required to maintain the skinrigid for the dinosaur and sine wave profiles are B˜cr = 0.776 and B˜cr = 1.86, re-spectively. The magnitude of extensional stress is clearly a function of the waveshape, but it is unclear to us why the sine wave should create larger extensionalstresses. The pattern of yielded/unyielded skin is similar in each case, with similarwave speeds. We also show the calculated inflow extensional stresses.Rigid skinLastly, we show some results from the rigid skin. As we have seen above we canestimate the minimal Bingham number (B˜cr) required for the skin to remain rigid.Figure 3.31 shows the results of solving the extensional flow model with rigid skinfor a sample set of parameters: (r2,0,r1,0,a,λ ,m/ε) = (0.9,0.8,0.5,20,0.01),using the previous sawtooth imposed interface profile and setting B˜ = ∞. Thecomputed interfaces are periodic in both z˜ and t, as expected.The input flow rates and plug velocity are shown in Fig. 3.32a & 3.32b. ForB˜=∞we have from (3.49) that Wp varies only in time. The pressure gradient in thelubrication and core layers is determined at each t by Eq. 3.41 & by the Reynoldsequation in the lubricating layer. Following Chapter 2 we can split the pressure intoa time varying pressure gradient G(t) (see Fig. 3.32c) and a lubrication pressure,say Pl(z˜, t), which is periodic in z˜ (see Fig. 3.32d). When the lubricating layerthickness increases, positive lubrication pressure is generated and for decreasinglubricating thickness, we have negative lubrication pressure. This is the basis ofthe model in Chapter 2, where the skin shape is designed to balance the densitydifference between core and lubricating fluids.101(a) (b)Figure 3.30: Effect of skin-lubrication interface shape on the solution. Toppanel shows the imposed shape, the middle panel indicates the spa-tiotemporal variations in the yielded and unyielded parts of the skin(grey region and white region show the plug and yielded skin, re-spectively.), and the bottom panel shows the inflow extensional stressτ¯zz(0, t) for yielded and unyielded skin. (a) Dinosaur shape with B˜ =0.65 and B˜cr = 0.776 and (b) sine wave with B˜ = 1.65 and B˜cr = 1.86;where (r2,0,r1,0,a,λ ,m/ε,m2) = (0.9,0.8,0.5,20,0.01,100).3.4 Discussion and conclusionsIn this chapter we have focused entirely on how to establish the shaped interface,within a triple-layer flow in which the intermediate skin fluid has a yield stress. Forfixed inner flow rate, we have shown there is a one-to-one relationship between theskin and lubricant flow rates and the two interface positions. This is not the casefor example in typical multi-phase flows in pipelining, where a pressure drop maybe imposed - allowing multiple interface/flow rate solutions.On understanding this, the basic method used here for shaping the interfacesis to control the flow rates of the individual fluid streams that combine to formthe triple-layer flow. Control of individual stream flow rates is also the methodcommonly used in co-extrusion flows. As the method is based on establishment102(a) (b)(d)(c)Figure 3.31: Evolution of interfaces in time; where(r2,0,r1,0,a,λ ,m/ε) = (0.9,0.8,0.5,20,0.01) and T = 0.64.of a multi-layer Poiseuille flow, intuitively we might expect that the method worksfor sufficiently long-wavelengths and for slow temporal variations. Here we haveexplored if this is true.The first part of the chapter has centred on the use of axisymmetric computa-tions. These have used a mixed finite element/finite volume method to discretizethe equations, with fine radial mesh to adequately represent the fine layered struc-ture. As a key component of the flow is the rigid skin layer, it is important to usea method that evaluates this feature correctly and hence an augmented Lagrangianmethod has been implemented. The two interfaces are captured using a VOF-PLICmethod, which has proven very effective (very little smearing of the interfaces isevident in our results).103(a) (b)(d)(c)Figure 3.32: (a) Inlet flow rates versus time; (b) Plug velocity; (c) Mean pres-sure gradient; (d) Lubrication pressure variation in z˜ for different times.Where (r2,0,r1,0,a,λ ,m/ε) = (0.9,0.8,0.5,20,0.01) and T = 0.64.The resulting computational model has been able to simulate developmentflows of the triple-layer type in model inflow manifold geometries. For largeenough yield stress (B) we have seen that on exiting the individual fluid stream,there results a development/shaping zone in which the interface deforms to its de-sired position. At the end of this zone an unyielded skin (plug) forms, which thenremains intact downstream as the flow evolves. Through varying the individualfluid stream we have been able to achieve rigid skin shapes that are close to thosein our design, e.g. we get approximately the designed sawtooth profile, but withsome rounding of corners.While shear dominates in the lubrication layer, the normal stress componentsdominate within the skin layer. We find that all three diagonal components of the104deviatoric stress are of comparable size in our computations, within the skin layer.Note however that the skin layer is artificially thick in our chosen examples (forbetter visualization); we would expect the hoop stress component to decrease insize for a thin skin.On reducing B we find partial yielding of the skin as the control flow rates os-cillate. The skin breaks mostly due to extensional stress. Where the skin breaksthere are no dramatic changes visible in the flow. However, the resulting interfaceis now a viscous-viscous interface which is able to deform and become unstable. Inour computations, we have a relatively short domain downstream (≈ 1− 2 wave-lengths) and for the fraction of the period when yielded the skin layer will have avery large effective viscosity. Thus, there is little opportunity for an instability togrow significantly within the duration of our simulations.We have used our computations to estimate the minimal yield stress required tomaintain a rigid skin, Bcr. As the viscosity ratio m increases we find Bcr increases,but the increase is modest within the range of m 1 that is anyway needed forreduction of frictional pressure drop. As the Reynolds number Re increases wealso see an increase in Bcr. Although the increases in Bcr are not extreme, thisneeds further investigation to determine if the larger Re also results in some formof flow transition within the lubricant layer.Our computations have used an inflow manifold in which the wall separatingskin and lubricant fluids is shorter than the core-skin wall. This allows for theunyielded skin to develop before the core-skin interface forms. On testing witha longer lubricant-skin wall, we have found that the core-skin interface developssignificant non-uniformities at the inflow. Our inflow manifold design is otherwisefairly crude. The separating walls are simply rectangular in shape, which we haveseen results locally in a form of stress singularity that propagates directly down-stream of the sharp corners, but eventually disappears as the skin develops. In adesign phase such issues could be reduced/eliminated.The concentric model has established that it is feasible to develop these triple-layer flows. It is however, not without problems. First it is an initial value problem,so needs a starting solution, which we have taken as a uniform flow. This thenneeds to cycle through many periods of controlled oscillation before we begin toapproach a periodic interface profile independent of the initial condition. The flows105computed are certainly sensitive to the manifold design, although we have used afunctioning setup that seems to work. In particular, when calculating Bcr the max-imal stresses are typically found close to the manifold walls and in the develop-ment/shaping region, rather than downstream. Assuming that a range of manifoldsand flows will converge to a shaped interface, we are probably more interested (asa design parameter) to estimate the size of extensional stresses generated withinthe skin, due to the influence of the skin shape variation on the lubrication layer.Lastly, the computational times required for these simulations are significant: theaugmented Lagrangian method converges slowly on each time step, a fine meshis needed radially for the interface structure and long domains are needed to testmultiple periods of oscillation.The above issues have prompted us to develop the extensional stress model,studied in the last part of the chapter. Derivation of this model requires scalingarguments that help to clarify that the long-wavelength assumption relates to theratio of interface amplitude to streamwise wavelength of the skin variation, i.e. notthe radius. This partially explains why we have found reasonable results in the ax-isymmetric computations, although the computational domain was not excessivelylong. The extensional flow model couples the two shear flows (lubrication andcore) with the extensional-stress dominated skin. Lubrication layer shear stressesgenerate significant pressures, which transmit to the skin via normal stress conti-nuity. This results in the extensional stresses developing within the skin. Inertiaplays a secondary role.The simplified extensional flow model results in two differential coupled equa-tions for the skin-averaged extensional stress τ¯s,z and plug velocity Wp. These areintegrated downstream at each fixed time. This allows computation of the localvolume flow rates in each layer, which are used to calculate evolution of r1 & r2.The procedure is relatively quick and results in computation of solutions that areperiodic in both time and space, as expected.We can observe the spatiotemporal evolution of patterns of yielded and un-yielded skin and explore sensitivity to the various flow parameters. In particular,we can use the computed solutions to rapidly estimate the minimal yield stress re-quired to maintain the skin rigid: here B˜cr. The estimates are of similar size tothose from the axisymmetric computations. This model also allows us to study106what happens when the skin yields. Partial yielding allows transmission of thenormal stresses between core and lubrication layers, which results in deformationof the interface r1. Visibly, we see a small notch appearing in the interface. Thisfeature will grow in time. The amplitude is controlled by m, but even for very smallm it is not eliminated (except when B˜ > B˜cr and the skin becomes rigid).To summarize the extensional flow model is useful for a variety of exploratorycomputations of parameter ranges, for design and implementation of sculpted in-terface profiles, etc.. This idea is the basis of a process design and control model.In essence, the extensional model fits well with the reduced model of Chapter 2,which assumed a rigid skin. Both models are quick to compute and can be used forprocess design. The estimates derived here for our extensional flow model give asecond estimate of the required yield stress to keep the skin rigid. A brief compar-ison between these two models is presented in §B.3.107Chapter 4Inertial Effects in Triple-LayerCore-Annular Pipeline Flow1In Chapter 2, we proposed a method for efficient heavy oil transportation via atriple-layer CAF and have established its feasibility for steady fully-developed flowin small-to-moderate diameter pipes with density differences and viscosity ratiostypical of those used in lubricated pipelining. The method combines 2 significantfeatures of yield stress fluid flows: (i) VPL; (ii) interface sculpting.In 2CAF the generated lubrication pressure leads to deformation of the inter-face, but in the triple-layer flows proposed the lubricant layer is bounded by ashaped rigid skin. Thus, the net lubrication force is imposed on the rigid skin(plus core fluid). In Chapter 3, we have studied interface sculpting numericallyand analytically in a concentric axisymmetric domain, representing one possibleconfiguration in which a triple-layer flow might be achieved industrially. We haveshown that this flow can be stably established in a controlled way by varying theindividual flow rates, and have estimated the size of extensional stresses generatedwithin the unyielded skin layer. This becomes an additional design requirement forthe minimum yield stress of the skin.Chapter 2 focused at laminar flows, neglecting the effect of inertia, as is validin long-thin geometries when the product of aspect ratio and Reynolds number issmall. As a key part of this method is to generate lubrication forces via shaping the1A version of this chapter has been published [156].108interface, asymptotically small aspect ratios are not possible, i.e. there would beno “shape”. Thus, inertial effects can become important both for large Reynoldsnumbers (eventually turbulent) and in trying to balance larger core fluid densitydifferences with larger amplitude sculpting. Both effects occur in larger pipelinesand with higher flow rates: common industrial objectives. Thus, extension of thetriple-layer concept (Chapter 2) to wider parameter regimes means that we needto understand the main effects that may arise from inertial effects and turbulencein the lubrication layer. The importance of these regimes and these constraintshas long been recognised and studied [124, 125] for 2CAF. The objective of thischapter is to study inertial and turbulent effects in the triple-layer configuration, inparticular on the lubrication force that is generated.Inertial effects on steady flow have been studied in similar flows to lubricatedpipeline flow, such as in journal bearings [26, 88, 166, 167, 192]. These stud-ies show, inertia has a significant effect on the bearing load. Other journal bear-ing studies include the effect of turbulence [27, 63, 115, 168]. In most of thesestudies, the mixing length method is used to find a closure for the eddy viscosityand lubrication layer-averaged momentum equations are then solved analyticallyor semi-analytically. We should note that there are two competing effects here aswe move to turbulent flows. First, since the shear stresses in turbulent shear flowsexceed the viscous stresses, we expect to generate progressively larger lubricationpressures. Second for fixed aspect ratios, as we increase the Reynolds number,eventually advective gradients of the mean flow will become significant and mayaffect lubrication pressures differently.In our triple-layer flows the lubrication layer is driven by both lubrication pres-sure gradients and by motion of the rigid skin. Thus, in terms of a turbulenceclosure model we need to estimate terms in a turbulent Couette-Poiseuille (C-P)flow. Lund and Bush [101] studied the asymptotic analysis of plane turbulent highReynolds C-P flow and proposed an eddy viscosity closure. Higher order modelsof C-P turbulent flow such as k− ε model have also been investigated [55], butare less easy for simple analysis, as here. The similarity of a logarithmic velocityprofile in C-P turbulent flow is also studied experimentally; see[113, 170]. Accu-rate direct numerical simulation studies have also been recently performed on C-Pflow by Kim and Lee [89]. They show that the extended logarithmic layer develops1091233) Lubrication Layer2) Skin(a) (b)1) Heavy OilzrFigure 4.1: (a) Shaped skin profile in 3D; (b) cross section of the pipe withtriple-layer configuration, illustrating the coordinates.from each wall to the centerline and that the stationary wall experiences higher tur-bulence intensity and Reynolds shear stresses than those in the Poiseuille flow; seealso Lee et al. [97]. In addition, many studies exist on the application of turbulentC-P flow, e.g. [100, 193]. Here, we use the simple zero order eddy viscosity modelto treat the turbulent flow, as the focus of this chapter is on estimating the effectsof inertia and added dissipation due to turbulence, rather than precise modeling ofthe turbulent flow itself.The outline of this chapter is as follows. We start with an introduction of theflow setup and notation (§4.1). As far as is possible, we consider a periodic sectionof developed triple-layer flow as studied previously in Chapter 2 and keep similarscaling to make the studies compatible. The governing equations of the inertiallaminar flow are derived in §4.2, i.e. the Reynolds equation. The modeling ofturbulent lubricant flow is discussed in §4.3. The results are presented in §4.2.1& 4.3.2 for laminar and turbulent flows, respectively. Finally, in response to ourobservations, in §4.4 we discuss what may be an appropriate wave shape design forinertial flows. Finally, the chapter closes with a discussion of the effect of inertiaand turbulence on generated lubrication force.1104.1 Flow descriptionOur goal in this chapter is to include the effect of inertia and turbulence withinthe lubrication layer of the triple-layer CAF method of Chapter 2 for typical core-annular configurations. For simplicity, we consider a horizontal pipe which is peri-odic in the streamwise zˆ-direction, with period lˆ. Figure 4.1 indicates schematicallythe positions of the 3 fluids: fluid 1 is the core fluid (viscous Newtonian heavy oil),with viscosity µˆ [1] and density ρˆ [1]. The skin layer is fluid 2, modeled as a Bing-ham fluid, with µˆ [2], τˆ [2]y , and ρˆ [2] denoting its viscosity, yield stress, and density,respectively. Fluid 3 is the lubricant layer, a low viscosity Newtonian fluid, withviscosity µˆ [3], and density ρˆ [3].The governing equations for the flow are the Navier-Stokes equations, in eachfluid domain. The traction and velocity vectors are continuous across each inter-face. For simplicity, throughout the chapter we ignore interfacial tension as theyield stress is dominant and bulk capillary effect are negligible in pipelining appli-cation (pipelining radii is in the range 5− 100cm). The flow is periodic in zˆ andno-slip conditions are satisfied at the pipe wall. Constitutive equations for the 3fluids are:τˆi j [k] = µˆ [k] ˆ˙γi j, k = 1,3, (4.1)τˆ [2]i j =[µˆ [2]+τˆ [2]y∣∣ ˆ˙γ∣∣]ˆ˙γi j⇐⇒ τˆ [2] > τˆ [2]y , (4.2)ˆ˙γ = 0 ⇐⇒ τˆ [2] ≤ τˆ [2]y , (4.3)whereˆ˙γi j =∂ uˆi∂ xˆ j+∂ uˆ j∂ xˆi,ˆ˙γ =[123∑i, j=1[ ˆ˙γi j]2]1/2, τˆ [2] =[123∑i, j=1[τˆ [2]i j ]2]1/2. (4.4)In the typical case where the oil density differs from that of the lubricant andskin fluid, there is a net transverse buoyancy force, so that the underlying con-figuration is unlikely to be concentric. A uniform eccentric annular flow generatesonly axial pressure gradients and cannot support density differences. As previously111recognised [125], it is necessary to have both eccentricity and axial variation in thelubrication layer thickness in order to generate transverse lift forces via viscouslubrication. As with any transport process there is an underlying constant pressuredrop along the pipeline. Superimposed on this is a periodic (in zˆ) variation in thepressure, which is governed by the local thickness of the lubricant layer. In thisway, differential lubrication pressures are generated that are able to counter thetransverse buoyancy force.Anticipating the above scenario, we use a classical lubrication scaling of theequations. We position our cylindrical coordinates at the centre of Ω1, which hasuniform radius rˆ1. The skin layer outer radius is rˆ2(zˆ), which has mean positionrˆ2,0 and axial variation. The pipe wall is denoted rˆ = rˆ3(θ), with variation due tothe eccentricity. We assume the mean thickness of the outer lubricant layer is thin,relative to circumferential and axial length-scales, piRˆ & lˆ respectively. In otherwords, δ = (Rˆ− rˆ2,0)/(piRˆ) 1 and assume that λ = lˆ/(piRˆ) ∼ O(1). Belowwe calculate the leading order shear stresses and pressure in the lubricant layer,i.e. δ → 0.We scale all lengths with Rˆ, velocities with the plug velocity Wˆp and the stressscale is the wall shear stress τˆw.(r,z) =(rˆ, zˆ)Rˆ, u =uˆWˆP.This leads to a problem governed by 2 dimensionless radii, r1 & r2,0, and 2further dimensionless groupsm =µˆ [3]µˆ [1], B =τˆ [2]yτˆw.Here m is the viscosity ratio and B is Bingham number. Provided that the skin layerremains unyielded, the second viscosity ratio µˆ [2]/µˆ [1] plays no role in the flow.4.1.1 Lubricating layerWe assume symmetry about a central vertical plane through the pipe, define z= zˆ/lˆ,and define a scaled azimuthal coordinate y = θ/pi with y ∈ [0,1], extending from112the top to bottom of the lubricant. A scaled radial coordinate x is defined by:rˆ = rˆ2,0+piRˆδx.Velocity components in axial and azimuthal directions are scaled with plugvelocity (Wˆp) and that in the radial direction with δWˆp. We break the pressure into3 parts: a constant axial pressure gradient, a periodic lubrication pressure (comingfrom the variation in layer thickness), and a hydrostatic pressure component:Pˆ =−Pˆ∗GGz+ Pˆ∗l Pl(y,z)+piδ ρˆ [3]gˆRˆxcos(piy). (4.5)The lubrication pressure scale (Pˆ∗l ) is chosen to balance the leading order shearstress gradients, Pˆ∗l = τˆw/δ , and the axial pressure gradient scale (Pˆ∗G) is chosento satisfy the force balance in axial direction, Pˆ∗G = piλ τˆw. Note the leading ordershear stresses scale with δ Pˆ∗l .With the above scaling the momentum equations are:δ 3f(∂u∂ t+wλ∂u∂ z+ v∂u∂y+u∂u∂x+O(δ ))=−∂Pl∂x+O(δ ), (4.6a)δf(∂v∂ t+wλ∂v∂ z+ v∂v∂y+u∂v∂x+O(δ ))=−∂Pl∂y+∂τxy∂x+O(δ ), (4.6b)δf(∂w∂ t+wλ∂w∂ z+ v∂w∂y+u∂w∂x+O(δ ))=− 1λ∂Pl∂ z+∂τxz∂x+O(δ ), (4.6c)O(δ ) =∂u∂x+∂v∂y+1λ∂w∂ z. (4.6d)Where f = τˆw/ρˆ [1]Wˆ 2p is the friction factor. In case of laminar flow, viscousscaling is the proper way to scale stresses and pressure, while, in turbulent flow,the appropriate stress scale is wall shear stress. If we use, viscous stress scale,τˆw = µˆ [3]Wˆp/(Rˆ− rˆ2,0) for stresses and Pˆ∗l to scale lubrication pressure, then f =m/δpiRe[1].We can see that Re[1]δpi/m is effectively the Reynolds number Re[3], relevantto the lubrication layer. With reference to (4.6), we see that 3 parameters dictatethe type of flow regime. First, Re[3] gives the overall flow regime of the lubricationlayer, i.e. laminar or turbulent shear flow. The lubrication layer varies in thickness,113Flow types Re[3] Re[3]δ δ/ fLaminar flow < Re[3]c << 1 -Inertial-laminar flow < Re[3]c & O(1) -Non-inertial turbulent flow > Re[3]c - << 1Inertial-turbulent flow > Re[3]c - & O(1)Table 4.1: Lubrication layer flow types based on Re[3], Re[3]δ , and δ/ f . HereRe[3]c is a nominal critical value for transition to turbulence.contains a stabilizing Couette component and is eccentric. Thus, the actual tran-sition to turbulent flow will be very complex and a critical Re[3] is not known. Toproceed, we will assume Re[3]c ∼ 103. Secondly, the varying layer thickness meansthat the inertial terms that include gradients of the (mean) flow will be non-zero.These contribute terms of size∼ Re[3]δ in laminar flow and size∼ δ/ f in turbulentflow. Thus, we can delineate 4 types of flow in lubrication layer, as presented inTable 4.1.Non-inertial laminar flow (Re[3] < Re[3]c & Re[3]δ << 1) is the classical viscouslubrication limit, studied in detail in Chapter 2. Non-inertial turbulent flow wouldrequire very small Re[3] and course pipe to increase f sufficiently, and practicallywould not happen for the sculpted interfaces intended in this application: practi-cally speaking we would expect δ/ f ∼O(1). This leaves us with 2 flow regimes tostudy here: inertial-laminar and inertial-turbulent. Evidently, “inertial-turbulent” isa misnomer as all turbulent flows are turbulent. Here it is explicitly meant to implythat the gradients of the advective derivatives of the mean velocity may be signifi-cant.1144.2 Inertial-laminar flowLeading order governing equations of steady state inertial laminar flow are:0 =∂Pl∂x, (4.7a)Re[3]δ(wλ∂v∂ z+ v∂v∂y+u∂v∂x)=−∂Pl∂y+∂ 2v∂x2, (4.7b)Re[3]δ(wλ∂w∂ z+ v∂w∂y+u∂w∂x)=− 1λ∂Pl∂ z+∂ 2w∂x2, (4.7c)0 =∂u∂x+∂v∂y+1λ∂w∂ z. (4.7d)Thus Pl is only function of (y,z). To simplify, we assume that the skin layermoves only axially. Boundary conditions for the lubricant are: (u,v,w) = (0,0,0)at x = x3(y) and (u,v,w) = (0,0,1) at x = x2(z). The functions x3(y) and x2(z)are derived from the pipe wall and outer skin positions, to leading order in δ . Thethickness of the lubricant layer is h(y,z) = x3(y)− x2(z), which is given to leadingorder by:h(y,z) = 1− ecospiy−aΦ(z)+O(δ ), (4.8)where the eccentricity is e= dˆ/(Rˆ− rˆ2,0) and the amplitude a= ∆rˆ2/piRˆδ . dˆ is theradial distance between the pipe and core centres. Note that the dimensionless axialskin thickness variation Φ(z) has zero mean and maximal amplitude 1. Integratingthe continuity equation along the lubrication layer gap, using Leibniz’s rule, andapplying boundary conditions lead to the following depth-averaged equation forthe conservation of mass:∂qy∂y+1λ∂qz∂ z− 1λ∂h∂ z= 0. (4.9)where qy =∫ x3x2 vdx and qz =∫ x3x2 wdx.Depth-averaged momentum equations are found by integrating the momentum115equations along the lubrication layer gap:Re[3]δ(∂∂y∫ x3x2v2dx+∂λ∂ z∫ x3x2wvdx)=−∂Pl∂yh+[∂v∂x]x3x2, (4.10a)Re[3]δ(∂∂y∫ x3x2wvdx+1λ∂∂ z∫ x3x2w2dx− 1λ∂h∂ z)=− 1λ∂Pl∂ zh+[∂w∂x]x3x2.(4.10b)where [F ]x3x2 = F(x3)−F(x2). Equation (4.10) contains inertial and friction terms.For the thin film flow of a lubrication layer, these terms can be determined explic-itly by approximating the velocity profile within the film to have the same self-similar form as a fully developed laminar flow, namely:v = 6¯¯vhξ (1+ξ ), (4.11a)w = 6¯¯whξ (1+ξ )−ξ . (4.11b)where ξ = (x− x3)/h. Note that, this velocity profile (4.11) satisfies boundaryconditions. Using (4.11) leads to following depth-averaged momentum equations.¯¯v =− h312∂Pl∂y− h212Re[3]δ Iy, (4.12a)¯¯w =− h3121λ∂Pl∂ z− h212Re[3]δ Iz. (4.12b)Where,Iy =65h(¯¯v∂ ¯¯v∂y+ ¯¯w∂ ¯¯vλ∂ z)+65¯¯v2∂∂y1h+65¯¯v ¯¯w∂λ∂ z1h− 3 ¯¯v5h∂hλ∂ z− 12∂ ¯¯vλ∂ z,Iz =65h(¯¯v∂ ¯¯w∂y+ ¯¯w∂ ¯¯wλ∂ z)+65¯¯w2∂λ∂ z1h+65¯¯v ¯¯w∂∂y1h− 3 ¯¯v5h∂hλ∂ z− 12∂ ¯¯wλ∂ z− 512∂hλ∂ z.Using (4.12) into continuity equation (4.9), the modified form of Reynolds116equation (incorporating the inertial effects) can be expressed as∂∂y(h3∂Pl∂y)+1λ∂∂ z(h3∂Pl∂ z)=− 6λ∂h∂ z−Re[3]δ[∂(h2Iy)∂y+1λ∂(h2Iz)∂ z].(4.13)the second term of the R.H.S. of (4.13) contributes to the effect of fluid inertia.This expression recovers the classical from of Reynolds equation when Re[3]δ ap-proaches zero.4.2.1 ResultsHere, results are presented for inertial-laminar flow in the lubrication layer in whichwould be mostly relevant for small-moderate pipe radii (Rˆ < 20cm) and very highviscous oil, resulting low Re[1] and consequently low enough Re[3]. Note the fullylaminar viscous lubrication regime that we have studied in Chapter 2 becomes lim-ited for Rˆ& 10cm or as the flow rates increase. For larger flow rates and pipes, withperhaps moderate viscosity of the heavy oil µˆ [1] < 1Pa.s, and very low viscositylubricant, we enter turbulent regimes; see §4.3 & §4.3.1.Equations (4.12) & (4.13) are solved in a periodic domain in the z-direction,i.e. Pl(y,0) = Pl(y,1), ¯¯v(y,0) = ¯¯v(y,1), and ¯¯w(y,0) = ¯¯w(y,1). The domain is sym-metric in the y-direction, at the top and the bottom of the pipe, i.e. ∂Pl/∂y(0,z) =∂Pl/∂y(1,z) = 0, ¯¯v(0,z) = ¯¯v(1,z) = 0, and ∂ ¯¯w/∂y(0,z) = ∂ ¯¯w/∂y(1,z) = 0. Thenumerical method is briefly described in appendix §B.1.In this study we use a sawtooth wave shape imposed on the skin/lubricant in-terface, as was done for the non-inertial viscous flows, see Chapter 2. Figure 4.2illustrates the variation of the lubrication layer thickness in y and z directions forparameters (e,a, l′,λ ) = (0.3,0.4,0.25,1). The narrowest gap is located at y = 0(top of the pipe) and the widest at the y= 1. The parameter l′ denotes the “breakingpoint” of the sawtooth wave.For the illustrative geometry of Fig. 4.2, the lubrication pressure colormap isgiven in Fig. 4.3 for different values of Re[3]δ . We can see that for inertialessflow positive lubrication pressure is generated where the lubrication layer thicknessincreases, e.g. in this case, z > 0.25. However, the inertial terms affect the pressurefield significantly as the value of Re[3]δ increases. We see that positive lubrication1170 0.5 100.511.52Figure 4.2: Lubrication layer thickness variation in z and y; where(e,a, l′,λ ) = (0.3,0.4,0.25,1).pressure shifts toward the other end of the periodic section. Figure 4.4 showsthe layer-averaged axial velocity ( ¯¯w) for the same eccentricity and range of Re[3]δas in Fig. 4.3. We can see, for inertialess flow, ¯¯w is maximal in thicker part oflubrication layer. As the thickness of the lubrication layer decreases with z, thevalue of ¯¯w increases. There are modest azimuthal variations. As inertial effects areincreased we see that the velocity field is also affected. We see larger velocitiesand some significant regions of negative ¯¯w, in the lower part of the annulus as thelayer thickness begins to increase.Both these effects are essentially Bernoulli effects, i.e. as the inertial terms be-come dominant over viscous, Bernoulli’s equation dictates lower pressures wherethe speed is larger and vice versa. For O(1) inertia (see Re[3]δ = 1, 3), inertia andviscous terms are comparable and the result is a competition of both effects, whilefor higher Re[3]δ , inertia is dominant and the pressure variation is governed by theBernoulli effect. For larger Re[3]δ we see that the azimuthal variations in lubrica-tion pressure and velocity are reduced. Assuming the streamlines are essentially inthe z-direction, the z-momentum equation reduces to:Re[3]δ2∂w2∂ z=−∂P∂ z+∂τxz∂x, (4.14)118  (a) (b)(c) (d)(e) (f)Figure 4.3: Lubrication pressure colormap for different Re[3]δ , (a) Re[3]δ = 0,(b) Re[3]δ = 1, (c) Re[3]δ = 3, (d) Re[3]δ = 30, (e) Re[3]δ = 50, and (f)Re[3]δ = 100; where (e,a, l′,λ ) = (0.3,0.4,0.25,1).119  (d)(e)(a) (b)(c)(f)Figure 4.4: Layer-averaged axial velocity ( ¯¯w) colormap in z and y fordifferent Re[3]δ , (a) Re[3]δ = 0, (b) Re[3]δ = 1, (c) Re[3]δ = 3,(d) Re[3]δ = 30, (e) Re[3]δ = 50, and (f) Re[3]δ = 100; where(e,a, l′,λ ) = (0.3,0.4,0.25,1).120integrating (4.14) in the flow direction results in:P+Re[3]δ2w2 =∫ z0∂τxz(x, t)∂xdz.The pressure generated in lubricating layer is thus a combination of the viscousterm and inertial terms. Note that the viscous term is also varying with Re[3]δ asthe momentum and continuity equations are fully-coupled. For the highly inertialflows such as Fig. 4.4d-f, the lubrication pressure progressively balances the in-ertial stress terms (the Bernoulli effect), but note that the viscous losses are everpresent and integrated along the streamlines.The layer-averaged azimuthal velocity ( ¯¯v) is not shown. It plays a passive rolehere as there is no azimuthal motion of the skin. Essentially ¯¯v adjusts to conservemass. The profiles of the axial velocity, across the lubrication layer, are shown inFig. 4.5 at y = 0, 1. The red lines depict the sawtooth wave. For Re[3]δ = 0, ve-locities have almost monotonic profiles and that is the reason of the predictabilityof the lubrication pressure variation. As inertia is added to the flow, the velocityprofiles starts showing non-monotonic behaviour. At the start of the periodic sec-tion we have adverse pressure gradients forming, leading to backflow near the pipewalls. Further downstream we can have strong pressure gradients helping the flowand hence maximal velocities exceeding the skin velocity.Figure 4.6 shows the effect of inertia on generated lubrication force for differ-ent values of eccentricity and l′, e = 0.1, 0.2, 0.3 and l′ = 0.25, 0.75; and caseparameters (a,λ ) = (0.4,1). We see that Fl initially decreases then increases asRe[3]δ is increased. The variation is bigger with larger eccentricity. For l′ = 0.25we see that Fl becomes negative for a modest value of Re[3]δ , and is always neg-ative for l′ = 0.75. Essentially this means that the operating window for finding avertical equilibrium position is significantly reduced once inertial effects becomesignificant in the laminar regime.4.3 Turbulent flowWe adopt the usual Reynolds decomposition to model the turbulent flows, splittingvariables into mean and fluctuating parts. We also have the same non-dimensionalization121  (a) (b)(c) (d)(e) (f)r2,0+πδ a≈ ≈≈≈≈ ≈Figure 4.5: Lubrication layer axial velocity profile (w(r,y,z)) variation alongthe pipe for different Re[3]δ , (a) Re[3]δ = 0, (b) Re[3]δ = 1, (c)Re[3]δ = 3, (d) Re[3]δ = 30, (e) Re[3]δ = 50,and (f) Re[3]δ = 100; where(e,a, l′,λ ) = (0.3,0.4,0.25,1). Note, red lines indicate the sawtoothwave shape.122(a) (b)Figure 4.6: Generated lubrication force variation with Re[3]δ , (a) l′ = 0.25and (b) l′ = 0.75 for e = 0.1, 0.2, 0.3; where (a,λ ) = (0.4,1).as for laminar flow except that now we use the wall shear stress τˆw to scale thestresses and τˆw/δ to scale the lubrication pressure of turbulent flow. This resultsin both viscous and turbulent stresses appearing at leading order in the turbulentshear flow. The viscous components τvi j come from ensemble averages of the vis-cous stress tensor. The turbulent (Reynolds) stresses are τ ti j =−ρˆ [3]uˆ′iuˆ′j, resultingfrom the fluctuating components of the velocity field. As turbulence closure weuse the simplest, zero equation eddy viscosity model.τ ti j = εγ˙i j.The following momentum and mass conservation equations result for the en-semble averaged velocity (u¯, v¯, w¯) and lubrication pressure P¯l .0 =∂ P¯l∂x, (4.15a)δf(w¯λ∂ v¯∂ z+ v¯∂ v¯∂y+ u¯∂ v¯∂x)=−∂ P¯l∂y+∂∂x((1+ εy)∂ v¯∂x), (4.15b)δf(w¯λ∂ w¯∂ z+ v¯∂ w¯∂y+ u¯∂ w¯∂x)=− 1λ∂ P¯l∂ z+∂∂x((1+ εz)∂ w¯∂x), (4.15c)0 =∂ u¯∂x+∂ v¯∂y+1λ∂ w¯∂ z. (4.15d)123here f is the friction factor; εy and εz are the eddy viscosities in y and z directions,respectively, i.e. azimuthally and axially in the lubrication layer. We treat theseseparately as the axial flow is mainly a Couette-Poiseuille flow and the azimuthalflow is a Poiseuille flow.For simplicity we first assume that δ/ f << 1, so the nonlinear inertial gradi-ents of the mean flow are negligible, i.e. we set the left-hand-side of (4.15) to zero.We relax this assumption later below in §4.3.1. Our intention is understandingonly the leading order effects of turbulence within a tractable model. We adopt themodel of Lund and Bush [101], who developed an asymptotic approximation ofturbulent plane Couette-Poiseuille flow for the high Reynolds number limit. Theyproposed an eddy viscosity model which includes Laufer core-flow distribution andVan Driest damping. They showed this model works reasonably for moderate tohigh Reynolds number.Lund and Bush developed the following expression for the eddy viscosity:ε(uw,Re[3],β ,ξ ) = κRe[3]uwξ (1+ξ )1+(2−β )ξ (1+ξ )+β M(ξ )2M(1+ξ )2, (4.16a)M(x) = 1− exp(−κRe[3]uwxA). (4.16b)where κ = 0.41, A = 6.9, uw =√f/2, and β = (dPˆ/dxi)/τˆw are von Ka´rma´nconstant, damping constant, wall shear velocity, and pressure gradient parameter,respectively. Here f is the friction factor, dPˆ/dxi is the imposed pressure gradientdriving the fluid, and τˆw = ρˆ [3]uˆ2w denotes the wall shear stress. Here β gives therelative importance of Poiseuille flow and Couette flow, defined as:β =dP¯dxiτw= 1− τw,mτw,s, (4.17)where τw,m and τw,s stand for shear stresses at moving and stationary walls. It canbe seen that for very small β → 0, τw,m ≈ τw,s and the flow approaches Couette flowand when β → 2, τw,m ≈−τw,s, the flow is mainly Poiseuille flow.In our flow β can be calculated for each directions, i.e, βy = (∂ P¯l/∂y)/τw =O(1) and βz = Gˆ/τˆw = δG/τw = O(δ ). Although β is the only parameter mod-124ifying ε in each direction, the values of eddy viscosities in the y and z directionsare still functions of the total wall shear stress, preventing us from decoupling twoflows completely. We also calculate friction factors in y and z directions followingLund and Bush[101]. In the z direction, where β = O(δ ) this results in:fz = 2[κ2log(κ2Re[3])(1+log log(κ2Re[3])+ log(2)−2.95log(κ2Re[3]))]2, (4.18)In the y-direction, the dominant flow is a Poiseuille flow. We use (4.16) withβ → 2, leading to (4.19) below.fy =2κ2log2(Re[3])[1+2loglog(Re[3])−2.6log(Re[3])]. (4.19)This is also in good agreement with other models [78, 100]. The wall shearvelocity is |uw| =√v2w+w2w =√fy/2+ fz/2, where vw and ww are wall shearvelocities in y and z directions. The friction factor is f =√f 2y + f 2z .The velocity profiles are calculated using (4.15) (with δ/ f = 0), equations(4.16), (4.18) and (4.19).v¯(ξ ,y,z) =∂ P¯l(y,z)∂y(V3(ξ )− V4V2V1(ξ )), (4.20)w¯(ξ ,y,z) =W1(ξ )W2+1λ∂ P¯l(y,z)∂ z(W3(ξ )−W4W2W1(ξ )), (4.21)where, V1(ξ ), V2, V3(ξ ), V4, W1(ξ ), W2, W3(ξ ), W4 are calculated based on theeddy viscosity equations. The details can be found in appendix §B.2; see (B.2) &(B.3).Using (4.20) & (4.21) in the momentum and continuity equations and inte-grating over lubrication layer thickness results in the following turbulent Reynoldsequation:∂∂y[C1∂ P¯l∂y]+1λ 2∂∂ z[C2∂ P¯l∂ z]=1λ(∂h∂ z− ∂∂ zC3). (4.22)125Values of C1, C2, C3 are reported in appendix §B.2; see (B.4). Note that (4.22)is similar to the classical Reynolds equation in structure, but with modified coeffi-cients.4.3.1 Inertial-turbulent flowWe now include the nonlinear inertial terms in the turbulent flow, i.e. resultingfrom advective derivatives of the mean flow. These can be the same size as theturbulent stress and pressure gradient terms, depending on whether δ/ f is O(1).The leading order equations are (4.15). To solve the system (4.15), we integrateall the equations over the lubrication gap thickness and (as with the laminar flow),adopt self-similar profiles for the velocities. Effectively this assumes that inertiadoes not affect the velocity profiles but does have a significant effect on the pressurefield. Therefore, we use the velocity profile from the turbulent shear flow of theprevious section, i.e.v¯ =¯¯vC1(V3(ξ )− V4V2V1(ξ )), (4.23a)w¯ =¯¯wC2(W3(ξ )−W4W2W1(ξ ))+W1(ξ )W2. (4.23b)Using (4.23) in the continuity equation and integrating over the lubrication gapthickness results in:∂ ¯¯v∂y+1λ∂ ¯¯w∂ z=1λ(∂h∂ z− ∂C3∂ z). (4.24)Substituting (4.23) into the momentum equations and integrating over the lu-brication gap thickness leads to:¯¯v =C1∂ P¯l∂y+δfIy, (4.25a)¯¯w =C2λ∂ P¯l∂ z+δfIz. (4.25b)The modified turbulent Reynolds equation is now calculated by using (4.24) &126(a) (b)Figure 4.7: Lubrication force Fl variation with a and l′, (a) inertialess lam-inar flow and (b) inertialess turbulent flow with Re[3] = 5000; where(e,λ ) = (0.3,1).(4.25).∂∂y[C1∂ P¯l∂y]+1λ 2∂∂ z[C2∂ P¯l∂ z]=1λ(∂h∂ z− ∂C3∂ z)− δf(∂ Iy∂y+1λ∂ Iz∂ z). (4.26)The values of Iy & Iz can be found numerically by integrating across the layer; seeequations (B.5) & (B.6). Note, if we replace the eddy viscosities by zero and δ/ fby Re[3]δ , (4.26) is the same as (4.13).4.3.2 ResultsHere, we present results for turbulent flow. Figure 4.7 shows the generated lubrica-tion force Fl , for inertialess laminar and turbulent flow (i.e. δ/ f → 0), for differentwave shapes with (e,λ ) = (0.3,1) fixed. The turbulent flow generates larger lu-brication pressures and hence larger Fl . Note that the turbulence model includesthe Reynolds stresses and so results in larger shear stresses, which in turn leads tolarger lubrication pressure variations.For this section we assume a constant value for δ/ f to explore the results.In practice f varies slightly with Re[3], within the range f ≈ O(10−2 − 10−3).However, the lubricant Reynolds number can varies in a broad range, e.g. 5000−127(a) (b)Figure 4.8: Generated lubrication force variation with Re[3] for l′ =0.25, 0.75 (a) δ/ f = 1 and (b) δ/ f = 10. The parameters are(e,a,λ ) = (0.3,0.4,1).106. Very high Re[3] would result for relatively low viscosity of oil, low lubricantviscosity, larger pipe, and larger δ , e.g. if (Rˆ, µˆ [1],Wˆp, µˆ [3],δ ) = (1m,0.1Pa.s,1m/s,0.001Pa.s,0.1), (Re[1],Re[3]) = (104,105).Figure 4.8 presents the variation of lubrication force (Fl) with Re[3] for l′ =0.25, 0.75 for δ/ f = 1, 10. Other parameters are (e,a,λ ) = (0.3,0.4,1). It showsthat for δ/ f = 1 moderate Re[3] has a negligible effect on the generated lubricationforce. As we increase Re[3] the amplitude of the lubrication force increases. Forlarger δ/ f = 10, the lubrication force is initially negative, then decreases withincreasing for Re[3]/ 105. For larger Re[3] the lubrication force eventually becomespositive for l′ = 0.25, but continues to decrease for l′ = 0.75.Figure 4.9 explores the effects of the shape of the sawtooth profile, (i.e. a, l′),on the lubrication force. We vary Re[3] = 20000, 300000 and δ/ f = 5, 10, 200.As expected, smaller Re[3] generates smaller lubrication force in general. The solidblack line and dashed black line show Fl = 0 and Fl = 0.3, respectively. A positiveFl is needed to balance buoyancy and the larger the value of Fl the larger the coreregion that is transported. Suitable lubrication force are found only for low valuesof l′ and high a, and these require large Re[3] and moderate δ/ f . However, smallerRe[3] do not generate sufficient lift force. Interestingly, the configuration with veryhigh δ/ f = 200, appears able to generate positive lubrication force for inverse128(a) (b) (c)(f)(e)(d)Figure 4.9: Generated lubrication force variation with wave amplitude (a)and wave breaking point (l′) for Re[3] = 20000, 300000 and differ-ent δ/ f ; (a) (Re[3],δ/ f ) = (20000,5), (b) (Re[3],δ/ f ) = (20000,10),(c)(Re[3],δ/ f ) = (20000,200), (d) (Re[3],δ/ f ) = (300000,5), (e)(Re[3],δ/ f ) = (300000,10), and (f)(Re[3],δ/ f ) = (300000,200). Thecase parameters are (e,λ ) = (0.3,1). Solid black line shows the Fl = 0and dashed black line presents Fl = 0.3 as a guide to eye.sawtooth shape, i.e. larger breaking point l′.Figure 4.10 shows the effect of Re[3] on lubrication pressure distribution. Here,the fixed parameters are (e,a, l′,λ ,δ/ f ) = (0.3,0.4,0.25,1,10): results are shownfor Re[3] = 20000, 50000, 100000, 300000. We can see, in general, the magnitudeof lubrication pressure is greater than that generated in laminar flow; compare withFig. 4.3 earlier. With increasing Re[3] the lubrication pressure variation also in-creases, although the distribution has not much changed in these high-Re[3] flows.Figure 4.11 shows the variation of axial velocity profile along the pipe at nar-rowest and widest part of the lubrication layer. The red lines depict the wave shape.We can see that turbulence results in a flat velocity profile and inertia imposes ad-ditional pressure gradients in the converging part of the layer for small Re[3], whenRe[3] / 20000, which results in negative lubrication pressure in this part of the129  (a) (b)(d)(c)Figure 4.10: Lubrication pressure colormap for different Re[3], (a) Re[3] =20000, (b) Re[3] = 50000, (c) Re[3] = 100000, (d) Re[3] = 300000;where (e,a, l′,λ ,δ/ f ) = (0.3,0.4,0.25,1,10).130  (a) (b)(d)(c)r2,0+πδ a≈≈ ≈≈Figure 4.11: Lubrication layer averaged axial velocity profile (w¯(r,y,z))variation along the pipe for different Re[3], (a) Re[3] = 20000, (b)Re[3] = 50000, (c) Re[3] = 100000, (d) Re[3] = 300000; where(e,a, l′,λ ,δ/ f ) = (0.3,0.4,0.25,1,10). Note, red lines indicate thesawtooth wave shape.layer.4.4 Redesigning the waveformLet us now recap some of the design features of the triple-layer method proposed.First, in this chapter we have isolated our attention to the lubrication layer. Thisis isolated from the rest of the flow by a yield stress skin. Previous studies in131Chapter 2 & Chapter 3 have indicated that the fluid used for the skin should havea minimal yield stress in the range 102− 103Pa, which is feasible for many poly-meric gels, emulsions, etc., although the minimal yield stress will increase for theturbulent flows studied here.Secondly comes the question of buoyancy. It is expected that the transportedcore fluid will have a different density than the lubricant (and skin). In order toachieve a steady flow the buoyancy of the core fluid must be balanced by the lubri-cation forces generated. These in turn vary with the eccentric position of the core.Here we have fixed the eccentricity e, but generally where we find e.g. Fl > 0, in-creasing e results in increasing Fl . In principle we may calculate an equilibriumeccentricity to balance buoyancy and lubrication forces, but in practice large ec-centricity (e∼ 1) means impractically thin lubricant layers (e.g. of the scale of anywall roughness). Therefore, it is of interest to increase Fl in other ways.Thirdly, we consider flow regimes. We have seen in both §4.2.1 & §4.3.2, thatthe advective gradients of the mean flow can have a significant effect on the pres-sure field (and consequent lubrication force), for both laminar flow and turbulentflow. In broad terms, the variations in annular gap width that we want to engi-neer, in order to promote lubrication pressures via shear stress, are countered byBernoulli effects. Thus, to have dominant lubrication forces we need to operatein regimes where shear stresses dominate the gradients of the mean flow, whichmeans either the laminar viscous lubrication limit or very high turbulent flow.The laminar viscous lubrication limit results in positive Fl and feasible flows,but only for relatively small pipes, e.g. typically / 20cm diameter, depending alsoon other parameters (as discussed in Chapter 2). Extremely large Reynolds num-bers are also not generally achievable in large pipelines, simply due to pumpingcapacity limitations. Thus, the majority of large-scale applications are unfortu-nately likely to be in regimes of inertial laminar flow or low-to-moderate Reynoldsnumber turbulent flow. These are precisely the flows for which our results haveshown that Fl > 0 is difficult to achieve.Consequently, we need to change the geometrical properties of the sculptedskin in order to make these regimes feasible. The viscous lubrication regimes workby having the largest variations at the top of the annulus, where narrowest, whichleads to positive lubrication pressure. In the wider part of the annulus we have a132Figure 4.12: Lubrication layer thickness (h) variation with y and z; where(e,a, l′,λ ) = (0.3,0.4,0.25,1).near constant negative lubrication pressure. This configuration has proven vulnera-ble to the Bernoulli effect, which is most significant where we have large variationsin annular gap. The alternative might be to design for negative varying lubricationpressures in the thicker part of annulus and positive almost constant pressures inthe upper narrower part of the annulus. To explore, we present brief results withthis idea.Fixed parameters are (e,a, l′,λ ,δ/ f ) = (0.3,0.4,0.25,1,10). Figure 4.12shows the designed lubrication layer thickness (h). The narrower upper part ofthe lubrication layer is not shaped and on the wider lower half, a sawtooth waveis imposed. The computed variation of the lubrication pressure along the pipe ispresented in Fig. 4.13a at different azimuthal positions. It shows, the lubricationpressure of the non-shaped half of the pipe has very small variation and is positive,while the other half generates mainly negative but varying lubrication pressures.Figure 4.13b shows the variation of axial velocity profile in lubrication layer atdifferent axial positions (z). On top of the pipe where there is no wave imposed,the velocity profile barely changes and on the bottom, velocity profile changes atthe decreasing gap thickness, in response to the pressure changes.The variations in lubrication pressure of Fig. 4.13a result in a positive lubri-133(a) (b)≈≈Figure 4.13: (a) Lubrication pressure variation along the pipe at different yand (b) lubrication layer velocity profile at y= 0, 1 at different z; where(e,a, l′,λ ,Re[3],δ/ f ) = (0.3,0.4,0.25,1,20000,10).cation force Fl . The effect of eccentricity on the generated lubrication force isshown in Fig. 4.14. The configuration generates positive lubrication force for allthe values of eccentricity (e) and over a wide range of wave breaking point (l′).However, now we see that higher l′ generates a bigger lubrication force, as mightbe expected. Indeed now we see that in order to have a monotonically increasing Flat larger e, we need larger l′. This monotonicity is required in order for the verticalbuoyancy equilibrium to be stable.To explore further the design configuration, we present the variation of lubri-cation force Fl with wave amplitude (a) and wave breaking point (l′) in Fig. 4.15,for both inertial laminar and turbulent flows. We see that the designed configura-tion generates positive lubrication force for both laminar and turbulent flow. Asexpected, higher values of a and l′ generate larger Fl and turbulent flow leads tohigher lubrication pressures and lubrication force due to the larger shear stresses.Furthermore, as in Fig. 4.14, Fl increases with e as the eccentricity increases suffi-ciently, suggesting a stable equilibrium can be achieved even for large pipes.1340.1 0.2 0.3 0.4 0.5-0.200.20.40.6Figure 4.14: Generated lubrication force (Fl) versus eccentricity e fordifferent wave breaking point l′ = 0.25, 0.5, 0.75; where(a,λ ,Re[3],δ/ f ) = (0.4,1,20000,10).4.5 ConclusionsThis chapter has extended our feasibility study of a three-layer lubrication flowaimed at heavy oil transport in Chapter 2. The new method eliminates interfacialinstabilities by ensuring that the core fluid is surrounded by a visco-plastic skinfluid that does not yield. It also generates lubrication forces transverse to the pipeaxis, which balance the buoyancy of the core fluid. These forces are generatedby shaping the skin. By eliminating interfacial instabilities this prevents dispersedand unstable flow regimes from developing. Thus, we have the possibility of triple-layer core-annular flows in regimes far outside those for 2-layer core-annular flows,as we will explore later in Chapter 6.In this chapter we have explored extending the feasibility of the method forlarger pipelines, higher density differences and faster flows, by studying inertialand turbulent flow effects. We investigated a single periodic wavelength of fullydeveloped flow, in which the buoyancy of the core fluid was balanced by the gen-eration of hydrodynamic lubrication pressures, conceptually similar to Chapter 2.Different scaling for the different flow regimes of the lubricating layer was dis-cussed and leading order governing equations were developed. We focused on two135  (a) (b)(d)(c)Figure 4.15: Generated lubrication force variation with wave shape (a &l′) for inertial-laminar and inertial-turbulent flow; (a) inertial-laminarRe[3]δ = 2, (b) inertial-laminar Re[3]δ = 15, (c) inertial-turbulent(Re[3],δ/ f ) = (20000,10), and (d) inertial-turbulent (Re[3],δ/ f ) =(300000,10). Fixed parameters are (e,λ ) = (0.3,1).types of flows in the lubricating layer: (i) inertial laminar flow and (ii) inertial tur-bulent flow. By inertial here is meant that the gradients of the gap-averaged meanflow play a significant role, which occurs since we are intentionally sculpting theskin shape to provide lubrication forces.We have used an analytical method to estimate inertial effects by assuming theself-similarity of the velocity profile in thin layer flow and integrating the equationsacross the lubrication layer. This study showed that inertia can have a significanteffect on the pressure field and consequently lubrication force. In inertial-dominantflow, the variation of pressure is influenced by a Bernoulli effect, which can to an136extent counter the lubrication effects of the shear flow. As we move to turbulentflows the Reynolds stresses become progressively dominant, generating higher lu-brication pressure than in laminar flow but still vulnerable to inertial effects fromgradients of the mean flow.The reduction in lubrication force as inertial effects increased and other in-sights gained have led us to redesign the imposed skin shape, in a way to generatefavorable lubrication force Fl > 0. We develop one such example, using simpleprinciples, showing that we can achieve a shape that results in monotone increasein Fl with the eccentricity e. This is needed for a stable equilibrium position to beachieved.Having explored how to extend the proposed triple-layer methodology to largepipelines and faster flows in this chapter, we move in the Chapter 6 to the explo-ration of different flow regimes, outside of the usual ranges of CAF.137Chapter 53D Simulation of Triple-LayerFlow Development1A triple-layer CAF was introduced in Chapter 2 that overcomes the difficulties ofinterfacial instabilities, commonly found in CAF. In Chapter 3, we have studiedthe sculpting of a desired streamwise interface shape in more detail. We have mod-elled the axisymmetric inlet manifold of a triple-layer flow both computationallyand analytically over broad ranges of parameters. Flow rate control is effective atproducing the desired interface shapes at least in the long wavelength limit. Wewere also able to estimate the extensional stress generated in the skin layer, whichtranslates into a design requirement for the yield stress of the skin fluid. The effectsof inertia and turbulence on the lubricant layer and generated lubrication force, asrelevant for larger pipelines were discussed in Chapter 4.In this chapter we study fully 3D computations. Although previous chaptershas established the feasibility of the triple-layer method of Chapter 2, a numberof questions remain. First, the semi-analytical approaches taken in Chapter 2 andChapter 3 assume established flows in which viscous lubrication theory can be ap-plied. Although the VPL studies suggest that interfacial instabilities should notarise, transient Navier-Stokes computations and/or experiments are needed to ver-ify this. In Chapter 3 the computed flows were axisymmetric, which are often lesssensitive to instability, but here we consider fully 3D simulations, i.e. we validate1A version of this chapter has been published [159].138the earlier axisymmetric and reduced order models within their regimes of appli-cability. Secondly and importantly, there are flow features that are fully 3D to bestudied. Here we study the effect of the density mismatch between fluid layers onthe developing flow and in particular as flow develops from axisymmetric inflowmanifold, rising due to buoyancy until its equilibrium position is established. Westudy this motion and the stability of the configuration using 3D computations. Wealso then develop a simplified semi-analytical model to estimate the same features.An outline of this chapter is as follows. In §5.1 we describe the flow and thegoverning equations. We briefly explain the mathematical model, the dimension-less numbers and leading order equations. The computational method and domainare introduced in §5.1.1. The code is validated in comparison with analytical so-lution and previous 2D axisymmetric simulations from Chapter 3. Section 5.2presents results from the 3D model. This shows that these flows can be establishedin fully 3D situations and the shaped interface can generate sufficient lubricationforce to counterbalance the buoyancy force. The effects of different parameterson the balanced configuration are studied. In §5.3 we develop a simplified semi-analytical model to resolve motion towards the balanced eccentricity: a lubricationapproximation is used for lubricant layer, inertia is neglected and the skin is as-sumed rigid. The chapter concludes with a brief summary.5.1 Modelling flow developmentIn this chapter, we aim to study development flows of the eccentric triple-layerflows with shaped interface. Figure 5.1 shows a schematic of the flow. Three fluidsare pumped into the pipe at specified time-dependent flow rates and we assume thatthe flow develops initially within an inflow manifold. Fluid 1 denotes the core fluid(a viscous Newtonian heavy oil), with viscosity µˆ [1] and density ρˆ [1]. The skin layeris fluid 2: a yield stress fluid, modeled most simply as a Bingham fluid with µˆ [2],τˆ [2]y , and ρˆ [2] denoting its viscosity, yield stress, and density, respectively. Fluid 3is the lubricant, assumed to be a low viscosity Newtonian fluid, with viscosity µˆ [3],and density ρˆ [3].For simplicity, throughout the chapter interfacial tension will be ignored. Thekey idea of the flow is to use the yield stress of fluid 2 to separate the other two139OutflowQ1=1Q2(t )Q3(t ) u=v=w=0u=v=w=0rzξ1ξ21      (core)2      (skin)3      (Lubricant)Lr1r2R=1LFigure 5.1: Schematic of the dimensionless triple-layer flow geometry.fluids by a rigid skin. This has the implication that the yield stress is the dominantstress. Additionally in terms of application area, pipelining radii tend to be in therange 5−100cm so that bulk capillary effects are anyway minimal.The incompressible Navier-Stokes equations are the governing equations forthe flow in each fluid domain. The traction and velocity vectors are continuousacross each interface and we assume that the interfacial tension may be neglectedas argued above. Constitutive equations for the three fluids are:τˆi j [k] = µˆ [k] ˆ˙γi j, k = 1,3, (5.1)τˆ [2]i j =[µˆ [2]+τˆ [2]yˆ˙γ]ˆ˙γi j⇐⇒ τˆ [2] > τˆ [2]y , (5.2)ˆ˙γ = 0 ⇐⇒ τˆ [2] ≤ τˆ [2]y , (5.3)whereˆ˙γi j =∂ uˆi∂ xˆ j+∂ uˆ j∂ xˆi,ˆ˙γ =[123∑i, j=1[ ˆ˙γi j]2]1/2, τˆ [2] =[123∑i, j=1[τˆ [2]i j ]2]1/2. (5.4)140In order to scale the equations, we use core flow rate Qˆ1. The superficial oilvelocity Wˆ0 = Qˆ1/piRˆ2 is defined as the velocity scale. We scale all lengths withRˆ. The stress scale is µˆ [3]Wˆ0/Rˆ, used for both the deviatoric stresses and pressure,representing the shear stress in the lubricating layer.(r,z) =(rˆ, zˆ)Rˆ, t =tˆWˆ0Rˆ, u =uˆWˆ0, p = pˆRˆµˆ [3]Wˆ0, τ = τˆRˆµˆ [3]Wˆ0.This leads to a problem governed by 2 dimensionless radii, r1 & r2,0, wherer2,0 is the intended mean value of r2(z), and 6 further dimensionless groups:m =µˆ [3]µˆ [1], m2 =µˆ [2]µˆ [3], B =τˆ [2]y Rˆµˆ [3]Wˆ0, Re =ρˆRˆWˆ0µˆ [3], φ2 =ρˆ [2]ρˆ [1], φ3 =ρˆ [3]ρˆ [1].Here m 1 is the viscosity ratio of lubricant to transported fluid. The viscos-ity ratio m2 plays little role as for the most part the skin layer is unyielded. TheBingham number B, is a measure of yield stress to viscous stress in the lubricantand Re is a Reynolds number, also based on the lubricant. We assume the core andskin have identical densities (φ2 = 1), i.e., ∆φ = |φ3−1|.In dimensionless form, the constitutive laws within each fluid areτ [1]i j =1mγ˙i j, (5.5)τ [3]i j = γ˙i j, (5.6) τ[2]i j =[m2+B|γ˙|]γ˙i j⇐⇒ τ [2] > B,γ˙ = 0 ⇐⇒ τ [2] ≤ B. (5.7)The dimensionless Navier-Stokes model considered in this chapter is as fol-lows:∇ ·u = 0, (5.8)Re[∂u∂ t+(u ·∇)u]=−∇p+∇ · τ, (5.9)where p = P+ ρg.x. To treat this system analytically we assume that the threefluids are separated by two interfaces, each of which evolve following a kinematic141condition.Equations (5.8) & (5.9) are solved over the domain 0 ≤ r ≤ 1, 0 ≤ θ ≤ 2piand 0 ≤ z ≤ L = Lˆ/Rˆ for various parameters. We impose no-slip conditions atthe walls and outflow conditions at the exit to the flow domain. To eliminate thedependency of the flow dynamics downstream of the manifold to the computationaldomain length, L is chosen to be significantly longer than the interior walls of theinflow manifold. Given the desired positions of the two interfaces, we are able todetermine the required flow rates for the 3 fluid streams: (1,Q2,Q3), following theprocedure explained in §3.1.1.5.1.1 Computational methodWe solve the model equations (5.8) & (5.9) numerically with the boundary con-ditions described to obtain the computational results of the chapter. As seen inFig. 5.1, the three inlet flows are separated with short thin walls, which allow theindividual streams to become developed before merging. The wall between flu-ids 2 & 3 is of length ξ2 and that between fluids 1 & 2 is of length ξ1. We haveset ξ1 > ξ2 so that the interface between fluids 2 & 3 forms before the interfacebetween fluids 1 & 2.As an initial condition we set a motionless uniform concentric flow. For t > 0we impose the prescribed shape of the interface on the flow, i.e. there is a “ramp”from zero to imposed flow rates for t > 0. For each t the desired interface ra-dius allows us to compute the relevant flow rates (1,Q2(t),Q3(t)), as described in[158]. Having determined the individual flow rates, we set a uniform inflow veloc-ity in each separated fluid stream, at z = 0, to give the inflow condition. At eachtime(step) this is repeated.The model equations are solved with Multilevel Pressure Schur Complement(MPSC) approaches and discretized using finite element method (FEM), see e.g. [174,175]. In this study, we use fully implicit Backward Euler method [112], with smallenough uniform timestep to discretize the equations in time. Discrete equationsin time are discretized in space by FEM. The main idea of MPSC is first to com-pute a velocity field without taking into account incompressibility and then per-form a pressure correction, which is a projection back to the subspace of diver-142gence free vector fields, i.e. pressure and velocity fields are solved in a decoupledway. The computations are carried out on a block-structured mesh, with high or-der Ladyzhenskaya-Babusˇka-Brezzi (LBB) stable conforming finite element pairQ2−P1 [47]. With this choice of elements, the inf-sup condition is satisfied.Two complexities are faced by the underlying computational method: (i) theskin layer contains a yield stress fluid, see e.g. [39, 153], and (ii) positions of themoving interfaces between fluids must be calculated. Two methods are popularfor dealing with yield stress fluid. Either one regularizes the effective viscosityfunctional to remove the singular behaviour; see e.g. [39]. This system can then besolved iteratively as a flow problem with a nonlinear viscosity. Alternatively, onesolves the equations using a relaxation-multiplier approach such as the augmentedLagrangian method, see e.g. [181]. In this study, the regularization method is usedin which stress tensor is replaced by (5.10):τ [2]i j (u) =(m2+B(ε2+ γ˙2(u))1/2)γ˙i j(u). (5.10)In (5.10) we follow [12], although other choices are equally effective.There are several methods to resolve time varying fluid domains of interest in-cluding interfaces, e.g. fictitious boundary, phase field (level-set method), or sharpinterface representation. Each of these methods has advantages and disadvantages,e.g. the first two methods allow topology changes, but cannot capture the interfaceexactly. The strongest advantage of the sharp interface representation is the orderof approximation, but it does not in general allow topology changes and extremedeformations. However, here we expect that topological challenges will be mi-nor. Thus, the sharp interface representation is applied as an Arbitrary LagrangianEulerian (ALE) based mesh deformation method, see e.g. [128]. In this method,the location of the mesh nodes representing the interfaces are updated based onthe normal fluid velocity. While the interface mesh motion is determined by thephysics, i.e. the kinematic equation, the mesh motion in the bulk fluid regions canbe chosen arbitrarily. To have a high quality mesh, Laplacian smoothing is succes-sively applied to the volume mesh. Laplacian smoothing consists in moving eachmesh point towards the barycenter of the surrounding mesh points [28]. Therefore,143new node positions x˜i are calculated as follows.x˜i = ΣNj1Nx j, (5.11)where N represents the number of neighbour points at point i. All the required CFDtechniques for 3D computations are implemented in the open source FEM-CFDFEATFLOW software (Finite Element Analysis and Tools for Flow problems).The package is utilized with multiphase flow numerical modules, such as flexibleand higher order FEM discretization schemes in space and time with flux correc-tion and edge-oriented stabilization techniques, unstructured meshes with adaptivegrid deformation, efficient Newton-Multigrid solvers and parallelization based ondomain decomposition. The description and exploitation of these numerical ingre-dients can be found in [29, 53, 94, 176] and brief introduction to FEATFLOW ispresented in appendix §A.2 .For the results presented below we set the length of computational domain toL = 15 and L = 30 for concentric and eccentric cases, respectively. The latter cor-respond to simulations with density differences and have longer flow developmentlengths. A few cases are computed on a domain with L = 60, however, we useL = 30 for most of the computations due to computational cost. Mostly we com-pute with modest Re. The main point here is that the flows considered have largeyield stress (B) and m 1, with the consequence that fluid layers 1 & 2 have large(effective) viscosity and the development lengths are consequently short. Once theunyielded skin forms, fluids 1 and 3 are separated and have development lengthsthat scale with a representative reduced width, i.e. with 1−r2,0 in fluid 3. Thus, theflows within each layer develop relatively rapidly. However, another developmentlength-scale arises which represents the required length over which a balanced ec-centric core position is achieved, i.e. at which the buoyancy (due to density dif-ference) is balanced with the lubrication force from the eccentricity and shapedinterface.5.1.2 Validation and benchmarkingAn analytical solution can be calculated for the uniform Poiseuille solution and isused to examine the convergence behavior of the discretized system. We integrate144mesh mesh 1 mesh 2 mesh 3 mesh 4numberofcells218880 449280 599040 1751040Table 5.1: Studied meshesthe model equations from initial conditions until steady, imposing only a constantflow rate. We compare the numerical solution at z = L with the analytical solution.For parameters (B,Re,m,r1,r2,0) = (10,1,0.1,0.4,0.75), Fig. 5.2a-d shows thedifference between the analytic solution and the computed solution at the exit, forvarious mesh sizes (the meshes used are illustrated in Table. 5.1). It can be seenthat the velocity, phase field (Ψ), and strain rate are well represented at these meshresolutions.For the same parameters as Fig. 5.2a-d, we show the effects of regularizationparameter (ε) in Fig. 5.3 & 5.4. For ε = 1 this is effectively just a shear-thinningfluid, but we observe that even for ε = 0.1 the exit velocity profile and strain rateare showing signs of converging. For rest of the simulations, we use ε = 0.001 as itleads to physically accurate results and the results barely changes for smaller val-ues of ε . Note, in Fig. 5.4, the top panels show the strain rates in logarithmic scale(log10(γ˙)) and the bottom panels present the stress (τ). We can see, as the regular-ization parameter decreases, the strain rate variation increases, i.e. smaller ε leadsto smaller strain rate in skin layer. However, the stress field changes negligiblywith regularization parameter (ε) for small enough ε .Figure 5.5 shows L2 norms of shear rate and axial velocity of the triple-layerfully developed flow. Here, the solution of ε = 0.00001 is chosen as the referencesolution and other results are compared to this one. We can see, smaller ε resultsin smaller errors and convergence rates of the shear rate behave like ε2/3, and for145(b)(d)(c)(a)Figure 5.2: (a) Fully developed velocity profile (W (r,L)), (b) L2 normof difference between exit velocity W (r,L) and analytic veloc-ity W (r), (c) phase property Ψ(r,L), and (d) strain rate fordifferent mesh sizes. Legends are same as Fig. 5.2a; where(B,Re,m,r1,r2,0) = (10,1,0.1,0.4,0.75).the velocity somewhat higher, like ε0.93.We validated current 3D results with axisymmetric results computed using themethods described in Chapter 3. Briefly, a mixed finite element finite volumemethod is used, with an augmented Lagrangian method for the yield stress fluid, allimplemented within the open source PELICANS software, see appendix §A.1. Fig-ure 5.6a shows the phase maps for parameters (B,Re,m,r1,r2,0,a,T,Tb)= (300,10,0.01,0.4,0.75,0.4,2,0.5) at t = Tb, T . In terms of the implemented waveform forthe skin shape, a sawtooth waveform is imposed with time period T , breaking pointat time Tb, and wave amplitude a = aˆ/Rˆ(1− r2,0). The black solid line indicates146(a) (b)Figure 5.3: (a) Fully developed velocity profile (W (r,L)) for different reg-ularization parameters, ε = 1, 0.1, 0.01, 0.001, 0.0001, 0.00001,and analytic solution. (b) Closer look at the core flowvelocity profile, legends are the same as Fig. 5.3a; where(B,Re,m,r1,r2,0) = (10,1,0.1,0.4,0.75).the position of interfaces for the same exact parameters and times computed by theaxisymmetric model in PELICANS. We can see a surprisingly good agreement be-tween two models, considering that the interface tracking methods in PELICANSand FEATFLOW are very different. Within PELICANS we use a PLIC Volume ofFluid (VOF) method [139] whereas FEATFLOW tracks the interface accurately bya mesh deformation method.Velocity magnitude colormaps for the same parameters are presented in Fig. 5.6bwhere the unyielded skin/plug is shown by the grey region on the colormaps (astress threshold within the FEATFLOW calculations). The plug computed by theaxisymmetric model using PELICANS is shown by the black solid line. The smalldifferences between 3D and 2D results may be because of different treatment ofyield stress fluid, i.e. the augmented-Lagrangian method is used to compute theskin layer in the axisymmetric computations.5.2 Computational resultsWe now present a number of examples that use the 3D simulation to explore flowdevelopment of eccentric triple-layer core-annular flow. In general we choose Blarge enough to maintain the skin layer completely unyielded when the periodic147  log10 γ˙τFigure 5.4: Strain rate colormaps in logarithmic scale log10(γ˙) and stresscolormaps (τ) for ε = 1, 0.1, 0.01, 0.001, 0.0001, 0.00001; where(B,Re,m,r1,r2,0) = (10,1,0.1,0.4,0.75).1480.650.93Figure 5.5: L2 errors for fully developed triple-layer flow computed via regu-larisation. We use the solution of ε = 0.00001 as the reference solutionand compute the error of other values, ε = 1, 0.1, 0.01, 0.001, 0.0001,with respect to that. Parameters are (B, Re, m, r1, r2,0) =(10, 1, 0.1, 0.4, 0.75).flow is established downstream of the inflow manifold, although we also look atwhat happens when the skin yields. We use thicker lubricant and skin layer forthe computed examples that would be normal, primarily to be able to visualize thevariations in sculpted skin shape more clearly. The main dimensionless parametersstudied are Re, m, B, ∆φ .5.2.1 Outflow boundary conditionWe first study the effect of outflow boundary condition before we investigate in-dividual flows. There are different choices for outflow boundary condition andan obvious one would be to implement a stress free outflow boundary condition(OBC), as we have used in Chapter 3. Although straightforward to implement, wenote that even if the flow were fully developed, slow axial gradients in the periodicflow (due to the shaped interface) mean that there should be small gradients in thenormal stresses at the outflow, i.e. the OBC is not strictly correct for the establishedflow. Imposing the intended periodic outflow might be an alternative option. It isnot completely clear how this would be done, but probably as a velocity condi-149(a)(b)Figure 5.6: (a) Phase maps; solid black line shows the border of skinlayer using axisymmetric model computed by PELICANS. (b) Ve-locity colormaps and the shaded grey region indicates the plugregion; black solid line shows the plug region using axisym-metric model computed by PELICANS at t = Tb, T ; where(B,Re,m,r1,r2,0,a,T,Tb) = (300, 10, 0.01, 0.4, 0.75, 0.4, 2, 0.5).tion rather than a stress condition. However, in doing this we would effectivelybe constraining the flow rather than allowing it to develop towards the establishedflow.A third option is using a free (Do-Nothing) outflow boundary condition (FBC).This natural outflow condition is obtained by integration by parts of the weak formof (5.8); see [59]. Figure 5.7 shows the effect of outflow boundary conditions(OBC vs FBC), on the velocity, development length (plug region), and stressesfor parameters (B,Re,m,r1,r2,0,a,T,Tb) = (300,1,0.01,0.4,0.75,0.4,2,0.5) att = 6T . The top panels present the stress free outflow condition (OBC) results andbottom panels show the free outflow condition (FBC) results. We have taken iso-dense fluids for this example. It can be seen that the results are essentially identical.150OBCFBC(a)(b)OBCFBC(c)Figure 5.7: Effects of outflow boundary condition on the flow: (a) phasemaps, (b) velocity colormaps and the shaded grey region indicates theplug region, and (c) stress colormaps in different directions, τzz, τrr, τrz;where (B,Re,m,r1,r2,0,a,T,Tb) = (300,1,0.01,0.4,0.75,0.4,2,0.5).Note, OBC and FBC stand for stress free outflow condition and freeboundary condition, respectively.Implementing either of these outflow conditions is relatively easy and we use stressfree outflow boundary conditions (OBC) for rest of the study, to be consistent withChapter 3.151Figure 5.8: Phase maps at t = 0, 2T, 5T, 10T, 15T and theblack line shows the border of unyielded skin/plug, where(B,Re,m,r1,r2,0,a,T,Tb,∆φ ,L) = (300,0.1,0.01,0.4,0.75,0.4,2,1.5,0.05,30).5.2.2 Development of eccentric flowsWe study the effect of density difference on the development of triple-layer core-annular flow. An example case is solved in a domain R×L = 1×60, with flow pa-rameters: (B,Re,m,r1,r2,0,a,T,Tb,∆φ) = (300,0.1,0.01,0.4,0.75,0.4,2,1.5,0.05).The interface variations in time and space at a slice of x = 0 are presented inFig. 5.8. We can see the outer interface is shaped and the inner interface remainsalmost constant. The initial concentric configuration changes to eccentric config-uration because of the density difference, i.e. the core and the skin have the samedensity, lighter than the lubricant. Thus, the core and the skin rise to a balancedeccentricity. The border of unyielded skin/plug is shown by black line. We can seethe plug is formed shortly after exiting the inflow manifold, due primarily to thehigh yield stress.We note that there are 2 flow developments here: temporal and spatial. Sincewe start from a uniform stationary flow, and since B is large, the layers down-stream of the inflow manifold cannot be shaped as the flow rate variations beginto be imposed. We see that uniform layers remain downstream and these must beadvected from the domain as the sculpted flow is established upstream in the inflowmanifold.Figure 5.9 shows the velocity colormap and mesh deformation at different152times and the black line depicts the interfaces for a single wavelength of the skin.We can see the plug velocity is constant at each time. Left column presents the re-sults at t = 2T, 5T, 10T, 15T and right column shows the results at t = 2.5T, 5.5T,10.5T, 15.5T . It shows, plug velocity has higher value in right column, however,the plug velocity barely changes within either right or left columns. The plug ve-locity variation is due to the time varying imposed flow rates and because of theperiodic inflow we expect periodic flow in downstream when flow develops. Wecan also see the mesh is deformed sharply to track the interfaces, i.e. deformationof the nodes presenting the interface exactly follows the physics of the flow and thebulk inner region nodes are deformed arbitrarily. Because of the eccentric config-uration, the lubricant flows faster in the wider gap and it imposes higher velocitygradient in this layer and consequently higher mesh deformation.The eccentricity variation in time at different cross sections of the flow is shownin Fig. 5.10a. It shows that sections further from inlet are more eccentric, as isnatural. Sufficiently far downstream and after sufficient time, the eccentricity ap-proaches a steady value and oscillates around this fixed value. The flow configu-ration thus reaches a balanced quasi-steady state. The oscillations are due to thewave shape imposed on the skin, which causes a consequent variation of lubricationpressure (force) to which the core responds. Figure 5.10b shows that oscillationsin e(t) follow the wave shape with period equal to the imposed wave period T .Figure 5.11a presents the shaped skin in 3D for the developed part of the flowat t = 20T , and also shows the velocity colormap of the lubricant. As the flowdevelops and eccentricity increases, the lubricant flows faster in the wider gap,i.e. a wider gap means larger wall shear stresses which then leads to larger shearrates and a larger flow rate (vice versa on the narrow side). Variation of lubricationpressure at different azimuths can be seen in Fig. 5.11b for the same section of thepipe as Fig. 5.11a. The narrowest lubrication layer generates the highest lubricationpressure. Note, the difference in lubrication pressure at narrowest and thickest partsof lubrication layer can be significantlly higher for narrower lubrication layers.In this study the average lubricant thickness is 0.25 which is artificially high aswe wish to present an example with easily visualised velocities. The azimuthalvelocity colormap is shown by Fig. 5.11c over a single wavelength. The origin ofazimuthal velocity is the eccentricity and this is generally small compared to the153  (a) (b)(d)(f)(h)(g)(e)(c)Figure 5.9: Velocity colormap and mesh deformation at t =2T, 2.5T, 5T, 5.5T, 10T,10.5T, 15T, 15.5T shown for a singlewavelength. The black line shows the interfaces. Flow parameters are(B,Re,m,r1,r2,0,a,T,Tb,∆φ ,L) = (300,0.1,0.01,0.4,0.75,0.4,2,1.5,0.05,30).154(b)T=2(a)Figure 5.10: (a) Eccentricity variations in time at z = 10, 20, 30, 40, 50.(b) Detailed of eccentricity variation in time at z = 20; where(B,Re,m,r1,r2,0,a,T,Tb,∆φ ,L) = (300,0.1,0.01,0.4,0.75,0.4,2,1.5,0.05,60).axial velocities.5.2.3 Parametric studyIn this section we study effects of different parameters on the flow. Figure 5.12shows the variation of eccentricity in time at z= 25 for different density differences(∆φ ). As expected, higher density difference leads to higher eccentricity and inthis sample case, the configuration is not able to support ∆φ ≥ 0.05. Variation ofthe balanced eccentricity is higher for high density differences because a narrowerlubrication layer and higher local lubrication pressure is required.Another important parameter in the flow development is the domain length. Wehave used L= 30 for most of the simulations. Figure 5.13a presents the eccentricitydevelopment along the pipe at different time for pipe with L = 60. It shows, fort ' 15T the eccentricity barely changes. In this case for z ' 33 the eccentricityhas almost constant value, i.e. the flow is fully developed and the core positiondoes not change with length of the domain. Note, the red solid line shows whereeccentricity reaches 90% of its balanced value.The eccentricity variation in t at z = 20 can be seen for two different domains,L = 30, 60 in Fig. 5.13b. We can see the pattern of the eccentricity variation andthe time that the point reaches the balance position is similar for the two lengths,155y=0y=1(b) (c)(a)Figure 5.11: (a) Fully developed shaped skin and the lubricant velocitycolormap, (b) lubrication pressure variation in z at different az-imuthal positions, and (c) azimuthal velocity colormap of lubricationlayer shown over a single wavelength at t = 20T . Parameters are:(B,Re,m,r1,r2,0,a,T,Tb,∆φ ,L) = (300,0.1,0.01,0.4,0.75,0.4,2,1.5,0.05,60).5 10 15 20 2500.10.20.30.40.5Figure 5.12: Variation of eccentricity with t at z = 25 for differentdensity differences ∆φ = 0.005, 0.01, 0.03, 0.02, 0.04; where(B,Re,m,r1,r2,0,a,T,Tb,L) = (300,1,0.01,0.4,0.75,0.4,2,1.5,30).156(a) (b)Figure 5.13: (a) Variation of eccentricity in z at t = 10T, 15T, 20T, 21Tfor very long domain (L = 60), the red solid line shows the pointwhere the eccentricity reaches 90% of its balanced value. (b) Eccen-tricity variation in t at z = 20 for L = 30, L = 60. Parameters are(B,Re,m,r1,r2,0,a,T,Tb,∆φ) = (300,0.1,0.01,0.4,0.75,0.4,2,1.5,0.05).however, the balanced eccentricity value is different by small amount; here the dif-ference is 6%. Although in a fully developed periodic flow the balanced eccentricposition is unique, here the flow is concentric on leaving the inflow manifold butdevelops eccentricity downstream and in time. Thus, lubrication pressures are dif-ferent in the two situations. For the remainder of the chapter we use the shorterdomain, L = 30. Since the computational time required is at least the advectivetimescale of the domain, the computational time is significantly shorter.The wave amplitude (a) is another parameter useful for governing the lubri-cation pressure and hence influencing the equilibrium eccentricity. Its effect oneccentricity is shown by Fig. 5.14 for a = 0.2, 0.4, 0.6. We can see the balancedeccentricity for a = 0.4 is slightly smaller than the one with a = 0.2 because of thehigher lubrication pressure for higher wave amplitude. Very high wave amplitude,a = 0.6 shows higher variations in eccentricity which could be due to the very nar-row gap and consequent computational limitation. It appears that the eccentricityis decreasing as time increases.Figure 5.15a shows the eccentricity development in t at z= 20 for different Re.Higher Re leads to higher balanced eccentricity, which is the result of inertia. Asstudied in Chapter 4 for high Re, inertial effects become comparable with viscous1575 10 15 20 250.260.280.30.320.340.360.38Figure 5.14: Variation of eccentricity in t at z = 20 for dif-ferent wave amplitude a = 0.2, 0.4, 0.6; where(B,Re,m,r1,r2,0,a,T,Tb,∆φ ,L) = (300,1,0.01,0.4,0.75,0.4,2,1.5,0.03,30).effects and reduce pressure variations through a Bernoulli effect, i.e. with decreas-ing lubrication layer gap, the velocity increases and pressure decreases. Figure5.15b shows the variation of eccentricity in t at z = 20 and Re = 0.1 for differentdensity differences. It shows that higher density differences result in higher ec-centricity and the configuration can support quite high density differences, i.e. theflow with smaller Re can generate higher lubrication pressure as it generates onlyviscous stresses; see Fig. 5.12 for comparison. We emphasize again that the lu-bricant and skin layers are relatively thick in these examples, the consequence ofwhich is that the (lubrication approximation) aspect ratio would be relatively largeand inertial effects come into play at relatively low Re.5.2.4 Yielding of the skinIn this study, we have typically choosen high enough Bingham number (B) to keepthe skin unyielded as the flow develops. Here, we examine yielding of the skin.Figure 5.16 presents the skin interface that results from a low Bingham number,B = 1, we can see the skin is initially shaped as it leaves the inflow manifoldbecause of its high effective viscosity. As the flow develops downstream and ec-centricity increases, higher pressure and consequently higher stresses are applied158(a) (b)Figure 5.15: (a) Variation of eccentricity in t at z = 20 for different coreReynolds number Re = 1, 0.1, 0.01 and ∆φ = 0.03. (b) Vari-ation of eccentricity in t at z = 20 for different density differ-ences ∆φ = 0.05, 0.1, 0.15, 0.2 and Re = 0.1. Parameters are(B,m,r1,r2,0,a,T,Tb,L) = (300,0.01,0.4,0.75,0.4,2,1.5,30).to the skin which leads to higher shear rates. For the low B the stresses exceed theyield stress and we effectively have a viscous-viscous interface at the interface ofthe skin and lubricating layer, which is known to be unstable even for low Reynoldsnumber. In the lower part of Fig. 5.16 we see in detail the small bumps that are ap-pearing at the top of the layer. Note that the growth timescale of such instabilitieswill be limited by viscosity, or in this case yield stress. Note here that the strainrate is presented in logarithmic scale on the skin, the bumps are completely yieldedand have relatively high strain rates. We have also tried the same simulation forhigher Bingham numbers, (B = 10, 100), both of which are less than needed tokeep the skin rigid. Both simulations also resulted in a misshaped interface, butless visible than here with B = 1. We note that the development of bamboo insta-bilities is common in core-annular flows with small viscosity ratios (as here), andthe bumps could represent the start of this. However, to simulate for the very longtimes needed to resolve this would also need a very long domain as the skin andcore advect downstream.159Figure 5.16: Strain rate colormap in logarithmic scalelog10(γ˙) on yielded shaped skin at t = 4T ; where(B,Re,m,r1,r2,0,a,T,Tb,∆φ ,L) = (1,1,0.01,0.4,0.75,0.4,2,0.5,0.03,30).d^Θθ r^1r^2r^3ω^ pX^ p , cW^ pL^(a) (b)Figure 5.17: Schematic of the periodic triple-layer eccentric core-annularflow a) cross section and b) side view.1605.3 Core rise modelFigure 5.17 illustrates the geometry of our model. The pipe includes core fluidwhich is encircled with completely unyielded visco-plastic skin layer. And, thesetwo layers are lubricated with low viscous Newtonian fluid. The layers’ configura-tion is eccentric, and the center of core and pipe are displaced from one another bya distance dˆ in a direction that can be specified by the angle Θ, shown in Fig. 5.17,which lies between the vertical line of centers. The skin layer acts as a solid layerwhich moves and rotates with constant axial velocity (Wˆp) and angular velocity(ωˆp), assuming its yield stress is big enough remains unyielded. Consequently,solid mechanics principles should be applied to determine its motion.The complete mathematical formulation of the model consists of equations ofmotion for the position and rotation of the skin layer, plus Navier-Stokes equationsfor the core and lubricating fluids, i.e. as we have solved in 3D. We do not solvethis complete formulation, but instead introduce Stokes approximation to eliminateinertial terms and then use a classical lubrication scaling to reduce the equationsfurther. To calculate forces we take a periodic flow in axial and azimuthal directionswith periods Lˆ and 2pi , respectively. We put the frame of reference at the centerof mass of the skin layer/plug, i.e, xˆ = Xˆ − Xˆ p,c(tˆ) and uˆ = Uˆ − Uˆ p,c(tˆ), where,Uˆ p,c = ( ˆ˙d, dˆ ˆ˙Θ,Wˆp) in (rˆ,θ , zˆ) directions. The core flow rate (Qˆ[1]) is constant andknown.5.3.1 Governing equationsThe configuration and motion of the skin is specified by four independent variables,Wˆp(tˆ), dˆ(tˆ), Θ(tˆ), and ωˆp(tˆ), as illustrated in Fig. 5.17. We derive momentumequations for core and skin fluids combined, as the latter is a rigid solid and thecore very viscous. Newton’s second law in the axial direction for the core and theskin layer leads to (5.12).ρˆ [2](Vˆ [1]+Vˆ [2]) dWˆpdtˆ= pi rˆ22,0[Pˆ(θ ,− Lˆ2)− Pˆ(θ , Lˆ2)]+∫ 2pi0∫ Lˆ2− Lˆ2rˆ2τˆ[3]rz |rˆ=rˆ2dθdzˆ,(5.12)where Vˆ [1] and Vˆ [2] are the volumes of core and skin layer, respectively. The posi-tion of the skin layer is determined by the momentum equations in r and θ direc-161tions (equations of motion). We write the equations of motion for core fluid andskin layer together, as these two layers move with each other.− mˆcp(dˆ ˆ¨Θ+2 ˆ˙d ˆ˙Θ)=−∫ 2pi0∫ Lˆ2− Lˆ2rˆ2[Pˆ|rˆ=rˆ2 sin(θ −Θ)− τˆ [3]rθ |rˆ=rˆ2 cos(θ −Θ)]dθdzˆ+(ρˆ [3]− ρˆ [2])gˆVˆ [2] sinΘ+(ρˆ [3]− ρˆ [1])gˆVˆ [1] sinΘ, (5.13)mˆcp(ˆ¨d− dˆ ˆ˙Θ2)=−∫ 2pi0∫ Lˆ2− Lˆ2rˆ2[Pˆ|rˆ=rˆ2 cos(θ −Θ)+ τˆ [3]rθ |rˆ=rˆ2 sin(θ −Θ)]dθdzˆ+(ρˆ [3]− ρˆ [2])gˆVˆ [2] cosΘ+(ρˆ [3]− ρˆ [1])gˆVˆ [1] cosΘ, (5.14)where mˆcp is the sum of the core and plug masses. Rotation of the skin layerfollows the angular momentum equation, which for the skin is:ΣTˆ = Iˆpαˆ = Iˆp ˆ˙ωp, (5.15)where ΣTˆ is the net torque which is applied by lubricating and core fluid on skinlayer and Iˆp is the moment of inertia of skin layer.ΣTˆ =∫ 2pi0∫ Lˆ2− Lˆ2rˆ21 τˆ[1]rθ |rˆ=rˆ1dθdzˆ+∫ 2pi0∫ Lˆ2− Lˆ2rˆ22 τˆ[3]rθ |rˆ=rˆ2dθdzˆ. (5.16)5.3.2 Lubrication approximation and reduced modelTo scale the equations, we use the scaling presented in Chapter 2 which is theclassical lubrication scaling. In other words, we assume that the mean thickness ofouter lubrication layer is thin relative to circumferential and axial length-scales, piRˆand Lˆ, respectively, and δ = (Rˆ− rˆ2,0)/piRˆ 1 and λ = Lˆ/piRˆ∼ O(1). We definez= zˆ/Lˆ, θ = piy, and r = r2,0+piRˆδx. Velocity components in axial and azimuthaldirections are scaled with the plug velocity Wˆp and that in the radial direction withδWˆp. We break the pressure into 3 parts: a constant axial pressure gradient, aperiodic lubrication pressure (coming from the variation in layer thickness), and a162hydrostatic pressure component:Pˆ =−Pˆ∗GGz+ Pˆ∗l Pl(y,z)− ρˆ [3]gˆ(rˆ2,0− rˆ)cos(θ). (5.17)The lubrication pressure scale (Pˆ∗l ), is chosen to balance the leading order shearstress gradients.Pˆ∗G =piλ µˆ [3]WˆpRˆ− rˆ2,0.Pˆ∗l =piRˆλ µˆ [3]Wˆp(Rˆ− rˆ2,0)2.For detailed information see Chapter 2.With aid of this scaling, the Navier-Stokes equations for the lubrication layerreduce to the Reynolds equation.∂∂y[h3∂Pl∂y]+1λ 2∂∂ z[h3∂Pl∂ z]= 12Up−6[h∂Vp∂y+Vp∂h∂y+1λ∂h∂ z], (5.18)where, h = 1− ecos(θ −Θ)− aΦ(z)+O(δ ) is the thickness of lubrication layer.And, e = dˆ/(Rˆ− rˆ2,0) is the eccentricity. Here, Up and Vp are the scaled plugvelocities in x and y directions, respectively.The other variable that needs to be scaled is time, tˆ. In this problem, there areseveral scales for time including viscous and advective time scales. To choose theproper timescale, we find the dominant timescale (smallest time) for each equationof motions (Eq. 5.12-5.14) and then select the smallest timescale. This is tˆb =√piδ 2Rˆ2/νˆ [3]√piδ Rˆ/λWˆp, which is a combination of the advective timescale of asqueeze flow and the dissipation timescale of the lubrication layer. Thus, we definet˜ = tˆ/tˆb.The dimensionless plug velocities are:Up =1δ√λRe[3][e˙cos(θ −Θ)+ eΘ˙sin(θ −Θ)] , (5.19)Vp =1δ√λRe[3](r2ωp−piδ[e˙sin(θ −Θ)− Θ˙ecos(θ −Θ)]) , (5.20)where Re[3] = δRe/m is the lubricant layer Reynolds number. The dimensionless163angular linear momentum equation can be written as:− ωppiδ√λRe[3]∫ 20∫ 12−121hdydz = Ipdωpdt, (5.21)and the non-dimensional equations of motion of the skin layer are:eΘ¨+2Θ˙e˙ =−2Fl,0 sinΘ, (5.22)e¨− eΘ˙2 =−∫ 20∫ 12−12(Pl cos(θ −Θ))dydz+2Fl,0 cosΘ, (5.23)where Fl,0 is the the ratio of buoyancy forces to lubrication pressure forces.Fl,0 =(ρˆ [3]− ρˆ [2]) gˆRˆ2Pˆ∗lr22,0. (5.24)The skin/lubricant interface moves axially with the dimensionless plug veloc-ity, which results in:∂h∂ t˜=δλ√Re[3]λ∂h∂ z. (5.25)5.3.3 Method of solutionReynolds equation (5.18) is linear as the lubricant is Newtonian. Consequently, wemay find the solution as Pl =P[0]l +ωpP[1]l + e˙P[2]l +Θ˙P[3]l . Where, P[0]l , P[1]l , P[2]l , P[3]lcorresponds to the pressure fields generated at unit plug velocity, unit angular plugvelocity, unit change in eccentricity e˙, and unit change in Θ˙. To find P[i]l , i =0,1,2,3, we have used a second order central finite difference approximation todiscretize (5.18) and then solve the algebraic system (5.21), (5.22), and (5.23) tofind ωp, e˙,Θ˙, and consequently Pl(y,z, t˜) with Matlab. By solving (5.25), we cantrack the interface. Note, in this model, plug velocity has been used as the veloc-ity scale, but if the superficial velocity of the core is instead chosen, we can findthe plug velocity Wp(t˜) and pressure gradient G(t˜) by adding axial linear momen-tum equation (5.12) and constant oil flow rate constraint to the above system ofequations; see Chapter 2.1645.3.4 ResultsFigure 5.18 presents the evolution of the eccentricity and variation of the lubrica-tion force (Fl), defined by (5.26), using both simplified core rise model and the 3Dcomputational model. We solve the core rise model for the exact same parametersand wave shape of 3D simulations, taking as parameters: (Re,m,λ ,r1,r2,0,a,∆φ) =(0.1,0.01,0.4,0.75,0.4,0.05). The eccentricity variation in time, calculated by thecore rise model, is shown by Fig. 5.18a and the inset presents oscillation in the bal-anced eccentricity, which we see has period T , as expected. For comparison, thetime has been scaled to have the same period as the 3D computational simulations.Figure 5.18b shows the variation of the lubrication force in time and its value canbe seen to oscillate around the required buoyancy force (Fl,0, which is indicatedby the dashed red line). The eccentricity variation in time for 3D simulation canbe seen in Fig. 5.18c, again with inset for the eccentricity variation when the flowis nearly developed. Figure 5.18d shows the lubrication force variation measuredover one wavelength of the domain.We observe that the dynamics are qualitatively similar but not identical. Bothmodels result in eccentricity that oscillates about a mean position, which here dif-fers by about 16%. The difference between these values is due to model differ-ences. The core rise model considers only a periodic domain with shaped interfaceexactly as designed. The 3D computations start from zero velocity and with aconcentric configuration. Although we can calculate the lubrication force on anyperiodic domain, the flow is developing in time and space. The finite domain alsoinfluences whichever section of the flow is considered. In the 3D simulations, theskin is immediately deflected (rises) due to buoyancy, behaving similarly to a solidbeam which is constrained at one end. It starts oscillating and adjusting to a bal-anced position because of the flow, In core rise model, only a fully developed flowcalculation is used. Therefore, the model does not predict the same developmentas the 3D computations, e.g. the development time looks different. however, bothmodels predict almost the same time if we use the eccentricity of the 3D model inwhich the skin starts reacting to the flow as the initial condition for the core rise165  TT bTT b(a) (b)(d)(c)Figure 5.18: (a) Calculated eccentricity with core rise model, (b) gen-erated lubrication force variation in time using core rise model,where red dashed line depicts the dimensionless buoyancyforce (Fl,0), (c) eccentricity variation in time using simula-tions, and (d) variation of lubrication force in time. Where(Re,m,λ ,r1,r2,0,a,∆φ) = (0.1,0.01,1,0.4,0.75,0.4,0.05).model instead of the initial concentric configuration.Fl =∫ 20∫ 12−12Pl cos(θ −Θ)dydz. (5.26)Figure 5.19a indicates the effect of density difference on eccentricity devel-opment using the core rise model. We can see, higher density difference leads tomore eccentric configurations. The comparison between core rise model eccentric-ity prediction (eCRM) and 3D computational results (e3D) are shown by Fig. 5.19bfor the same study parameters and waveform. Note, the eccentricity development166  (a)(b)‖e3D−eCRM‖2=0.045Δ ϕ=0.005 Δ ϕ=0.01Δ ϕ=0.02Δ ϕ=0.03Δ ϕ=0.04Figure 5.19: (a) Variation of eccentricity in time calculated by corerise model for various density differences. (b) Computed bal-anced eccentricity by 3D computations (e3D) versus calculatedeccentricity by core rise model (eCRM). The red dashed lineshows the linear function fitted to these data. Parameters are(Re,m,λ ,r1,r2,0,a) = (1,0.01,1,0.4,0.75,0.4).in time for different density differences are shown by Fig. 5.12. The red dashedline indicates a linear function fitted to these data and it shows the data are in goodagreement. However, the difference between predictions by 3D computations andcore rise model increases for very high and low density differences. The differencebetween data is also presented by computing the L2 norm of these differences:||e3D− eCRM||2 = 0.045.167Figure 5.20 presents the effect of initial condition on the configuration devel-opment. We solve the core rise model for three cases with identical parameters,(Re,m,λ ,r1,r2,0,a,∆φ) = (1,0.01,1,0.4,0.75,0.4,0.03), but different initial con-figurations. The variation of eccentricity in time is shown in Fig. 5.20a for thesecases and the variation of planar angle (Θ) is presented in Fig. 5.20b. For the casewith initial concentric position, the eccentricity increases to its balanced value andno variations in planar angle (Θ) can be seen. In the other two cases with ini-tially eccentric positioning, initially Θ decreases quickly then it oscillates aboutzero, however, the oscillations for a case with Θ|t=0 = pi/2 are larger than the onewith Θ|t=0 = pi . Note, the oscillations in both eccentric cases have same frequencythat may be estimated from (5.22) and is equal to fΘ ≈√2Fl,0/e. Also, the corereaches the concentric position initially when Θ|t=0 = pi and then the eccentricityincreases. We can also see, the predicted balanced eccentricity for the case withΘ|t=0 = pi/2 is higher than the the other 2 cases but the difference is almost negli-gible, i. e. the balanced solution is independent from the initial condition, however,the balanced configuration development depends on initial conditions.We compute the same 3 cases of Fig. 5.20 by 3D model and the results arepresented by Fig. 5.21. As predicted by model, the case with initial concentricconfiguration shows very small variation in Θ about zero, Θ ≈ 0 and the eccen-tricity increases to balanced value. For the initially eccentric case with Θ|t=0 = pi ,Θ decreases to 0 while the eccentricity decreases then Θ oscillates about zero andeccentricity increases to its balanced value. The initially eccentric positioned casewith Θ|t=0 = pi/2 shows a very different behavior than predicted by the core risemodel. Θ decreases to some intermediate value: here Θ ≈ pi/8 while the eccen-tricity increases to its balanced value. This value is approximately equal to themaximum of the magnitude of oscillating Θ in core rise model, i.e. core rise modelneglect any inertial effect, however, computational modeling solves fully Navier-Stokes equations. Consequently, in core rise model both Θ≈ pi/8 and Θ≈−pi/8can be the balanced value of Θ and there is no constraint to impose any preferencein the model, however, 3D model picks only one answer, here Θ ≈ pi/8. Insetsof Fig. 5.21b show the axial velocity on the cross section of the pipe and the greyshaded region shows the unyielded skin. We can see, the eccentric cases needlonger time than initial concentric configuration to develop. concentric and the ec-168  (a)(b)Figure 5.20: Effect of initial condition on the solution calculated by core risemodel. (a) Variation of eccentricity in time and (b) variation of pla-nar angel (Θ) in time (legends are the same as Fig. 5.20a); where(Re,m,λ ,r1,r2,0,a,∆φ) = (1,0.01,1,0.4,0.75,0.4,0.03).centric case with Θ|t=0 = pi/2 predict almost the same balanced configuration butthe other case does not reach the balanced position in this time span and it needsmore time to develop. Consequently, the simplified core rise model is a quick toolto predict the balanced eccentricity and the approximate magnitude of balanced Θ,however, balanced values of Θ cannot be necessarily predicted by core rise model.169  (a)(b)Figure 5.21: Effect of initial condition on the solution using 3D computa-tions. (a) Variation of eccentricity in time and (b) variation of pla-nar angel (Θ) in time (legends are the same as Fig. 5.21a). In-sets show the axial velocity colormap at the intersection of thepipe and the grey shaded region is unyielded skin. Parameters are(Re,m,λ ,r1,r2,0,a,∆φ) = (1,0.01,1,0.4,0.75,0.4,0.03).1705.4 ConclusionsFully 3D computations of a triple-layer core-annular flow have been presented,successfully benchmarked against previous axisymmetric computations and usedto investigate different features of this novel flow. As previously mentioned, thismethod allows reduction in pressure drops required to transport high viscosity flu-ids and does so over a wide range of parameters, including those not attained insimpler two-layer core-annular pipelining flows (2CAF); see Chapter 6. This isachieved by the elimination of interfacial instabilities by positioning an unyieldedyield stress fluid at the interface. A shaped skin then provides a lubrication forceto counter buoyancy effects.Three-dimensional CFD is often advanced as the pinnacle of numerical mod-elling. In reality, it is one of a number of different tools that should be employed indeveloping a new processing flow. Here it has been indispensable in verifying thatour prior axisymmetric models (Chapter 3) and lubrication approximation models(Chapter 2) are being applied to flow situations that are hydrodynamically stable.This is important as a key advantage of this triple-layer method over 2CAF is theavoidance of instability that leads to e.g. dispersed phase flow configurations.The second area where 3D simulations have been useful here is in studying theeccentric flow development. In Chapter 3 we neglected the effect of density mis-match and we focused on interface evolution in an axisymmetric geometry. Herewe have considered buoyancy effect due to density differences and these are feltdirectly within the inflow manifold as the interfaces are forming. We have beenable to establish steady eccentric flows, as we hoped, and the sculpted wave shapesare not very different from those in the axisymmetric iso-density cases. Probablythis is because, for the large yield stresses required in the skin layer, the lengthof pipe over which the skin is yielded is relatively short, i.e. there is insufficienttime/distance for buoyancy to have a large effect on the interface dynamics be-fore the skin becomes rigid. This validates the approach in Chapter 3 of studyingthe inflow manifold (plus a few pipe diameters) as an axisymmetric flow, but ne-glects the question of rise of the core fluids into an eccentric equilibrium. Thishas been resolved here via our core rise model, which is a quick computationalmethod for estimating these transients. Essentially we now have a suite of reduced171(and computationally fast) models to study different parts of the flow. The viscouslubrication model for the steady established flow (Chapter 2), the extensional flowmodel to estimate required yield stresses (Chapter 3) and now a further model forcore-rise. These models are appropriate for flow control and design.Computationally, we are pleased at the surprisingly good correspondence be-tween axisymmetric results from Chapter 3 and those here, computed with differentvolumetric discretizations, different treatment of the interfaces and different treat-ment of the yield stress. This is a validation of the consistency of both codes.Whether this good correspondence would extend to less structured flows than hereis unclear. Using the 3D code we have then studied the effects of different param-eters on the flow development, e.g. Re, a, etc.. Although each parametric studyshown has some physical interest, the value of these is more in demonstrating theutility of the 3D code. As mentioned earlier, our core fluid radius is relatively smalland both skin and lubricant layers are relatively thick, so our results might not berepresentative of practical flows.Apart from the present use for flow development, the 3D code appears attrac-tive for studying instabilities, asymmetries and other limits of the triple-layer core-annular flow. With respect to instabilities, some caution needs to be exercised.Due to the regularization method, the skin fluid is in fact always very viscous asopposed to rigid, i.e. with viscosity B/ε . Hence restricting computational times sothat viscous effects in the plug cannot manifest is one limitation. Other limitationsare more process and computation oriented. We have seen that flow developmentneeds a long time and also a long pipe L, both combining to give long computa-tional times regardless of code efficiency. Indeed the 3D FEATFLOW simulationsusing a viscosity regularization are efficient, taking less time than the axisymmetriccomputations in Chapter 3 using an augmented Lagrangian method within PELI-CANS.172Chapter 6Stable Triple-Layer Flow inInaccessible Domains1As reviewed in §1.1, oil-water two-phase flows have been studied extensively. Theaim of many of these studies is to produce maps that delineate where flows ofdifferent identifiable flow regimes may be found, in terms of the flow control pa-rameters. As an example Fig. 6.1 indicates the flow regime map for a horizontalpipe flow studied in [10], with different oil and water superficial velocities (J1 andJ2, respectively). Core-annular flow configurations are found only for relativelyhigh oil superficial velocities. However, this configuration is only stable for highoil input ratios, i.e. the top right corner of the flow map (designated AP or AOE).Many of the flows have bubbles or dispersed phase and we see that large regionsof the parameter space are simply inaccessible to the core-annular configuration inpractice. We say inaccessible because of course there is a concentric core-annularPoiseuille flow solution at every part of Fig. 6.1.As a second example, Sotgia et al. [161] investigated horizontal pipe flow con-figurations of mineral oil and water for various oil-water input ratios and pipes.As with [10] many regimes are observed: dispersed, dispersed annular, annular,wavy annular, wavy stratified etc. Figure 6.2 shows the flow map for the horizon-tal pipe in [161]. Their study shows a significant operating window where core-annular configurations are achieved. However, this configuration bounded by other1A version of this chapter has been published [155].173Figure 6.1: Horizontal flow map reproduced from [10], with permission,showing observed flow regimes for oil and water superficial velocities(J1 and J2, respectively) and parameters as in Table 6.1; see [10] forregime descriptions.regimes that are not as favourable. The pressure drop reduction is also measuredand shows a maximum with respect to the superficial oil velocity Jo. In practice,transitions between the regimes, unfavourable pressure drop reduction and variousother physical constraints discussed in [161] limit the operational range for core-annular flows to a relatively small parameter domain.Figures 6.1 & 6.2 are intended only as illustrative examples of the complexityof the flow regimes observed in 2-fluid flows and the practical difficulty of achiev-ing a core-annular flow with reasonable pressure drop reduction. In this chapter weuse the examples of Figs. 6.1 & 6.2 to show that the triple-layer method proposedin this thesis allows us to achieve stable core-annular flows over a much wider setof flow parameters and hence achieve pressure drop reductions closer to those thatare designed. Of course, nothing is free. The method entails a trade-off in termsof increased complexity of rheological design and flow establishment, and muchwork is still needed. Here we target what may be achievable.The method proposed has been described in depth in Chapter 2 and involvesthe triple-layer configuration of Fig. 6.3. Three fluid-mechanical features are com-bined. First, in order to eliminate interfacial instabilities we use ideas from vis-174Figure 6.2: Horizontal flow map reproduced from [161], with permission,showing observed flow regimes for oil and water superficial velocities(Jo and Jw, respectively) and parameters as in Table 6.1; see [161] forregime descriptions.coplastically lubricated flows (VPL), as studied by [110] and [70]. In these flows ayield stress fluid is positioned in such a way that it is unyielded at the interface.The second important features of the triple-layer method is that a modest den-sity difference, such as that typical in pipelining, can be accommodated. Core-annular flows are often observed to adopt wavy/corrugated interfaces e.g. [161].Although these initially attracted interest as a hydrodynamic instability, e.g. bam-boo instabilities, perhaps the most relevant mechanical feature is that the inter-face shape generates a lift force via hydrodynamic lubrication. Indeed without anyskewing of the wave shape and an eccentrically positioned core, the flow cannotsupport a density difference between the fluids. In our case, the interface is rigiddue to the yield stress. In Chapter 2 we studied a steady fully-developed periodiceccentric triple-layer flow, establishing that a shaped rigid skin could generate suf-ficient lift force from viscous lubrication to balance the buoyancy force of the corefluid, and estimating the yield stresses required in the skin to remain unyielded.The final fluid mechanical aspect of the triple-layer method concerns shaping1751231) Heavy Oil3) Lubrication Layer2) Skin(a) (b)Figure 6.3: a) Shaped skin profile in 3D; b) cross section of the pipe withtriple-layer configuration, illustrating the coordinates.of the interface. In Chapter 3, we used both computational and analytical meth-ods to study how we may sculpt a desired streamwise interface shape in the inletmanifold of a triple-layer flow. We show that this flow can be stably established ina controlled way and estimate the extensional stresses generated in the skin layer.This becomes another design requirement for the yield stress of the skin.Whereas previous chapters has targeted only the mechanical feasibility of theprocess, here we seek to illustrate its advantages. In this chapter, we demonstratethat otherwise inaccessible parameter regimes may have stable, shaped interfacetriple-layer core-annular flows by using the model established in Chapter 2. Thisallows us to separate attaining the desired core-annular regime from other designobjectives such as high oil hold-up ratio, high pressure drop reduction, and minimalrequired yield stress to maintain the skin unyielded.An outline of this chapter is as follows. In §6.1 we describe the flow and thegoverning equations. We briefly explain the mathematical model, the dimension-less numbers and leading order equations. The minimal yield stress is approxi-mated by estimation of the order of magnitude of normal stresses acting on theskin-lubricant interface. In §6.2 we present the results of the triple-layer core-annular model calculated using the same oil input ratios and other parameters as in[10, 161]. The effects of fluids input ratios and interface shape properties on thepressure reduction, minimal yield stress and balanced core position are explored.176We demonstrate the significantly larger operational windows for these two specificcases. We conclude the chapter with a brief discussion on operation design andother aspects of the method, see §6.3.6.1 Flow descriptionOur aim in this chapter is to demonstrate the advantages of the triple-layer lubri-cated pipeline method of Chapter 2 for typical core-annular configurations. Herewe use the model and methodology of Chapter 2 to solve it. We consider a hori-zontal pipe which is periodic in streamwise zˆ-direction, with period lˆ. Figure 6.3indicates schematically the positions of the 3 fluids, within a cross-section of thepipe at fixed zˆ. The domain consists of 3 individual fluids. Fluid 1 denotes thecore fluid (viscous Newtonian heavy oil), with viscosity µˆ [1] and density ρˆ [1]. Theskin layer is fluid 2, modeled simply as a Bingham fluid, with µˆ [2], τˆ [2]y , and ρˆ [2]denoting its viscosity, yield stress, and density, respectively. Fluid 3 is the lubri-cation layer (a low viscosity Newtonian fluid) with viscosity µˆ [3], and density ρˆ [3].Constitutive equations for the 3 fluids are:τˆi j [k] = µˆ [k] ˆ˙γi j, k = 1,3, (6.1)τˆ [2]i j =[µˆ [2]+τˆ [2]y∣∣ ˆ˙γ∣∣]ˆ˙γi j⇐⇒ τˆ [2] > τˆ [2]y , (6.2)ˆ˙γ = 0 ⇐⇒ τˆ [2] ≤ τˆ [2]y , (6.3)whereˆ˙γi j =∂ uˆi∂ xˆ j+∂ uˆ j∂ xˆi,ˆ˙γ =[123∑i, j=1[ ˆ˙γi j]2]1/2, τˆ [2] =[123∑i, j=1[τˆ [2]i j ]2]1/2. (6.4)The outer radius of the skin may vary with zˆ: rˆ2 = rˆ2,0 +∆rˆ2Φ(zˆ), i.e. Φ(zˆ)captures the sculpted streamwise skin profile. The inner radius (rˆ = rˆ1) is assumedto be uniform. The skin fluid is assumed to have a sufficiently high yield stress thatit remains rigid (unyielded), except during a flow development region in which theinterface shape is sculpted; see Chapter 3 & Chapter 5. Thus, fluids 1 and 3 remain177separated. In general, we might assume that the required yield stress is moderatelylarge (estimated later) and surface tension effects are negligible in comparison tothe other stresses, as the radii of curvatures are relatively large.The governing equations for the flow are the Navier-Stokes equations, in eachfluid domain. However, since the lubricant and skin layers are assumed relativelythin and slowly varying in zˆ, with common scaling arguments made for pressureand stresses, the Navier-Stokes equations are first reduced to leading order shearflow equations and then averaged across the skin and lubricant to give Reynoldslubrication equation; see Chapter 2.The core fluid is assumed lighter than the lubricant. As shown by Oomset al. [125], both eccentricity and axial variation in lubrication layer are necessaryto generate a transverse lift force via viscous lubrication. The lubrication pressuregenerated is larger where the gap is smaller and hence eccentricity towards the topof the pipe results in a net downward force to balance the buoyancy. We split thepressure into two parts: Pˆ = −Gˆzˆ+ Pˆl , where the constant axial pressure gradientGˆ drives the flow and Pˆl is the lubrication pressure. For the latter we use a classicallubrication scaling of the equations. We also assume the mean thickness of theouter lubricant layer is thin, relative to circumferential and axial length-scales, piRˆand lˆ respectively. In other words, δ = (Rˆ− rˆ2,0)/(piRˆ) 1 and for simplicity wetake λ = lˆ/(piRˆ)∼O(1). The domain of the problem (rˆ,θ , zˆ) is mapped to (x,y,z):rˆ = rˆ2,0+piRˆδx, y =θpi, z =zˆlˆ. (6.5)We calculate the leading order shear stresses and pressure in the lubricant layer (asδ → 0), and average in x to find the following Reynolds equation for the lubricationpressure Pl:∂∂y[h3∂Pl∂y]+1λ 2∂∂ z[h3∂Pl∂ z]=−6Wpλ∂h∂ z. (6.6)Note that all dimensional variables are denoted by the ·ˆ accent and scaled vari-ables without. In (6.6) Wp is the plug velocity and h(y,z) is the lubrication layer178thickness, given to leading order by:h(y,z) = 1− ecospiy−aΦ(z)+O(δ ). (6.7)The wave amplitude is a = ∆rˆ2/piRˆδ . The dimensionless axial skin thickness vari-ation Φ(z) has zero mean and maximal amplitude 1.The Reynolds equation is linear and parameterized by 3 dimensionless scalars:the mean frictional pressure gradient along the pipe (G), the plug velocity (Wp) andthe eccentricity of the core (e = dˆ/(Rˆ− rˆ2,0)). These 3 parameters are determinedby satisfying 3 integral constraints. (i) The flow rate of fluid 1 is specified. (ii) Thepressure drop along the pipe is balanced by the wall shear stresses (z-momentum).(iii) The static pressure and viscous shear forces acting on the skin in the verticaldirection balance the weight of liquid. The first two of these determine G and Wp:1 =Wpr21 +mδr418piG. (6.8)0 =G+2∫ 10∫ 10r3∂w∂x(x3,y,z) dydz. (6.9)These are 2 linear equations, parameterized only by the sculpted geometry and theviscosity ratio m= µˆ [3]/µˆ [1]. To determine the vertical displacement or eccentricitye, we resolve a scaled vertical force balance. To leading order in δ the balance isas follows:∫ 10∫ 10r2Pl(y,z)cospiy dydz−Fl(1−ρ[1− r21r22,0])= 0. (6.10)In the above equation Fl is the ratio of buoyancy forces to lubrication pressureforces. The parameter ρ = (ρˆ [1]− ρˆ [2])/(ρˆ [3]− ρˆ [1]) arises if there is a densitydifference between skin and core fluids. For simplicity we shall assume that ρ = 0.The derivation of above equations can be found in detail in §2.2.1.1796.1.1 Solution methodWe assume that the sculpting is controlled to produce an axisymmetric skin thick-ness, periodic in z. The Reynolds equation is thus solved with boundary conditionsof symmetry at y = 0, 1, together with periodicity in z. As the lubricant is New-tonian (6.6) is linear. Consequently we may find the solution P[0]l that would begenerated by a unit plug velocity and simply multiply by the plug velocity to findPl =WpP[0]l . To find P[0]l we have used a second order central finite difference ap-proximation to discretize (6.6) and then solve the resulting linear algebraic systemwith Matlab. The plug velocity Wp can be found by eliminating G between (6.8) &(6.9):Wp =1r211+mδr214pi∫ 10∫ 10[1h− h2λ∂P[0]l∂ z]dydz. (6.11)Equation (6.10) is found to be monotone with respect to the eccentricity e, withsolution where the buoyancy and lubrication forces balance. More detail of themethod can be found in Chapter 2.6.1.2 Estimating a minimal yield stressIf we consider the x-momentum equation in the skin layer:∂τ [2]xy∂y+1λ∂τ [2]xz∂ z=∂P∂x+O(δ 2), (6.12)we see that the shear stresses in the skin may adjust to accommodate normal stressgradients. The axial pressure gradient −Gz transmits across the skin, driving theflow. The lubrication pressure should however vanish on the inner radius r = r1.Therefore by integrating the shear stressed throughout the skin, in (y,z), we mayroughly estimate:τ [2] ≈ |Pl|x2− x1 .The normal stresses are typically one order larger than the shear stresses in lubri-cation problems and here we again amplify by insisting that the normal stresses180vanish at r = r1. Thus, the above estimate is generally much larger than that basedon the shear stresses alone. Consequently, we require the minimal yield stress tosatisfyB > Bmin = max(y,z)( |Pl|x2− x1). (6.13)Where B is the dimensionless yield stress, the Bingham number. We believe thatthis should give a reasonable estimate of the maximal stresses in the skin. Note thatwe have also developed other estimates of minimal yield stress, via the extensionalstress model of §3.3. In principle, these estimates could used instead, or in addition.6.2 ResultsFor our results we compare against the same fluids and pipe properties as were usedin [10] and [161], for the results presented in Figs. 6.1 & 6.2. The fluids properties,pipe dimensions and flow rate ranges are reported in Table 6.1 for these studies. Tobe able to compare our proposed method with two-layer method used in [10, 161],we use the same oil input ratio εo = Jˆo/(Jˆo+ Jˆw) as in these studies. Here Jˆo is oilsuperficial velocity, which is the velocity of the oil if transported alone in the pipe.However in our studies, in place of water we have 2 fluids: the lubricant (water)and the skin fluid (with a yield stress). Thus, on keeping the same εo and Jˆo as in[10, 161], we use a combined superficial velocity of lubricant and skin together:Jˆls = Jˆl + Jˆs, defined such that:εo =JˆoJˆo+ Jˆls. (6.14)In other words, our Jˆls matches the Jˆw used in [10, 161]. However, in our flows wemay split Jˆls as we like between lubricant and skin. We define the relative lubricantinput ratio εl,R by:εl,R =JˆlJˆl + Jˆs=JˆlJˆls. (6.15)In this way, a number of different triple-layer flows may be compared to the two-layer core-annular flow.The key dimensionless variables are listed in Table 6.2. For this study, we181Variable Symbol [10] [161]Crude oil viscosity µˆ [1] 0.488Pa.s 0.919Pa.sCrude oil density ρˆ [1] 925.5kg/m3 889kg/m3Water viscosity µˆ [3] 0.00126Pa.s 0.00126Pa.sWater density ρˆ [3] 1000kg/m3 1000kg/m3Pipe radius Rˆ 28.4mm 40mmCrude oil superficial velocity Jˆo 0.07−2.5m/s 0.01−0.8m/sWater superficial velocity Jˆw 0.04−0.5m/s 0.01−1.4m/sTable 6.1: Flow properties for Fig. 6.1 from [10] and Fig. 6.2 from [161].Note that Jˆo = J1 and Jˆw = J2 on Fig. 6.1.Variable Symbol [10] [161]Relative viscosity m 0.0025 0.0014Wave length λ 10 10Wave amplitude a 0.3 0.3Wave breaking point l′ 0.25 0.25Table 6.2: Key dimensionless variables for [10] and [161].use the same sawtooth wave Φ(z) as we used in Chapter 2 at the skin/lubricationinterface. The parameters of the waveform are the wave amplitude (a) and wavebreaking point (l′), as illustrated in Fig. 6.4. In Chapter 2, we showed that in orderto generate a force that can counterbalance a reasonable density difference, werequire l′ < 0.5 and a > 0.2. The density difference is given, from Table 6.1, andin each case we find the balanced core position (i.e. the eccentricity e), as part ofour calculation.To initially illustrate the characteristics of the triple-layer flow, we select 3sample cases from Figs. 6.1 and 6.2, corresponding to small, moderate and highoil input ratio, as summarized in Table 6.3. These 3 cases are marked on Figs. 6.1and 6.2 by the symbols introduced in Table 6.3. All parameters are otherwise asfor these figures (i.e. Table 6.1).182Figure 6.4: Sawtooth outer skin profile Φ(z).Case Symbol Jˆom/s Jˆlsm/s εoI  0.1 1.2 0.077II N 0.8 0.8 0.5III  2.5 0.1 0.96Table 6.3: Sample cases used for this study. Symbols presented in the tableare shown on Fig. 6.1 and 6.2.6.2.1 Geometric features and oil hold up ratioFigure 6.5 shows the variation of core and mean skin radii with relative lubricantinput ratio εl,R = Jˆl/Jˆsl , for the three sample cases. From this we first get an ideaof the oil hold-up ratio (Ho = r21) which is the ratio of heavy oil volume to thepipe volume. For the small oil input ratio (case 1), although we would be able toachieve the flow the oil hold-up is small (< 10%), which might not be desirablefrom the perspective of robustness of the flow. Note that this case in the two-fluidsystem results in a dispersed flow. For the other cases the oil hold-up ratio is morereasonable. Note that provided that the viscosity ratio is very small (m 1), Hois not very sensitive to m. We also mark on Fig. 6.5 where (r2,0− r1) = 0.5(1−r2,0), (vertical red broken line). The skin/lubricant interface needs to be shaped tosupport the density difference and for this purpose skin should be thick enough,183(a) (b)(c)Figure 6.5: Variations of core and mean skin radii (r1,r2,0) with relativelubricant input ratio (εl,R) for the different lubricant-skin and oil su-perficial velocities of cases I-III: (a) Jˆo = 0.1m/s, Jˆls = 1.2m/s, (b)Jˆo = 0.8m/s, Jˆls = 0.8m/s, and (c) Jˆo = 2.5m/s, Jˆls = 0.1m/s. Nota-tion: (−) : r2,0, (−−) : r1, and the vertical red-dotted line shows where(r2,0− r1) = 0.5(1− r2,0).e.g. at least larger than the wave amplitude. Thus, practically we should select εl,Rsmaller than this indicated limit.As mentioned, our proposed method is not limited to the flow regimes of thetwo-layer flow, i.e. we can achieve stable core-annular flow for any given inputs. Itis helpful to see how the geometric features of the flow vary with oil and relativelubricant input ratios. For this we fix on the viscosity ratio of [10]. Figure 6.6shows the variations of core and skin radii with oil and relative lubricant input184(a) (b)Figure 6.6: (a) Core radius (r1) and (b) mean skin radius (r2,0) variations withrelative lubricant input ratio (εl,R) and oil input ratio (εo). Dashed blackline indicates (r2,0− r1) = 0.5(1− r2,0).ratios. As expected, higher εo results higher r1 and with εl,R . 0.6, the skin is thickenough for an imposed wave with amplitude a < 0.5.6.2.2 Feasibility studyWe now consider other design outputs for the same 3 cases of fluids input ratio.The main advantage of the lubricated pipeline flow is pressure drop reduction.Figure 6.7a presents the pressure drop reduction factor (G0/G) versus εl,R for the3 different cases where the sawtooth wave is applied to skin/lubricant interfacewith (a, l′) = (0.3,0.25) and the data is otherwise based on Table 6.1 (from [10]).Here G0 is the pressure gradient required to flow the oil alone in the pipeline andG is the required pressure gradient for proposed lubricated pipeline method. Thebroken line indicates where G0/G = 20. We can see for small value of εl,R orvery thin layer of lubricant, pressure drop reduction factor is quite small althoughalways G0/G> 1. As εl,R increases the required pressure gradient decreases. Also,it seems that there is an optimum value of oil input ratio, as the pressure gradientis smaller for moderate oil input ratio than both high and low values of εo. Asεl,R→ 0, there is no lubrication layer and the skin layer is in contact with the pipewall. This can result in two scenarios, neither of which is desirable: (a) skin could185(a) (b)(c)Figure 6.7: (a) Pressure drop reduction factor (G0/G), (b) minimal skin yieldstress (τˆy,minPa), and (c) balanced eccentricity variations with relativelubricant input ratio (εl,R), for different oil input ratios (εo) where theflow parameters are as in [10] (legends are the same as Fig. 6.7a).stick to the wall and be motionless if the yield stress is high enough, (effectivelyreducing the pipe radius and flowing the oil alone in the pipe); (b) the skin canslightly yield at the wall and the relative pressure gradient will then be function ofthe skin effective viscosity at the pipe wall. Because the yield stress of the skinlayer is assumed high, the effective viscosity is likely also high and thus ineffectiveat pressure drop reduction. Scenario (b) is essentially the VPL studied by [70, 110].Another important factor is the minimal yield stress required to maintain theskin completely unyielded (τˆy,min). The variation of minimal required skin yieldstress with εl,R is shown in Fig. 6.7b for 3 sample cases. It shows, for higher oil in-186put ratio, required minimal yield stress is smaller and its value increases when εl,Rincreases because of the decrease in skin thickness. Eccentricity of the core alsoplays role in the value of minimal required yield stress, i.e. a higher lubricationpressure is generated for a more eccentric core. Figure 6.7c presents the eccentric-ity of the core at which the buoyancy is balanced. We can see, for smaller oil inputratio, a higher eccentricity is needed, which may seem strange. Note however thatthe buoyancy force decreases with Ho, the thickness of lubricant layer is generallyalso thicker, which reduces the balancing lubricating force. As discussed earlier,there are practical restrictions limiting the upper range of εl,R, in order to keep athick enough lubricant layer. The inset of Fig. 6.7b focuses on the lower range ofεl,R, where we can see that yield stresses < 1000Pa are sufficient to maintain theskin rigid. These are quite attainable for various polymeric hydrogels or emulsions.It is important also to note that these values are conservative, based on the maximalratio of lubrication pressure to skin thickness.The variations of same 3 parameters (G0/G, τˆy,min, e) are presented for thedifferent flow parameters of Table 6.1 (relevant to [161]) in Fig. 6.8. We can seethe same qualitative effect of εl,R and εo as before. These inputs result an increasedpressure drop reduction factor because of the smaller viscosity ratio, but require ahigher minimal yield stress because of the bigger pipe radius.Waveform sensitivityThe right choice of wave shape and properties can affect the flow design, as weshow here. We use the same sawtooth wave for skin/lubricant interface and wepresent the effect of its wavelength and amplitude on the outputs (G0/G, τˆy,min,e),fixing on case II for brevity. Figure 6.9 shows the effect of the wavelength (λ ).The longer wavelengths are easier to impose practically, i.e. sculpting of the inter-face requires local flow rate control of skin and lubricant flow rates. We see thatthe shorter wavelengths have slightly better pressure drop reduction and smallerminimal yield stress (due to lower eccentricity). However, the variations are slight.Effect of wave amplitude is presented in Fig. 6.10, this parameter shows moresignificant changes in pressure drop reduction and eccentricity for same εl,R atlarger a. Note that e+ a < 1 to avoid contact of the skin with the wall. Although187(a) (b)(c)Figure 6.8: (a) Pressure drop reduction factor (G0/G), (b) minimal skin yieldstress (τˆy,minPa), and (c) balanced eccentricity variations with relativelubricant input ratio (εl,R) for different oil input ratios (εo) where theflow parameters are as in [161] (legends are the same as Fig. 6.8a).increasing a appears beneficial in this way, note that earlier we have argued that forrobustness we should keep a minimal relative skin thickness, leading to the upperlimits on εl,R as shown in Fig. 6.5. Thus, this trade-off needs to be managed forany particular shape. In Chapter 2 we arrived at a value of a ≈ 0.55 to maximizethe pressure drop reduction and minimize required yield stress, while retaining areasonable skin thickness.188(a) (b)(c)Figure 6.9: (a) Pressure drop reduction factor (G0/G), (b) minimal skin yieldstress (τˆy,minPa), and (c) balanced eccentricity variations with relativelubricant input ratio (εl,R) for different wavelengths where the fluids andpipe properties are as in [10] (legends are the same as Fig. 6.9a).Feasible design windowIn this section, we elaborate on the previous section, illustrating one method forhow a design might achieved. Let us first assume that the waveform shape andwavelength have been fixed. We wish to balance two important considerations.First, the pressure drop reduction should be significant to make the most of thismethod. Second, the skin should be thick enough in order to be shaped to generatelubrication pressure. Using the criterion G0/G≥ 20, we define a lower bound εl,R,1on εl,R, where G0/G = 20. Secondly, the mean thickness of the skin should be atleast 50% of the mean lubrication layer thickness, so that we may include waves189(a) (b)(c)Figure 6.10: (a) Pressure drop reduction factor (G0/G), (b) minimal skinyield stress (τˆy,minPa), and (c) balanced eccentricity variations with rel-ative lubricant input ratio (εl,R) for different wave amplitude (a) wherethe fluids and pipe properties are similar to [10] (legends are the sameas Fig. 6.10a).with amplitude 0.2≤ a < 0.5. This leads to an upper bound εl,R,2. Admissible de-signs can be defined as any relative lubricant input ratio between these two bounds,i.e. εl,R,1 ≤ εl,R,d ≤ εl,R,2.For simplicity, so that we can explore the full range of feasibility of this method,we fix our design εl,R = εl,R,d :εl,R,d =εl,R,1+ εl,R,22, (6.16)as our desired design value of relative lubricant input ratio. Figure 6.11 uses the190flow properties from Tables 6.1 and 6.2 relevant to [161]. We see that, for almostany oil and lubricant-skin flow rates we can have an admissible triple-layer flow,i.e. the regions shaded in Fig. 6.11 - meaning that there is an admissible value ofεl,R,d . Only for 2 small regions, shaded in red, we are unable to have G0/G ≥ 20while r2−r1≥ 0.5(1−r2). This figure can be compared with Fig. 6.2 earlier, wherethe 2-layer core-annular flow region is significantly reduced; the axes ranges arethe same.In more detail, Fig. 6.11a & b show the variations of oil and mean skin radii,with oil and lubricant-skin superficial velocities. As mentioned before, higher oilinput ratio results in higher oil hold-up ratio. For most of the illustrated region,we have r1 > 0.5 which is desirable. Figure 6.11c shows the designed relativelubricant input ratio (εl,R,d), the black line shows the approximate boundary thatleads to core-annular flow for two-layer oil-water flow of [161]: “2CAF” marksthe region with stable core-annular flow and “U2CAF” indicates the unstable core-annular flow, i.e. see Fig. 6.2. The pressure drop reduction factor (G0/G) is givenin Fig. 6.11d and the dashed black line indicates G0/G = 50, as a guide to the eye.As we optimize the design with G0/G≥ 20, significant pressure gradient reductionis achievable for almost any given oil and lubricant-skin superficial velocities. Thevalue of minimal skin yield stress is shown in Fig. 6.11e. We can see, its valueis less than 1000Pa for most of the region which is achievable. Also, the distribu-tion of minimal yield stress is similar to the distribution of equilibrium eccentricity(see Fig. 6.11f) as might be expected, i.e. higher eccentricity results in higher lu-brication pressures and consequently higher minimal yield stress. To conclude, itappears that the triple-layer method results in stable core-annular flows with goodpressure drop reduction far outside the region where two-layer core-annular flowsare found in [161].We now repeat the exercise using the input data listed in Table. 6.1 and 6.2,which relates to the study of [10]. The results are shown in Fig. 6.12. Figure 6.12ashows the design relative lubricant input ratio εl,R,d for given oil and lubricant-skininput ratios. The black line depicts the approximate region where core-annular flowis found for the two-layer oil-water method (marked as “2CAF”); see Fig. 6.1. Thisshows we should avoid using very high or very low oil input ratios as these resultin higher pressure gradients; see Fig. 6.12b. Figure 6.12c shows minimal skin yield191  U2CAF 2CAF(a) (b)(c) (d)(e) (f)Figure 6.11: Feasible triple-layer flows for the flow parameters of [161],showing variations with Jˆo and Jˆls; (infeasible triple layer flows shadedsolid red). (a) r1. (b) r2,0. (c) εl,R,d . Note, “2CAF” and “U2CAF” markthe stable and unstable two-layer oil-water core-annular flow, respec-tively; from [161]. (d) G0/G and dashed line indicates G0/G= 50. (e)τˆy,minPa. (f) Balanced core eccentricity position.192stress. As expected, the design is very effective in the sense of a good pressure dropreduction and reasonable values for the required skin yield stress. Figure 6.12dpresents the variation of core balanced eccentricity (e) with Jˆo and Jˆls. Smallereccentricity is required for this flow in comparison with Fig. 6.11f, as the densitydifference is smaller in this study. Again we conclude that the triple-layer methodresults in stable core-annular flows with good pressure drop reduction far outsidethe region of admissibility of [10].6.3 ConclusionWe have presented a new triple-layer lubricated pipeline method and explored thefeasibility of the method over 2 data sets taken from recent experimental studies.In both cases the main advantage of this method is that the unyielded skin assuresthe linear stability of the interface, allowing stable core-annular flow far outside theranges of two-layer core-annular flows. We have shown that the method producesreasonable pressure drop reduction, in line with that of the two-layer configura-tions, and that the flows are not overly sensitive to process parameter variations.193(a) (b)(c) (d)2CAFFigure 6.12: Feasible triple-layer flows for the flow parameters of [10], show-ing variations with Jˆo and Jˆls; (infeasible triple layer flows shaded solidred). (a) εl,R,d . Note, “2CAF” marks the stable two-layer oil-watercore-annular flow region in [10]. (b) G0/G. (c) τˆy,minPa. (d) Balancedcore eccentricity position.194Chapter 7Summary and Future ResearchDirectionIn this chapter, results and contributions of individual chapters and a synopsis ofcontributions are presented first in §7.1. Then we look back at the research motiva-tions and discuss some limitations and possible improvements in §7.2. Finally, thethesis closes with our recommendations for future research directions in this areain §7.3.7.1 Contributions of the thesisFirst, results and contributions of individual chapters are presented. Then wepresent a synopsis of contributions in this section.7.1.1 Contributions of the individual chaptersTriple-layer visco-plastically lubricated pipe flow (Chapter 2)A novel triple-layer lubrication flow aimed at heavy-oil transport was developed.We used classical lubrication theory to estimate the leading order pressure distri-bution in the lubricating layer and calculate the net force on the skin. We exploredthe effects of skin shape, viscosity ratio, and geometry on the pressure drop, theflow rates of skin and lubricant fluids, and the net force on the skin. We showed195that the viscosity ratio and the radius of the core fluid are the main parameters thatcontrol the pressure drop and consumption of outer fluids, respectively. The shapeof the skin and the eccentricity mainly affect the lubrication pressure. These pre-dictions are essential in designing a stable transport process. Finally, we estimatedthe yield stress required in order that the skin remains unyielded and ensures inter-facial stability. Some of the observations regarding the proposed method includethe following.• Critical parameters (G,Q2,Q3) are relatively insensitive to the geometricalparameters.• The proposed method is suitable for stable lubricated pipelining of a rangeof heavy crude oils in moderate sized pipelines, e.g. Rˆ/ 20cm.• The skin material can be easily designed as the required yield stress tomaintain the skin completely unyielded is in an accessible range, e.g. 10−1000Pa.Flow development and interface sculpting in stable lubricated pipelinetransport (Chapter 3)In Chapter 2, we introduced a novel methodology for efficient transport of heavyoil via a triple-layer core-annular flow. We positioned a shaped unyielded skinof a visco-plastic fluid between the transported oil and the lubricating fluid layer.The shaping of the skin layer allows for lubrication forces to develop as the coresettles under the action of transverse buoyancy forces: adopting an eccentric posi-tion where buoyancy and lubrication forces balance. In Chapter 2 we focused ona steady periodic length of established flow, to establish feasibility for the pipelin-ing application. In Chapter 3, we addressed the equally important issue of how inpractice to develop a triple layer flow with a sculpted/shaped viscoplastic skin, allwithin a concentric inflow manifold.First, we used a simple 1D model to control layer thickness via flow rates ofthe individual fluids. This was used to give the input flow rates for an axisymmetrictriple-layer computation using a finite element discretization, with the augmented196Lagrangian method to represent the yield surface behavior accurately and a Piece-wise Linear Interface Calculation (PLIC) method to track the interface motion.This showed that these flows may be stably established in a controlled way witha desired interface shape. The shaped interface induces extensional stresses in theskin layer. We studied this directly by developing a long-wavelength/quasi-steadyanalysis of the extensional flow. This allowed us to predict the minimal yield stressrequired to maintain the skin rigid, for a given shape, all while maintaining a con-stant flow rate of the transported oil. The results can be summarized as follows.• For fixed inner flow rate, there is a one-to-one relationship between the skinand lubricant flow rates and the two interface positions.• Using the VOF-PLIC method to capture the interfaces was very effective andleads to negligible smearing of the interfaces.• Rigid wavy skin close to the desired design shape are achieved using themethod of controlling the flow rates.• The development zone of skin layer is fairly short due to high yield stress.• Normal stress components are dominant in the skin while the shear stresscomponents are dominant in lubrication layer. Thus, reducing yield stresscan result breaking the skin due to extensional stresses.• Increasing both viscosity ratio and Re number lead to an increase in criticalyield stress to maintain the skin completely unyielded.• Core-skin interface in the development zone creates significant non-uniformities.Consequently, the inlet manifold is made longer for the core-skin barrier thanthe skin-lubricant barrier. It allows for the unyielded skin to develop beforethe core-skin interface forms.• The extensional flow model couples the two shear flows (lubrication andcore) with the extensional-stress dominated skin. Lubrication layer shearstresses generate significant pressures, which transmit to the skin via normalstress continuity. This results in extensional stresses along the skin.197• We can observe the spatiotemporal evolution of patterns of yielded and un-yielded skin and explore sensitivity to the various flow parameters. In par-ticular, we can use the computed solutions to rapidly estimate the minimalyield stress required to maintain the skin rigid.• The extensional model also allows us to study what happens when the skinyields. Partial yielding allows transmission of the normal stresses betweencore and lubrication layers, which results in deformation of the interface r1.• The extensional flow model is useful for a variety of exploratory computa-tions of parameter ranges, for design and implementation of sculpted inter-face profiles, etc. With the model from Chapter 2, it is the basis of a processdesign and control model.Inertial effects in triple-layer core-annular pipeline flow (Chapter 4)In Chapter 4, we extended the feasibility of triple-layer core-annular method intro-duced in Chapter 2 to large pipes and higher flow rates by considering the effectsof inertia and turbulence in the lubrication layer. We showed that the method cangenerate enough lubrication force to balance the buoyancy force for wide range ofdensity differences and pipe sizes if the proper shape is imposed on the unyieldedskin. The results can be summarized as follows.• The gap-averaged mean flow “inertia” plays a significant role in calculat-ing pressure field and consequently affects the lubrication force required tobalance the buoyancy of the core fluid.• In inertial-dominant flow, the variation of pressure is influenced by a Bernoullieffect, which can counter the lubrication effects of the shear flow to some ex-tent.• In turbulent flow, Reynolds stresses become progressively dominant, gener-ating higher lubrication pressure than in laminar flow but still vulnerable toinertial effects from gradients of the mean flow.198• The above effects do affect the lubrication pressure generated, making someof the wave shapes used for viscous lubrication ineffective. However, thetarget skin shape can be designed in a way to generate lubrication force (Fl >0) based on our insight from inertial and turbulent effects.3D simulation of triple-layer flow development (Chapter 5)In Chapter 5, we presented three dimensional triple-layer computations which cap-ture the buoyant motion of the core to reach its equilibrium position. The 3Dcomputations were successfully benchmarked against axisymmetric computationsfrom Chapter 3. The study showed that these flows may become stably establishedwith control over interface shape, but development lengths (times) for the core toattain equilibrium are relatively long, meaning extensive 3D computation. We alsopresented a simplified analytical model using the lubrication approximation andequations of motion for the lubricant and skin layers. This model allowed us toquickly estimate motion to the balanced configuration for a given shape and initialconditions. The results can be summarized as follows.• 3D numerical modelling verifies the validity of our prior axisymmetric mod-els [158] and lubrication approximation [157] in hydrodynamically stableflow.• The buoyancy effect due to the density differences is immediately felt asthe interfaces are forming and 3D simulation is a useful tool in studying theeccentric flow development.• Large yield stresses are required in the skin layer to maintain the skin com-pletely unyielded leading to short development length of the skin before itbecomes unyielded.• Short unyielded skin development length leads to sculpted waveshapes whichare negligibly different from those in the axisymmetric iso-density cases.• Considering the axisymmetric flow is therefore a valid approach in studyingthe inflow manifold and shaping the skin.199• Simple semi-analytical core rise model is a quick method for estimating thebalanced eccentricity and it can be used for flow control and design.• The numerical codes used for axisymmetric simulations and 3D computa-tions are consistent and the results are in very good agreement. However,different volumetric discretizations, different treatment of the interfaces anddifferent treatment of the yield stress are applied in these codes.• 3D codes are attractive for studying instabilities, asymmetries, and other lim-its of the triple-layer core-annular flow.Stable triple-layer flow in inaccessible domains (Chapter 6)In Chapter 6, we showed an explicit advantage of the triple-layer core-annularmethod in crude heavy oil lubricated transport industry: namely that we are ableto produce stable core-annular flows in regimes where conventional core-annularflows are not found, i.e. either dispersed phases or instability result. Essentially themethod can give stable flows for a very wide range of fluid input ratio, although notall will produce the desired reduction in frictional pressure losses. The summarizedresults are as follows.• The feasibility of the method is explored over two data sets taken from recentexperimental studies.• The main advantage of the method is allowing stable core-annular flow faroutside the ranges of two-layer core-annular flows by using unyielded skinat the interfaces and it produces reasonable pressure drop reduction, in linewith that of the two-layer configurations.• In this method, the flows are not overly sensitive to process parameter varia-tions.• For the studies that we compared with, we would be able to establish goodpressure drop reduction in wide parameter regimes with skin yield stressesin the range: 10−1000Pa.2007.1.2 Synopsis of contributionsThe work carried out in this thesis investigates a novel stable multi-layer pipe flowmethod. The method has broad application in lubricated transport of heavy vis-cous oils to reduce the frictional pressure gradient and ensure the continued flow.In this method, we purposefully positioned a shaped unyielded visco-plastic fluid(skin layer) at the interfaces to eliminate interfacial instabilities. Specifically, theskin layer is shaped which allows for lubrication force to develop as the core risesunder the action of transverse buoyancy forces due to density differences betweenlayers. First, the feasibility of the method for the pipelining application was inves-tigated for a steady periodic length of established flow. Second, we showed thattriple-layer flows with a sculpted/shaped visco-plastic skin are stably establishedin a controlled way with a desired interface shape and the minimal yield stressrequired to maintain the skin rigid was calculated for a given shape by the exten-sional model. Third, the feasibility of the method was extended to larger pipes andhigher flow rates by considering the effects of inertia and turbulence. The buoyantmotion of the core was studied by three-dimensional computations and a simpli-fied model. Finally, we showed an explicit advantage of the proposed method inproducing stable core-annular flows in regimes where conventional core-annularflows are unsuitable.In summary, the thesis is based on results derived from both theoretical andcomputational methodologies. The theoretical aspect introduced fast and cheapdesign methods. Effects of various fluid properties, geometries, and flows wereinvestigated on the flow development which helped us to identify limitations ofthe method. Additionally, computational studies provided thorough results of flowdevelopment including the interface sculpting and the buoyant motion of the core.7.2 LimitationsInevitably, the methodology adopted for the present study includes inherent limi-tations. In what follows, we outline these limitations.2017.2.1 Rheological characterizationIn this study, we adopted the Bingham model to describe the yield stress fluidand retain the key feature of plasticity. Although, the model illustrates interestingfeatures of interfacial stabilizing effects of the yield stress fluid, high yield stressfluids usually show shear-thinning and sometimes elastic behaviors as well. Theseeffects are important when the fluid yields, e.g. at entrance of the flow. Also, weconsidered the other two layers to be Newtonian while complex fluids can be used,e.g. highly viscous crude oil can have elastic properties or visco-elastic fluids havebeen used in transport applications for drag reduction purposes.These flows need more investigation due to their broad application in industry.They can be investigated computationally and analytically to some extent and thegap between these studies can be filled by designing proper experiments.7.2.2 Computational limitationsAll the simulations presented in Chapter 3 are carried out on a structured rectan-gular mesh and a constant time step size for each simulation. Simulations take avery long time to compute the solution for high yield stress fluid with augmented-Lagrangian method and high viscosity difference between layers. Consequently,the computational domain was chosen relatively short (L = 15×R) which we be-lieve is long enough for the studied application. In general, faster algorithms couldimprove the efficiency of computational methods.The 3D simulations benefited from very robust and efficient algorithm in linewith very precise interface tracking method. Density differences are taken into ac-count in these simulations so the gap between theory (e.g. Chapter 2) and Chapter 3is fully covered by 3D simulations. We used a regularization method to solve theyield stress layer which physically can give us insight but very small deformationand asymmetry can be seen in the skin which can affect the generated lubricationforce. In present study, this effect is negligible as the domain and simulation timeare relatively short, however, this can affect the final results for problems with longsimulation run time. Quite low Re has been studied in this thesis due to both com-putational limits and the nature of application, but higher Re flow can be interestingto be investigated for other applications.202We neglect the surface tension in this thesis as the surface tension betweenoil/water is negligible and the pipe radius is quite high. In addition, high yieldstresses used are likely to dominate surface tension effects. For applications withlower yield stress, smaller geometry, and bigger deformation, the capillary effectcould become important and the surface tension is crucial to be taken into account.Overall, a robust and efficient algorithm equipped with augmented-Lagrangianmethod can improve the efficiency and precision of the computational methods.7.2.3 Industrial applicabilityWe developed this modified lubricated pipeline method by exploiting VPL strat-egy. We can be sure that achieving stable core-annular flow is feasible for widerange of flow rates, see Chapter 6, and the method can accommodate even highdensity differences for properly shaped skin. The method imposes very high crit-ical Bingham number which is achievable by available hydrogels, emulsions, etc..However, pumping these gels at the first place needs a high pumping pressure. Inaddition, shaping the skin asymmetrically is not achievable by conventional pumpsso the wave shape should be designed in a way to be axisymmetric. Finally, wehave not made any cost estimates for the process: primarily material costs of theskin and lubricant.7.3 Future directionsApart from addressing limitations from the previous sections, a number of futureavenues appear interesting.7.3.1 Experimetal investigationIn this thesis, we introduced a stable multi-layer flow which has broad industrial ap-plications. We also tried to investigate all the aspects of problem computationallyand theoretically but the experimental investigation of the problem can improvethe study by answering many questions, such as the effects of different parame-ters on shaping the skin and the critical Bingham number. Experiments can alsointroduce more complicated problems such as complicated fluid rheologies, turbu-lence, etc.. We should mention, visualization of experiments with three layers can203be very challenging as well. In future, this is an area which is within our interest todesign an experimental setup with advanced control on flow rates, long pipe withconcentric manifolds, and robust visualization method and in this regard:• Other research groups at UBC have been collaborating to establish this typeof flow experimentally on the micro-scale. Results are not reported in thisthesis.• The next natural step, prior to industrial implementation, would be to test thetriple-layer flow idea in an experimental loop collaboratively with one of thegroups operating pilot scale CAF facilities.7.3.2 Optimization problemIn previous sections, we addressed different aspects of triple-layer CAF designwith application in heavy oil pipelining. We showed that stable core-annular flowis achievable far outside the ranges of two-layer core-annular flows, see e.g. Chap-ter 6. In Chapter 4, we showed that wave shape can have a significant effect ongenerated lubrication force. Also, the effects of input conditions and waveform onpressure reduction, required yield stress, and flow configuration were investigatedin Chapter 6 & Chapter 3. In Chapter 5, the flow parametric study was considered.Further study could be directed at optimizing the wave shape for specific inputconditions and requested outputs. For example, one might wish to achieve a givenlubrication force over a range of flow rates, while minimizing the usage of the skinmaterial - perhaps due to cost considerations. The simplified models that we havedeveloped are suitable for addressing that type of design consideration.7.3.3 Visco-plastically lubricated flows of visco-elstic fluidsMany industrial multi-layer flows involve fluids with visco-elastic properties. Visco-elastic materials have both viscous and elastic properties in varying degree, as im-plied by the name. The material which can be described by a constitutive equationrelating the stress components to the strain rates is purely viscous. And the materialfor which the stress components are determined by only the strain is purely elastic.204If the material exhibits more complex time dependent properties beside thestrain and strain rate dependency, the material is visco-elastic. In other words,for a visco-elastic material, internal stresses are not only a function of instanta-neous deformation but also depend on the past history (usually the recent history)of deformation [30]. Flows of visco-elastic fluids exhibit very interesting prop-erties which are the consequences of extra normal stresses that are developed inthe direction normal to the direction of the flow when the fluid is sheared. Someof the examples are: Weissenberg effect, jet swell, reverse circulation in a case ofimmersed rotating disks in a fluid, etc.. In case of small deformation, the linearvisco-elastic models can be used to represent the behavior of visco-elastic mate-rial and nonlinear visco-elastic materials can be presented by linear models if thedeformation is sufficiently small.For large deformation, linear visco-elastic models are not useful. In addition,the rheological equations are associated with a specific element of material. Hence,the objective deformation is measured relative to coordinate system of materialpoint and a constitutive equation of state is formulated relative to these convectedcoordinates [30]. The upper convected Maxwell model (UCM) is the nonlin-ear modification of Maxwell linear model which considers convected coordinates.More popular models have been introduced by adding additional nonlinearities toUCM model. The Oldroyd B [119] and FENE (finitely extensible nonlinearly elas-tic) models are the superposition of the UCM and a Newtonian model. Anothernonlinear model is the Giesekus model [45] which adds quadratic nonlinearity anddivides the deviatoric stress into a solvent contribution and a polymer contribution.We refer the reader to [77, 95, 129, 172, 173] for more details.Feasibility of establishing multi-layer stable flows with visco-elastic fluids isinherently of interest due to its broad industrial applications. The linear stability ofplanar multi-layer flow of viso-plastic and visco-elastic fluids is studied in [111].The feasibility of VPL with visco-elstic fluid as the core fluid is studied by Hormoziet al. [68, 69]. These studies extend the VPL methodology to visco-elastic fluids.Although, an elastic instability may appear at the entry of the flow, they freezeat the interface of the unyielded plug around the visco-elastic core and this effectcan be amplified by increasing elasticity. We can extend this idea to triple-layercore-annular flow by using visco-elastic core fluid. In this case, the normal stresses205appear in core layer while, Newtonian core results only shear dominant flow in corelayer. The normal stresses are the same order of magnitude of core shear stresses(τrr ∝ γ˙2rz) but still the extensional stresses generated in high yield-stress visco-plastic fluid are dominant and are persistent in the skin. Therefore, the elasticityshould have a negligible effect on critical Bingham number to maintain the skincompletely unyielded. But, for high elasticity and lower Bingham, the balance ofelastic instability and freezing is evidently delicate and can lead to stable wavyinterface which can be useful in polymer processing applications.The co-extrusion is a great example of multi-layer flow of complex fluids whichhas a wide application in plastic industries. The process mainly suffers from inter-facial instabilities and swelling after extrusion. The VPL method can stabilize theinterfaces and reduce the swelling.Composite materials and coating are other industrial applications of multi-layerflows. Regarding composites, layers are usually shaped to reinforce the strength ofmaterials. Consequently, the VPL sculpting can be used to stably shape the layers.For these applications, other geometries need to be studied, such as rectangularslots for co-extrusion processes. Also, more complex fluids should be consideredas these applications involve generally visco-elastic and shear-thinning fluids. Inaddition, boundary conditions are different for some of these applications, such ascoating.7.3.4 Turbulent drag reduction in the lubrication layerPresence of elasticity in the lubricating layer can result even more pressure dropreduction in the case of turbulent flows, however, it may lead to higher flow rates.Adding polymers to a Newtonian solvent even at minute concentrations can spec-tacularly reduce the turbulent energy loss and consequently reduce the friction fac-tor. The drag reduction is affected by polymer relaxation time, flow rate, and geo-metric length scale which can lead to a design problem. This area can be interestingto be investigated from industrial point of view. However, we should be cautiouswhile using this method for a lubricated pipeline flow. 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In more detail, a fixed time step is usedfor the Navier-Stokes equations, advancing from time step N to N + 1. We use asemi-implicit method to implement non-linear terms, i.e. the convective velocityis approximated at the time step N while linear spatial derivatives of the velocityare approximated implicitly at the time step N+1. In case of using the regularizedmodel for a yield-stress fluid, the regularized viscosity is computed using the ve-locity field from the previous step, and iteration is continued until the velocity fieldconverges. Using the augmented-Lagrangian method introduces additional vari-ables to solve iteratively. The pressure is approximated at the time step N+1. Theconcentration equation is solved via a finite volume method, in which the concen-tration is approximated at the center of each regular mesh cell. Piecewise Linear1PELICANS is distributed under the CeCILL license agreement(http://www.cecill.info/licences/Licence CeCILL V2-en.html). PELICANS can be downloadedfrom https://gforge.irsn.fr/gf/project/pelicans/.225Interface Calculation method (PLIC) is implemented to deal with advective terms(brief introduction is provided A.1.1). On each (Navier-Stokes) time step a split-ting method is used to advance the concentration equation over a number of smallersub-time steps. This time advance is explicit and a CFL condition is implementedfor sub-time steps to ensure the numerical stability.All of the algorithms are implemented in C++ as an application of PELI-CANS, an object oriented platform developed at IRSN, France, to provide gen-eral frameworks and software components for the implementation of PDE solvers.PELICANS is distributed under the CeCILL license agreement.A.1.1 PLIC advection schemePiecewise Linear Interface Calculation method (PLIC) is a volume of fluid (VoF)based method which is used to solve the advection of the fluid volume C,∂C∂ t+u.∇C = 0. (A.1)PLIC scheme track the interface more accurately than MUSCL method as itavoids the numerical diffusion of the interface. The scheme is 2nd order accuratein space and a forward Euler scheme is used for the temporal discretization suchas:Cn+1 =Cn−un∆t.∇Cn. (A.2)There are two sub-steps in every time step as the scheme is decomposed into xand y directions. Details of the method can be found in [139]. Figure A.1 presentsthe interfaces of three-layer flow in a plane which are tracked by PLIC method andMUSCL method. It shows, the interface is highly smeared in MUSCL method andPLIC improves the interface tracking. The color function (C) value at the end of theplane can be seen for fully developed flow, see Fig. A.2. It can be seen, C changesonly in one or two cells with PLIC method but C < 1 for the red fluid captured byMUSCL method and its value changes gradually over number of cells. Note, themesh used for these simulations are quite coarse.226PLICMUSCLFigure A.1: The interfaces of three-layer flow tracked by PLIC and MUSCLmethods.0 0.5 100.20.40.60.81Figure A.2: Color function C variation across the domain of fully developedflow.A.2 FEATFLOWThis section briefly discuss the features of the open source FEM-CFD FEATFLOWsoftware (Finite Element Analysis and Tools for Flow problems). More informa-tion can be found at http://www.featflow.de2. The code structure mainly includes:(i) discretization schemes, (ii) linear solvers, (iii) discrete projection scheme, and(iv) nonlinear solvers.As mentioned before, for spacial discretization, the code uses Q2−P1 elements,adaptive FE-upwinding, and adaptive streamline-diffusion techniques are availablefor convective terms of the Navier-Stokes equations. Stress free, natural do noth-2FEATTFLOW can be downloaded from http://www.featflow.de/en/software.html.227ing, flux, and pressure drop boundary conditions can be found in the package. Im-plicit Euler scheme, Crank-Nicolson scheme, Fractional-step-θ 2nd order scheme,and adaptive time stepping are offered by software for temporal discretization.Similar to most of the solvers, there are two possible structures: (i) Galerkinscheme and (ii) Projection scheme. The version used in this thesis is based on theprojection scheme and the introduction to Multilevel Pressure Schur Complementmethod as the “solver engine” is given in the following section (§A.2.1).A.2.1 MPSCHere, we briefly review the “Discrete Projection Method” as a special variant ofMultilevel Pressure Schur Complement (MPSC) approaches for the solution ofincompressible flow problems, and we combine it with FEM discretization tech-niques. We will explain some characteristics of high-resolution FEM schemes asapplied to incompressible flow problems and discuss computational details regard-ing the efficient numerical solution of the resulting nonlinear and linear algebraicsystems.The Navier-Stokes equations need to be discretized in time initially by anyavailable usual method, for instance the Crank-Nicolson, Forward Euler, Back-ward Euler, or Fractional-Step-θ scheme. The implicit scheme allows adaptivetime stepping due to accuracy reasons only and does not depend on CFL-like re-strictions. In this study, we use fully implicit Backward Euler method with smallenough uniform time step.Discrete equations in time are discretized in space by finite element method(FEM). FEM is well-known for its flexibility in dealing with any complex PDEsand geometry. Hence, it becomes favorable for many industrial purposes. It worksin a weak formulation and it is integrated over domain which results a system ofequations. [S(un+1) kB−BT 0][un+1pn+1]=[g0], (A.3)228where,S(un+1) := M+ k[K(un+1)+ 1Re L],g := Mun+ kFn+1,k := ∆t. (A.4)And, M, K, L, B, BT and F are mass, transport, Laplacian, gradient, diver-gence, and external force matrices, respectively.The main idea of MPSC is first to compute a velocity field without takinginto account incompressibility, and then perform a pressure correction, which is aprojection back to the subspace of divergence free vector fields. If we reformulatethe system of equation (A.3), we get one scalar equation that contains the pressure:BT S−1Bp =1kBT S−1g. (A.5)Consequently, pressure is updated with a proper preconditioner C. For this study,reactive preconditioner (BT M−1B) is used which requires small time steps.pl+1 = pl +C−1(BT S−1Bp− 1kBT S−1g). (A.6)The main idea of this method is to apply a decoupling step for u and p asouter iteration. Therefore, equations solving the velocity field u and the pressure preduce to:Sul+1 = g− kBpl, (A.7)Cpl+1− 1kBT ul+1 =Cpl. (A.8)The convergence criteria is BT u = 0. For the details of this method, we referthe reader to [175].229Appendix BSupporting MaterialsB.1 Method of solution used in Chapter 4Equations (4.25) & (4.26) are solved in a periodic domain in the z-direction, i.e.Pl(y,0) = Pl(y,1), ¯¯v(y,0) = ¯¯v(y,1), and ¯¯w(y,0) = ¯¯w(y,1). The domain is symmet-ric in the y-direction, at the top and the bottom of the pipe, i.e. ∂Pl/∂y(0,z) =∂Pl/∂y(1,z) = 0, ¯¯v(0,z) = ¯¯v(1,z) = 0, and ∂ ¯¯w/∂y(0,z) = ∂ ¯¯w/∂y(1,z) = 0. Thegoverning equations are discretized by second order accurate Finite DifferenceMethod in a rectangular domain (y,z) ∈ (0,1)× (0,1) using a staggered grid ofsize ∆y×∆z. The lubrication pressure, Pl , are located at cell centers , (i, j), and theaveraged velocity components, ¯¯v, ¯¯w, are at the cell faces, (i+1/2, j), (i, j+1/2).This avoids the well-known checkerboard instability. The mesh Pe´clet number,Pe, stability condition, defined as the ratio of the convection to friction term in themomentum equation, is given by:Pe =δ/ f max[( ¯¯v2, ¯¯w2)/(∆y,∆z)](1+ ε)min [( ¯¯v, ¯¯w)/(∆y2,∆z2)], (B.1)For laminar flow, this condition leads to:Pe =Re[3]δ max( ¯¯v, ¯¯w)10min(∆y,∆z)≤ 2 ⇒ Re[3] ≤ Re[3]cr = 20min(∆y,∆z)δ .For typical mesh spacing of ∆y = ∆z = 0.05 and δ = 0.05 the value of critical230Reynolds number is small Re[3]cr = 20. To alleviate this restriction, the advectiveoperator is discretized using a second order accurate total variation diminishingscheme [56] and using the van Albada flux limiter [180].B.2 Coefficients of modified Reynolds equation forturbulent flow used in Chapter 4Velocity profiles are calculated using (4.15), (4.16), (4.18), and (4.19) for turbulentflow.v¯(ξ ,y,z) =∂ P¯l(y,z)∂y(V3(ξ )− V4V2V1(ξ )), (B.2a)V1(ξ ) =∫ ξ011+ εy(uw,Re[3], t)dt, (B.2b)V2 =∫ 1011+ εy(uw,Re[3], t)dt, (B.2c)V3(ξ ) =∫ ξ0t1+ εy(uw,Re[3], t)dt, (B.2d)V4 =∫ 10−t1+ εy(uw,Re[3], t)dt. (B.2e)w¯(ξ ,y,z) =W1(ξ )W2+1λ∂ P¯l(y,z)∂ z(W3(ξ )−W4W2W1(ξ )), (B.3a)W1(ξ ) =∫ ξ011+ εz(uw,Re[3], t)dt, (B.3b)W2 =∫ 10−11+ εz(uw,Re[3], t)dt, (B.3c)W3(ξ ) =∫ ξ0t1+ εz(uw,Re[3], t)dt, (B.3d)W4 =∫ 10−t1+ εz(uw,Re[3], t)dt. (B.3e)231The coefficients used in (4.22) are:C1 =∫ 10(V3(ξ )− V4V2V1(ξ ))dξ , (B.4a)C2 =∫ 10(W3(ξ )−W4W2W1(ξ ))dξ , (B.4b)C3 =∫ 10W1(ξ )W2dξ . (B.4c)Values of Iy & Iz in (4.26) are:Iy = ∂∂y(¯¯v2C1∫ 10(V3(ξ )− V4V2 V1(ξ ))2dξ)+1λ∂∂ z(¯¯v ¯¯wC2∫ 10(W3(ξ )− W4W2 W1(ξ ))(V3(ξ )− V4V2 V1(ξ ))dξ)+1λ∂∂ z(¯¯v∫ 10(V3(ξ )− V4V2 V1(ξ ))W1(ξ )W2dξ). (B.5)Iz = ∂∂y(¯¯v ¯¯wC1∫ 10(W3(ξ )− W4W2 W1(ξ ))(V3(ξ )− V4V2 V1(ξ ))dξ)+ ∂∂y(¯¯v∫ 10(V3(ξ )− V4V2 V1(ξ ))W1(ξ )W2dξ)+1λ∂∂ z(¯¯w2C2∫ 10(W3(ξ )− W4W2 W1(ξ ))2dξ + ¯¯w∫ 10(W3(ξ )− W4W2 W1(ξ ))W1(ξ )W2dξ)+1λ∂∂ zC2∫ 10(W1(ξ )W2)2dξ . (B.6)B.3 Required minimal yield stressThe estimation of the required yield stress to maintain the skin completely un-yielded is investigated in this thesis by mainly two methods, (i) the reduced modelin Chapter 2 and (ii) the extensional model in Chapter 3. Both models are in a goodagreement. They are quick to compute and can be used for process design.The reduced model assumes a rigid skin and choose locally the maximum ra-tio of lubrication pressure to the skin thickness as the minimal yield stress. Butthe extensional model is derived based on the long-wavelength assumption and theminimal yield stress is computed by integrating the variation of extensional stresses232across the skin. Consequently, the extensional model predicts a higher value for re-quired Bingham number than the reduced model. As an example, for a sawtoothwave shape and case parameters (r2,0,r1,a,λ ,m, l′) = (0.9,0.8,0.5,1,0.001,0.25),the reduced model predicts B˜ = 1.215 and the extensional model estimates B˜ =1.642.233

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