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Interfacial effects in visco-plastic lubrication flows Dunbrack, Geoffrey E. 2013

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Interfacial Effects in ViscoplasticLubrication FlowsbyGeoffrey E. DunbrackBASc, The University of British Columbia, 2011A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinThe Faculty of Graduate Studies(Mechanical Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)August 2013c? Geoffrey E. Dunbrack 2013AbstractPoiseuille flows with yield stress fluids produce an unyielded central plugwhich can act as a solid conduit surrounding central (core) flows of Newto-nian or power law fluids. Effectively, the annular yield stress fluid acts asa lubricant that isolates the core flow from wall friction. Stable flows witha yield stress annular fluid and a Newtonian or power law core fluid aretermed viso-plastic lubrication (VPL) flows. This study examined interfa-cial effects in vertical VPL Poiseuille flows using a carbopol solution as theannular (yield stress) fluid and xanthan (inelastic shear thinning fluid) orpolyethlyeneoxide (PEO; an elastic shear thinning fluid) as the core fluids.Experiments with the inelastic core fluid (xanthan) involved introducingstepped (high to low) or pulsed (high to low to high) changes in the coreflow to an established stable VPL flow. Step changes produced a ?yieldfront? (narrowing of the core flow or ?interfacial radius?) that propagatedupward at a velocity considerably greater than the velocity of the annularcarbopol plug but close to the average velocity of the xanthan core flowfollowing the step change. Pulsed changes in the core flow produced oneof three outcomes depending on the magnitude of the flows preceding andfollowing the step change: (1) a stable (?frozen in?) deformation in the car-bopol/xanthan interface that moved upward at the velocity of the carbopolplug,(2) no persistent deformation of the interface, or (3) a breakdown of thestable VPL flow characterized by extensive mixing of the core and annularflows. Experiments with the elastic core fluid (PEO) involved introducingmultiple pulsed changes (high/low/high, high/low/high, ...) in the core flowto an established VPL flow. These pulsed changes typically produced linkedmultiple diamond shaped stable deformations (?diamond necklace?) in theinterface that moved upwards at the velocity of the carbopol plug. Thefrequency and amplitude (maximum radius) of the diamond deformationscould be controlled by the timing of pulses and the respective flow rates,but not the diamond shape itself which appears to be a consequence of thecomplex rheology of the fluids.iiPrefaceI carried out all the fluid preparation, experiments, rheological testing, ex-perimental data analysis, Arduino programming, and built the solenoid valvebypass system. All experimental results have been presented at the 2012Nordic Rheology Society conference in Oslo, Norway and at the 2013 Tech-nical Forum on Unconventional Natural Gas in Victoria, BC, Canada. Pre-liminary results have been published in [6].iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii1 Introduction to Subject . . . . . . . . . . . . . . . . . . . . . . 11.1 Interfacial Instabilities . . . . . . . . . . . . . . . . . . . . . . 21.2 Stability of Multilayer Flows . . . . . . . . . . . . . . . . . . 31.3 Multilayer Viscoplatic Lubrication Flows in Experimentation 51.4 Interfacial Patterns in Experimental and Numerical Studies . 51.5 Applications of VPL . . . . . . . . . . . . . . . . . . . . . . . 61.6 Problem Description and Outline . . . . . . . . . . . . . . . 72 Materials and Methods . . . . . . . . . . . . . . . . . . . . . . 82.1 Physical Description of Apparatus . . . . . . . . . . . . . . . 82.2 Description of Flow Loop . . . . . . . . . . . . . . . . . . . . 92.3 PID Control System . . . . . . . . . . . . . . . . . . . . . . . 92.4 Solenoid and Throttle Valves . . . . . . . . . . . . . . . . . . 162.5 Anti-distortion Tank . . . . . . . . . . . . . . . . . . . . . . 182.6 Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.6.1 Carbopol EZ2 Polymer Solution . . . . . . . . . . . . 192.6.2 Xanthan Gum . . . . . . . . . . . . . . . . . . . . . . 202.6.3 PEO . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.7 Rheometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.7.1 Carbopol . . . . . . . . . . . . . . . . . . . . . . . . . 20ivTable of Contents2.7.2 Rheometry Tests Done on Xanthan . . . . . . . . . . 262.7.3 Rheometry Tests Done on PEO . . . . . . . . . . . . 262.7.4 Rheometry Tests Done on Glycerol . . . . . . . . . . 272.7.5 Further Description of Rheometry Tests and ValuesUsed . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.7.6 Summary of Rheometry Parameters . . . . . . . . . . 302.8 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.8.1 Cameras . . . . . . . . . . . . . . . . . . . . . . . . . 302.8.2 Flow Rate Measurement . . . . . . . . . . . . . . . . 312.9 Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 342.9.1 Qualitative Stability Observations . . . . . . . . . . . 342.9.2 Yield Front Speed Analysis . . . . . . . . . . . . . . . 352.9.3 Image Sequences and Spatiotemporal Plots . . . . . . 402.10 Experimental Procedures . . . . . . . . . . . . . . . . . . . . 432.11 Differential Analysis of Visco-Plastic Lubrication . . . . . . . 462.11.1 Simplified Flow Problem . . . . . . . . . . . . . . . . 462.12 VPL Flow Regimes . . . . . . . . . . . . . . . . . . . . . . . 472.12.1 Integrating to Obtain a Velocity Equation . . . . . . 482.12.2 Yielded and Static Regimes . . . . . . . . . . . . . . . 522.13 Applying the Analytical Model to Experimental Design . . . 562.13.1 Inversion of the Flow Equations . . . . . . . . . . . . 603 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.1 Calibration and Benchmarking . . . . . . . . . . . . . . . . . 613.1.1 Observing the Plug . . . . . . . . . . . . . . . . . . . 613.1.2 Establishing VPL Flows . . . . . . . . . . . . . . . . 623.1.3 Calibration and Comparison with Model . . . . . . . 633.1.4 Efforts to Use PID Control System to Create Frozenin Effects . . . . . . . . . . . . . . . . . . . . . . . . . 643.2 Scope of Experiments . . . . . . . . . . . . . . . . . . . . . . 663.3 Experiments with Carbopol and Xanthan . . . . . . . . . . . 663.3.1 Step Changes in the Flow Rate . . . . . . . . . . . . 663.3.2 Freezing in of Pulses . . . . . . . . . . . . . . . . . . 713.4 Experiments with Visco-elastic Fluids . . . . . . . . . . . . . 753.4.1 Pulse Duration and Spacing . . . . . . . . . . . . . . 783.4.2 Other Experiments . . . . . . . . . . . . . . . . . . . 824 Contributions and Future Research Directions . . . . . . . 864.1 Contributions of the Thesis . . . . . . . . . . . . . . . . . . . 864.2 Future Research Directions . . . . . . . . . . . . . . . . . . . 87vTable of ContentsBibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88AppendicesA List of Experiments . . . . . . . . . . . . . . . . . . . . . . . . 90A.1 Step Change Experiments . . . . . . . . . . . . . . . . . . . . 90A.2 Single Pulse Experiments . . . . . . . . . . . . . . . . . . . . 90A.3 Experiments with Visco-Elastic Fluids . . . . . . . . . . . . . 90B Detailed Experimental Procedures . . . . . . . . . . . . . . . 97B.1 Startup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97B.2 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . 97B.3 Fluid Preparation . . . . . . . . . . . . . . . . . . . . . . . . 98C Computer Programs . . . . . . . . . . . . . . . . . . . . . . . . 100C.1 Mutilayer Flow Curves . . . . . . . . . . . . . . . . . . . . . 100C.2 Photo Sequence . . . . . . . . . . . . . . . . . . . . . . . . . 107C.3 Inverted Flow Problem . . . . . . . . . . . . . . . . . . . . . 108C.4 Spatiotemporal Plot . . . . . . . . . . . . . . . . . . . . . . . 109C.5 Yield Front Speed Analysis . . . . . . . . . . . . . . . . . . . 112viList of Tables2.1 Rheometry data for 0.5% carbopol. . . . . . . . . . . . . . . . 232.2 Rheometry parameters for fluids used in this thesis. . . . . . 30A.1 Table of step change experiments performed for 0.5% Xan-than and 0.5% carbopol. Q1 is the core fluid flow rate beforethe step change in introduced, Q?1 is the core fluid flow rateafter the step change in introduced, Q2 is the annular fluidflow rate that remains constant during these experiments. . . 91A.2 Table of single pulse experiments performed for 0.5% xanthanand 0.5% carbopol. T ? is the time spent at Q?1 during thereduced flow pulse. . . . . . . . . . . . . . . . . . . . . . . . 94A.3 Table of experiments performed for 0.5% PEO and 0.5% car-bopol. T ? is the time spent at the reduced flow (Q?1) and T isthe time spent at the base flow (Q1). . . . . . . . . . . . . . . 95A.4 Table of experiments performed for 0.4% PEO and 0.5% car-bopol. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96viiList of Figures1.1 Schematic core annular pipe flow. . . . . . . . . . . . . . . . . 11.2 Classic viscous-viscous instability. . . . . . . . . . . . . . . . . 32.1 Main apparatus used to perform experiments . . . . . . . . . 102.2 Injector used to inject core fluid (internal diameter is 13.7mm, external diameter is 14.5 mm). . . . . . . . . . . . . . . 112.3 Arrangement of tripods (one camera absent for taking thisimage). See appendix ?? for specification. . . . . . . . . . . . 122.4 Flow loop schematic. The schematic shows the loop used toconduct fluid to the apparatus. . . . . . . . . . . . . . . . . . 122.5 Moyno 1000 pump used to pump outer fluid. See appendix?? for specifications. . . . . . . . . . . . . . . . . . . . . . . 132.6 Variable frequency drives (left SP500 for fluid 1, right BaldorVector Drive for fluid 2) used to control pump motor speeds.See appendix ?? for specifications. . . . . . . . . . . . . . . . 132.7 Moyno 500 pump used to pump inner fluid. See appendix ??for specifications. . . . . . . . . . . . . . . . . . . . . . . . . . 142.8 Labview PID control interface showing the control optionsavailable for the flow loop. The readouts on the two graphscorrespond to the flow rates of the two fluids. The outputwas also saved to an excel spreadsheet that could be lookedat during analysis to determine accurate flow rates. . . . . . . 152.9 Asco 8210 solenoid valve and generic gate valve installed inseries and used to introduce rapid changes in flow rate to thesystem. See appendix ?? for specifications on solenoid valve. 17viiiList of Figures2.10 Arduino microcontroller and MOSFET transistor switchingcircuit. See appendix ?? for specifications of the Arduino.This circuit was used to boost the 5 v signal from the micro-controller by switching a transistor and sending 24 v DC toactuate the solenoid valve. Much of the accuracy and repro-ducibility of the experiments was based on the performanceof this switching circuit. . . . . . . . . . . . . . . . . . . . . . 172.11 Optical abberation caused by a curved surface. The part ofthe diagram shows the distortion that occurs as a result ofthe curved pipe surface. The lower part of the diagram showshow this distortion is eliminated by the anti-distortion tank. 182.12 Bohlin CS-10 rotational rheometer. . . . . . . . . . . . . . . 212.13 Typical stress shear strain rate curve for 0.5% carbopol. . . . 242.14 Shear Stress vs effective viscosity for 0.5% carbopol. Theyield stress is taken as the shear stress that corresponds tothe maximum effective viscosity. . . . . . . . . . . . . . . . . 252.15 Typical stress strain for 0.5% carbopol with the least squaresfit shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.16 0.4 and 0.5% PEO stress vs strain curve with the power lawfit shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.17 A summary of 2 PEO concentrations, carbopol and visco-elastic properties of PEO(a) Shear stress vs shear rate for0.5% concentration of xanthan and 0.5 and 0.4% concetra-tions of PEO. (b) Shear stress vs shear rate for 0.5% concen-tration of carbopol.(c) Loss Modulus G?? and elastic modulusG? for 0.5% PEO. (d) Complex modulus phase angle vs fre-quency (angle between G? and G??). . . . . . . . . . . . . . . . 282.18 Glycerol stress vs shear rate curve with the power law fit shown. 292.19 Omega milliamp output flow meter. See appendix ?? forspecifications. . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.20 Flow Rate vs RPM test data for inner fluid (pump 1). . . . . 322.21 Flow Rate vs RPM test data for outer fluid (pump 2). . . . . 322.22 Flow meter output (mA)vs RPM test data for inner fluid(pump 1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.23 Flow meter output (mA)vs RPM test data for outer fluid(pump 2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.24 Sequence of a VPL flow being perturbed and rapidly progress-ing to an unstable flow (1 second intervals between images insequence) Q1 = 21 mls , Q1?, Q2 = 9 mls ,T = 5 s. . . . . . . . . 36ixList of Figures2.25 Example of a spatiotemporal plot used to estimate the yieldfront velocity. There is 100 cm of test pipe is shown. Q1 = 31mls , Q1? = 15 mls , Q2 = 17 mls . The slope of the line is usedto estimate the velocity of the yield front. Here the velocityis about 0.55 m/s. . . . . . . . . . . . . . . . . . . . . . . . . 372.26 Plot used to more accurately determine the yield front veloc-ity. Q1 = 31 mls , Q1? = 15 mls , Q2 = 17 mls . . . . . . . . . . . . 382.27 Example of a photo sequence. A high to low step changein the core flow is introduced at time t=0 and can be seento advect upwards. To report results of an experiment, inthe results section, axis are put on a photo sequence to showwhen during the experiment the image was captured and thedimensions of the photos. . . . . . . . . . . . . . . . . . . . . 412.28 Spatiotemporal plot of a image sequence. Spatiotemporalplots are used to illustrate the concentration of dyed (core)vs clear fluid (annular fluid). In the results section, axis areclarified to convey the vertical dimensions. . . . . . . . . . . . 422.29 Flow loop used to vary and adjust flows in both fluids. Seetext for details. . . . . . . . . . . . . . . . . . . . . . . . . . . 442.30 Schematic of a VPL flow velocity profile. . . . . . . . . . . . 502.31 Schematic of an yielded flow velocity profile. . . . . . . . . . 532.32 Schematic of a static layer flow velocity profile. . . . . . . . 552.33 Flow rate contours plotted on the Ri,|?P?Z | plane for 0.5% xan-than and 0.5% carbopol fit with a Herschel-Bulkley model.The regime of VPL flow is shown as the unshaded region be-tween the shaded yielded and static regions. The red linesare flow rate iso lines (in mls ) for annular fluid (carbopol) andthe blue lines are flow rate iso lines of core fluid (xanthan). . 572.34 Flow rate contours on Ri, |?P?z |. for 0.5% PEO and 0.5%carbopol fit with a Herschel-Bulkley model. The red lines areflow rate iso lines (in mls ) for annular fluid (carbopol) and theblue lines are flow rate iso lines of core fluid (PEO). . . . . . 582.35 Flow rate contours on Ri, |?P?z |. for 0.4% PEO and 0.5%carbopol fit with a Herschel-Bulkley model. The red lines areflow rate iso lines (in mls ) for annular fluid (carbopol) and theblue lines are flow rate iso lines of core fluid (PEO). . . . . . 59xList of Figures3.1 Sequence showing the presence of an unyielded plug. Imagesare taken 1 second apart, Q = 20mls . Camera captures 1.2meters of test pipe. Fluid os 0.5% carbopol. Only the lefthalf of the pipe is shown in each frame for clarity. . . . . . . . 623.2 Spatiotemporal plot of a startup. The flow rates are Q1 =48and Q2 =14 mls . 120 cm of test pipe are shown. . . . . . . . . 633.3 Theoretical and experimental values of ri for 50 experiments.Fluids were 0.5% xanthan, 0.5% carbopol. . . . . . . . . . . 643.4 Preliminary experiment with 0.5% concentrations of PEO andcarbopol. PID was set to make 5 second period square wavepulses at 0.5 relative amplitude to inner fluid. Flow rate isQ1 =28, Q2 =14 mls . . . . . . . . . . . . . . . . . . . . . . . . 653.5 Example of a step change experiment using 0.5% carbopoland 0.5% xanthan. Flow rates: (Q1 ? Q?1): Q1 = 27.5 ml/s,Q?1 = 5.5 ml/s, Q2 = 30.3 ml/s. (a) A spatiotemporal plotof a 150 cm section of the test pipe shows the transition ininterfacial radius after the step change in flow rate. Thisplot is constructed by averaging the image intensity acrossthe pipe at each fixed height, with images taken from a highdefinition video at 30 frames/second. (b) Sequential imagesare shown at 15 s intervals a 90 cm section of the test pipe. . . 673.6 Control volume moving axially at speed wY f . . . . . . . . . . 693.7 Plot of w?in against wY f for each of our step change exper-iments. Error is +? 0.1 m/s for wY F and +? 0.15 m/s forw?in based on image capture and analysis error and flow ratemeasurement error. . . . . . . . . . . . . . . . . . . . . . . . . 703.8 (a) No frozen structure: Q1 = 21.9 ml/s, Q?1 = 7.7 ml/s,Q2 = 20.1 ml/s, T ? = 1 s. Photos show 90 cm of the test pipe.Images are 2 seconds apart. (b) An example of a single pulseexperiment that resulted in flow instability: Q1 = 20.6 ml/s,Q?1 = 2.6 ml/s, Q2 = 8.6 ml/s, T = 3 s. Photos show 100 cmof test pipe and images are 2 second apart. . . . . . . . . . . 723.9 Stable single pulse photo sequence (a) and spatiotemporalplot (b) of a single pulse experiment. Q1 = 15.4 ml/s, Q?1 =2.6 ml/s, Q2 = 15.2 ml/s, T ? = 1 s. 100 cm of test pipe isshown in (a) and 120 cm is shown in (b). . . . . . . . . . . . 733.10 Plot of the measured speed of frozen-in structures dividedby the speed of the fluid-fluid interface, for approximately 80experiments. . . . . . . . . . . . . . . . . . . . . . . . . . . . 74xiList of Figures3.11 Observed flow regimes for different durations of pulse (T ?),plotted against Q?1/Q1 and Q2 for different single pulse ex-periments. (a) 1 second, (b) 2 second, (c) 4 seconds (d) 6second. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753.12 Observed flow regimes for different durations fixed Q2, plot-ted against Q?1/Q1 and T for different single pulse experi-ments. (a) 10 ml/sec, (b) 20 ml/sec, (c) 30 ml/sec.(d) presentsthe entire experimental matrix. . . . . . . . . . . . . . . . . 763.13 (a), (b), (c). The flow regimes observed during single pulseexperiments with 3 different concentrations of PEO as theinner core fluid and T ? =1 s. (a) 0.5% PEO, (b)0.4% PEO,(c) 0.25% PEO. (d) Flow regimes observed for 0.5% PEO andT ? = 6 s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 773.14 Sequences of images showing the continuous formation of astable ?diamond necklace? pattern. (a) 0.5% PEO, T = 2s,T ? = 1s, relative amplitude Q?1/Q1 = 0.167, Q1 = 10.0mls ,Q2 = 10.0mls , images show 45 cm section of test pipe. (b)0.4% PEO,T = 10s, T ? = 1s, relative amplitude Q?1/Q1 =0.17, Q1 = 15.4mls , Q2 = 8.6mls , images show 100 cm sectionof test pipe. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 793.15 Frequency effects on the formation and retention of ?diamondnecklace? patterns: a) T = 10s; b) T = 5s; c) T = 2s; d)T = 1s. Other parameters: Q1 = 22mls , Q2 = 15mls , rela-tive amplitude Q?1/Q1 = 0.1, 0.5% PEO is used, T ? = 1s.Approximately 100cm of the test pipe is shown. . . . . . . . . 803.16 Plot of the measured wavelength between diamonds, ?, andthe simple prediction of wiT +w?iT ?, for 34 experiments con-ducted with 0.5% PEO and 0.5% carbopol. . . . . . . . . . . 813.17 Images showing diamond shapes with (a) relative amplitudeof 0.36 and (b) relative amplitude of 0.17. (c) Structure sizeresponse to relative amplitude for two flow rates. . . . . . . . 833.18 (a) An example of another effect; displacing the core fluid toproduce a ?country road? shape with 0.5% PEO and carbopol.(b) Formation of the country road. Q1 =12, Q2 =8 mls . . . . . 843.19 Experiment done using glycerol as the core fluid. Q1 =15,Q?1 =3, Q2 =15 mls , T = 10, T ? =1 second. . . . . . . . . . . . 85xiiAcknowledgementsI would like to thank Dr. Sarah Hormozi for her training in the lab, helpfulinsights, and logical suggestions, my supervisor, Professor Ian Frigaard, forhis research vision, funding, and clear and helpful suggestions over the pasttwo years, and my father, Professor Robert Dunbrack, for his interest, wis-dom and support. The experimental apparatus was built by Billy Fung andRyan Yee in Sept 2010- Sept 2011 under supervision of Dr Sarah Hormozi.The following people were also very helpful to me during this research:? Dr. Kamran Alba? Dr. Mohammed Taghavi? Gustavo MoisesThis research was made possible by the funding of NSERC and Slumberger.xiiiChapter 1Introduction to SubjectFigure 1.1 shows a schematic of the type of fluid flow studied in this thesis.The main features of this flow are:1. Poiseuille flow, driven by a pressure gradient.2. Pipe flow, inside a cylindrical pipe.3. Core-annular flow, also called concentric flow, and multilayer pipe flow.4. Iso density, both fluids have the same density.Figure 1.1: Schematic core annular pipe flow.11.1. Interfacial Instabilitiesw(r) is the axial component of velocity as a function of distance fromthe center (r). The slope of the axial velocity profile changes slope abruptly.This point is known as the ?interface? and is located at the ?interfacial radius?Ri. The abrupt change in slope is caused by the discontinuity in fluidproperties across the interface.A variety of different models exist to express the properties of fluids.These models are called constitutive equations. It must be acknowledgedthat no constitutive model can fully explain the behaviour of a fluid underall conditions. Constitutive models must be selected that explain the fluidbehaviour within the region of interest for the study and that do not com-plicate the problem to the point of being so complex as to cloud the mainphenomena. In this thesis, we use the ?Herschel-Bulkley? model, equation(1.1).|?ij | = |?rz| = ? = ?y + ???n (1.1)In the 1D pipe flow model we use to model the flow studied, the non zeroterm of the stress tensor is stress is |?rz|. The strain rate in this scenariois ?? = ?w?r . ?y is the yield stress (defined here as the stress at highesteffective viscosity). The indices i,j correspond to directions of the stresstensor, defined in cylindrical co-ordinates in this thesis:???11 ?12 ?13?21 ?22 ?23?31 ?32 ?33?? =???rr ?r? ?rz??r ??? ??z?zr ?z? ?zz??The stress tensor is not used in derivations since we are dealing with a singlenon zero stress direction. The Herschel-Bulkley model becomes the powerlaw model when ?y ? 0, which is what is used for the non-yield stress fluidswe have studied.1.1 Interfacial InstabilitiesA fluid fluid interface with viscosity stratification across it is inherentlyunstable in many situations[3],[4],[13]. Perturbations will grow over timeand lead to mixing of the two fluids. In terms of industrial applicationsof multi-layer flows, mixing of the two fluids lowers the quality of productsand limits the production rate, as flow rates would have to be kept lowto minimize mixing. The mechanism of instability across an interface withviscosity stratification is as follows:1. An initially stable interface is perturbed by a fluctuation in flow rate.21.2. Stability of Multilayer Flows2. The interface is perturbed away from the original equilibrium position.3. Fluids on either side of the boundary experience a change in shearstress and accelerate according to their different viscosities.4. The less viscous fluid responds to the perturbation by acceleratingfaster than the more viscous fluid.5. The difference in velocities caused by unequal acceleration across theinterface promotes vortices which cause further instability growth.This is a classic instability and is shown in figure 1.2. Additional mechanismsof instability exist but are left out of this discussion because stability analysisis not a main focus of this thesis.Figure 1.2: Classic viscous-viscous instability.1.2 Stability of Multilayer FlowsHickox [3]focused on concentric (core annular) multilayer flows in a pipewith two fluids of different viscosity and density. The stability was studiedby introducing a change in flow rate that results in an alteration of theinterfacial radius. By linearizing the Navier Stokes equations and applyinga method that introduced long wave, small perturbations, Hickox foundthat even at small Reynolds numbers ? O(1), there was no evidence thatthe interface could remain stable indefinitely. Axisymetric (and asymetric)perturbations introduced to viscous-viscous core annular flows will initiallygrow at an exponential within the area close to equilibrium. This study wasdone using two newtonian fluids of different viscosity.31.2. Stability of Multilayer FlowsIf a yield stress fluid is used as the lubricating fluid, the long waveinstabilities studied in [3] can be eliminated. Essentially, the fluid-fluidinterface is transformed into a fluid-solid interface that stabilizes the flow.The idea of using a yield stress lubricating fluid was first introduced in [2]in a paper that studied the linear stability of core annular ?visco-plasticallylubricated? (VPL) flows. It has been demonstrated experimentally that VPLflows would be stable up to the point where the shear stress at the interfaceexceeds the yield stress of the lubricating fluid [10]. Compared to theirpurely viscous analogues, VPL flows are resistant to linear instabilities, evenat moderate Reynolds number. This was a promising result suggesting thatmulti layers flows can be made stable. Frigaard ([2]) studied VPL flows inchannel flows but the experimental work in this thesis is done on pipe flows.In addition, the practical use of linear stability analysis is low if this stabilityanalysis cannot be extended into the non-linear regime.The non-linear stability of VPL flows in ducts was studied in [11] whereit was suggested that VPL flows can be stable under perturbations thatdisplace the position of the interface a small distance away from an equi-librium position. This stability was seen to persist in some situations upto the same order of Reynolds number commonly seen in Newtonian pipeflows. Although not directly applicable to this research, this is an interestingresult because it provides strong evidence that VPL flows can remain stablein experimental and industrial applications, where perturbations cannot becompletely eliminated. An additional study in [12] showed that VPL flowsare stable up to very high Reynolds numbers when a Carreau fluid is usedas the core fluid. The effective viscosity of a Carreau fluid is:? = ?inf + (?0 ? ?inf)(1 + (???)2)n?12where ?inf is the viscosity at infinite shear rate, ?0 is the viscosity at zeroshear rate, and ? is the relaxation time. A Carreau fluid behaves as both apower law and a newtonian fluid depending on the shear rate. A wide rangeof core fluids were also expected to be able to produce stable core annularflow when surrounded by an un-yielded plug. It was also noted in [12] thatby choosing a high enough value for the lubricating fluid flow rate, the Cou-ette component superimposed on the core fluid by the resulting interfacialvelocity can increase the stability. It was demonstrated experimentally thata power law, shear thinning core fluid (n < 1) will produce stable flowswithin a pipe [10] that persist for 100s of pipe diameters.Further work on the stability of VPL flows in pipes was done by [8].In this extensive study, start-up, development length, and perturbations41.3. Multilayer Viscoplatic Lubrication Flows in Experimentationwere studied numerically for a VPL pipe flow. Perturbations of variousamplitudes were introduced to a VPL flow and no instabilities were seento develop at moderate Reynolds numbers ? O(100). This is evidence thatVPL pipe flows are robustly stable in practice.1.3 Multilayer Viscoplatic Lubrication Flows inExperimentationThe flows studied in this thesis are concentric flows with a power law orvisco-elastic core fluid and a Herschel Bulkley lubricating fluid. The corefluid is surrounded by a ?plug? of fluid (unyielded fluid) where the Binghamnumber (B = ?yL?w(r)) > 1. As mentioned above, Frigaard introduced thismethod of stabilizing fluid-fluid interfaces in [2]. In [10], Huen provided thefirst experimental evidence that these flows can be stable by using xanthanas the core fluid and carbopol as the yield stress lubricating fluid (the samefluids that will be used in this thesis). In [10], a 25 mm radius verticalpipe was filled with carbopol and then carbopol and xanthan were injectedinto the pipe simultaneously, xanthan in the center of the pipe and carbopolin the outer annulus of the pipe. The experimental results showed thatafter an initial period of mixing at startup, there was good agreement withthe expectation that VPL flows remain stable up to the point where thelubricating fluid is yielded at the interface. Further experimental work wasdone in [5] where stable flows were observed with carbopol as the lubricatingfluid and polyethylene oxide (PEO) and xanthan as the core fluids. In [5],the use of a visco-elastic fluid as the core fluid was experimented with andseen to produce an unexpected ?frozen wavy interface? pattern. In addition,multilayer VPL flows with viscoelastic core fluids were shown to be non-linearly stable in [8].1.4 Interfacial Patterns in Experimental andNumerical StudiesExperimental and numerical studies have been carried out that suggest rel-atively large interfacial patterns can exist and even persist at the interfacewithout causing extensive mixing. d?Olce [1] observed naturally occurringinstability patterns in concentric pipe flow of two Newtonian fluids up toReynolds number of 100. Generally, these instability patterns were classi-fied as either ?mushroom? or ?pearl?. The mushroom patterns were observed51.5. Applications of VPLto form at low Reynolds number ? O(1) and lower interfacial radius, com-pared to the pearl patterns that formed at higher Reynolds numbers? O(10)and higher interfacial radius. The patterns did not persist unchanged, ratherevolving over time.In [8], Hormozi found in a numerical study that at small Reynolds num-ber ? O(10), perturbations introduced will decay in amplitude at a ratethat depends on the Reynolds number. When the Reynolds number andamplitude of the perturbations are increased, the structures persist in theinterface. This is supported by the non-linear stability analysis in [11]. If avelocity perturbation does not completely decay before the shear stress atthe interface drops below the yield stress, the effect of the perturbation willbe ?frozen into? the interface at the point where the interface Bingham num-ber increases above 1. Interestingly, in [8],the interfacial stress was lower forhigher frequency perturbations after they were frozen in. Experimentally,[7],[10], [5] recorded observations of perturbations that were frozen into theinterface and persisted unchanged for lengths of the order of 100 ? Ri. Itwas suggested in [5] that the phenomena of perturbations being frozen intothe interface could be utilized for polymer engineering, manufacturing, andfood processing if the shapes can be precisely controlled.1.5 Applications of VPLVPL has possible applications in lubricated pipelining, slurry transport, co-extrusion processes in manufacturing and surface coating to name a few. Thelubricated pipelining and slurry applications present challenges because ofthe increased energy required to shear the yield stress fluid compared withusing a fluid like water, which can be used here because precise control overthe interface is not required, only reduction in the required pressure dropfor a given production rate (flow rate). It is possible that applications couldexist in the transport of heavy oil products where pumping is too energyintensive. In this case, the oil is usually mixed with water or solvents to di-lute it and alter it?s rheology. Such an emulsion is expensive to separate andthe dilution process often uses harmful chemicals. VPL flow as a lubricationmethod could allow for a large reduction of the energy required to overcomepipe wall friction if a suitable shear thinning, yield stress lubrication fluid isused. This would also allow pure, undiluted product to be transported.VPL is promising for several manufacturing techniques. A co-extrusionprocess is where a product is made by extruding multiple layers of poly-mer/metal melts from the same die. These processes do require accurate61.6. Problem Description and Outlinecontrol over the shape and position of the interfaces by modifying the flowrates and rheologies of each layer. An example of a co-extrusion process in apipe is the extrusion of tubing used in medical devices which require differ-ent properties on the inner and outer diameter of the tube. In this example,layers of different resins are co-extruded to meet the design requirements.Another application of VPL is the possibly of ?near net shape manufac-turing?, where perturbations frozen into the interface form the geometry ofproducts in molding processes. If our knowledge of VPL is extended to in-clude the control of the ?freezing in? phenomena noted in [6],[8], this methodcould accurately define complex internal geometries by replacing either thecore or outer fluid with, for example, a polymer melt. Investigating thisphenomenon is one of the main objectives of this thesis.1.6 Problem Description and OutlineVPL flows are studied experimentally in this thesis. The primary researchobjective is to investigate the possibility of freezing shapes into the interfaceof VPL flow by introducing perturbations to initially stable flows. Theconditions varied include flow rate parameters and fluid parameters. Thefollowing three chapters describe the experimental approach to the problem,the results, and a short discussion of the contributions and future directionsof this research.7Chapter 2Materials and Methods2.1 Physical Description of ApparatusFigure 2.1 show the main apparatus used to perform all experiments. Theflow of interest in the apparatus is through the 50.8 mm inner diameter,63.5mm outer diameter transparent cast acrylic pipe. The pipe is 2.0 m inlength, oriented vertically. At the top of the pipe a flange is fastened toa PVC ring on the bottom outside surface of a catch basin that containsthe discharged fluid and transmits it to a drain through a 25.4 mm innerdiameter vinyl hose. Hidden from view inside the catch basin is a 305 mmsection of the same cast acrylic pipe with a flange on one end fastened tothe same acrylic ring through the top surface of the catch basin. Fluid musttravel through this extra section of pipe before overflowing into the catchbasin so, although it is not part of the flow of interest, this extra section addshydrostatic pressure to the system. The bottom of the pipe is terminated ina flange which butts to an annular manifold with a 50.8 mm diameter, 110mm deep blind through-hole aligned with the center line of the main pipe.There are 8 x 6.4 mm ports evenly spaced around the manifold with thecenter lines of adjacent ports entering the through-hole intersecting at 45degree angles. These ports are injection ports for the outer fluid and ensurethat the outer fluid is injected into the main pipe evenly. At the bottomof the manifold, there is a PVC flange fastened with a threaded neck. Theneck threads allow a central injector tube to be installed (Figure 2.2). Theinjector screws into the bottom of the manifold. Around the main 1524.0mm long cast acrylic pipe is an ?anti-distortion tank?. The anti-distortiontank has a square profile in plan view and a rectangular profile in elevationview and is formed from acrylic sheets butted together with beveled edges.The basin, main pipe, anti-distortion tank, manifold, and central injectorare supported on an aluminum frame. Four 20 watt fluorescent lights aremounted vertically on the aluminum frame on each corner of the framebetween the floor and the roof. Mirrors are arranged to allow viewing of thethree hidden sides of the anti-distortion tank. Three sides of the aluminiumframe are filled in with sheets of white plastic to reflect light toward the82.2. Description of Flow Loopexperimental apparatus. Two tripod mounted Nikon D800 DSLR camerasare positioned in front of the apparatus (Figure 2.3). The rear camera fieldof view includes the entire 1524 mm length of the apparatus and the smallerfield of view of the front camera captures the development region of the flowand is adjusted for each experiment as needed.2.2 Description of Flow LoopFigure 2.4 shows a schematic of the flow loop used in all experiments. Eachfluid is stored in a separate reservoir prior to pumping. The feed from thereservoir to the apparatus is provided by two pumps. The inner fluid ispumped by a Moyno 500 progressive cavity pump driven by a 12 HP, 3 phasemotor (Figure 2.7) which is controlled by a Baldor 240v 0-500 Hz variablefrequency drive (VFD) (Figure 2.6). The same pump/motor combination isused for the outer fluid but the motor is controlled by a VF Drive sp500 240v0-500 Hz VFD. The VFDs chosen can operate the pumps over their ratedoperating range and can be finely adjusted to a maintain a single motorRPM. Progressive cavity pumps were chosen because they do not alter therheology of the fluids during pumping. Both fluids travel from the reservoirsthrough a 25.4 mm vinyl hose to the suction side of the pump. After leavingthe pumps, but before entering the apparatus manifold, both fluids flowthrough separate Omega FPD1204 flow meters. These flow meters outputa current from 4 to 20 milliamps depending on the flow rate and can becalibrated to overlap the flow rate operating ranges used in the experiments(see below). After exiting the flow meter an 8 way splitter separates theouter fluid flow with one flow going to each of the 8 injection ports on themanifold. After exiting its flow meter, the inner fluid is piped to a 13.7mm inner diameter stainless steel tube that passes through the PVC flangeat the bottom of the manifold and extends 305 mm up from the bottomof the pipe (pipe/manifold interface). The inner fluid is injected into theexperimental pipe at the top of the stainless steel injection tube. The twofluids flow through the main pipe and the 305 mm top section and spill intothe catch basin from which they drain via a 25.4 mm vinyl tube.2.3 PID Control SystemFigure 2.8 shows the Labview control interface of the control system basedon a proportional/integral/derivative (PID) controller which allows fixedflow rates to be maintained throughout an experiment. A National Instru-92.3. PID Control SystemFigure 2.1: Main apparatus used to perform experiments102.3. PID Control SystemFigure 2.2: Injector used to inject core fluid (internal diameter is 13.7 mm,external diameter is 14.5 mm).112.3. PID Control SystemFigure 2.3: Arrangement of tripods (one camera absent for taking this im-age). See appendix ?? for specification.Figure 2.4: Flow loop schematic. The schematic shows the loop used toconduct fluid to the apparatus.122.3. PID Control SystemFigure 2.5: Moyno 1000 pump used to pump outer fluid. See appendix ??for specifications.Figure 2.6: Variable frequency drives (left SP500 for fluid 1, right BaldorVector Drive for fluid 2) used to control pump motor speeds. See appendix?? for specifications.132.3. PID Control SystemFigure 2.7: Moyno 500 pump used to pump inner fluid. See appendix ??for specifications.142.3. PID Control SystemFigure 2.8: Labview PID control interface showing the control options avail-able for the flow loop. The readouts on the two graphs correspond to theflow rates of the two fluids. The output was also saved to an excel spread-sheet that could be looked at during analysis to determine accurate flowrates.152.4. Solenoid and Throttle Valvesthat were created by a solenoid and throttle valve system (described below).2.4 Solenoid and Throttle ValvesTo produce the more rapid changes in flow rate required for some of the ex-perimental treatments, an additional flow control system was incorporatedconsisting of solenoid and throttle valves. Figure 2.9 and Figure 2.10 showphotos of the Solenoid and throttle valve in series and the arduino micro-processor and switching MOSFETs used respectively. An Omega 24V 12inch solenoid valve and a 12 inch hand operated throttling gate valve wereplaced in series on a bypass loop at a T junction with each fluid upstreamof the flow meters. The bypass loop is opened or closed by the position ofthe solenoid valve and throttled by the position of the gate valve. Preciseflow control was needed later on in the experiments to produce an accurateexperimental matrix. This was achieved by having 2 throttling valves inseries. The two valves increase the precision by allowing twice the handoperated rotation between completely blocked and completely open. Onevalve was used to give a rough flow rate slightly higher than the desiredflow rate while the second throttling valve was used to trim the flow rateto the desired value. The solenoid valve is controlled by an Arduino mi-croprocessor running a program that controls the timing of its opening andclosing. When the solenoid in the bypass loop is switched from closed toopen the pressure in the feed loop drops significantly. It was found that thiscause a transitory reverse flow in the inner fluid supply to the manifold. Toeliminate reverse flows, a brass ball and spring type one way valve (checkvalve) was added between the flow meter and manifold .162.4. Solenoid and Throttle ValvesFigure 2.9: Asco 8210 solenoid valve and generic gate valve installed inseries and used to introduce rapid changes in flow rate to the system. Seeappendix ?? for specifications on solenoid valve.Figure 2.10: Arduino microcontroller and MOSFET transistor switchingcircuit. See appendix ?? for specifications of the Arduino. This circuitwas used to boost the 5 v signal from the microcontroller by switchinga transistor and sending 24 v DC to actuate the solenoid valve. Muchof the accuracy and reproducibility of the experiments was based on theperformance of this switching circuit.172.5. Anti-distortion Tank2.5 Anti-distortion TankFigure 2.11: Optical abberation caused by a curved surface. The part of thediagram shows the distortion that occurs as a result of the curved pipe sur-face. The lower part of the diagram shows how this distortion is eliminatedby the anti-distortion tank.The space between the anti-distortion tank and the main pipe is filledwith distilled, vacuum treated water. This is done to largely eliminate theoptical magnification of the interfacial radius (Figure 2.11) that would becaused by refraction at the interface of the curved pipe and the air. Thismagnification would be large in the absence of the anti-distortion tank be-cause of the substantial refractive index mismatch between acrylic (RI=1.49)and air, combined with the non-normal angle of incidence of the interfacialimage edge with the curved acrylic/air interface. This magnification is muchsmaller with the anti-distortion tank in place primarily because the angleof incidence of the interfacial image edge with the flat interface of the anti-distortion tank and the air is nearly normal, producing only small refractive182.6. Fluidsmagnification The visual effect of the anti-distortion tank is essentially toallow viewing as if the experiments took place in a tube with a plane faceperpendicular to the viewing axis. Although the anti-distortion tank largelyeliminates horizontal refractive magnification of the interfacial radius inde-pendent of vertical position, magnification could still effect measurementsbetween vertically separated points because the vertical viewing angle be-comes less normal as the distance from the pipe to the camera lens increases.To account for this effect a measuring tape was lowered into the center ofthe experimental pipe and its markings equated to vertically equivalent con-stant points marked on the outside of the apparatus. The actual verticalposition of a point on the interfacial radius could then be obtained by notingits position relative to these points and then converting this position to thecorresponding position on the calibration tape. The term ?fish tank? effectis often used in literature to explain this phenomena and the anti distortiontanks are often called fish tanks.2.6 FluidsThere are three fluids that are used in these experiments:1. Carbopol EZ2 polymer + NaOH Solution (carbopol)2. SIgma Xanthan Gum (xanthan)3. Sigma Polyethlyeneoxide Solution (PEO)2.6.1 Carbopol EZ2 Polymer SolutionCarbopol was prepared from Carbopol EZ2 Polymer powder and waterwhich was then neutralized with NaOH pellets. A digital scale was usedto measure out 40 liters of tap water in a mixing bucket, assuming a waterdensity of 1000 kg/m3. A predetermined mass of Carbopol EZ2 powderwas then measured with a digital analytic balance and added to the water.The concentrations used for all experiments was 0.005 mass fraction of dryCarbopol EZ2 powder to the mass of water. The Carbopol EZ2 + water wasmixed for 8 hours with a stainless steel fin type mixing blade rotating at 350RPM. The mixture was then allowed to stand for 6-8 more hours to de-gasand come to room temperature. At this stage, the mixture has Newtonianrheology, is milky in colour, and has a pH of around 5.0. Increasing the pHto 7.0 through the addition of NaOH allows the polymer to form bonds anddevelop a yield stress. A digital pH meter was used to measure the pH of the192.7. Rheometrysolution as NaOH was gradually mixed in until the pH reached 7.0. The pHis measured with a hand held digital pH meter. At pH=7 the solution hasdeveloped a yield stress and becomes transparent. Mixing is stopped afterthe neutralized carbopol has a homogeneous consistency. The yield stressdoes not change very much between pH=7-10 but a pH of 7 was chosen forsafety in the event of a spill and to prevent corrosion damage to equipment.2.6.2 Xanthan GumSigma Xanthan Gum (G1253) powder was used to create the xanthan so-lutions used in experiments. The xanthan preparation procedure was es-sentially identical to that of carbopol without the need to adjust pH. Theresulting solution is a thick transparent fluid. To produce visual contrastbetween the xanthan and carbopol, the xanthan was colored by the additionof 30 ml of black fountain pen ink to 40 L of fluid. The concentration ofxanthan used in this thesis by mass weight of xanthan to water was 0.005.2.6.3 PEOThe PEO was mixed in an identical method to xanthan. The resultingsolution is a thick fluid that has an apparent high extensional viscosity(when a gloved finger is dipped into the reservoir of PEO, strands of fluidform as the finger is removed). Concentrations of PEO used in this thesisby mass weight of PEO to water are 0.005 and 0.0025.2.7 Rheometry2.7.1 CarbopolCarbopol, the outer fluid or lubricating fluid, is characterized as a yield stressfluid. The yield stress of the lubricating fluid is an important parameter inthe experiments because it largely determines the transition between thestable and unstable multilayer flows.Rheometry of a Yield Stress FluidA yield stress is the minimum stress that must be applied to a fluid to cre-ate a permanent deformation (ie. flow). In theory, below the yield stress,the fluid acts like a solid and obeys Hooke?s law with a stress resulting ina proportional strain, but a zero shear rate. Above the yield stress, thefluid displays a non-zero shear rate. The yield stress is also known as the202.7. RheometryFigure 2.12: Bohlin CS-10 rotational rheometer.212.7. Rheometryproportional limit because it is the limit of stress that will result in a pro-portional strain. Determining the yield stress of a fluid is difficult becausein real fluids, the transition from solid behaviour to fluid behaviour is oftengradual rather than abrupt. A common method of measuring yield stressis to perform a ?ramped stress test? and calculate the ?effective viscosity?after each successive increase in shear stress. The ?effective viscosity? ofa yield stress, Herschel-Bulkley fluid is here defined as the shear stress di-vided by the ?transitory shear rate?. ?Effective viscosity? is calculated fromthe transitory rate of deformation (transitory shear rate) in the unyieldedfluid shortly after the increase in shear stress, but before the fluid reachesan equilibrium strain. The ?effective viscosity? will start low because, asHooke?s law states, the resistance to deformation is proportional to the in-stantaneous deformation. So the unyielded fluid will strain easily because itis not yet deformed. The ?effective viscosity? will be the instantaneous ratioof stress/strain rate. The shear rate here is not truly a shear rate becauseit is transitory and will decay to zero given enough time since the stress isbelow the yield stress. If each transitory shear rate is measured shortly afterthe ramped increase in stress (5 seconds for these tests), it will be non-zero,allowing for the calculation of the effective viscosity. As the ramped stressincreases the unyielded fluid strains more but at a decreasing rate, resultingin an increase in the ?effective viscosity?. The yield stress is taken as thestress corresponding to the highest ?effective viscosity? measured using theramped stress protocol. This is because once the fluid yields, the transi-tory strain rate will increase and the ?effective viscosity? will decrease. Themethod of highest ?effective viscosity? was used to determine the yield stress(used later for the analytical model).Rheometry Tests Done on CarbopolThe carbopol rheometry tests were performed using a pneumatic BholanCS10 digital controlled shear stress and strain rate rotational rheometer(Figure 2.12). The tests were done immediately after mixing to preventthixotropic effects from causing error . The shear is between two parallelserrated plates 40mm in diameter with a gap thickness of 1.00 mm. Car-bopol was measured with 40 mm parallel serrated plates because surfaceserration is required to enforce the ?no slip? condition that is critical in theserheometry measurements. During the tests, carbopol was loaded onto thebottom plate and the top plate is lowered down. The gap is then set tothe 1.00 mm standard using a screw adjuster. The excess carbopol is thentrimmed away manually. The fluid is pre-sheared for 120 seconds. A linear222.7. RheometryShear Stress Shear Rate Effective ViscosityPa 1/s Pa.s2.55E+00 5.68E-04 4.48E+037.48E+00 3.65E-03 2.05E+031.25E+01 3.91E-03 3.20E+031.75E+01 4.16E-03 4.20E+032.25E+01 4.43E-03 5.07E+032.75E+01 4.77E-03 5.76E+033.25E+01 5.15E-03 6.31E+033.75E+01 5.55E-03 6.75E+034.25E+01 5.97E-03 7.11E+034.75E+01 6.36E-03 7.46E+035.25E+01 6.82E-03 7.69E+035.74E+01 7.38E-03 7.78E+036.25E+01 8.29E-03 7.54E+036.75E+01 9.96E-03 6.77E+037.26E+01 1.33E-02 5.48E+037.75E+01 1.94E-02 4.00E+038.25E+01 3.13E-02 2.64E+038.76E+01 6.92E-02 1.27E+039.25E+01 2.07E-01 4.47E+029.75E+01 5.18E-01 1.88E+021.02E+02 9.41E-01 1.09E+021.07E+02 1.53E+00 7.03E+011.12E+02 2.33E+00 4.83E+011.17E+02 3.29E+00 3.57E+011.22E+02 4.45E+00 2.75E+011.27E+02 5.77E+00 2.21E+01Table 2.1: Rheometry data for 0.5% carbopol.232.7. RheometryFigure 2.13: Typical stress shear strain rate curve for 0.5% carbopol.controlled stress ramp test is then performed (see above) which starts belowthe yield stress and increases to well above the yield stress. The samplingrate and the time between the stress ramp increase and sampling were cho-sen to eliminate thixotropic effects (5 seconds/sample). Using this method,a stress is introduced for 5 seconds after which the transitory shear ratecorresponding to that shear stress (used to calculate effective viscosity) isrecorded. Figures 2.13 and 2.14 show the resulting curves from this test.Actual values are given in Table 2.1. The yield stress was determined tobe 57.4 Pa (highlighted in bold in Table 2.1). In general, the yield stress ofall carbopol batches used in experiments was very consistent in this value,differing by +/? 1 Pa.Shear stress versus shear rate curves are of primary interest in this thesisbecause it is the yield stress that determines the transition between stableand unstable flows. The Hershel-Bulkley model was used as a constitutivemodel for all fluids:?=?y + ???n.Fitting the rheological data to the Hershel-Bulkley model was done usingMatlab least squares fit. Figure 2.15 shows the carbopol stress strain datafrom Figure 2.13 fit to the Herschel-Bulkley model. The resulting fit was? = 57.4 + 45.0??0.28242.7. RheometryFigure 2.14: Shear Stress vs effective viscosity for 0.5% carbopol. The yieldstress is taken as the shear stress that corresponds to the maximum effectiveviscosity.Figure 2.15: Typical stress strain for 0.5% carbopol with the least squaresfit shown.252.7. Rheometry2.7.2 Rheometry Tests Done on XanthanXanthan is used as the inner fluid during experiments. It is a pseudoplas-tic (shear thinning) fluid that has no yield stress (it flows at any non-zeroshear stress) and exhibits long-term thixotrophy in the form of a reducedconsistency coefficient if it is allowed to sit for a period of weeks [10]. Forthis reason, the xanthan was mixed immediately before the experiment soits behaviour would not be affected by thixotropy. The rheometry was per-formed using the same ramp test described for carbopol to produce stress vs.shear rate data. The one difference in testing xanthan was that a 40mm, 4degree cone and plate was used to conduct the rheometry tests on xanthan.The cone plate was used because it is a more accurate way of measuringviscosity than parallel plates. This is because the shear rate in independentof radius in the cone and plate method. Xanthan does not require serratedplates because it will shear even on a smooth surface since it has no yieldstress.and fit this data to the power law constitutive equation (Herschel-Bulkley model with a zero yield stress). For the 0.5% concentration used,the rheometry data was fit to the constitutive equation by a least squarestest. The constitutive model was found to be:? = 1.39??0.392.7.3 Rheometry Tests Done on PEOThe rheology of PEO was quantified in a similar fashion to Xanthan. Themain difference was that, since PEO is very viscoelastic, the sample was heldat a constant shear stress until an equilibrium shear rate was reached. Thereading was then taken by the rheometer. The equilibrium shear rate waschosen such that the value of shear rate did not change more than 0.05%in the second before the reading was taken. The readings taken thereforecorrespond to the rheological behaviour of the fluid in a fully developedsteady flow in which the fluid is in a relaxed, equilibrium state. Shown arethree plots of data obtained from 0.5% PEO. Figure 2.16 shows shear stressand plotted against the shear rate with the power law fit for 0.4 and 0.5%concentrations. The constitutive equations were found to be:? = 1.5?0.49and? = 1.37?0.50for 0.5 and 0.4% respectively using the Matlab non linear fit tool.262.7. Rheometry0 50 100 150 200 2500510152025Shear Rate (1s )ShearStress(Pa)  0.5%0.5% fit0.4%0.4% fitFigure 2.16: 0.4 and 0.5% PEO stress vs strain curve with the power law fitshown.To quantify the visco-elastic behaviour of PEO, an oscillatory rheometrytest was done to obtain plots for the viscous and elastic components of thefluid as a function of frequency. The visco-elastic behaviour of the PEO canbe seen in Figure 2.17 c. The viscous modulus (G??), dominates the behaviourof the fluid at low frequencies and the phase angle, ?, of the complex modulus(G = G? + iG??) is close to 90 degrees (ie. the maximum stress occurs atthe maximum strain rate as would be expected in a Newtonian fluid). Inthis region, the elastic stresses relax more quickly than the strain is applied.The phase angle of the complex modulus, G, decreases at higher frequenciesand G?, the elastic modulus, becomes greater than G?? at slightly less than 1Hz. In this region, the elastic stresses caused by the visco-elastic property ofthe PEO release slower than the strain is applied and the phase angle dropsbelow 45 degrees, meaning that the peak stress occurs closer to maximumdisplacement that to the maximum shear rate.2.7.4 Rheometry Tests Done on GlycerolGlycerol was tested with the Bohlin CS-10 rheometer using a 4 degree 40mm cone and plate. Figure 2.18 shows a stress strain curve for the glycerolused in experiments. The Newtonian behaviour of glycerol is confirmed bythe linear stress strain profile. The viscosity was found to be 1.21 Pa.s at25 degrees (temperature of the lab at UBC during glycerol experiments).272.7. Rheometry0 50 100 150 200 2500510152025?? (1s )?(Pa)  0.5% PEO0.4% PEO0.5% Xanthan0 50 100 150 200 250020406080100?? (1s )?(Pa)0 1 2 3 4 50123Frequency (1s )G? (x),G??(o)(Pa)10?2 10?1 100 101101.3101.6101.9log10(Frequency )(1s )log 10(?)?(a) (b)(c) (d)Figure 2.17: A summary of 2 PEO concentrations, carbopol and visco-elasticproperties of PEO(a) Shear stress vs shear rate for 0.5% concentration ofxanthan and 0.5 and 0.4% concetrations of PEO. (b) Shear stress vs shearrate for 0.5% concentration of carbopol.(c) Loss Modulus G?? and elasticmodulus G? for 0.5% PEO. (d) Complex modulus phase angle vs frequency(angle between G? and G??).282.7. Rheometry0 50 100 150 200 250 300 350 400 450050100150200250300350400Shear Rate (1s )ShearStress(Pa)Figure 2.18: Glycerol stress vs shear rate curve with the power law fit shown.292.8. CalibrationFluid ?y ? n0.5% Carbopol 57.4 45.0 0.280.5% Xanthan 0 1.39 0.390.5% PEO 0 1.5 0.490.4% PEO 0 1.37 0.500.25%PEO 0 0.54 0.77Glycerol 0 1.21 1.0Table 2.2: Rheometry parameters for fluids used in this thesis.2.7.5 Further Description of Rheometry Tests and ValuesUsedTo ensure the rheometry curves are relevant for the experiments, a realisticrange of experimental shear rate and shear stress was estimated. Selectingthis range of shear rates and shear stresses to test was done by using themaximum value of the pressure gradient. This is obtained from the pumpspecifications and flow curves (figures 2.33,2.34,2.35). Using equation (2.7)(see below), the shear stress can be obtained as a function of radius for agiven forward flow pressure gradient. The outer fluid (carbopol) is tested toa minimum shear stress of 150 Pa and the inner fluid is tested to a minimumshear stress of 12 Pa to ensure the experimental range of shear stresses iscovered.2.7.6 Summary of Rheometry ParametersTable 2.2 shows the Herschel-Bulkley model fit parameters of all the fluidsused in this thesis.2.8 Calibration2.8.1 CamerasTwo cameras were used to record still images and video of the apparatusduring experiments. Measurements of the fluid inside the pipe, includingthe core fluid radius and position of development fronts and perturbationswithin the pipe, were extracted from these still images and video. Themeasurements of interest are all inside the main pipe and the measurementreference points used are outside the pipe. To account for this, a series ofhigh resolution images were taken of a measuring tape inserted into the pipe302.8. CalibrationFigure 2.19: Omega milliamp output flow meter. See appendix ?? for spec-ifications.while it was filled with water. The image pixel distance between incrementson the tape were then compared to the image pixel distance between perma-nent lines on the apparatus to produce a scaling factor for converting pixeldistance in images to actual distance. The single image sequences takenfrom videos have a lower resolution than discrete single images (1920 x 1080vs 5000 x 7000, respectively). Because more precise measurements can betaken from higher resolution images, full size discrete images were used forthis purpose whenever possible. However, because of the limited frame ratepossible using full size images, video was used to capture high speed phe-nomenon. The result is lower resolution in spatial measurements when usingvideo because 1 pixel represents a greater proportion of the image field atthe lower pixel density.2.8.2 Flow Rate MeasurementAs mentioned previously, two Omega flow meters were used, one for eachfluid (Figure 2.8.2). The flow meters output a current between 1.5mA and25mA depending on the rotation speed of the internal oval gears, which aredriven by fluid flowing through the meter. High and low set points can beentered by moving fluid through the meter at a steady rate and pushing highand low set point buttons. The flow meters are advertised as maintaining312.8. CalibrationFigure 2.20: Flow Rate vs RPM test data for inner fluid (pump 1).Figure 2.21: Flow Rate vs RPM test data for outer fluid (pump 2).322.8. CalibrationFigure 2.22: Flow meter output (mA)vs RPM test data for inner fluid (pump1).Figure 2.23: Flow meter output (mA)vs RPM test data for outer fluid (pump2).332.9. Data Analysisgood linearity over the range 4-20mA. Because a linear relationship betweenthe mA current output and the flow rate will make analysis of experimentseasier, the low set point was chosen to be 4mA for zero flow rate in both flowmeters. The upper limit of the flow rates were based on the theoretical flowcurves of the system and were chosen such that the system would be stablepast the upper set point. To measure flow rate, buckets of the experimentalfluids (carbopol and xanthan or PEO) were placed on a digital scale with0.1 gram resolution. The pumps were turned on and their output adjustedto a constant RPM using the motor controllers to produce constant flowrates somewhat in excess of the maximum used in the experiments. Digitalscale readings were then taken 120 seconds apart and the flow meters setto an upper set point of 20 mA at the halfway point (60 seconds) in thisinterval. The actual flow rates (ml/s) corresponding to a 20 mA flow meterreading were calculated by dividing the change in the volume of liquid (ml)over the 120 second interval by 120. This gave high set point (20 mA) flowrates of 41.2 and 45.3 ml/sec for xanthan and carbopol, respectively. Toverify that the flow meter output increased linearly with flow rate, a flowrate vs. motor RPM plot was constructed for each pump and fluid usingthe mass change over 120 s technique described above. Flow rate was foundto be a linear function of RPM (figures 2.20,2.21), as indicated the pumpmanufacturer. Next the flow meter output was monitored while the pumpswere run at constant RPM . This was repeated for a range of RPM valuesfor each pump (figures 2.22, 2.23). The results indicated that the flow ratefrom the flow rate vs. RPM plots corresponded to the flow rate obtainedfrom the calibrated flow meters because the flow meter output was linearwith motor RPM as can be seen in the previous 4 figures. The flow metercalibrations were verified whenever there was an extended period betweenexperiments, using the flow rate vs. pump RPM curves. These curves werealso used to preform re-calibrations of the flow meters that were requiredperiodically due to power outages and wear in the oval flow meter gears.2.9 Data Analysis2.9.1 Qualitative Stability ObservationsPrevious experiments using a similar apparatus [10] provided general condi-tions under which VPL flows may be established with constant flow rates. Asmall number of initial experiments were carried out to identify the bound-ary conditions within which VPL flows could be reliably produced in theapparatus. This allowed the subsequent variable flow rate experiments to342.9. Data Analysisbe performed well within this stable regime. The classification of exper-imental outcomes as stable or unstable was qualitative and based on thefollowing criterion:1. In steady flow experiments, the flow remained stable while being mon-itored for at least the transit time of the plug. The interface showsno signs of mixing and instabilities introduced in the development re-gion shrink or remain the same size during the entire transit time (nomixing).2. Unstable steady flow experiments showed signs of mixing within thetransit time.3. In step change experiments, stable flows showed no signs of mixingduring the initial establishment of the VPL flow, the introduction ofa step change, and one subsequent transit time.4. Unstable step changes showed signs of mixing after the introductionof a step change. Flows that became unstable before the introductionof the step change were classified as unstable steady flow experiments.5. Stable Pulse experiments showed no signs of mixing before, during orone transit time after the introduction of a pulse.6. Unstable Pulse experiments showed signs of mixing after the introduc-tion of a pulse. Figure 2.24 shows one such experiment that met thiscriteria. Note the growth of the step change and the mixing.In all variable flow rate experiments, a stable flow was first establishedand confirmed by visual inspection (see above criterion). Variable flow rateexperiments were initiated once the stability of the flow was confirmed andthe initial region of stability had fully transited and cleared the experimentalapparatus.2.9.2 Yield Front Speed AnalysisA ?yield front? is a visible deformation of the interface produced during stepchange experiments and is the results of a transition between two base flows.The ?yield front? advects upwards following a step change in the flow rate ofthe inner fluid. During step change experiments, the VPL flow was first es-tablished. The cameras were turned on approximately 10 seconds before thestep change was introduced to provide a background reading of the ambient352.9. Data AnalysisFigure 2.24: Sequence of a VPL flow being perturbed and rapidly progressingto an unstable flow (1 second intervals between images in sequence) Q1 = 21mls , Q1?, Q2 = 9 mls ,T = 5 s.362.9. Data AnalysisFigure 2.25: Example of a spatiotemporal plot used to estimate the yieldfront velocity. There is 100 cm of test pipe is shown. Q1 = 31 mls , Q1? = 15mls , Q2 = 17 mls . The slope of the line is used to estimate the velocity of theyield front. Here the velocity is about 0.55 m/s.372.9. Data AnalysisFigure 2.26: Plot used to more accurately determine the yield front velocity.Q1 = 31 mls , Q1? = 15 mls , Q2 = 17 mls .382.9. Data Analysislight in the room used for error estimates (see below). The video sequencestaken during high/low step change experiments capture the yield front asit advects upwards and are used to estimate its velocity. It is important toget an accurate estimate of the yield front speed under different conditionsto see how it is related to the flow rates inside the pipe and determine if theintroduction of the step change breaks the unyielded interface (plug inter-face). For each step change experiment, a spatiotemporal plot was produced(described below). A spatiotemporal plot can be an effective tool for exam-ining yield fronts because changes in the radius of the inner fluid (containingblack dye) alter the average pixel intensity in fixed-width areas overlappingthe fluid-fluid interface. The subsequent filtering and re-scaling of the pixeldata used to prepare the spatiotemporal plot can allow for a more preciseidentification of the yield front whose velocity can be approximated fromthe slope of the sharp contrast line between dark and light areas of the plot.These slopes were estimated for all step change experiments by fitting astraight line to the spatiotemporal contrast lines (Figure 2.25).To get a better estimate of yield front velocity, a method here termed?intensity propagation analysis? (IPA) was used, and is described below. Tostart with, consider a single image of a VLP flow. The image is made of anarray containing columns and rows of pixels, and each pixel is a 3x1 matrixof RGB values. For IPA, the columns of pixels that include the entire coreflow and flow outside, but near the fluid-fluid interface, are located in thepixel matrices of this single image. These columns span the entire imagefrom top to bottom (the length of the test pipe). The result is an n x m(rows x columns) array of RGB pixel values. The m columns were dividedinto 10 equal groups of n/10 rows. The pixels of the top 10 rows of eachgroup (a subgroup) were then converted to grey scale using the rgb2greycommand in Matlab. The resulting subgroups were then averaged usingthe ?avg? command in Matlab to produce a single greyscale value for thatsubgroup. The result of this was a vector of 10 greyscale intensity valuescorresponding to the 10 subgroups. This was repeated for every photo in asequence to obtain an 10xN matrix, where N is the number of images in asequence.At this point in the IPA analysis, as the yield front propagates, thechange between high (light) and low (dark) intensity values occurs grad-ually rather than abruptly. This is analogous to the blurred division be-tween light and dark in the spatiotemporal plots of step changes (Figure2.25). To better estimate yield front speed the intensity values in the firstIPA modified image before the step change were subtracted from each ofthe corresponding values in each of the subsequent IPA image (subtractive392.9. Data Analysisnormalization). A single intensity cut-off value was then chosen that wasroughly two times the average standard deviation of the intensity of lightin the 10 seconds before the step change was introduced. To obtain thisvalue, the ?stddev? function in matlab was applied to each of the 10 valuesin each of the 300 IPA modified images from the last 10 seconds before thestep change was introduced ( 30 frames/second * 10 seconds=300 imagesx10intensities/image=3000 iintesities). The resulting value was multiplied by 2to get an appropriate cutoff value that would avoid confounding noise witha step change. The cutoff value of 2xsd was determined by inspection. A?while? loop in Matlab was applied to the 10xN matrix from step changeimages to find the point in the image sequence where the relative intensitydropped below the threshold at each of the 10 locations. The resulting datais presented in Figure 2.26. Each curve in the figure corresponds to the av-erage intensity falling over time within a single subgroup, with the numberof points in each curve being equal to the number of photos and the numberof curves being 10.The residual errors in measuring the front velocity in this way are causedmainly by small fluctuations in flow rate and flickering of the fluorescentillumination as well as the limited resolution of the cameras. These naturalfluctuation can be seen in Figure 2.26 as the relative intensity fluctuatesslightly before dropping off as the yield front passes. If the light flickeringis assumed to be stochastic, then there is a 4.4% chance that a change inrelative intensity of twice the standard deviation is a random occurrence.If the error is 1 image frame, the error is large. At 30 measurements persecond, the error becomes 130 seconds for each 10 cm section. The highestmeasured velocities in experimental results are approximately 0.7 m/s. Thisgives and error of (1/30)/(0.1/0.7) = 20%. If the error is taken to be 4.4%of this (the probability of being out by a single frame), the error is reducedto 1%.The reported speed of the yield front in the results section is the averagetransit time between the 10 sequential segments across the cutoff line. Thespatiotemporal plots are used to check verify the values reported from theabove described algorithm to prevent errors caused by, for example, theturning on of a light, opening of a door, or movement in the lab.2.9.3 Image Sequences and Spatiotemporal PlotsMuch of the data in this thesis is extracted from single image sequences andconsecutive video frames. To compress the data presentation and becausesome phenomena are often not easily identified in conventional imagery,402.9. Data AnalysisFigure 2.27: Example of a photo sequence. A high to low step change inthe core flow is introduced at time t=0 and can be seen to advect upwards.To report results of an experiment, in the results section, axis are put on aphoto sequence to show when during the experiment the image was capturedand the dimensions of the photos.image sequences and spatio-temporal plots are often used to present andanalyze these sequences. An image sequence is simply a sequence of imagesarrange to show the progression of an experiment. A code was written inmatlab to extract the area of interest from the raw images and to adjustthe colour to make the core fluid more visible. A spatio-temporal plot is acontour plot of image intensity in the time (X) and space (Y) plane. Thespatio-temporal plots presented were constructed using data extracted from10 to 400 sequential photos or video frames. To construct spatio-temporalplots from sequential images, the ?area of interest? is first identified. Thearea of interest generally includes the full vertical dimensions of a frame (allpixel rows) but restricts the horizontal dimensions to the experimental pipe(eliminates pixel columns outside the pipe). The area of interest is identifiedin the first image in the sequence and the pixel values from this area are readinto an array which then contains the red/green/blue (RGB) values for eachpixel in the area of interest. This process is repeated for all images in thesequence using the same area of interest (identical pixel rows and columns).This RGB information for each pixel is initially stored as 3 values, eachbetween 1 and 256, which represent the relative intensity of red, green, orblue in the pixel. The three RGB values for each pixel are then converted into412.9. Data AnalysisFigure 2.28: Spatiotemporal plot of a image sequence. Spatiotemporal plotsare used to illustrate the concentration of dyed (core) vs clear fluid (annu-lar fluid). In the results section, axis are clarified to convey the verticaldimensions.422.10. Experimental Proceduresa single greyscale intensity value using the rgb2bw function in MATLAB.In all the experiments presented here, the inner fluid contains a black inkdye and the outer fluid is transparent, so that this greyscale conversionprovides excellent resolution of the flows while at the same time facilitatingadditional analyses. To produce the spatiotemporal plot, the greyscale datais smoothed by taking a running average of the values for adjacent pixelsusing smoothing function in matlab called ?medfilt2?, to reduce the effect ofsmall random intensity variations. Sequential images of the area of interestare then plotted side by side along the x (time) axis to produce what isessentially a filtered time lapse record of the area of interest. To account fordifferences in light intensity and interfacial radius between experiments, thegrey scale is normalized to make the darkest parts of the image completelyblack (grey scale = 0) and the lightest parts of the image completely bright(greyscale=1) and with 50 intensity bins in between keep the initial imageresolution. This is done after the smoothing process described above. Thisalgorithm allows even small changes in the interfacial radius to be pickedup by adjusting the resolution and the threshold values. ?Thresholding? thevalues essentially normalizes them by using the same darkest and lightestvalues for all the images in a sequence. Figure 2.27 and Figure 3.6 showan image sequence and a spatiotemporal plot, respectively. In the resultssection of the thesis, the vertical dimensions of the spatiotemporal plots andimage sequences are given in the captions. Not all spatiotemporal plots andimage sequences are the same scale due to noise and distortion near thetop of the test pipe that caused irregularities and blurring in images whichmade it difficult to produce clear figures. This noise and distortion had to beeliminated by cropping the images on an experiment by experiment basis.2.10 Experimental ProceduresA typical experimental run is begun by turning on the two cameras whichwere set to record either 1080X1920 video at 30 frames/s or time lapse photosat 1 frame/s. The Labview program, used to record flow data, was thenactivated and the flow loops were powered up to begin the flow of carbopoland xanthan to the experimental pipe. Experimental flow manipulationswere not initiated until the two flows reached a stable core annular flow.The same flow loop was used to adjust and vary the flows in both fluids (seeFigure 2.29).432.10. Experimental ProceduresFigure 2.29: Flow loop used to vary and adjust flows in both fluids. See textfor details.442.10. Experimental ProceduresStable Flow ExperimentsConstant flow rates were established by closing the solenoid valve and ad-justing the VFD to desired flow rate, which was maintained for the durationof the experiment. These lasted, at a minimum, for the time required for apoint on the interface to transit this pipe (transit time).Step Change ExperimentsStep change experiments were conducted by establishing a stable flow, thenintroducing a rapid and permanent change to either the inner flow, theouter flow, or the inner and outer flows simultaneously. The step changewas controlled by the VFDs and the throttling and solenoid valves. Forexample, for a high to low step change, the desired high flow rate was setby first closing the solenoid valve and then adjusting the VFD to give thedesired high flow. The low flow was then set by opening the solenoid valveand adjusting the flow by using the throttling valve on the bypass loop. Thestable flow was then established by closing the solenoid valve prior to thebeginning of an experimental run. The Arduno microcontroller is turnedon before the experiment is started and runs a program that keeps thesolenoid valve closed for 30 seconds, then opens the solenoid valve until theexperimental run is terminated. For low to high step change experiments,this process is reversed.Pulse ExperimentsPulse experiments were conducted by establishing a stable flow then intro-ducing a pulse change (high/low/high) to the inner core flow. To achieve a?near net shape? pattern the interfacial radius must be altered. The reasonthe inner flow was changed is because the interfacial radius is more sensi-tive to the core flow rate then to the annular flow rate. The pulse changewas controlled by the VFDs and the positions of the solenoid and throttlingvalves. For example, for a high/low/high pulse experiment, the high and lowflows were set as for a step change experiment. The initial change from highto low flow also used the procedure described for the step change. However,in the pulse experiments the low flow was only allowed to proceed for a setamount of time between 1 and 6 seconds, after which the solenoid valve wasclosed to reestablish the high flow. The solenoid valve remained closed forsingle pulse experiments, but the pulse procedure was repeated for multiplepulse experiments.452.11. Differential Analysis of Visco-Plastic Lubrication2.11 Differential Analysis of Visco-PlasticLubricationIn order to initially identify VPL flow regimes, a simplified one dimensionalmodel of flow is used [9]. This section will derive the equations used inthis model and detail the computations used to predict flow regime andinterfacial radius (ri) based on a inner and outer flow rate (Q1, Q2).2.11.1 Simplified Flow ProblemThe starting point for deriving this model will be the continuity equation(2.1) and Navier Stokes equations (equations (2.2),(2.3),(2.4)) in cylindricalcoordinates (origin is top center of pipe). The subscripts 1 and 2 (on ? and?) are for the core and annular fluid, respectively. The pressure, P is themodified pressure (i.e. P = p? ?gz).1r??r (rur) +1r?u??? +?w?z = 0 (2.1)?(?u?t + u?u?r +vr?u?? ?v2r + w?u?z)=[1r??r (r?k,rr) +1r??k,r??? +??k,rz?z ??k,??r]? ?P?r(2.2)?(?v?t + u?v?r +vr?v?? +vur + w?v?z)=[ 1r2??r (r2?k,r?) +1r??k,???? +??k,?z?z]? 1r?P??(2.3)?(?w?t + u?w?r +vr?w?? + w?w?z)=[1r??r (r?k,zr) +1r??k,z??? +??k,zz?z]? ?P?z(2.4)In the case of fully developed, iso-density, 1 dimensional flow inside a pipe,there is no convective acceleration and the velocity is assumed to be nonzeroonly in the Z direction and depend on the distance from the center of thepipe only. So (u, v, w) = (0, 0, w(r)). Applying these simplifying assump-tions to the continuity and Navier Stokes equations, the continuity equationbecomes zero and equations (2.2) and (2.3) become:462.12. VPL Flow Regimes?P?r = 0, (2.5)?P?? = 0, (2.6)and equation (2.4) becomes:0 = 1r??r (r?k,zr)??P?z . (2.7)Equation (2.7) can be integrated and solved keeping in mind ?rz = 0 atr = 0:? ( ??r (r?k,zr))dr =? (r?P?z)drr?k,zr =? (r?P?z)dr?k,zr =1r(r22?P?z + C)?k,zr =r2?P?z (2.8)Equation (2.8) shows that the shear stress ?zr at any point in the pipefor either fluid is necessarily linear and is proportional to the distance awayfrom the center of the pipe. The pressure gradient is negative when the flowis upwards. This allows calculations of wall shear stress and interfacial shearstress to be performed easily if the pressure gradient is known and gives anidea of the pressure head required to overcome the yield stress of a fluid in apipe at the wall, however, the parameters that can be easily altered duringan experiment are only Q1, Q2. A model that takes flow rate as input istherefore needed.2.12 VPL Flow RegimesThe primary interest in this thesis is to examine VPL flows. This section willuse the equations described above to analytically solve the velocity profileand flow rates for VPL flow. VPL flow occurs when the plug thicknessis non zero and there are three distinct regions: the inner core fluid, theun-yielded region of the lubricating fluid (the plug), and the yielded region472.12. VPL Flow Regimesof the lubricating fluid. When the annular fluid is entirely yielded, the flowbecomes yielded flow. Finally, if the annular fluid is entirely unyielded,a static layer flow will result. In the following sections, these three flowregimes are shown to be three cases of the parallel flow solution of theNavier-Stokes equations that governs multilayer hydraulics.2.12.1 Integrating to Obtain a Velocity EquationTo derive equations for flow rate, the velocity equations must be integratedover both the inner and the outer fluid layer in the pipe. This requiresseparate solutions for the inner and outer fluids, so that the final velocitysolution is piecewise smooth.Constitutive Equation and Velocity ProfileThe Herschel-Bulkley constitutive model is used to model the stress/strainrate of both the core and lubricating fluid?k = ?k,y + ?k??nk ,?? = |?w?r |.This equation can be substituted into an equation for the velocity gradient.Keeping in mind U? = (0, 0, w(z)), the velocity profile can be written as twointegrals. One for the core fluid (2.9) and one for the lubricating fluid (2.10).w(r) =? Rirdwdr dr = ?? Ri0??(r)dr Ri ? r ? 0 (2.9)w(r) =? rRdwdr dr = ?? rR??(r)dr R ? r ? Ry (2.10)Integrating to obtain the Velocity equation of the Core FluidEquation (2.9) is integrated by substituting the Herschel Bulkley equation(which is expressed in terms of ??). Note the ?y is dropped from the HerschelBulkley equation when solving for core flow because the core fluid used inthis thesis has zero yield stress (Xanthan).w(r) =? Rir( ??1) 1n1 dr Rint ? r ? 0 (2.11)Using the result of equation (2.8), (2.11) can be rewritten as:482.12. VPL Flow Regimesw(r) =? Rir1?1[r2 |?P?z |] 1n1 dr Rint ? r ? 0 (2.12)Pressure gradient, ?1, and n1 are constants, so (2.12) can be written as:w(r) =( 12?1|?P?z |) 1n1? Rirr1n1 drw(r) =( 12?1|?P?z |) 1n1[r1n1+11n1+ 1]Rirw(r) =( 12?1|?P?z |) 1n1[(Ri ? r)1n1+11n1+ 1]Ri ? r ? 0 (2.13)The inner fluid velocity profile has boundaries with the outer fluid and thevelocity must be continuous across the interface. The velocity at r = Ri isnon zero in a core annular flow if the outer fluid has a non zero flow rate. The?plug?, in this case, will be travelling at wi. To make the equation accrossthis equation continuous across the interface, wi is added to equation (2.13)w(r) =( 12?1|?P?z |) 1n1[(Ri ? r)1n1+11n1+ 1]+ wi Ri ? r ? 0 (2.14)Integrating to obtain the Velocity equation of the LubricatingFluidIn an identical method to that of the core fluid (above), the lubricatingfluid?s velocity equation is integrated:w(r) =(12?2|?P?z |) 1n21n2+ 1[(R?Ry)1n2+1 ? (r ?Ry)1n2+1](2.15)Summary of Velocity ProfileFor two Herschel-Bulkley Fluids flowing in a pipe with the inner fluid havinga zero yield stress, the velocity function is piecewise smooth:492.12. VPL Flow RegimesFigure 2.30: Schematic of a VPL flow velocity profile.w(r) =?????????????(12?1|?P?z |) 1n1[(Ri?r)1n1+11n1+1]+ wi Ri ? r ? 0wi Ry ? r ? Ri(12?2| ?P?z |) 1n21n2+1[(R?Ry)1n2+1 ? (r ?Ry)1n2+1]R ? r ? Ry(2.16)It can be seen from inspection that these equations are continuous acrossboth the fluid-fluid interface and the un-yielded-yielded interface. Figure2.31 shows a schematic of the velocity profile.Integrating the Velocity Equation to Obtain a Core Flow RateequationIntegration is over the area of the core fluid. Since the flow is assumed tobe one dimensional, the area integral is a single integral over infinitesimal502.12. VPL Flow Regimesrings of radius r (2.17)Q1 =? Ri02pir[( 12?1|?P?z |) 1n1[(Ri ? r)1n1+11n1+ 1]+ wi]dr (2.17)= 2pi??????(12?1|?P?z |) 1n11n1+ 1???? Ri0(Ri ? r)1n1+1dr + wi? Ri0rdr???Using integration by parts.= 2pi??????(12?1|?P?z |) 1n11n1+ 1?????(r(Ri ? r)1n1+21n1+ 2)Ri0?? Ri0(Ri ? r)1n1+21n1+ 2dr?????+2pi(+wi? Ri0rdr)Q1 = piwiR2i + 2pi(12?1|?P?z |) 1n11n1+ 1??R1n1+3i(1n1+ 2)(1n1+ 3)?? (2.18)Integrating the Velocity Equation to Obtain the LubricatingFluid Flow RateThis has to be done over the plug region and yielded region of the lubricatingfluid separately because of the discontinuity that exists between the yieldedand un-yielded regions. In the yielded region (R ? r ? Ry), the integral isQ2,yielded =? RRy???(12?2|?P?z |) 1n11n1+ 1[(R?Ry)1n1+1 ? (r ?Ry)1n1+1]??? drQ2,yielded = 2pi(12?2|?P?z |) 1n21n2+ 1(R?Ry)1n2+2(R?Ry1n2+ 3+ Ry1n2+ 2)512.12. VPL Flow RegimesIn the un-yielded region, the velocity will be equal to wi everywhereQ2,unyielded =? RyRi2piwidrQ2,unyielded = piwi(R2 ?R2i )Q2 = Q2,unyielded +Q2,yieldedQ2 = piwi(R2 ?R2i ) + 2pi(12?2|?P?z |) 1n21n2+ 1(R?Ry)1n2+2(R?Ry1n2+ 3+ Ry1n2+ 2)(2.19)wherewi = w(Ry) =(12?2|?P?z |) 1n21n2+ 1(R?Ry)1n+1 (2.20)andRy =2?y|?P?z |(2.21)The flow rate equations for Q1, Q2 are kept separate for both fluids sincethe flow rates can be controlled individually in the experiments.2.12.2 Yielded and Static RegimesYielded RegimeIn a yielded VPL flow, the velocity profile will different from in (2.16) be-cause Ry < Ri:w(r) =?????????(12?1|?P?z |) 1n1[(Ri?r)1n1+11n1+1]+ wi Ri ? r ? 0(12?2| ?P?z |) 1n21n2+1[(R?Ri)1n2+1 ? (r ?Ri)1n2+1]R ? r ? Ri(2.22)where wi is the same as in (2.20). Figure 3.8 shows a schematic of theyielded velocity profile and a lack of plug region at the interface. The flowrates of the two fluids are:522.12. VPL Flow RegimesFigure 2.31: Schematic of an yielded flow velocity profile.532.12. VPL Flow RegimesQ1 = piwiR2i + 2pi(12?1|?P?z |) 1n11n1+ 1??R1n1+3i(1n1+ 2)(1n1+ 3)?? (2.23)which is the same as in the VPL regime since the core fluid profile is unaf-fected by the absence of a plug. The flow rate of the outer fluid will be thesolved the same way as for the unstable portion in (2.28):Q2unstable =? RRi???(12?2|?P?z |) 1n11n1+ 1[(R?Ri)1n1+1 ? (r ?Ri)1n1+1]??? drQ2unstable = 2pi(12?2?P?z) 1n21n2+ 1(R?Ry)1n2+2(R?Ry1n2+ 3+ Ry1n2+ 2)(2.24)Static RegimeIn cases of a low pressure gradient, the shear stress at the pipe wall can bebelow the yield stress of the lubricating fluid. This case is easily solved anda schematic of the velocity profile is shown in Figure 2.32:w(r) =???(12?1|?P?z |) 1n1[(Ri?r)1n1+11n1+1]+ wi Ri ? r ? 00 R ? r ? Ri(2.25)Q1 = piwiR2i + 2pi(12?1|?P?z |) 1n11n1+ 1??R1n1+3i(1n1+ 2)(1n1+ 3)?? Ri ? r ? 0(2.26)Q2 = 0542.12. VPL Flow RegimesFigure 2.32: Schematic of a static layer flow velocity profile.552.13. Applying the Analytical Model to Experimental Design2.13 Applying the Analytical Model toExperimental DesignThe motivation for the analytical solution was to develop a method fordesigning experimental matrices that would fall into the stable region. Theflow rate equation derived above are dependant on ?k, nk where k = 1, 2.These values are found for the fluids as discussed in the rheometry sectionof materials and methods. The value R is the inner radius of the pipe (25.4mm). The structure of the flow rate equations is then:Q1 = f1(Ri, |?P?z |)Q2 = f2(Ri, |?P?z |)Because Q1, Q2 are monotonic functions of the same two parameters, themapping (Q1, Q2) ? (Ri, ?P?z ) is one to one, which means that a pair offlow rates will produce a unique pair of values for the pressure gradient andinterfacial radius.562.13. Applying the Analytical Model to Experimental Design0 10 20051015Ri(mm)|?P ?z|(kPa/m)111606060601111606060VPLRegionYieldedStaticFigure 2.33: Flow rate contours plotted on the Ri,|?P?Z | plane for 0.5% xan-than and 0.5% carbopol fit with a Herschel-Bulkley model. The regime ofVPL flow is shown as the unshaded region between the shaded yielded andstatic regions. The red lines are flow rate iso lines (in mls ) for annular fluid(carbopol) and the blue lines are flow rate iso lines of core fluid (xanthan).572.13. Applying the Analytical Model to Experimental Design0 10 20051015Ri(mm)|?P ?z|(kPa/m)111606060601111606060YieldedVPLRegionStaticFigure 2.34: Flow rate contours on Ri, |?P?z |. for 0.5% PEO and 0.5%carbopol fit with a Herschel-Bulkley model. The red lines are flow rateiso lines (in mls ) for annular fluid (carbopol) and the blue lines are flow rateiso lines of core fluid (PEO).582.13. Applying the Analytical Model to Experimental Design0 10 20051015Ri(mm)|?P ?z|(kPa/m)111606060601111606060YieldedVPLRegionStaticFigure 2.35: Flow rate contours on Ri, |?P?z |. for 0.4% PEO and 0.5%carbopol fit with a Herschel-Bulkley model. The red lines are flow rateiso lines (in mls ) for annular fluid (carbopol) and the blue lines are flow rateiso lines of core fluid (PEO).A Matlab code was written to produce an array of values by using the?meshgrid? function on vectors of pressure gradient and interfacial radiusvalues that were assumed to give flow rates that lie within the operationalenvelope of the flow loop based on previous studies with similar flow loops(this required some iterations) [9]. The MATLAB program applied the flowrate equations to this array which produced a new array containing thecorresponding flow rate. The VPL regime flow equation was used by thecode. Contour plots of the resulting array with level lines corresponding to1 and 60mls for both Q1 and Q2 is shown in figures 2.33 and 2.34 and 2.35.The level lines of Q2 are distinguished from Q1 in that they do not enterthe static region (Q2 is zero over this entire region). In the experimentsperformed in this thesis, a thick plug was needed to to prevent instabilities592.13. Applying the Analytical Model to Experimental Designduring rapid changes. The plug thickness at a given pair of flow rates is thedistance between the intersection of the flow rate contours and the boundarybetween stable and non stable. There is a region enclosed by the 4 flowcontours in the graph. It can be seen that if the experimental matrix iscontained within the regions 60 ? Q1, Q2 ? 1 than the plug thickness willbe a minimum of 4 mm.2.13.1 Inversion of the Flow EquationsTo set the limits of the experimental matrix, the flow curves in Figures2.33,2.34, and 2.35 were used. To generate theoretical predictions of theinterfacial radius and pressure gradient based on the controllable inputs tothe system (the flow rates), an explicit method is needed. These predictionsare generated by inverting the flow problem so that:Ri = g1(Q1, Q2)?P?z = g2(Q1, Q2)This was done by using the fsolve function in matlab (which finds zerosof functions) on the following equations to solve for interfacial radius andpressure gradient for a given pair of flow rates:0 = piwiR2i + 2pi(12?1|?P?z |) 1n11n1+ 1??R1n1+3i(1n1+ 2)(1n1+ 3)???Q1 (2.27)0 = piwi(R2?R2i )+2pi(12?2|?P?z |) 1n21n2+ 1(R?Ry)1n2+2(R?Ry1n2+ 3+ Ry1n2+ 2)?Q2(2.28)The results of the inversion are used in the results section to compare theo-retical and actual values of ri in chapter 3.60Chapter 3Results3.1 Calibration and Benchmarking3.1.1 Observing the PlugEstimates of plug thickness are theoretical and are obtained by subtractingthe yield radius from the interfacial radius in Figures 2.33,2.34,2.35 in Chap-ter 2. Experiments were conducted within a range of flow rates that werepredicted to have a plug thickness greater than 5 mm. To confirm that theplug thickness is roughly in this range, a flow visualization technique wasused. In Figure 3.1, a small amount of dye was added to a pure carbopol flowjust upstream of the suction side of the pump. The plug thickness can beapproximated by measuring the section of displacement front that remainshorizontal. This is measured to be 11.0 mm at the point shown by the whitearrow, in Figure 3.1. This approximates the predicted plug thickness.613.1. Calibration and BenchmarkingFigure 3.1: Sequence showing the presence of an unyielded plug. Images aretaken 1 second apart, Q = 20mls . Camera captures 1.2 meters of test pipe.Fluid os 0.5% carbopol. Only the left half of the pipe is shown in each framefor clarity.3.1.2 Establishing VPL FlowsPreliminary experiments showed that the apparatus was capable of sustain-ing VPL flows. Figure 3.2 shows a spatiotemporal plot of a startup of 0.5%solutions of carbopol and xanthan. The colour intensity is initially high(white left side of spatiotemporal plot) when Q1 =0. Dyed xanthan is thenpumped into the apparatus and the colour intensity drops (dark black band)near the inlet because of mixing throughout the area of interest. The mixedarea is advected leaving a VPL flow (grey in spatiotemporal plot) behind.623.1. Calibration and Benchmarking10 20 30 4010.750.500.250Figure 3.2: Spatiotemporal plot of a startup. The flow rates are Q1 =48and Q2 =14 mls . 120 cm of test pipe are shown.3.1.3 Calibration and Comparison with ModelThe interfacial radius, r, was measured in 50 experiments and comparedwith the theoretical value using the rheometry parameters listed above. Asin [10], the experimental results match the theoretical predictions closely.Figure 3.3 shows theoretical predictions plotted with measured results. Asmall error is also introduced in the predicted values since they are basedon a fit to the Herschel-Bulkley model.633.1. Calibration and BenchmarkingFigure 3.4: Preliminary experiment with 0.5% concentrations of PEO and

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