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Particles in a yield-stress fluid : yield limit, sedimentation and hydrodynamic interaction Chaparian, Emad 2018

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Particles in a yield-stress fluid: yield limit,sedimentation and hydrodynamic interactionbyEmad ChaparianBSc Mechanical Engineering, Isfahan University of Technology, 2011MSc Mechanical Engineering, University of Tehran, 2013a thesis submitted in partial fulfillmentof the requirements for the degree ofDoctor of Philosophyinthe faculty of graduate and postdoctoralstudies(Mechanical Engineering)The University of British Columbia(Vancouver)June 2018c© Emad Chaparian, 2018The following individuals certify that they have read, and recommend tothe Faculty of Graduate and Postdoctoral Studies for acceptance, the dis-sertation entitled:Particles in a yield-stress fluid: yield limit, sedimentation andhydrodynamic interactionsubmitted by Emad Chaparian in partial fulfillment of the require-ments forthe degree of Doctor of Philosophyin Mechanical EngineeringExamining Committee:Ian A. Frigaard, Mechanical EngineeringCo-supervisorAnthony Wachs, MathematicsCo-supervisorJames J. Feng, Chemical and Biological EngineeringUniversity ExaminerReza Vaziri, Civil EngineeringUniversity ExaminerAdditional Supervisory Committee Members:Neil J. Balmforth, MathematicsSupervisory Committee MemberMark Martinez, Chemical and Biological EngineeringSupervisory Committee MemberiiAbstractA theoretical and numerical study of yield-stress fluid creeping flow about aparticle is presented motivated by theoretical aspects and industrial appli-cations. Yield stress fluids can hold rigid particles statically buoyant if theyield stress is large enough. In addressing sedimentation of rigid particlesin viscoplastic fluids, we should know this critical ‘yield number’ beyondwhich there is no motion. As we get close to this limit, the role of viscositybecomes negligible in comparison to the plastic contribution in the leadingorder, since we are approaching the zero-shear-rate limit. Admissible stressfields in this limit can be found by using the characteristics of the govern-ing equations of perfect plasticity (i.e., the sliplines). This approach yields alower bound of the critical plastic drag force or equivalently the critical yieldnumber. Admissible velocity fields also can be postulated to calculate theupper bound. This analysis methodology is examined for different familiesof particle shapes. Numerical experiments of either resistance or mobilityproblems in a viscoplastic fluid validate the predictions of slipline theoryand reveal interesting aspects of the flow in the yield limit. For instance,the critical limit is not unique and here we show that for the same criti-cal limit we may have different shaped particles that are cloaked inside thesame unyielded envelope. The critical limit (or critical plastic drag coeffi-cient) is related to the unyielded envelope rather than the particle shape.We show how to calculate the unyielded envelope directly. Here we alsoaddress the case of having multiple particles, which introduces interestingnew phenomena. Firstly, plug regions can appear between the particles andconnect them together, depending on the proximity and yield number. Thisiiican change the yielding behaviour since the combination forms a larger (andheavier) “particle”. Moreover, small particles (that cannot move alone) canbe pulled/pushed by larger particles or assembly of particles. Increasing thenumber of particles leads to interesting chain dynamics, including breakingand reforming.ivLay SummaryYield-stress fluids are common as we encounter them on a daily basis:toothpaste, shaving gels, whipped cream and cement paste. One can eas-ily appreciate that these type of materials are neither solids nor ‘simple’fluids like water—they will flow just when we put enough force on them:brushing toothpaste or beating whipped cream. Because of this exceptionalbehaviour—yield stress—they can hold solid objects statically within them,e.g., concrete is coarse aggregates in cement paste, chocolate chips in cookiedough, etc. This happens when the particle is not too buoyant and thisproperty is of great industrial interest.In this study we will, in detail, address the stability of particles in yield-stress fluids. Moreover, sedimentation/motion of particles and also the hy-drodynamic interaction (i.e., how the motion of adjacent particles may affectstability/motion of others) are investigated.vPrefaceThe contents of this thesis are the results of the research of the author,Emad Chaparian, during the course of his PhD studies at UBC, under thesupervision of Prof. Ian A. Frigaard and co-supervision of Prof. AnthonyWachs. The following papers have been published and/or are in progress:• E. Chaparian, I.A. Frigaard, Yield limit analysis of particle motionin a yield-stress fluid, Journal of Fluid Mechanics 819 (2017) 311–351.(DOI: 10.1017/jfm.2017.151)This paper was co-authored with I.A. Frigaard and I did the imple-mentation of the code, running, data compilation and contributed toanalysis of the results. I.A. Frigaard supervised the research and con-tributed to manuscript edits.• E. Chaparian, I.A. Frigaard, Cloaking: Particles in a yield-stressfluid, Journal of non-Newtonian Fluid Mechanics 243 (2017) 47–55.(DOI: 10.1016/j.jnnfm.2017.03.004)This paper was co-authored with I.A. Frigaard and I did the imple-mentation of the code, running, data compilation and contributed toanalysis of the results. I.A. Frigaard supervised the research and con-tributed to manuscript edits.• E. Chaparian, A. Wachs, I.A. Frigaard, Inline motion and hydrody-namic interaction of 2D particles in a viscoplastic fluid, accepted forpublication in Physics of Fluids and to appear, 2018.viThis paper was co-authored with I.A. Frigaard and A. Wachs. I didthe implementation of the code, running, data compilation and con-tributed to analysis of the results. I.A. Frigaard and A. Wachs super-vised the research and contributed to manuscript edits.• E. Chaparian, B. Nasouri, L-box test: a theoretical study, underreview.This paper was the outcome of MATH 519 Fluid Mechanics courseproject at UBC. I did the computations of slipline analysis and thepaper was authored primarily by myself. B. Nasouri provided theasymptotic analysis and contributed to manuscript edits.viiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiNomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . xxiii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Industrial applications/motivations . . . . . . . . . . . . . . . 21.1.1 Fractionation of suspensions . . . . . . . . . . . . . . . 21.1.2 Concrete industry . . . . . . . . . . . . . . . . . . . . 51.1.3 Drilling oil wells . . . . . . . . . . . . . . . . . . . . . 71.1.4 Other applications: suspensions in food, drug, andcosmetic industries . . . . . . . . . . . . . . . . . . . . 91.2 Yield-stress fluids . . . . . . . . . . . . . . . . . . . . . . . . . 101.3 Theoretical and computational tools . . . . . . . . . . . . . . 151.3.1 Regularization methods . . . . . . . . . . . . . . . . . 15viii1.3.2 Yield surface tracking techniques . . . . . . . . . . . . 171.3.3 Variational principles and numerical methods foundedon that . . . . . . . . . . . . . . . . . . . . . . . . . . 181.4 Resistance and Mobility problems . . . . . . . . . . . . . . . . 231.4.1 Mobility formulation . . . . . . . . . . . . . . . . . . . 251.4.2 Resistance formulation . . . . . . . . . . . . . . . . . . 281.5 Rigid perfect-plasticity . . . . . . . . . . . . . . . . . . . . . . 291.5.1 Variational principles and uniqness of solution . . . . 321.5.2 Lower and upper bound theorems . . . . . . . . . . . 331.5.3 Plane strain problem (2D flow case) . . . . . . . . . . 351.6 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . 411.6.1 Particle motion in yield-stress fluids–early work . . . . 421.6.2 Beris et al. [15]–a robust framework . . . . . . . . . . 441.6.3 Drag and flow around single particles . . . . . . . . . 451.6.4 Finding Yc for particles in a yield-stress fluid . . . . . 461.6.5 Yc in other yield-stress fluid flows . . . . . . . . . . . . 481.6.6 Multiple-particle flows in yield-stress fluids . . . . . . 491.6.7 Open problems, issues and gaps . . . . . . . . . . . . . 541.7 Objectives and outline of the thesis . . . . . . . . . . . . . . . 571.7.1 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . 582 Stability of particles and the yield limit . . . . . . . . . . . 602.1 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . 612.1.1 Yielding of flows . . . . . . . . . . . . . . . . . . . . . 642.1.2 Mapping between problems [M] & [R] . . . . . . . . . 652.1.3 Computational method . . . . . . . . . . . . . . . . . 672.1.4 The yield limit and perfect plasticity . . . . . . . . . . 702.1.5 A benchmark problem: a 2D disk particle . . . . . . . 712.2 Ellipse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 782.3 Rectangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 862.4 Diamond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 922.5 General features of limiting viscoplastic solutions . . . . . . . 942.5.1 The limit χ→ 0 . . . . . . . . . . . . . . . . . . . . . 95ix2.5.2 A general framework . . . . . . . . . . . . . . . . . . . 992.5.3 Long particles . . . . . . . . . . . . . . . . . . . . . . . 1002.6 Summary and discussion . . . . . . . . . . . . . . . . . . . . . 1073 Cloaking and unyielded envelope rule . . . . . . . . . . . . . 1133.1 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . 1143.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1163.2.1 Cloaking and its consequences . . . . . . . . . . . . . 1163.2.2 Finding the unyielded envelope: two lines of symmetry 1183.2.3 Particles with only left-right symmetry . . . . . . . . . 1233.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1244 Effect of particle orientation . . . . . . . . . . . . . . . . . . 1284.1 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . 1294.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1304.2.1 Yield limit . . . . . . . . . . . . . . . . . . . . . . . . 1304.2.2 Settling velocity . . . . . . . . . . . . . . . . . . . . . 1354.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1385 Inline motion of particles and hydrodynamic interaction . 1405.1 Inline particle motion . . . . . . . . . . . . . . . . . . . . . . 1425.1.1 Example results for two particles (χ2 < 1) . . . . . . . 1435.2 Results for uniform disks . . . . . . . . . . . . . . . . . . . . . 1495.2.1 N = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 1495.2.2 N = 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 1505.2.3 N > 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 1555.3 Dynamics of motion . . . . . . . . . . . . . . . . . . . . . . . 1565.3.1 N = 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 1575.3.2 Chain dynamics for N > 3 . . . . . . . . . . . . . . . . 1595.4 Locality of the stress . . . . . . . . . . . . . . . . . . . . . . . 1615.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1656 Summary and conclusions . . . . . . . . . . . . . . . . . . . . 1686.1 Results and contributions from the individual chapters . . . . 169x6.1.1 Stability of particles and yield limit (Chapter 2) . . . 1696.1.2 Cloaking and unyielded envelope rule (Chapter 3) . . 1726.1.3 Effect of particle orientation (Chapter 4) . . . . . . . . 1736.1.4 Inline motion of particles and hydrodynamic interac-tion (Chapter 5) . . . . . . . . . . . . . . . . . . . . . 1736.2 Thesis limitations and future directions . . . . . . . . . . . . 1756.2.1 Rheological idealizations . . . . . . . . . . . . . . . . . 1766.2.2 Computational challenges . . . . . . . . . . . . . . . . 1776.2.3 Applicability to industrial flows . . . . . . . . . . . . . 177Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179A Particle geometries considered in Chapter 3 . . . . . . . . . 192B Details of upper-bound calculation in Chapter 3 . . . . . . 193C Details of slipline field calculations in Chapter 3 . . . . . . 197D Details of lower-bound calculations in Chapter 5 . . . . . . 200xiList of TablesTable 2.1 Dimensional and dimensionless parameters for the 3 ge-ometries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78Table 2.2 Critical plastic drag coefficients for a square particle . . . . 92Table 3.1 Computed critical plastic drag coefficients compared tolower and upper bounds, for each of Figure 3.6. . . . . . . 122xiiList of FiguresFigure 1.1 A demonstration of the fractionation of a bidisperse sus-pension of spherical particles. In (a) an image of the sus-pension is given before the commencement of the cen-trifuge. (b) is the state of the suspension after the appli-cation of the centrifugal force. It should be noted thatmost of the darker particles are on the periphery of thecentrifuge. Reproduced from [86]. . . . . . . . . . . . . . . 5Figure 1.2 Slump test for measuring workability of concrete . . . . . 6Figure 1.3 Chocolate chips in cookies as an example of particles inyield-stress fluids in food industry. . . . . . . . . . . . . . 10Figure 1.4 Some examples of yield-stress fluids. . . . . . . . . . . . . 11Figure 1.5 Evidence of elasticity in carbopol gel. Left panel is theflow curve of carbopol and right panel shows PIV velocityfield and stream lines for a moving sphere in Carbopol.Sphere is moving from right to left. Reproduced from [113]. 14Figure 1.6 Avalanche flow of a clay suspension over an inclined plane.The pictures are taken at the critical angle for which thesuspension just starts to flow visibly. Reproduced from [30]. 14Figure 1.7 The regularisations models compared against the exactBingham model at  = 0.01, B = 1: (a) stress vs. strainrate; (b) effective viscosity vs. strain rate. Reproducedfrom [47]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 16xiiiFigure 1.8 Steady-state rheology. Solid straight line: Bingham liquidwith yield stress τy and slope ηB. Dashed line: in a bi-viscosity model the yield stress is mimicked by a very highviscosity (ηB0) liquid for low shear. Reproduced from [36]. 17Figure 1.9 Circles show the analytic yield surface and other linesare yield surfaces from Papanastasiou regularization withincreasing value of m. The main difference is that theconvexity of yield surfaces in analytical solution is notcaptured in regularized model. Reproduced from [22]. . . 18Figure 1.10 Example of an adaptation cycle. Left column: four meshesof a neighbourhood of the cylinder (cross section in white);Right column: corresponding to each mesh rigid zones(light grey), deformed zone (dark grey), stream lines. Re-produced from [115]. . . . . . . . . . . . . . . . . . . . . . 24Figure 1.11 A curvilinear element bounded by sliplines. . . . . . . . . 36Figure 1.12 Segments of slip lines of two family . . . . . . . . . . . . . 40Figure 1.13 Slipline field geometries for establishing Hencky’s theo-rems. Reproduced form [23]. . . . . . . . . . . . . . . . . 40Figure 1.14 Flow patterns about a falling sphere: (a) proposed byAnsley and Smith [4], (b) proposed by Yoshioka et al.[144], (c) computed by Beris et al. [15]. Reproduced from[28]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44Figure 1.15 Flow past different objects. Reproduced from [21]. . . . . 44Figure 1.16 Change in the computed plastic drag coefficient as a func-tion of Od for a 2D circle. Reproduced from [134]. . . . . 47Figure 1.17 Measured Yc for different particle shapes by Jossic andMagnin [72]. Reproduced from [72]. . . . . . . . . . . . . 48Figure 2.1 Schematic of the particle motion. . . . . . . . . . . . . . . 62Figure 2.2 Finite element approximation. Reproduced from [116]. . . 67Figure 2.3 Settling velocity of a square particle for increasing Y . . . 68Figure 2.4 L2 and L∞ norm of the error versus cycles of adaptationfor Poiseuille flow . . . . . . . . . . . . . . . . . . . . . . 69xivFigure 2.5 (a) j(u∗) for square case (dissipation in computationaldomain, i.e., one quarter of the physical domain) ver-sus number of adaptation cycles. Symbols are computedquantities and red broken line is the limiting value (b)Part of the mesh generated after 8 cycles of adaptationfor square particle. . . . . . . . . . . . . . . . . . . . . . . 69Figure 2.6 Characteristic network adjacent to a circle . . . . . . . . . 73Figure 2.7 Normalized shear stresses: left half from computation ofthe [R] problem with B = 104, i.e. τ∗xy/B; right half fromthe characteristics prediction τ˜xy/B. Rigid regions areplotted gray. . . . . . . . . . . . . . . . . . . . . . . . . . 77Figure 2.8 Computed velocity in the y-direction ([R] problem withB = 104), measured with distance r from the cylindersurface along the x-axis. . . . . . . . . . . . . . . . . . . . 77Figure 2.9 Dimensional geometries considered. . . . . . . . . . . . . . 78Figure 2.10 Characteristic network around an ellipse: (a) χ = 10; (b)zoom of frontal plug region for χ = 10; (c) χ = 0.2. . . . . 79Figure 2.11 Geometrical parameters for lower and upper bound cal-culations . . . . . . . . . . . . . . . . . . . . . . . . . . . 81Figure 2.12 Schematic of mechanism II. . . . . . . . . . . . . . . . . . 84Figure 2.13 Dependency of plastic dissipation on the Bingham num-ber for ellipse: (a) χ = 10; (b) χ = 1; (c) χ = 0.2. TheBlue symbols are data from the [R] problem. Red discon-tinuous lines are the characteristics predictions and blackones are the asymptotic values of the blue symbols forlarge Bingham numbers. . . . . . . . . . . . . . . . . . . . 86Figure 2.14 Critical plastic drag coefficient of 2D elliptical particles . 87Figure 2.15 Velocity magnitude colour map, computed from problem[R] at B = 104, (the white lines are the yield surfaces):(a) χ = 10, (b) χ = 1, (c) χ = 0.2. . . . . . . . . . . . . . 87Figure 2.16 (a) schematic of characteristic network adjacent to a rect-angle, (b) geometrical parameters for calculation of upperbound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88xvFigure 2.17 Critical plastic drag coefficients for a rectangular particle. 91Figure 2.18 Velocity magnitude colour map, computed from problem[R] at B = 104, (the white lines are the yield surfaces):(a) χ = 10; (b) χ = 1; (c) χ = 0.2. . . . . . . . . . . . . . 91Figure 2.19 Characteristic network around the diamond: (a) χ = 10;(b) χ = 1; (c) χ = 0.2. . . . . . . . . . . . . . . . . . . . . 93Figure 2.20 Critical plastic drag coefficient of diamond . . . . . . . . . 94Figure 2.21 Velocity magnitude contours around the diamond: (a)χ = 10; (b) χ = 1; (c) χ = 0.2. . . . . . . . . . . . . . . . 94Figure 2.22 Critical plastic drag coefficient for χ→ 0, computed withB = 104. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97Figure 2.23 Geometric behaviour at large χ, computed for the rect-angular particle at B = 104 using problem [R]: (a) ωp;(b) `p: (symbols from computation and solid lines fitted,with the indicated slopes). . . . . . . . . . . . . . . . . . . 102Figure 2.24 Contributions to the plastic dissipation at large χ, com-puted for the rectangular particle at B = 104 using prob-lem [R]: (a) ji(u∗); (b) jo(u∗); (c) jt(u∗): (symbols fromcomputation and solid lines fitted, with the indicated slopes).103Figure 2.25 Convergence at large B for the 3 geometries at χ = 100:symbols are computes and lines are power law fits withexponent m (the slope of the fitted lines in log-log). Thisgives ν ≈ 1.61, 1.32, 1.60 for ellipse, rectangle and dia-mond, respectively. . . . . . . . . . . . . . . . . . . . . . . 105Figure 2.26 Velocity field averaged over a plate displacement directlyobtained from PIV measurements for two different posi-tions of the plate with regards to the window of observa-tion: (left) plate tip at 3 cm above the window bottom,(b) plate tip at 10 cm below the window bottom. Repro-duced from [20]. . . . . . . . . . . . . . . . . . . . . . . . 110Figure 2.27 Effective velocity profile (in the frame of the container)along the plate direction during the penetration througha Carbopol gel. Reproduced from [20]. . . . . . . . . . . . 111xviFigure 3.1 Schematic of the problem considered. . . . . . . . . . . . 115Figure 3.2 Speed contours and yield surfaces (white): (a) Ellipse, (b)‘Batman’. . . . . . . . . . . . . . . . . . . . . . . . . . . . 117Figure 3.3 Speed contours and yield surfaces (white): (a) Tiltedsquare, (b) Triangle. . . . . . . . . . . . . . . . . . . . . . 118Figure 3.4 Speed contour: (a) Rounded-end rectangle (a = 1, b =1.5), (b) ‘Dumbbell’ (a = 1, b = 1.5, b′ = 0.5). See Fig-ure A.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119Figure 3.5 Schematic of quarter of the unyielded envelope (particle+ attached plugs): (a) quarter of the bounding surface Γ,illustrating the angles η1 and η2; (b) quarter of the slipline(characteristic) network generated from the unyielded en-velope. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120Figure 3.6 Speed contour for 3 different particles close to yielding. Itshould be noted that only a quarter of the whole domainis presented here. . . . . . . . . . . . . . . . . . . . . . . . 121Figure 3.7 Characteristic network adjacent to a ‘kite’ (see Figure A.1,a = b′ = 1, b = 2): (a) two separate centred fans; (b)calculated network ([CpD,c]L= 11.85). Only half of thedomain is shown. . . . . . . . . . . . . . . . . . . . . . . . 124Figure 3.8 Characteristic network adjacent to a ‘parachute’ (seeFigure A.1d,a = b = 1): (a) two separate networks; (b) calculated net-work ([CpD,c]L= 11.70). Only half of the domain is shown.125Figure 3.9 Elongated ‘Kite’ (see Figure A.1e, a = b′ = 1, b = 5): (a)Slipline network ([CpD,c]L= 14.79); (b) speed contourcomputed from the [R] problem (CpD,c = 14.79). . . . . . . 126Figure 3.10 Elongated ‘parachute’ (see Fig. A.1d, a = 1, b = 4): (a)Slipline network ([CpD,c]L= 13.87); (b) speed contourfrom the [R] problem (CpD,c = 13.97). . . . . . . . . . . . . 127Figure 4.1 Schematic of the problem considered. . . . . . . . . . . . 129xviiFigure 4.2 Velocity contours for χ = 2: (a) φ = 30, Y = 0.122, (b)φ = 60, Y = 0.095, (c) φ = 75, Y = 0.0925. The bluearrow shows the steady direction of motion of the particleand white lines are yield surfaces. . . . . . . . . . . . . . . 131Figure 4.3 Velocity contours for χ = 5: (a) φ = 30, Y = 0.13, (b)φ = 60, Y = 0.075, (c) φ = 75, Y = 0.06. The blue arrowshows the steady direction of motion of the particle andwhite lines are yield surfaces. . . . . . . . . . . . . . . . . 131Figure 4.4 Schematic of the decomposition. . . . . . . . . . . . . . . 132Figure 4.5 Yc versus φ for a rectangular particle of χ = 1. . . . . . . 133Figure 4.6 Yc versus φ for a rectangular particle of χ = 2. The blackcircles are results computed by Wachs and Frigaard [142]. 134Figure 4.7 Yc versus φ for a rectangular particle of χ = 15. . . . . . . 134Figure 4.8 Yc versus φ for a rectangular particle of χ = 50. . . . . . . 135Figure 4.9 Yc versus φ for a 2D ellipse with χ = 2. . . . . . . . . . . 136Figure 4.10 Yc versus φ for a 2D ellipse with χ = 15. . . . . . . . . . . 136Figure 4.11 Velocity of the particle versus yield number (χ = 2, φ = 30).138Figure 4.12 Velocity of the particle versus yield number (χ = 50, φ =15). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138Figure 4.13 Velocity of the particle versus yield number (χ = 2, φ = 60).139Figure 5.1 Schematic of the problem considered. . . . . . . . . . . . 142Figure 5.2 Example resutls for Y = 0.11, `1 = 4, χ2 = 1/2: (a)Speed colourmap and yield surfaces (white); (b) settlingvelocities of the two particles. . . . . . . . . . . . . . . . . 144Figure 5.3 Speed colormap and yield surfaces (white) for an assemblyof two particles. Top panels show the flow before breakingthe plug bridge and bottom panels are associated withslightly increase the inter-particle distance, which resultsin the plug breaking: Y1 = 0.11. (a,e) χ2 =12 , (b,f) χ2 =13 , (c,g) χ2 =14 , (d,h) χ2 =15 ; `1 = 2.55, 1.85, 1.62, and 1.5. 145xviiiFigure 5.4 Critical distances and flow regimes for different χ2 and `1:Y = 0.11. The discontinuous blue line represents wherethe discs make contact. Blue crosses indicate points atwhich the flow has been computed and characterized. . . 147Figure 5.5 Schematic of the unyielded envelope around assembly oftwo particles (the red discontinuous line shows the sym-metry line). . . . . . . . . . . . . . . . . . . . . . . . . . . 148Figure 5.6 Comparison of Yc for assembly of two disks: (a) χ2 =1/2, (b) χ2 = 1/4. Blue line shows the slipline theoryprediction (Equation 5.5) and red symbols are numericalresults. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149Figure 5.7 Speed colourmap and yield surfaces (white lines). `1 = 10:(a) Y = 0.12, (b) Y = 0.14; `1 = 6.5: (c) Y1 = 0.12, (d)Y1 = 0.14; `1 = 3: (e) Y1 = 0.12, (f) Y1 = 0.14. Pleasenote that 0.12 < Y ∗c = 0.1316 < 0.14. . . . . . . . . . . . . 151Figure 5.8 Different flow regimes in the (`1, Y )-plane for two disks ofthe same size (χ2 = 1). The colormap indicates the set-tling speed of the disks. The solid blue line marks the bor-der between two separate moving particles and two staticparticles regimes, while the dotted blue line just showsY ∗c . Broken black line is Yc computed by Tokpavi et al.[135]. Broken green line is the Yc calculated by expression(5.5). Diamond symbols are experimental measurementsby Jossic and Magnin [74]. . . . . . . . . . . . . . . . . . 152Figure 5.9 Triple disks map and examples (Y = 0.15): (a) `1 =10.5, `2 = 2.1 (region (VII)), (b) `1 = 9.75, `2 = 4.9(region (III)), (c) `1 = 6.5, `2 = 6.5 (region (II)), (d)Different regions in `1 − `2 map, (e) Zoom of the panel(d) around `1 ∼ `2 ∼ 2. Computational points are notmarked in this figure in order to avoid cluttering it. . . . 154Figure 5.10 Regime (I) flows: (a) Y = 0.15 `1 = 2.065, `2 = 2.05; (b)Y = 0.25, `1 = 2.01, `2 = 2.07. . . . . . . . . . . . . . . . . 155xixFigure 5.11 Five disks `1 = `2 = 2.1: (a) Y = 0.15, (b) Y = 0.25,(c) Y = 0.35. . . . . . . . . . . . . . . . . . . . . . . . . . 156Figure 5.12 Velocity at the centerline versus y-coordinate which isfixed at the centre of the middle particle. The filled greyboxes show inside of particles and numbers on that areindex of the particles. . . . . . . . . . . . . . . . . . . . . 157Figure 5.13 Phase paths of the system (5.6) and (5.7) with Y1 = 0.15.Phase paths are superimposed upon the colourmap ofdV1(`1, `2) map, Y1 = 0.15. The small panel in the righttop represents T × 10−3 versus `2 where T is the orbit‘time’ for each path. Please note that in all paths theinitial condition is `1 = 12. . . . . . . . . . . . . . . . . . 159Figure 5.14 Different snapshots of three particle dynamics. In all pan-els (a-f) y-axis shows the y coordinate. Asterisks in panel(f) mark the ‘times’ of panels (a-e). Also same symbolsare used in Figure 5.13 to mark the time-lapses of panels(a-e) in (`1, `2) domain. . . . . . . . . . . . . . . . . . . . 160Figure 5.15 Speed contour around a single disk, Y = 0.11. . . . . . . . 162Figure 5.16 Second invariant of stress: (a) Y = 0.12 < Y ∗c , (a) Y =0.15 > Y ∗c . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163Figure 5.17 Speed contours (Y = 0.12): (a) `1 = `2 = 20, (b)`1 = `2 = 14, (c) `1 = `2 = 11. Panel (d) shows velocityon the centerline versus y-coordinate which is fixed atthe center of the middle disk. In this panel the red linecorresponds to (a), blue discontinuous line to (b), and theblack dashed-dotted line to (c). . . . . . . . . . . . . . . . 166Figure A.1 Schematic of particle geometries. . . . . . . . . . . . . . . 192Figure B.1 Deformation mechanism. . . . . . . . . . . . . . . . . . . 194Figure C.1 Schematic of the slipline network: (a) aft of the ‘kite’, (b)fore of the ‘kite’. . . . . . . . . . . . . . . . . . . . . . . . 198xxNomenclatureτˆ Deviatoric stress tensor;τˆY Yield stress;µˆp Plastic viscosity;ˆ˙γ Strain rate tensor;γˆ Relaxed strain rate tensor;Kˆ Consistency of a Herschel-Bulkley fluid;n Flow index of a Herschel-Bulkley fluid;Φ Velocity potential function;ρˆ density;Ψ Stress potential function;σˆ Cauchy stress tensor;pˆ Pressure;n Normal vector;tˆ Traction vector;fˆ Body force vector;Sˆt Part of the boundary with traction boundary condition;Sˆv Part of the boundary with velocity boundary condition;fˆ Body force vector;Tˆ Lagrange multiplier;a Augmentation parameter;uˆ Velocity vector;gˆ Gravity acceleration;Lˆ Length scale;xxiY Yield number;Y ∗c Critical yield number for a single disk;a(., .) Viscous dissipation;j(.) Plastic dissipation;B Bingham number;Uˆp Particle velocity vector;ωˆp Particle rotation vector;γe Elastic strain;γp Plastic strain;Gˆ Elasticity shear modulus;Eˆ Young elasticity modulus;tˆ time;Vp Volume of the particle;Ap Area of the particle;ˆ`⊥ Width of the particle;ˆ`‖ Length of the particle;CpD Plastic drag coefficient;χ Aspect ratio of the particle;φ Orientation of the particle;xxiiAcknowledgmentsThis research has been carried out at the University of British Columbia,supported financially by Natural Sciences and Engineering Research Coun-cil of Canada (NSERC) via their Strategic Projects programme (grant no.STPGP 447180-13) and via discovery grants program (grant no. RGPIN-2015-06398). I really appreciate this support. Also awards and scholarshipscoming from the Department of Mechanical Engineering of the Universityof British Columbia are really appreciated.Firstly, I would like to express my sincere gratitude to my PhD supervi-sor, Prof. Ian Frigaard for all of his supports during this research. Ian, doingthis study would not have been possible without your guidance, support, pa-tience, and constant feedback. Your scientific perspective and knowledge areinspiring examples.Moreover, my sincere thanks goes to my advisory committee members,Prof. Neil Balmforth, Prof. Anthony Wachs, and Prof. Mark Martinez.Their guidance helped me in all the time of research and writing of thisthesis. Neil, we had various discussions during the course of this workwhich were extremely interesting and helpful for me. Anthony, your sup-port, specifically in the last parts of this research as a co-supervisor, was aninvaluable help.Also I am grateful to my previous supervisors and professors at theIsfahan University of Technology and University of Tehran, in particular,Prof. Ebrahim Shirani and Prof. Kayvan Sadeghy for enlightening me thefirst glance of research.I thank my labmates in Complex Fluids Lab., UBC, for the constructionxxiiidiscussions and for all the fun we have had in these years.Last but indeed not the least, I would like to say a heartfelt thank you tomy parents and my brother for supporting me throughout my life in general.Emad ChaparianVancouver, CanadaJune 2018xxivTo my parents, Mehdi,and · · ·xxvChapter 1IntroductionThis thesis concerns particle motion in yield stress fluids. Although the con-tributions are mostly advances in fundamental mechanical understanding,the underlying motivations for the thesis arise from industrial applications,e.g., fractionation of pulp fibers in the paper industry, stability of aggregatesin concrete, static stability of cuttings in the wellbore when the drilling hasbeen stopped. By yield stress we mean a threshold in stress which mustbe exceeded in order to deform: a fluid that behaves in this way is a yieldstress or viscoplastic fluid. Intuitively we can think of small/light particles asbeing suspended quiescently in yield-stress fluids and large/heavy particlesthat can sediment/settle through the fluid. Interpolating between these twobehaviours we see that there should be a limiting parameter set that cap-tures the boundary between flow and no-flow conditions, i.e., a yield limit.It might also be intuitive that the shape of particles and also the proximityof particles to each other can affect this scenario. This thesis provides amechanical analysis of these intuitive situations and surrounding questions.In this chapter we first briefly discuss industrial applications and moti-vations of the study (Section 1.1). Then we introduce rheological models ofviscoplastic fluids (Section 1.2) and the mathematical methods and tools weneed to attack flow problems associated with this type of non-Newtonianfluids (Section 1.3). We continue with an overview of the mathematical for-mulation of the present topic (Section 1.4) and how plasticity can help us in1the present context (Section 1.5). Finally we close the chapter with an out-line of the thesis, the research questions and the objectives of the followingchapters (Section 1.7).1.1 Industrial applications/motivations1.1.1 Fractionation of suspensionsOne of the applications of more direct interest to this thesis is the use ofyield-stress fluids as an aid to the fractionation of suspensions [86]. Thesemethods rely on our being able to use “flow—no flow” to differentiate parti-cles based on a combination of shape, size, and density. The motivation hasits origin in one particular suspension: a pulp fibre suspension as used inpaper making. There are two main reasons behind using fractionation in thepulp and paper industry. Firstly, the low quality fraction can be processedto improve product performance while minimizing the energy, chemical, andsubstance consumption. Secondly, by means of fractionation, papermakerscan produce papers with variety of different properties: short fibres in a pa-per result in specific type of papers which are perfect for printing while longfibered papers exhibit excellent strength in applications with high tensileforces. A spectrum of paper products with wide range of characteristics canbe produced by mixing different portions of short and long fibres as well.Most paper industries use two well-known methods to fractionate: pres-sure screens and hydrocyclones.Pressure screenIn the pressure screen method, an annulus is filled with the pulp fibre suspen-sion. The inner cylinder is a rotor and the outer cylinder (screen) containssmall apertures. The screen apertures could be either narrow slots or holeswith small diameter based on the desired fractionation quality. As the smallparticles can pass through the apertures, fractionation here is based uponlength of the fibres. In another words, the most efficient and industriallypractical means of fractionating fibres by length is with pressure screens.2Several studies have been performed to understand how the design factors(e.g., volumetric reject ratio, aperture type (slots or holes), aperture size,aperture fluid velocity, feed consistency, and feed fibre length distribution)can affect fibre fractionation efficiency. They can be found in general reviewsof the topic, e.g., such as that by Sloane [127].HydrocyclonesWhile the pressure screen method mostly works for separating particlesbased on size, the second well-known fractionation method, the hydrocy-clone, is more efficient in separating fibres based on their ‘surface area pergram’. This has been shown by a number of experimental studies, e.g., Paav-ilainen [106], in which a successful separation of earlywood fibres (large di-ameter thin walled fibres) from latewood fibres (small diameter thick walledfibres) was performed.Multiple stages can also be used in this method to achieve a desiredquality of fraction. For example, a system of five hydrocyclone stages hasbeen used to separate a chemical pulp in [56]. The resulting two distinctfractions were refined separately and one of the fractions contained 89%earlywood fibres, while the other one has 46% earlywood fibres. This resultsin tensile strength in the paper of 3.5 times that of the other coarse fibrefraction.Particle motion is induced by the swirling motion of a suspension withina hydrocyclone: the circular motion induces a radially increasing pressure.The flow produces a differentiation in response between the liquid mediumand the suspended particles. Properties such as particle and fluid densities,particle size, particle concentration, and fluid viscosity can affect the relativemotion between particles and the suspending fluid. The non-trivial taskof measuring the motion of particles and also the high levels of chaoticmotion within the flow lead to limitations in investigating this type of flowexperimentally.On idealizing fibres as cylinders, one can say that for isolated particlesin an unbounded bath of fluid the terminal velocity is related to the density,3size, aspect ratio and orientation of the cylinder. Under creeping flow condi-tions with extremely dilute suspensions, one can thus separate fibres basedon the terminal velocity of fibres, supposing that orientation distribution re-mains constant. However, at elevated concentrations, in inertial conditions(non-zero Reynolds numbers), long range hydrodynamic disturbances leadto undesirable floc formation and less predictable behaviours.Toward a novel techniqueThe efficiency of fractionation in either hydrocyclones or pressure screens isknown to be relatively low. Improving the efficiency in these methods seemsvery hard if not impossible, due to the complexity of the flow within thesedevices.Recently, Madani et al. [86] demonstrated a novel methodology to sortparticle suspensions. The methodology involves the use of a complex fluidto replace water as the carrier fluid. In this context, the complex fluidwas a yield-stress material that can transform between two physical states,in this case between a solid and a liquid, depending upon the stress stateapplied. Madani et al. [86] have shown that under very ideal conditionsseparation can indeed be achieved with these types of yield-stress gels (seeFigure 1.1). This happens because the hydrodynamic interactions and long-range disturbances will be decreased by means of controlling the appliedstress and the regions in which the carrier fluid—the yield-stress fluid—is inthe solid or liquid state.There are limitations to this proposed technique as well. Firstly, againthe hydrodynamic interactions exist in this technique, however, the inter-actions are on a smaller scale since the fluid remains rigid away from theyielded envelope around the fibres. In addition, it has been reported that theseparation principle is based upon the mass per unit area of the unyieldedenvelope around the particle. However, the shape of this envelope is still anopen question in the literature. Later in this thesis we consider the shapeof unyielded envelope around particles (Chapter 3) and the hydrodynamicinteraction of particles in yield-stress fluids (Chapter 5).4Figure 1.1: A demonstration of the fractionation of a bidisperse sus-pension of spherical particles. In (a) an image of the suspensionis given before the commencement of the centrifuge. (b) is thestate of the suspension after the application of the centrifugalforce. It should be noted that most of the darker particles areon the periphery of the centrifuge. Reproduced from [86].1.1.2 Concrete industryThe rheology of fresh concrete is mostly controlled by that of the cementpaste (since gravel and sand are non-colloidal particles) which, by its own,is a yield-stress fluid [117, 118]. That is why civil engineers have proposedmany techniques to measure the yield stress of concrete accurately, andcontinually improve them. Most of these techniques can be categorized asstoppage tests, e.g., L-box test [26, 98], LCPC box test [119], and slump test[121]. Rheometers are also used to quantify the behaviour of concrete butare less interesting/applicable in a sense that using rheometers for in situmeasurements is not practical and there is also an uncertainty in measure-ments using different type of rheometers [43, 44].In recent decades, there is widespread agreement that “more liquifiedconcrete” is desirable. Indeed, the most important nature of “workable”concrete is its “lubrication”. If a concrete shows more fluid nature, then itwill have advantages, such as:• Exhibiting low friction between internal ingredient aggregates/parti-5Figure 1.2: Slump test for measuring workability of concretecles.• Overcoming frictional resistance between the surface of the formworkand reinforcement elements.• Becoming consolidated with minimum effort: concrete will fill theformwork homogeneously by its own weight as the driving force.It is clear that one of the key factors that determine the quality of concrete,“workability”, is yield stress. For example, if the yield stress of the concreteis high then air bubbles will be trapped in the cement and extensively weakenthe quality of the concrete.On the other hand, low yield stress of concrete can be a source of somedrawbacks as well. For instance, gravity can induce sedimentation of thesolid aggregates (the coarse particles) between the casting and setting of theconcrete. It is the yield stress of the cement that must hold the particlesbuoyant and not to settle, i.e. preventing segregation in concrete. Roussel[120] has nicely put together these two “contradictory” objectives (low yield-stress concrete and large yield-stress concrete): “On one hand, the concretehas to be as fluid as possible to ensure that it would fill the formwork under itsown weight. On the other hand, it has to be a stable mixture as the high strainrates generated by a flow in such a confined zone as a reinforced formwork6are able to separate the components of the concrete during casting.”Hence, defining a safe yield stress zone for concrete seems a non-trivialjob. One of the missing parts of this puzzle is a rational model for the sta-bility of concrete. Surprisingly, in this context the literature is significantlylacking. There are just some models based on empirical studies which arelimited to a small number of cases and special geometries, e.g., [129]. Inone of very few theoretical studies, Roussel [117] proposed a framework forstability of fresh concrete. Indeed, there are two parts in this framework.The first part is an attempt to find the critical diameter for a single spherebeyond which the sphere can settle in a yield-stress fluid medium. This hasbeen well documented in the literature of yield-stress fluids. We will reviewthese studies later in Section 1.6, in more detail. In the second part of [117],Roussel has tried to develop a more general framework for a multi-sphereflow. One of the first and principal assumptions made in [117] is that allthe spheres are moving if their diameter is larger than the critical diameter,otherwise they are quiescent. This assumption should be validated first,however. The stability of adjacent objects in a yield-stress fluid is a largelyunder explored problem area, even one decade after [117]. We will partlyaddress this problem (stability of multiple particles) later in Chapter 5.1.1.3 Drilling oil wellsIn drilling oil and gas wells, yield-stress fluids (drilling muds) are circulatedaround the wellbore, used to lubricate the drillpipe and drillbit, and to aidin removal of rock cuttings from the well to the surface. When the drillingmud is circulating, the yield stress contributes to the viscous drag on therock cuttings, enhancing transport. However, when circulation is stopped(as routinely happens), the more critical role of the yield stress is in pre-venting settling of cuttings in the well. For instance, Okrajni et al. [101]have experimentally investigated the effects of field-measured mud rheolog-ical properties on cuttings transport in directional well drilling, using aninclined annular geometry and transporting cuttings using water and ben-tonite/polymer muds. They categorized the observations into three main7regimes:• Laminar flow dominates cuttings transport in low-angle wells (i.e., 0◦to 45◦ inclinations from vertical).• Turbulent flow dominates cuttings transport in high-angle wells: 55◦to 90◦.• In an intermediate zone (i.e., 45◦ to 55◦) turbulent and laminar flowhave similar effects.Perhaps the most relevant observation for this thesis is that they haveclaimed that in laminar flows, muds with larger yield stress values pro-vide better cuttings transport. Since the effectiveness of laminar transportis mostly for low-angle wells, this suggests that the effect of mud yield stressbecomes small or even negligible in strongly inclined wells. The main pointhere is that in a strongly inclined well, settling particles/cuttings do nothave to settle far before they form a bed on the low side of the well. Inthis situation, a mud yield stress becomes rather detrimental to removal,i.e. the yield stress in the mud also acts to bind the particles together ina cuttings bed. Therefore, to remove cuttings in a horizontal well requiresturbulent flow in order to suspend the cuttings. However intuitively, in afully turbulent flow the flow characteristics should not depend critically onthe mud rheological properties.Other fluids have been proposed for cuttings transport. Elgaddafi et al.[41] presented a study on the settling behaviour of spherical particles infibre-containing fluids. These fluids have also been proposed for hydraulicfracturing operations in which proppant particles are transported into thefractures. Performance in this type of application largely depends on thesettling behaviour of suspended particles. Here instead of relying on yieldstress of the carrier fluid, which can support the particle weight under staticconditions, the idea is to increase the drag on particles by means of the fibres.These create additional drag mechanisms that can resist/reduce the motionof the particles. But the problem in this type of ideas is that the hydrody-namic effect of fibres is high at large settling velocities, but diminishes as8the settling velocity deceases, i.e. it is less effective for static situations andthis is why yield-stress drilling muds remain of interest. A general reviewon cuttings transport in the drilling context can be found in [143].In drilling and transport applications, proximity of other particles/cut-tings affects both static stability and drag, as studied, for example in [82, 90]for model situations. Equally, particle shape is a significant influence [73].We will review these studies later in detail and study the effect of multipleparticles in Chapter 5.1.1.4 Other applications: suspensions in food, drug, andcosmetic industriesMotion of large particles in yield-stress fluids is also important in industrialapplications such as the flow of food suspensions and the mixture of additivesin the cosmetic and drug industries [10]. For instance, there are three maintoothpaste ingredients that may be made of particles: hydroxyapatite, silver(bacteria killer), and titanium dioxide. The stability of these particles isan important concern, e.g., Liu et al. [84] studied the stability of chitosanand toothpaste active (cetylpyridiniumchloride and NaF) nanoparticles byscanning electron microscopy, transmission electron microscopy, and X-rayphotoelectron spectroscopy. Specifically in these type of applications, thepH of toothpaste is very important since particles size is increased at higherpH value.Particle stability in cosmetics is also important since adding particles incosmetics is very common. The particle size affects how the product feels onthe skin and hence is important for product development/invention: particlesizes smaller than 10 µm produce a reduction in skin feeling and viscositybenefits whereas particle sizes exceeding 200 µm form silicon elastomer gelballs on the skin [85].Food industries are also dealing with yield-stress fluids and particle mo-tions, for example chocolate chips in cookies (Figure 1.3)!There are a series of empirical models that describe the behaviour ofemulsions or suspensions after the well-known Einstein’s theoretical modelfor dilute suspensions. However, the literature is not so rich when it comes to9Figure 1.3: Chocolate chips in cookies as an example of particles inyield-stress fluids in food industry.yield-stress fluid as the suspending fluid. Recent interest has been devotedto suspensions of yield-stress fluids (mostly experimental studies, e.g., [33,105]), with far fewer in-depth theoretical and/or numerical studies.1.2 Yield-stress fluidsThe term “yield-stress fluid” is used to categorize a range of non-Newtonianfluids which behave as a rigid solid if the imposed stress on them is belowa threshold (called the yield stress). Otherwise these materials flow (i.e. de-form) when the level of imposed stress on them is beyond the yield stress[37] (see Figure 1.4 for examples). The description of materials which ex-hibit yield stress behaviour is one of the most difficult tasks in rheology,since one of the aims is having minimum number of material parameters,yet be respectful of the complex rheological behaviour of these type of mate-rials. Another goal, if one wishes to study non-viscometric flows, is that theconstitutive model should be computationally manageable. In other words,an ideal constitutive equation needs to predict the actual properties of suchmaterials, yet be simple enough to use theoretically and numerically in com-plex flows. It is likely that the satisfaction of all these ideals seems almostimpossible. Thus, in order to gain a reasonable estimation of the behaviour10Figure 1.4: Some examples of yield-stress fluids.of yield stress materials, several simplifying assumptions have been made inthe formulation of constitutive equations.In particular, about a century ago, Bingham in his seminal work onplasticity [16] assumed a threshold pressure drop below which a pipe full ofviscoplastic material would not flow, and a linear increase in pressure dropwith flow rate above this threshold. This developed into the Bingham fluidconstitutive equation: τˆ =(µˆp +τˆY‖ˆ˙γ‖)ˆ˙γ if ‖τˆ‖ > τˆY ,ˆ˙γ = 0 if ‖τˆ‖ 6 τˆY .(1.1)This model assumes hat the yield-stress materials are rigid when they areunyielded and a linear flowcurve when yielded. The two physical parametersare µˆp, which represents the plastic viscosity, and τˆY , which is the yield stressof the fluid. In (1.1), which is in tensorial form, τˆ is the deviatoric stresstensor and ˆ˙γ is the so-called rate of deformation tensor. It should be notedthat in (1.1), the von Mises yield criterion has been used implicitly. Indeed,it states that if the stress does not exceed τˆY , then the material is unyielded11(i.e. rate of deformation tensor vanishes in the Bingham model).Regarding notation, here and later in the thesis ‖ · ‖ denotes the normassociated with the tensor inner product:c : d =12∑ijcijdij ,i.e. ‖τˆ‖ = (τˆ : τˆ ) 12 . We use subscripts in the conventional Einstein summa-tion notation. Throughout this thesis, quantities with a ‘hat’ symbol (ˆ·) aredimensional and others are dimensionless.After Bingham, a few years later Herschel and Bulkley [63] added apower-law dependency to the Bingham model to explicitly fit shear-thinningbehaviour in rheological measurements of yield-stress materials: τˆ =(Kˆ‖ˆ˙γ‖n−1 + τˆY‖ˆ˙γ‖)ˆ˙γ if ‖τˆ‖ > τˆY ,ˆ˙γ = 0 if ‖τˆ‖ 6 τˆY .(1.2)This allowed the rheology of more materials to be fitted since many yield-stress fluids exhibit shear-thinning behaviour. Other models proposed overthe years include the Casson and Robertson-Stiff models, both of whichexhibit a yield stress and shear-thinning. These are not as widely usedas the Bingham and Herschel-Bulkley fluids but do have their own niceapplications.The Bingham and Herschel-Bulkley (HB) models have been extensivelyused to study the flow problems of yield-stress fluids until the present day,with many thousands of papers published [10, 29]. Drawbacks of these twomodels can be categorized into two main points: 1. the singularity in theeffective viscosity makes it hard to handle within conventional computationalschemes, and 2. the rigidity of fluid in the sub-yield stress parts of the flowis not physical from the rheological point of view. We will cover in depth thenumerical approaches that have been developed to overcome the difficultiesin modelling these type of fluids flow in Section 1.3, but here we reviewimprovements in rheological modelling of yield-stress fluids.12First, some effort has been made by scholars to improve Bingham andHB models by allowing more physical behaviours than rigidity in the sub-yield state of these two models. For instance, Oldroyd relaxed the zero-deformation assumption in the Bingham model by assuming perfect Hookeansolid behaviour [70]. Another aim was to include elastic effects even afteryielding, indeed once these materials yield and start to flow. In this category,we can point to Saramito’s model [123], in which material behaves as Kelvin-Voigt viscoelastic solid before yielding, and after yielding it becomes anOldroyd-B viscoelastic fluid. This type of extension of simple yield stressfluids leads us to question how apparent are elastic effects in flows of thesematerials.In recent years advancement in flow visualization techniques and con-ducting experiments for more complex flows has resulted in observationswhich cannot be explained by Bingham and/or HB models. For instance,Putz et al. [113] observed fore-aft asymmetry around a sphere sedimentingin a Carbopol solution, even in the creeping flow regime. Moreover, theyobserved a negative wake behind the sedimenting particle. These obser-vations have not been addressed in the numerical work of Beris et al. [15]which solved the same problem by using the Bingham model as the constitu-tive equation. Careful rheometric measurements showed that Carbopol gel(which is basically a yield-stress fluid) can behave differently from the HBfluid model, especially close to yielding (see Figure 1.6). Shear banding [38]and ‘avalanche behaviour’ [30] are other interesting complex flow observa-tions which have suggested that thixotropic behaviour is important in someyield-stress fluids. Such observations have motivated rheological researchersto study yield-stress materials in more detail and to propose more sophis-ticated elastoviscoplastic and thixo-elastoviscoplastic models for these typeof fluids. We can mention Dimitriou and Mckinley’s model [37] for waxycrude oils as one of the more complicated proposed models so far with ninematerial parameters. This has been derived based on a framework adoptedfrom plasticity theory: it decomposes the additive strain into characteris-tic reversible (elastic) and irreversible (plastic) contributions, coupled withthe physical processes of isotropic and kinematic hardening. Other recent13field and provided the filtering and smoothing algorithms forthe obtained velocity data.The rheological properties of yield stress fluids includingCarbopol® solutions have been discussed by a number ofdifferent groups; e.g., by Kim et al.,23 Møller et al.,24 andrecently by Piau.25 A study of the microrheology ofCarbopol® can be found in Ref. 26 comparing the microscaleresults with the bulk rheology. It is widely accepted thatwhen neutralized, these solutions are thixotropic and have ayield stress, at least over the length and time scales of ourexperiment. A controversial discussion of the existence of asby some definition trued yield stress can be found in Refs.27 and 28. In addition, when fully yielded they behave mac-roscopically as a shear thinning fluid. The rheological prop-erties of these solutions were determined using a Bohlinsnow Malvernd C-VOR rheometer at 25 °C. To eliminatethixotropic effects, we applied 60 s of preshear followed by a60 s resting period before we started any rheological mea-surements. In order to minimize wall slip effects a vane toolgeometry was used, which, according to Stokes andTelford,29 does not exhibit any wall slip effects. Additionallywe compared the vane tool data with measurements usingrough, serrated plates. The datasets acquired with both ge-ometries were quite similar, displaying precisely the samedistinct flow features ssee the discussion belowd, except for aslight stress underestimation in the range of small appliedstresses.As we are interested in slow flows close to the yieldingregime, our main interest lies in the transition from fluidliketo gel-like behavior and vice versa. We determined the strainrate-stress dependence sflow curved of the Carbopol®-940 so-lutions by operating the rheometer in the controlled stressmode sas our sedimentation experiment is in a sense a stresscontrolled experiment; i.e., the stress is set by the buoyancyof the sphere and its surface aread. We set the range of thestresses from 0.01 Pa up to a value far enough above theyield stress to ensure a good fit with a Herschel–Bulkleyconstitutive model. Corresponding to each value of the ap-plied stress, the rate of strain, averaged over 6 s, has beenmeasured for both increasing scircles in Fig. 3d and decreas-ing sopen squares in Fig. 3d values of the stress. As illus-trated in this representative figure for C=0.08% Carbopol®,we delineate three regimes:s1d With 0.01 Pa,t,1 Pa, the strain increases linearlywith the stress ssee the inset in Fig. 3d according tog ~ t . s5dWe interpret this as an elastic response of the material inthis range of applied stresses.s2d With 1 Pa,t,3.8 Pa, an abrupt increase of the rate offluid deformation is clearly visible. The response of thefluid in this regime is less understood, but we assumethat it is dominated by both shear banding and a compe-tition between breakage and rejuvenation of thematerial.18,19 A hysteresis is evident in regions s1d ands2d.s3d Above these limits, i.e., in region s3d, the data are con-sistent with a Herschel–Bulkley model and the three pa-rameters were estimated through regression ssee TableId. No hysteresis was observed in this region.A similar observation can be made for the lower concentra-tion solution; however, the regions occur at different thresh-olds.In addition to the controlled stress ramps, we have con-ducted oscillatory stress measurements for different values ofthe stress amplitude and measured the elastic and viscousmoduli of the material ssee Fig. 4d. We note that the sameflow regimes as illustrated in Fig. 3 are visible in Fig. 4 aswell. For low stress amplitudes, the elastic modulus is largeand approximately independent on the stress value. This cor-responds to the elastic solid flow regime discussed above. Asthe stress amplitude is gradually increased, the elastic modu-lus decreases sharply but it is still larger than the viscousmodulus. In this range of stresses, both elastic and viscouscontributions are significant and this regime corresponds toFIG. 3. sColor onlined Flow curve of the 0.08% Carbopol® solution. ssdIncreasing stress; shd decreasing stress, positive rates of strain; sjd decreas-ing stress, negative rates of strain. The solid sblued line denotes theHerschel–Bulkley fit, the dashed black line denotes the linear fit. The fitvalues are summarized in Table I. The inset shows the dependence of thestrain g on the applied stress. The full line is a guide for the eye; g=0.042t.FIG. 4. sColor onlined Stress dependence of the elastic ssquaresd and viscousscirclesd moduli for an oscillating stress measurement for 0.08% Carbopol®at 2 Hz. Empty symbols refer to increasing stresses and filled symbols todecreasing stresses.033102-4 Putz et al. Phys. Fluids 20, 033102 ~2008!the particle. We would like to point out that the fluids in-volved in these studies had no apparent yield stress. A studyof the influence of the flow conditions has been presented byKim et al.34 Recent simulations using a lattice Boltzmannapproach combined with a Maxwell model can be found inRefs. 35 and 36. Harlen37 used the finite element method tosimulate the flow around a sphere using the PeterlinsFENE-Pd and alternatively the Chilcott and RallisonsFENE-CRd closure approximations to the Finite ExtendibleNonlinear Elastic model ssee Refs. 38 and 39d. He concludedthat the shape of the downstream velocity wake is governedby the competition of two forces. The decay of the velocity islengthened by high extensional stresses which are opposedby an elastic recoil of the shear stress. It is the latter forcethat is claimed to be responsible for the appearance of thenegative wake.To help characterize the magnitude of the asymmetry, weplot the velocity component along the centerline of thesphere ssee Fig. 10d. From the velocity profile we can con-struct a similar picture to the flow curve. The stresses closeto the sphere are greater than the yield stress except in thesmall region in front of the sphere and we are in the fullyyielded state of the material fregion s3dg. Far away from thesphere, at values far below the yield stress, only a pure elas-tic contribution is felt by the material and the material is in apurely unyielded state. In region s2d, we claim to be in thetransition region of the flow curve and we claim that stressrelaxation is responsible for the negative wake. This alsoallows us to draw a parallel between the flow around a set-tling sphere and the flow in the rheometer. The increasingstress curve of the flow curve corresponds to the downstreampart of the flow of the sphere and consequently the decreas-FIG. 8. sColor onlined Flow for cases s1d–s4d. Note:Particles move from right to left. The color map refersto the modulus of velocity and the full lines are stream-lines. For clarity, we display only a fraction 1 /25 of thetotal velocity vectors.033102-7 Settling of an isolated spherical particle Phys. Fluids 20, 033102 ~2008!Figure 1: plots of....1Figure 1.5: Evid nce of l sticity in carbopol gel. Left panel is theflow urve of carbopol and right panel shows PIV velocity fieldand stream lines for a moving sphere in Carbopol. Sphere ismoving from right to left. Reproduced from [113].Figure 1.6: Avalanche flow of a clay suspension over an inclined plane.The pictures are taken at the critical angle for which the sus-pension just starts to flow visibly. Reproduced from [30].models of the same genre include [35, 89], which may be effective for specificmaterials and flows. More detailed review on elastoviscoplastic constitutivemodels can be found in [46], where the authors comp red th advantagesand drawbacks of each model and examined them in rheometric flows andthe complex flow around a sedimenting sphere.141.3 Theoretical and computational toolsApart from classical solutions for Bingham and HB models in simple geome-tries (e.g., Poiseuille flow), computing more complex flows is not a trivialtask because of singularities in the effective viscosity for these type of rhe-ological models. Hence, researchers started to think about different ways ofdealing with this issue. We will review three main strategies that have beendeveloped so far.1.3.1 Regularization methodsThese methods rely on eliminating the singularity of the effective viscosity inthe Bingham/HB model by adding a small parameter, , to the constitutiveequation. For instance,τˆ =(µˆp +τˆY+ ‖ˆ˙γ‖)ˆ˙γ, (1.3)which is a modified version of Bingham model and is easily differentiable.This is known as ‘simple’ regularization [47]. Two other regularization mod-els are,τˆ =µˆp + τˆY√2 + ‖ˆ˙γ‖2 ˆ˙γ, (1.4)due to Bercovier and Engleman [14], and,τˆ =(µˆp +τˆY‖ˆ˙γ‖[1− e−m‖ˆ˙γ‖])ˆ˙γ, (1.5)due to Papanastasiou [107]. Maybe the most commonly used model sofar is the one proposed by Papanastasiou. A quick comparison of thesethree models with the exact Bingham model is given in Figure 1.7, wherethe regularization parameters have been fixed to give the same zero-sheareffective viscosity.The other proposed technique which has been used widely in numeri-cal computations (especially Lattice-Boltzmann solvers) is the bi-viscosity15Figure 1.7: The regularisations models compared against the exactBingham model at  = 0.01, B = 1: (a) stress vs. strain rate;(b) effective viscosity vs. strain rate. Reproduced from [47].model. We can categorize this model as regularization method as well, butthe idea behind it is slightly different as it considers the Bingham fluid as afluid with two constant viscosities: a high viscosity fluid at low shear ratesand a fluid with plastic viscosity at large shear rates [36]; see discontinuousline in Figure 1.8.The regularized models are known to be fast and easy to implement instandard Navier-Stokes solvers such as those coded in commercial software.As long as only the velocity field is of concern, these methods should convergeto the exact Bingham model velocity solution if we go to small values → 0.Nevertheless, these models do not work well in some specific types of prob-lems. Frigaard and Nouar [47] have put it: “For a range of hydrodynamic(and static) stability problems we have found that regularization methodsproduce physically spurious results, i.e., they predict instability where theexact Bingham fluid model predicts stability. Of course it is evident that formany of the stability problems, e.g., static stability, decreasing  will slowthe growth rates of an instability . . . probably these methods should beavoided for hydrodynamic stability problems.” Not only in stability prob-lems, but also in some other problems, the regularized methods result inunphysical solutions. For instance, Figure 1.9 shows the problematic results16Figure 1.8: Steady-state rheology. Solid straight line: Bingham liquidwith yield stress τy and slope ηB. Dashed line: in a bi-viscositymodel the yield stress is mimicked by a very high viscosity (ηB0)liquid for low shear. Reproduced from [36].of regularized models in Poiseuille flow in a duct with square cross section.In this case the bi-viscous model fails whereas the Papanastasiou model iseffective [22].1.3.2 Yield surface tracking techniquesSzabo and Hassager [131] used an interesting approach to compute thesteady Poiseuille flow of a Bingham fluid within an eccentric annular geom-etry. Basically they guessed a reasonable initial shape for the yield surfacesand then solved the problem only in the yielded regions. The location of theyield surfaces was then corrected by an iterative approach based on the ob-tained solutions in each step. A similar idea was used earlier in the numericalwork of Beris et al. [15] where they solved the creeping flow around a sphereby guessing and mapping the yielded region to another ‘smooth sphere’ andthen iterating the yielded regions successively until convergence. Beris andhis co-workers used regularization technique to solve the governing equationswithin the yielded region. A similar idea has been used by Smyrnaios and17Figure 1.9: Circles show the analytic yield surface and other lines areyield surfaces from Papanastasiou regularization with increasingvalue of m. The main difference is that the convexity of yieldsurfaces in analytical solution is not captured in regularizedmodel. Reproduced from [22].Tsamopoulos to study squeeze flow of Bingham fluids [128].1.3.3 Variational principles and numerical methodsfounded on thatSince the work of Prager [110], in which two variational principles werestated for the Stokes flow of Bingham fluids, this type of formulation of vis-coplastic flow problems has received much attention. Prager derived velocityminimization and stress maximization principles for Stokes flow of Binghamfluids. These principles can be generalized to the Herschel-Bulkley and Cas-son rheological models (and others) which are of this general type: τˆ =(φ(‖ˆ˙γ‖)+τˆY‖ˆ˙γ‖)ˆ˙γ if ‖τˆ‖ > τˆY ,ˆ˙γ = 0 if ‖τˆ‖ 6 τˆY .(1.6)18Here φ(‖ˆ˙γ‖)represents the effective plastic viscosity of the rheologicalmodel. Let us assume governing equations:∇ · σˆ + ρˆfˆ = 0, in Ω, (1.7)∇ · uˆ = 0, in Ω, (1.8)with boundary conditions σˆ · n = tˆ and uˆ = uˆs on surfaces St and Sv,respectively. Here σˆ is the Cauchy stress tensor; ρˆ is the fluid density;ρˆfˆ is the body force (per unit volume - but for convenience we will usethe term ‘body force’ in this thesis); uˆ is the velocity vector. We maydecompose Cauchy stress tensor into pressure and deviatoric stress tensorparts: σˆ = −pˆI + τˆ .We define the velocity potential function, Φ(‖ˆ˙γ‖), as, Φ = 0, if ‖ˆ˙γ‖ = 0,dΦd‖ˆ˙γ‖ = ‖τˆ‖, if ‖ˆ˙γ‖ > 0.(1.9)Then, the velocity minimum principle states that the functional J (vˆ), ofthe form,J (vˆ) =∫ΩΦ dVˆ −∫Ωρˆfˆ · vˆ dVˆ −∫Sttˆ · vˆ dSˆ, (1.10)will reach its minimum value among all kinematically admissible velocityfields vˆ for the solution velocity field uˆ to Eqs. (1.7) and (1.8). In a similarway, if we define stress potential function, Ψ (‖τˆ‖), as,{Ψ = 0, if ‖τˆ‖ 6 τˆY ,dΨd‖τˆ‖ = ‖ˆ˙γ‖, if ‖τˆ‖ > τˆY ,(1.11)then the maximum principle expresses that of all statically admissible stressfields σˆ′ (= −pˆ′I + τˆ ′), the true stress field σˆ = −pˆI + τˆ will maximize thefunctional,K(σˆ′)= −∫ΩΨ dVˆ +∫Svσˆ′ · n · uˆs dSˆ. (1.12)19It should be mentioned that both functionals J andK will take same limitingvalue when evaluated with the actual fields solution. Indeed,K(σˆ′)6 K (σˆ) = J (uˆ) 6 J (vˆ) . (1.13)Proofs of these two principles and Equation 1.13 could be found in [68].To proceed further conveniently, here we expand the terms containing thetwo potential functions (first terms on the RHS of (1.10) and (1.12)) forBingham fluids since for the main parts of this study we will consider theBingham model as the constitutive equation. These are:∫ΩΦ dVˆ =12µˆp∫Ωˆ˙γ(vˆ) : ˆ˙γ(vˆ) dVˆ + τˆY∫Ω‖ˆ˙γ(vˆ)‖ dVˆ , (1.14)and,−∫ΩΨ dVˆ = − 18 µˆp∫Ω(‖τˆ ′‖ − τˆY )2+ dVˆ , (1.15)where (f)+ denotes the positive part of f .Augmented Lagrangian methodGlowinski and his coworkers in a series of works [45, 53–55] proposed arobust numerical method (founded on the variational principles mentionedabove) which can handle the yield-stress models exactly. Glowinski et al.[55] showed that the solution uˆ of problem (1.6 to 1.8) may be expressed asa minimizer of (1.10) on V, where,V = {v :∇ · v = 0, in Ω},is the space of admissible velocity fields which also satisfy the essentialboundary conditions of the problem. Since (1.10) is non-differentiable onV when τˆY is non-zero, due to the second term in RHS of (1.14), we cannotproceed with conventional gradient-type algorithms. However, the func-tional J (vˆ) is convex and this opens other approaches to the optimization.20In a general form, Glowinski formulated the optimization problem,minx{f(Ax) + g(x)},as,minx,y{f(y) + g(x)}, (1.16)where y = Ax. A Lagrange multiplier is then introduced to impose theequality of y and Ax, i.e. replacing J (vˆ) with a Lagrangian functional. In thenext step the Lagrangian functional (a saddle point problem) is augmentedwith an additional penalty terms to stabilize the computations. In [45], a setof augmented Lagrangian methods have been proposed to solve (1.16) whichhave been called ALG1-ALG4 later. Among them ALG2 has found moreattention in engineering applications and specifically in solving yield-stressfluid flows.To discuss ALG2, let us consider the problem (1.7 and 1.8) and Binghammodel as the constitutive equation. In order to talk about minimizationproblem (1.16), we set x = uˆ and operator A =∇+∇T . Then, y would bea relaxed form of strain rate tensor. From now on we denote y by γˆ. Hence,(1.16) will become:L(uˆ, γˆ, Tˆ ) = 12µˆp∫Ωγˆ : γˆ dVˆ + τˆY∫Ω‖γˆ‖ dVˆ −∫Ωρˆfˆ · uˆ dVˆ−∫Sttˆ · uˆ dSˆ +∫Ω(γˆ − ˆ˙γ(uˆ)): Tˆ dVˆ+a2µˆp∫Ω(γˆ − ˆ˙γ(uˆ)):(γˆ − ˆ˙γ(uˆ))dVˆ . (1.17)Here Tˆ is the Lagrange multiplier and a is the augmentation parameter. Ithas been proven [45, 53] that finding the saddle-point of L is the same asthe minimizer of J . As can be seen, L is a function of three variables and inALG2, optimization process will be followed by three main steps that takethe final form:21Algorithm 1 ALG21: procedure2: n← 03: γˆ0, Tˆ0 ← 0 (or any other initial guess)4: loop (Uzawa algorithm):5: if error < convergence then close.6: find uˆn+1 and pˆn+1 satisfying−a µˆp∆uˆn+1 = −∇pˆn+1 +∇ ·(Tˆ n − a µˆpγˆn)+ ρˆfˆ , in Ω,∇ · uˆn+1 = 0, in Ω,with given BC.7: γˆn+1 ← 0, if ‖Σˆ‖ 6 τˆY ,(1− τˆY‖Σˆ‖) Σˆ1+a , if ‖Σˆ‖ > τˆY .where Σˆ = Tˆn+ a µˆp ˆ˙γ(uˆn+1).8: Tˆn+1 ← Tˆ n + a µˆp(ˆ˙γ(uˆn+1)− γˆn+1)9: n← n+ 1.10: goto loop.We have used,max(∫Ω|uˆn+1 − uˆn| dVˆ ,∫Ω‖γˆn+1 − γˆn‖ dVˆ ,∫Ω‖γˆn+1 − ˆ˙γn+1‖ dVˆ ),as the ‘error’ measure for convergence check, both above and in the thesisbelow.With this numerical method we can manage to consider the exact Bing-ham model. However, one more refinement is to use an adaptive meshingmethod in order to recover a fine resolution of the yield surfaces. Thishas been proposed by Saramito and Roquet for the Finite Element Method(FEM) [115]. Basically, they have used an anisotropic mesh adaptation as-sociated to a metric which can be computed from the Hessian of square rootof the dissipative energy field:√µˆp ˆ˙γ(uˆ) : ˆ˙γ(uˆ) + τˆY ‖ˆ˙γ(uˆ)‖.As an example, for the flow around a cylinder, the different meshes which22they have generated are shown in Figure 1.10. Our adaptivity implemen-tation in this study is exactly the same as Saramito and Roquet, althoughwe use FreeFEM++ [62] to generate the meshes. More details on numericalmethods for yield-stress fluids could be found in [125].1.4 Resistance and Mobility problemsTwo classic formulations of particle motion within a bath of fluid could befound in the literature [57]. One is the Resistance formulation, [R], whenthe translational and rotational velocities of the particle are specified (asDirichlet boundary condition) and the hydrodynamic force and torque haveto be determined. However, it is of more interest physically to have thereverse case: the force and/or torque is specified and the translational/rota-tional velocities of the particle are calculated. This case is referred to as themobility problem, [M]. For instance, sedimentation problem is of [M] type.The particle is denoted by X; ∂X is the boundary of the particle; Ωand ∂Ω represent the entire domain (fluid+particle) and its outer boundary,respectively. In what follows, without loss of generality, we assume that forthe [R] problem, there is no additional body force in the fluid, and for [M]problem, the body force has constant magnitude and direction: ρˆfˆ = ρˆfˆef ,e.g. gravitational acceleration.In terms of boundary condition in the far field, for yield-stress fluids(BC on ∂Ω), usually uˆ = const. is specified. This might be either zero, ifsurrounded by static walls or Uˆ0 for moving walls (or a background uniformflow). Generally, sufficiently far from the particle the stresses fall below theyield stress and the fluid becomes unyielded. Typically there is a boundedyielded region formed near to the particle which is surrounded by outer rigidfluid (unyielded). Although the extent of the yielded region is not knowna priori, the fact that it exists means that practically speaking, we mayconsider uˆ = const. to be imposed at a rigid wall. Choosing an appropri-ate ∂Ω to be just far enough from the particle can significantly reduce thecomputational cost of the problem.23Figure 1.10: Example of an adaptation cycle. Left column: fourmeshes of a neighbourhood of the cylinder (cross section inwhite); Right column: corresponding to each mesh rigid zones(light grey), deformed zone (dark grey), stream lines. Repro-duced from [115].241.4.1 Mobility formulationWithout loss of generality, we formulate this problem based on the motionbeing driven by buoyancy (fˆef = gˆeg) and we assume the particle is denserthan the fluid: ρˆp > ρˆf . We use τˆb to scale the flow, starting with definitionof the velocity scale Uˆb, selected to balance τˆb with the viscous stress:µˆpUˆbLˆ= τˆb ⇒ Uˆb = (ρˆp − ρˆf )gˆLˆ2µˆp. (1.18)On scaling the Stokes equations with τˆb/Lˆ, we find:∇ · σ + ρ1− ρeg = 0, ∇ · u = 0 in Ω \ X¯, (1.19)where σ and u are the non-dimensional Cauchy stress tensor and velocityvector, respectively, and ρ = ρˆf/ρˆp. In this type of formulation, the Binghammodel is,  τ =(1 +Y‖γ˙‖)γ˙ if ‖τ‖ > Y,γ˙ = 0 if ‖τ‖ 6 Y,(1.20)where Y is the yield number, which represents the ratio of the yield stressto the buoyancy stress:Y =τˆY(ρˆp − ρˆf )gˆLˆ.Note that Lˆ is the length scale which we will talk about more in the nextchapters when we consider specific examples.Firstly, lets consider the space of admissible velocity solutions:V[M ] = {v :∇·v = 0, in Ω\X¯;v = 0, on ∂Ω\∂X;v → V +ω×x, on ∂X}.For arbitrary v, w ∈ V[M ], the viscous dissipation, plastic dissipation, andbuoyancy work functionals, respectively are:a(v,w) =∫Ω\X¯γ˙(v) : γ˙(w) dV, (1.21)25j(v) =∫Ω\X¯‖γ˙(v)‖ dV, (1.22)L[M ](v) =ρ1− ρ∫Ω\X¯v · eg dV +∫∂X(V + ω × x) · σ · n dA. (1.23)The variational inequality, which is equivalent to the minimum principle, is:a(u,v − u) + Y [j(v)− j(u)] > L[M ](v − u), (1.24)and so the minimum principle in the new notation is to minimize J[M ](v):J[M ](v) =12a(v,v) + Y j(v)− L[M ](v). (1.25)The corresponding maximum principle is:K[M ](σ′) = −18∫Ω\X¯(‖τ ′‖ − Y )2+dV + L[M ](u), (1.26)where the space of admissible stresses S is,S = {σ′ :∇ · σ′ + ρ1− ρeg = 0 in Ω \ X¯;∫∂Xσ′ · n dA = Vp1− ρeyand∫∂Xx× (σ′ · n) dA = M b},i.e., symmetric stress tensors (σ′ = τ ′− p′I) satisfying the Stokes equationsand two supplementary conditions: the linear force balance and the angularmomentum balance. Note that M b is the scaled buoyancy moment.An interesting feature of formulating a particulate flow in a yield-stressfluid as a [M] problem is the question of whether or not a given particlemoves. Intuitively we expect that for a sufficiently large yield stress a particlewill not move under the influence of finite body force. This is equivalent topostulating a critical value of yield number, Yc, beyond which the particleis static. We can consider this yield limit mathematically by looking at howthe solution varies with Y . Suppose that Y2 > Y1 and that u2 and u1 are26the velocity field solutions associated with Y2 and Y1, respectively. Since u2and u1 are both test functions for each other, we may substitute each in(1.24) and sum to give,(Y1 − Y2) [j(u2)− j(u1)] > a(u2 − u1,u2 − u1) > 0. (1.27)We may note from above inequality that the plastic dissipation decreasesmonotonically as Y increases, which proves that Yc is unique, i.e., sincej(u) = 0 at Yc implies that j(u) = 0 for any yield number larger than theYc. Now suppose that Y2 = Yc in (1.27), which becomes:(Yc − Y ) j(u) > a(u,u) > 0. (1.28)Using Cauchy-Schwarz inequality, we may claim that j(u) 6 C√a(u,u),where C is a domain-dependent quantity only. Now, supposing that Y →Y −c , we find:a(u,u) = O([Yc − Y ]2), j(u) = O (Yc − Y ) , (1.29)a(u,u)j(u)= O (Yc − Y ) . (1.30)These bounds are not sharp, but (1.30) allows us to claim that a(u,u)always decays faster than j(u) as we are reaching the yield limit. Theenergy equation isa(u,u) + Y j(u) = L[M ](u), (1.31)and is also useful in this limit since we can see that L[M ](u) ∼ Y j(u) asY → Y −c . Hence, considering velocity minimization problem, we can saythat J[M ](v) ∼ Y j(v) − L[M ](v), which suggest a minimizer in the spaceof bounded deformations. As a result (see (1.13)) we can conclude thatthe associated stress field with that minimizer has to be close to Y almosteverywhere in the domain, or ‖τ ′‖ = Y . We will investigate this case in moredepth below in Section 1.5, as it is suggestive of rigid-perfect plasticity.271.4.2 Resistance formulationIn the resistance formulation [R], as mentioned earlier , we specify the ve-locity boundary condition on the particle surface (∂X),uˆ = Uˆp + ωˆp × xˆ.Together with the far-field velocity conditions we are able to calculate thefluid velocity and as a result we can calculate the force and torque:Fˆ =∫∂Xσˆ · n dSˆ, Mˆ =∫∂Xxˆ× (σˆ · n) dSˆexerted on the particle surface. Here Uˆp is the translational velocity ofthe centre of gravity of the particle and ωˆp is the angular velocity aboutthat centre. We formulate problem [R] in the Stokes regime for a particleinside a Bingham fluid in the absence of any body force. Thus the governingequations will be,∇ · σˆ = 0, ∇ · uˆ = 0, in Ω \ X¯, (1.32)and (1.1) is the constitutive equation. To make the equations non-dimensional,we may choose particle velocity to scale the velocity field and also is used todefine a viscous stress scale τˆp = µˆpUˆp/Lˆ, with which the Stokes equationsare scaled to,∇ · σ∗ = 0, ∇ · u∗ = 0, in Ω \ X¯, (1.33)and Bingham model to, τ ∗ =(1 +B‖γ˙∗‖)γ˙∗ if ‖τ ∗‖ > B,γ˙∗ = 0 if ‖τ ∗‖ 6 B,(1.34)whereB =τˆYτˆp=τˆY LˆµˆpUˆp,28is the Bingham number. The ‘asterisk’ is used to distinguish variables fromthe [M]-scaled variables, where needed.We may write the variational inequality as,a(u∗,v∗) +B [j(u∗ + v∗)− j(u∗)] > 0, u∗ ∈ VR, ∀v∗ ∈ VR,0 (1.35)where V[R] is the subspace of [H1(Ω \ X¯)]d consisting of divergence freevelocity fields that satisfy the far-field boundary condition (u∗ = 0 on ∂Ω)and the no-slip boundary condition on the particle surface, while V[R],0 isthe same as V[R] except that satisfying zero boundary condition at ∂X. Theminimum principle isJ[R](w∗) =12a(w∗,w∗) +Bj(w∗), w∗ ∈ V[R]. (1.36)The maximum principle for problem [R] isK[R](σ′) = −18∫Ω\X¯(‖τ ′‖ −B)2+dV +∫Ω\X¯τ ′ : γ˙(u∗) dV, (1.37)in which τ ′ is the deviatoric part of σ′, which satisfies Stokes equations.One may write the second term in RHS of maximum principle as,∫Ω\X¯τ ′ : γ˙(u∗) dV =∫∂X(U∗p + ω∗p × x) · σ′ · n dA.1.5 Rigid perfect-plasticityThe mechanical behaviour of any continuous medium is described in termsof stress and strain. Hence, if an elastic body is loaded and the load is thenremoved, the body will return to its primary shape. If we assume that thereis a linear relation between the strain tensor (γˆ) and stress tensor (σˆ), thenwe end up with Hooke’s law [80] for an elastic medium which can be simplywritten as,σˆ = KˆI + Gˆγ, (1.38)29where Kˆ and Gˆ are the bulk and the shear modulus, respectively, and  = γii.We can alternatively write the bulk and shear modulus with the well-knownYoung modulus (or modulus of elasticity), Eˆ, and Poisson ratio, ν, as,Kˆ =Eˆ3(1− 2ν) , Gˆ =Eˆ2(1 + ν). (1.39)In the plastic range (when the von Mises criterion has been satisfied),the total strain can be decomposed into elastic (γe) and plastic (γp) strains.This plastic strain is the permanent strain: if the load is removed, the plasticstrain will remain in the medium. Here we just consider incompressibleplasticity: p = γpii = 0. We can assume that the rate of plastic strain atany instant is proportional to the deviatoric stress, hence,Gˆˆ˙γp = λˆτˆ , (1.40)where λˆ is the factor of proportionality. According to (1.38), the rate of theelastic strain can be written as,Gˆˆ˙γe = ˆ˙τ . (1.41)Assembling both strains, we will get,Gˆ(ˆ˙γe + ˆ˙γp)= Gˆˆ˙γ = ˆ˙τ + λˆτˆ . (1.42)Having in mind that (1.42) is valid when the von Mises yield criterion issatisfied: ‖τˆ‖ = τˆY or τˆ : τˆ = const., we can claim that,d (τˆ : τˆ )dtˆ= 2(τˆ : ˆ˙τ)= 0. (1.43)The dyadic product of (1.42) and the stress tensor can be written as,τˆ :(Gˆˆ˙γ)= Gˆ(τˆ : ˆ˙γ)= τˆ :(ˆ˙τ + λˆτˆ)= τˆ : ˆ˙τ + λˆ (τˆ : τˆ ) = λˆτˆ2Y , (1.44)30or,Gˆ ˆ˙W = λˆτˆ2Y , (1.45)where ˆ˙W = τˆ : ˆ˙γ is the local ‘rate of work dissipation’. Therefore bysubstituting expression (1.45) in (1.42), we can find the stress-strain relation,ˆ˙τ = Gˆ(ˆ˙γ −ˆ˙Wτˆ2Yτˆ), (1.46)which is well-known as the Prandtl-Reuss model [111].Finding an elastic-plastic solution for most of the problems is very com-plicated. For the sake of simplicity and also when the plastic strain is muchlarger than the elastic one in a class of practical problems, the elastic defor-mation is being ignored either in the plastic or elastic regions. It means thatwe are dealing with unrestricted plastic flow. Whenever this assumption isvalid, we can substitute the plastic strain in (1.40) with the total strain.Now, again the factor of proportionality can be eliminated by using the vonMises yield criterion like before, which results in,ˆ˙γ : ˆ˙γ =(λˆGˆ)2(τˆ : τˆ ) =(λˆGˆ)2τˆ2Y , (1.47)and by substituting into (1.40),τˆ =τˆY‖ˆ˙γ‖ˆ˙γ. (1.48)Expression (1.48) constitutes the relation between deviatoric stress and rateof deformation tensors when we have plastic flow in rigid-perfectly plasticmediums. If the second invariant of the deviatoric stress tensor is less thanτˆY , then the material is considered as rigid.To proceed more conveniently, we used non-dimensional form of (1.48)as,τ˜ =B‖˜˙γ‖˜˙γ. (1.49)31in the following sections. Note that it is the form that we have in expression(1.34), except that in the absence of any viscosity in this model, the firstterm of RHS (1.34) has been dropped and again the non-dimensional yieldstress of the material is denoted by B. The ‘tilde’ sign will be used as wellfor rigid-perfect plasticity problem from now on to prevent any potentialmisinterpretation with viscoplastic problem.1.5.1 Variational principles and uniqness of solutionConsider interior of a body, Ω, is bounded by surface St which is undertraction force t˜. The stress tensor (σ˜) then should satisfy the equilibriumequation ∇ · σ˜ = 0. We can write the rate of work done by the surfacetraction as,∫Stt˜ · u˜ dS =∫St(σ˜ · n) · u˜ dS =∫Ω∇ · (σ˜ · u˜) dV=∫Ω(∇ · σ˜) · u˜ dV +∫Ωσ˜ :∇u˜ dV (1.50)where the velocity field u˜ is chosen independently of stress field. The firstterm in the very RHS of (1.50) is zero because of Stokes equation. Since,σ˜ij∂u˜j∂xi= σ˜ji∂u˜i∂xj= σ˜ij∂u˜i∂xj=12σ˜ij(∂u˜i∂xj+∂u˜j∂xi)= σ˜ij ˜˙γij , (1.51)we can rewrite (1.50) as,∫Stt˜ · u˜ dS =∫Ωσ˜ : ˜˙γ dV, (1.52)where ˜˙γ is the rate of deformation tensor associated with the velocity u˜.Now, we consider a more general problem: the interior of the body, Ω,is bounded by a surface S, that is subdivided into a surface St on which atraction t˜ is specified, and a surface Sv on which a velocity u˜s is specified(problem P).Now to prove that the solution to the problem P is unique, assume that(σ˜1, u˜1) and (σ˜2, u˜2) are two different consistent solutions of the stress and32velocity fields of the problem P, with t˜1 and t˜2 as the surface traction onSt, respectively. Because of absence of viscosity in the rigid–perfectly plasticmodel, across certain internal surfaces, we may have tangential discontinuityin the velocity field (the physical interpretation of this discontinuity is asharp transition or boundary layer in the velocity field of a real materialwith viscosity). In what follows, |∆u˜(1,2)| is the magnitude of the velocityjump along any velocity discontinuity, denoted by Sd,(1,2)(u˜(1,2)). Using(1.52), we can claim that,∫S(t˜1 − t˜2) · (u˜1 − u˜2) dS = ∫Ω(σ˜1 − σ˜2) :(˜˙γ1 − ˜˙γ2)dV+∫Sd,1(B − τ˜2) |∆u˜1| dSd+∫Sd,2(B − τ˜1) |∆u˜2| dSd, (1.53)where τ˜1 and τ˜2 are the shear stresses associated with stress tensors σ˜1 andσ˜2, respectively. Since τ˜(1,2) 6 B, the last two terms of the RHS of (1.53)are non-negative and rewriting the first term,(σ˜1 − σ˜2) :(˜˙γ1 − ˜˙γ2)= (σ˜1 − σ˜2) : ˜˙γ1 + (σ˜2 − σ˜1) : ˜˙γ2,which is also non-negative due to the maximum work inequality. Moreover,the LHS of (1.53) is zero because across St, t˜1 = t˜2 = t˜ and across Sv,u˜1 = u˜2 = u˜s. Hence, each term of RHS of (1.53) should be also zerowherever ˜˙γ1 or ˜˙γ2 is nonzero: σ˜1 = σ˜2. In other words, stress field isunique in the flowing regions.1.5.2 Lower and upper bound theoremsTo discuss the lower and upper bound theorems, the actual stress and veloc-ity field of problem P are denoted by σ˜ and u˜, respectively, with associatedstrain rate tensor ˜˙γ. Admissible stress tensors σ˜′ satisfy the Stokes equa-tions in Ω, the traction condition on St, and have shear stress: τ˜′ 6 B. The33variational principle gives us:∫Su˜ · (σ˜ · n− σ˜′ · n) dS = ∫Ω(σ˜ − σ˜′) : ˜˙γ dV +∫Sd(u˜)(B − τ˜ ′) |∆u˜| dSd,(1.54)where |∆u˜| is the magnitude of the velocity jump along any velocity discon-tinuity, denoted by Sd(u˜). The RHS of (1.54) is non-negative: the first termis zero in the rigid zones and is non-negative in the yielded parts, in thesense that∫Ω σ˜ :˜˙γ dV >∫Ω σ˜′ : ˜˙γ dV (from the maximum work principle).The second term is always non-negative because the shear stress is less thanor equal to B everywhere. Hence, the LHS must also be non-negative. Thetraction terms cancel on St, leading to:∫Svu˜s · σ˜ · n dS >∫Svu˜s · σ˜′ · n dS. (1.55)In other words, the work done by the actual traction σ˜ · n on Sv is greaterthan the work done by the traction of any other admissible stress field:σ˜′ · n. When the boundary condition on Sv is a constant velocity (i.e. the[R] problem in the context of this thesis), it follows that we can calculate alower bound to the actual traction force from any admissible stress field.Now, consider an admissible velocity field u˜′ which gives us strain ratefield ˜˙γ ′, and stress field σ˜′. Note that σ˜′ is now not necessarily an admissiblestress field. Using the Stokes equations we can write,∫Ωσ˜ : ˜˙γ ′ dV +∫Sd(u˜′)τ˜ |∆v˜| dSd =∫Su˜′ · σ˜ · n dS=∫Svu˜s · σ˜ · n dS +∫Stu˜′ · σ˜ · n dS.(1.56)We rearrange (1.56), noting that τ˜ 6 B, and use the maximum work prin-34ciple as before:∫Svu˜s · σ˜ · n dS 6∫Ωσ˜′ : ˜˙γ ′ dV +∫Sd(u˜′)B|∆u˜′| dSd −∫Stu˜′ · σ˜ · n dS.(1.57)Using Cauchy-Schwarz inequality for the first term of the RHS of (1.57), wehave,∫Svu˜s · σ˜ · n dS 6 B(∫Ω‖˜˙γ ′‖dV +∫Sd(u˜′)|∆u˜′| dSd)−∫Stu˜′ · σ˜ · n dS. (1.58)Equation (1.58) states that the rate of work done by the actual surfacetractions on Sv is less than or equal to the internal work dissipated by anyadmissible velocity field.1.5.3 Plane strain problem (2D flow case)We call the flow of a material ‘planar’ if a Cartesian coordinate system canbe found in which the flow is parallel to xy plane (the velocity in the z-direction is zero) and hence the velocity field is independent of z-direction.Therefore, the governing equations and von Mises yield criterion reduce to,∂σ˜x∂x+∂τ˜xy∂y= 0,∂τ˜xy∂x+∂σ˜y∂y= 0, (1.59)and,τ˜2xx + 2τ˜2xy + τ˜2yy = 2B2. (1.60)The equilibrium equations for this unrestricted 2D plastic flow with theassumption of incompressibility of the material form a closed set of hy-perbolic equations. This can be easily proved by considering a curvilinearelement shown in Figure 1.11. The element is bounded by orthogonal linesα and β which show the direction of maximum shear stress. So we can write35xyϕ𝐵𝐵𝐵𝐵 𝑝 𝑝 𝑝 𝑝𝛼𝛽Figure 1.11: A curvilinear element bounded by sliplines.the stresses as,σ˜x = −p˜−B sin(2φ), σ˜y = −p˜+B sin(2φ), τ˜xy = B cos(2φ), (1.61)where φ is the counterclockwise orientation of the α-lines with the x-axis.If we substitute expressions (1.61) in (1.59), we will end up with,∂p˜∂x+ 2B[∂φ∂xcos(2φ) +∂φ∂ysin(2φ)]= 0, (1.62)∂p˜∂y+ 2B[∂φ∂xsin(2φ)− ∂φ∂ycos(2φ)]= 0. (1.63)Now if we assume that p˜ and φ are given along a curve C, then the differ-entials of these quantities could be written as,dp˜ =∂p˜∂xdx+∂p˜∂ydy, dφ =∂φ∂xdx+∂φ∂ydy.The determinant of the coefficients of derivatives of p˜ and φ in the x and36y directions is∣∣∣∣∣∣∣∣∣∣1 0 2B cos 2φ 2B sin 2φ0 1 2B sin 2φ −2B cos 2φ0 0 dx dydx dy 0 0∣∣∣∣∣∣∣∣∣∣=(dx2 − dy2) sin(2φ)− 2dxdy cos(2φ),which vanishes at,dydx= tanφ,dydx= − cotφwhich are the directions of α and β lines. Hence, the derivatives may bediscontinuous across curve C and so the characteristic lines of the hyperbolicequation of stress coincide with the α and β lines which show the directionof maximum shear stress. These lines are well-known as sliplines.The analysis proceeds conveniently if we arrange the differential equa-tions along the sliplines. It is not complicated since we can set φ = 0 in(1.62) and (1.63),∂p˜∂sα+ 2B∂φ∂sα= 0,∂p˜∂sβ− 2B ∂φ∂sβ= 0,where sα and sβ are coordinates along α and β lines, respectively. Hence,we can claim that,p˜+ 2Bφ = const. along an α-line, (1.64)p˜− 2Bφ = const. along a β-line, (1.65)which are known as Hencky’s equations [23]. These two equations are indeedequilibrium equations which are simply transferred to sliplines coordinates.The same procedure can be followed for velocity field by introducing thecomponents u˜α and u˜β along the α and β lines, respectively. Then theCartesian components of velocity u˜ = u˜xex + u˜yey are expressed as,u˜x = u˜α cosφ− u˜β sinφ, u˜y = u˜α sinφ+ u˜β cosφ. (1.66)37Since in an isotropic material the principal axes of stress and strain ratemust coincide,2˜˙γxy˜˙γxx − ˜˙γyy=2τ˜xyτ˜xx − τ˜yy ,and also we are interested in incompressible plastic flow,∂u˜x∂x+∂u˜y∂y= 0,If now we consider the velocities along the sliplines, where the total normalstresses are equal and the shear stress is equal to B, then one can claimthat,∂u˜α∂α=∂u˜β∂β= 0,and after some algebra,du˜α − u˜β dφ = 0, along an α-line, (1.67)du˜β + u˜α dφ = 0, along a β-line, (1.68)which are well-know as Geiringer equations [23]. In other words, it says that,the conservation of mass requires that the normal component of velocity tobe continuous across any curve. The tangential component may, however, bediscontinuous. The physical interpretation of this discontinuity is the limitof a narrow region (e.g., a boundary layer) in which the rate of shearing inthe tangential direction is infinitely large. The tangent to the discontinuitytherefore coincides with a direction of the maximum shear stress: the dis-continuity should be a slipline. For instance, If the velocity is discontinuousacross an α-line, u˜β is continuous while u˜α changes by a finite amount, ∆u˜αacross the discontinuity. It should be noted that ∆u˜α = const. along thediscontinuity. We will use these equations in the next chapters, within thecontext of this thesis.38Properties of sliplinesThe field of orthogonal characteristics have interesting geometrical featuresthat easily can be proved by simple mathematics/algebra. These propertiesare frequently used in the solution of 2D plastic flow problems.It can be found from Hencky equations that along a straight slipline (φ =const.) the pressure is constant (see Hencky equations (1.64) and (1.65)).Also, it can be easily proved that if a segment of one slipline of one familyis straight then all other corresponding segments of sliplines of the samefamily are straight. It should be noted that it means that when one sliplineis totally straight, then all of the sliplines of the same family are straight.To establish it, consider two sliplines of α-family (AB and CD) and twofrom β-family (AC and BD) in Figure 1.12. Assume that AB is a straightsegment of the slipline: so φA = φB and from Hencky equation for α lineswe find that p˜A = p˜B. Hence,p˜A − 2BφA = p˜B − 2BφB. (1.69)Also considering (1.65), we have,p˜C − 2BφC = p˜A − 2BφA, (1.70)and,p˜D − 2BφD = p˜B − 2BφB. (1.71)The RHS’s of (1.70) and (1.71) are equal (see (1.69)), hence,p˜C − 2BφC = p˜D − 2BφD. (1.72)Combining the previous equation with the Hencky relation along line CD(i.e., p˜C + 2BφC = p˜D + 2BφD) results in:p˜C = p˜D and φC = φD. (1.73)Therefore, it can be observed that CD is also a straight line.39Figure 1.12: Segments of slip lines of two familytheory of the slipline field 427Figure 6.2 Slipline field geometries for establishing Hencky’s theorems.obtain the following relationship between the nodal values of φ and p:φC − φD = φA − φB pC − pD = pA − pB (17)This is known as Hencky’s first theorem,† which states that the angle between thetangents to a pair of sliplines of one family at the points of intersection with a sliplineof the other family is constant along their lengths. In other words, if we pass fromone slipline to another of the same family, the angle turned through and the changein hydrostatic pressure are the same along each intersecting slipline.It follows that if a segment AC of a slipline is straight, the corresponding segmentBD of any slipline of the same family will also be straight (Fig. 6.2b). The straightsegments must be of equal lengths, since the intersecting curved sliplines are theirorthogonal trajectories. Indeed, the curved sliplines have the same evolute E, whichis the locus of the centers of curvature along either of them. Both AB and CD maytherefore be described by unwinding a taut string from the evolute. The ends of thestring in the two cases will be separated from one another by a distance equal tothe length of each straight segment. It follows from (12) that the normal componentof velocity changes by a constant amount in passing from one straight slipline toanother along any intersecting curved slipline.Let the radii of curvature of the α and β lines be denoted by R and S respec-tively. These radii will be taken as positive if the α and β lines, regarded as aright-handed pair of curvilinear axes, rotate in the counterclockwise and clockwise† H. Hencky, Z. angew. Math. Mech., 3: 241 (1923).Figure 1.13: Slipline field geometries for establishing Hencky’s theo-rems. Reproduced form [23].Almost the same approach can be followed to capture some other featuresof sliplines. Consider a curvilinear quadrilateral ABCD bounded by the α-lines AB and CD and the β-lines AC and BD (Figure 1.13a). The pressuredifference between C and B may be written as,p˜C − p˜B = (p˜C − p˜A) + (p˜A − p˜B) = 2B(φB + φC − 2φA), (1.74)or alternatively as,p˜C − p˜B = (p˜C − p˜D) + (p˜D − p˜B) = 2B(2φD − φB − φC). (1.75)40Because the LHS’s of the above equations are equal,p˜C − p˜D = p˜A − p˜B, (1.76)and,φC − φD = φA − φB. (1.77)In other words, the angle formed by the tangents of two fixed shear lines ofone family does not depend on the choice of the intersecting shear line ofthe second family. It also means that if we pass from one slipline to anotherof the same family, the angle turned through and the change in pressureare the same along each intersecting slipline. The second consequence ofthis expression is pointed to earlier about straight segments of a family ofsliplines. More interesting features of sliplines can be found in [23, 66, 111].For instance, Chakrabarty [23] in explanation of Figure 1.13b mentionedthat: “the curved sliplines have the same evolute E, which is the locus of thecentres of curvature along either of them. Both AB and CD may thereforebe described by unwinding a taut string from the evolute”. These featuresare used to construct the slipline network around particles to study yieldlimit in the following chapters.1.6 Literature reviewAmongst the classical problems of fluid mechanics is the Stokes flow abouta particle settling in a viscous fluid. Particles of higher density (ρˆp) thanthe fluid (ρˆf ) settle at a speed proportional to the density difference, grav-itational acceleration (gˆ) and particle diameter squared (Dˆ2), but inverselyproportional to the fluid viscosity. The sedimentation of particles in an in-finite domain is well understood/documented in this case for low Reynoldsnumbers [12, 58]. Closed form solutions have been proposed and symme-try/reversibility properties in a very general framework have been proved forsimple particle geometries. For more complex particle geometries, Faxen’sprinciple can be helpful in finding a lower and upper bound on the sedimen-tation velocity. More complex configurations also have been studied, e.g.,41bounded domains can have significant effect on the velocity [59] and closedform solutions have been derived for many cases.As already discussed, viscoplastic fluids are generalized Newtonian fluids,i.e., the effective viscosity varies with the strain rate. As the strain rate isinduced by the buoyancy force, it is natural that the settling speed shouldvary nonlinearly with (ρˆp − ρˆf ). An additional feature of viscoplastic fluidsis that they exhibit a yield stress. Consequently, in the particle settlingcontext, we intuitively expect that if density difference is not sufficientlylarge, the buoyancy stress (ρˆp − ρˆf )gˆDˆ in insufficient to overcome the yieldstress and there is settling motion. In this section we review studies inthe literature which have been devoted to particle motion/sedimentation inyield-stress fluids.1.6.1 Particle motion in yield-stress fluids–early workThe study of particle motion in yield-stress fluids dates back to the 1950swhen Volarovich and Gutkin [140], in the very first attempts in this context,found that if a particle moves within a viscoplastic medium then it shoulddo so within a bounded subset, since the stress falls below yield stress ifwe go sufficiently far from the particle. Following that, Andres [3] defineda ‘sphere of influence’ terminology to describe the yielded fluid around asphere which is moving within a yield-stress fluid. Almost at the same time(early 60s), Whitmore and his colleagues started to look at the same prob-lem in a series of experiments. For instance, Boardman and Whitmore [19]performed experiments using glass blocks of different sizes moving within aclay suspension which exhibits yield stress. Their main concern was to mea-sure the limiting force (they called it Fs) that should be exceeded on theparticle to move in a yield-stress fluid. Rather than any in-depth quantita-tive study, they wanted to get an idea of how the yield stress contributes toFs. They mentioned contradictory ideas proposed by different scholars up tothat time: “For a cube falling with its faces horizontal and vertical, Yanceyet al. have assumed that the minimum value of Fs would result from theyield stress acting over the bottom face only of the cube. Meerman suggested42that yield stress would act at least over the total vertical area of the cube,and Whitmore that it would act over the whole surface area of the cube.” Toshow that yield stress does not act over the whole surface, Boardman andWhitmore [19] used a nice trick and measured Fs for different orientationsof glass blocks and found that the values are different. Five years later,in another experimental study, Valentik and Whitmore [139] measured theterminal velocity of a falling sphere in a yield-stress fluid. They tried to finda model expression for the drag force and terminal velocity of the sphereby generalizing the well-known Stokes drag coefficient for a sphere in a vis-cous fluid, using the ‘sphere of influence’ concept that had been introducedby Andres [3]. Although their proposed model was not a successful theory,a new finding of their study was that there is “an envelope of suspensionattaching itself to a moving sphere”. Two years later, Ansley and Smith[4] neglected the observation of ‘attached yield-stress fluid’ to the spheresurface (although they also cited Valentik and Whitmore work) and triedto postulate the flow pattern about a falling sphere (see Figure 1.14a) tocalculate the drag force. To the best of our knowledge, Ansley and Smithwere first to point to plasticity theory as an important/helpful tool for at-tacking this type of problem. Although years after [4], different studies haveshown that their postulated flow was not accurate (consequently also theirproposed expression for the drag force), their approach was an improvementon past work. In their derivation they considered the pressure distributionabout the sphere surface rather than only a yield stress contribution: thepressure had been neglected by other researchers previously.Later experiments and usage of flow visualization techniques by Brookesand Whitmore [21] clearly revealed the presence of a “stagnant zone” abouta moving particle in a yield-stress fluid (see Figure 1.15). Indeed, theytested cylinder, plate, and wedge-shaped particles in different yield-stresssuspensions.A few years later, in a series of studies, Adachi and Yoshioka built up avariational framework for investigating such problems. Included in the seriesof papers is a wide range of different qualitative results, pertaining more gen-erally to viscoplastic flows. With regard to particles, they extended partly43Figure 1.14: Flow patterns about a falling sphere: (a) proposed byAnsley and Smith [4], (b) proposed by Yoshioka et al. [144], (c)computed by Beris et al. [15]. Reproduced from [28].History Volarovich, Colloid J., 1953 ‘Sphere of influence’{Andres, Dokl. Akad. Nauk. SSSR, 1960Valentic & Whitmore J. Appl. Physc. 1965 Brookes & Withmore, Rheol. Acta, 1969Reproduced from Brookes & Withmore, Rheol. Acta, 1969.Chaparian, Wachs, Frigaard Particles in a viscoplastic fluid December 18, 2017 6 / 50Figure 1.15: Flow past different objects. Reproduced from [21].the classical results of Faxen, and developed a framework for estimating thedrag force using test functions within the variational framework. In thisway they calculated lower and upper bounds of the drag force experiencedby a sphere [144] and by a 2D circular cylinder [1]. The test function for thesphere is shown in Figure 1.14b. Although the flow pattern is quite differentfrom the solution, this was not the intention of the method, i.e. this was atest function used to bound the drag and no more.1.6.2 Beris et al. [15]–a robust frameworkAlmost a decade after Adachi and Yoshioka’s studies, a numerical study byBeris et al. [15] uncovered, beyond the shadow of a doubt, the flow pattern44about a moving sphere in a yield-stress fluid (see Figure 1.14c). Two capsof unyielded fluid were found to form at the leading and trailing parts ofthe sphere (close to the stagnation points where the stress goes below theyield stress). The authors tried to form a more general picture of [R] and[M] problems for yield-stress fluids. The accuracy of their solutions wasverified both by mesh refinement and by calculation of the two functionalscorresponding to the maximum and minimum variational principles, whichwe have seen should be equal for the solution: these showed a nice agreementin [15]. They also calculated the critical limit (Yc) for a sphere and identifythe formation of a plastic boundary layer next to the sphere in this yield limit(the very first publication on the plastic boundary layer goes back to Oldroyd[102]). Also the Stokes limit of zero yield stress was been investigated byBeris et al. [15]. Apart from the fluid mechanics discussions, their numericalmethod was novel, as has been reviewed in Section 1.3.1.6.3 Drag and flow around single particlesFollowing Beris et al. [15], many scholars have worked on single particlemotion in yield stress fluids, both computationally and experimentally. Forinstance, Mitsoulis and co-workers [13, 17] investigated the wall effects onthe flow of Bingham and Herschel-Bulkley fluids over a sphere in cylindricaltubes. Identical results have been reported by Liu et al. [83]. In these nu-merical studies, the discontinuity in the constitutive equations was managedby the regularization method. The first augmented Lagrangian numericalstudy of the creeping flow of Bingham fluid past a 2D cylinder is due toRoquet and Saramito [115]. Many other 2D computations can also be men-tioned on this topic. For instance, Mitsoulis [93] has studied the size ofvarious unyielded regions around a 2D cylinder as a function of Binghamnumber and also developed drag coefficient relationships based on his nu-merical results. Similarly, Tokpavi et al. [134] studied the same problemand focused on the viscoplastic boundary layer about a 2D cylinder, ex-pressing the boundary layer thickness as a function of Od (equivalent to theBingham number here). The same problem for Herschel-Bulkley fluid was45investigated by De Besses et al. [34], finding the drag coefficient as a functionof the shear-thinning index and the Oldroyd number. They also studied theeffect of partial slip on the drag force and confirmed that the slip conditioncan reduce drag on the cylinder. Some studies have addressed the Poiseuilleflow [147] and inertial flows [96] of yield-stress fluids past particles.Performing experiments on these problem is not a trivial-task as thereproducibility of the terminal settling velocity of particles falling in a vis-coplastic fluid is poor [60, 113]. This finds its root in rheological complexityof yield-stress fluids, as pointed to briefly in Section 1.2. For instance, insome of the experimental studies [5, 6, 60, 132], the objective was to find thedependence of drag coefficient on the Bingham number (or Oldroyd number)and power-law index for falling spheres in yield-stress fluids in unconfinedand confined configurations.1.6.4 Finding Yc for particles in a yield-stress fluidAs mentioned previously, finding the limiting force which should be exceededin order to move a particle in a yield-stress fluid was an open questionfrom the time of Boardman and Whitmore [19] It can be calculated fromnumerical studies and also can be measured in experiments.There are two ways of studying particle motion, and hence the limit-ing flows. One may impose the particle velocity and calculate the forceon the particle: the resistance problem [R]. Alternatively, one imposes the(buoyancy) force on the particle and calculates the (settling) velocity. If theparticle velocity is non-zero there is a 1-to-1 correspondence between thesetwo problems [112]. Either method may be used to calculate the limitingforce. The plastic drag coefficient (CpD = FˆD/τˆYA⊥) approaches a con-stant value when the particle motion is stopped (CpD → CpD,c correspondsto B →∞ in a [R] problem and Y → Y −c in a [M] problem). For example,Tokpavi et al. [134] have used the [R] problem to find the critical plasticdrag coefficient for a 2D circle by going to a very large Bingham numberlimit (Oldroyd number in their notation; see Figure 1.16). Beris et al. [15]used same method to calculate it for a sphere and Nirmalkar et al. [100] for4674 D.L. Tokpavi et al. / J. Non-Newtonian Fluid Mech. 154 (2008) 65–76Fig. 20. Change in tangential stress r on the cylinder for Od=10; 104; 105 and2×105 .at the equator ( =0), remains higher than the yield stress in thefluid zone, quickly decreases on entering the rigid zone Zr2 anddisappears on the axis of flow ( =/2).The distribution of the maximum shear stress on the cylinderbecomes independent of Od in the case of very high-Od values. Thehighest value of the maximum shear stress is then of the order of0.4.4.2. Drag coefficientFig. 23 shows the changes in drag coefficients C∗das a functionof Od in the case of the present study and other studies carried outby other authors. The results obtained in the present study showa decrease in the drag coefficient C∗dwhen Od increases, with anasymptotic tendency at very high values of Od.This trend is similar to those obtained by Adachi and Yosh-ioka [9] with the maximum principle and by Mitsoulis [11] withPapanastasiou’s regularisation model.Fig. 21. Pressure profiles on the cylinder for Od=10; 104; 105 and 2×105 .Fig. 22. Maximum shear stress on the cylinder for Od=10; 104; 105 and 2×105 .By separating viscous effects from plasticity effects, the dragcoefficient C∗dcan be written as a function of Od as follows:C∗d= C1+ C2OdC3, (16)inwhichC1,C2andC3are constants andC1 thedragwhenplasticityeffects becomepreponderant. Throughnon-linear regression of theresults, the following are obtainedC∗d= 11.98+ 20.43Od−0.68, Od∈ [10; 2× 105], (17)andC∗d= 11.94+ 13.64Od−0.55, Od∈ [7.5× 103; 2× 105]. (18)When plasticity effects become preponderant, i.e. when Odbecomes very high, the drag coefficient is constant, and in this caseis C∗d∞= 11.94.Fig. 23. Change in drag coefficients C∗das a function of Oldroyd number. C∗disobtained by the present study and others’ studies.Figure 1.16: Change in the computed plastic drag coefficient as afunction of Od for a 2D circle. Reproduced from [134].a 2D square. However, Putz and Frigaard [112] computed directly Yc for a2D ellipse for a wide range of aspect ratios by solving the [M] problem.Measuring CpD,c also has been the objective of many experimental stud-ies. Examples can be found in the work of Magnin and his co-workers:Ahonguio et al. [2] measured CpD,c for cones with different aspect ratios andJossic and Magnin [72] for various shapes (see Figure 1.17). Moreover, Tok-pavi et al. [136] validated the previously calculated CpD,c ≈ 11.94 for a 2Ddisk by conducting experiments with a very long cylinder moving in a highyield-stress Carbopol gel. Note that is particularly difficult to have quan-titative results that agree with theory/computation. First, experimentallythe particle surface roughness can have a significant effect. Secondly, the lowstrain limit is where idealized rheological descriptions of yield stress fluidsare the least effective at describing the rheology of real fluids.The limit Y → Yc is often studied through a steady Stokes flow and has47Figure 1.17: Measured Yc for different particle shapes by Jossic andMagnin [72]. Reproduced from [72].the meaning of a yield number limit above which the steady velocity solutionis zero (no motion). This is sometimes referred to as a static stability limit.However for single particle systems, Y > Yc also corresponds to a dynamicstability limit above which (even inertial) particle motion decay to zero in afinite time as shown theoretically by Wachs and Frigaard [142]. In [142] thisis also illustrated for two computed examples: a 2D disk and a rectanglewith aspect ratio 2. The particles are initially moving with an assignedvelocity but because Y > Yc, the yielded regions progressively shrink withtime until only thin layers remain and finally both particle and fluid motionapproach zero.1.6.5 Yc in other yield-stress fluid flowsCritical yield numbers are not only a feature of particle settling in yield-stress fluids, as many other flows involve a balance between an imposedforce (stress) and the yield stress. Thus flow onset below a suitably definedcritical Y is an essential feature of very many viscoplastic flows, includingthe most simple 1D flows, such as Poiseuille flows in pipes and plane channelsor uniform film flows, e.g., paint on a vertical wall.The first systematic study of critical yield numbers was carried out byMosolov & Miasnikov [94, 95] who considered anti-plane shear flows, i.e.,48flows with velocity u = (0, 0, w(x1, x2)) in the x3-direction along ducts (in-finite cylinders) of arbitrary cross-section. These flows driven by a constantpressure gradient only admit the static solution (w = 0) if the yield stress issufficiently large. Amongst the many interesting results in their works, thekey contributions relate to exposing the strongly geometric nature of calcu-lating the critical yield number Yc. Firstly, they show that Yc can be relatedto the maximal ratio of area to perimeter of subsets of the cross-section(say Ω). Secondly, they develop an algorithmic methodology for calculatingYc for specific symmetric Ω, e.g., rectangular ducts. This methodology isextended further by Huilgol [67].The context of settling particles in anti-plane shear flows (meaning in-finitely long solid cylinders within Ω) has recently been analyzed by Frigaardet al. [49]. As with Mosolov & Miasnikov [94, 95] these limiting flows may beevaluated geometrically and calculated exactly for many simple particle andfluid domain shapes. Unlike the flows considered in this thesis, the limitingflows are non-local, affecting the entire fluid domain.Critical yield numbers have also been studied for many other flows, usinganalytical estimates, computational approximations, and experimentation.Critical yield numbers to prevent bubble motion are considered in [39, 138].Natural convection is studied in [75, 77]. Slope stability is also a classicalproblem in geotechnical engineering, and the failure of a vertical embank-ment is one model problem to address that [64, 108]. The onset of landslidesare studied in [61, 65, 69] (where the terminologies ‘load limit analysis’ and‘blocking solutions’ have also been used). As mentioned for single particleflows, one stepping away from the Stokes flow context, the critical Yc canalso serve as a stability limit for transient flows, including those with inertia.This has been investigated for wide classes of interior flows by Karimfazliand Frigaard [76]. Thus, determination of Yc has additional significance.1.6.6 Multiple-particle flows in yield-stress fluidsThe flow around multiple particles has been addressed many times in lit-erature as well. For instance, inline configurations have been studied both49computationally [71, 82, 135] and experimentally [74, 90]. Liu et al. [82]were first to study multiple particles, considering 2 co-axial spheres in Stokesflow ([R] problem). At fixed B they identified 3 regimes: the spheres mov-ing in separate yielded envelopes for (separation distance to particle radiusratios) ` = Lˆ/Rˆ > 5.5; the spheres moving unattached in the same envelope(4.5 < ` < 5.5); the spheres connected by a bridge of unyielded fluid ` < 4.5.Jie and Zhu [71] computed drag coefficients for pairs of (and arrays of upto 7) co-axial spheres moving in a Herschel-Bulkley fluid, including initialexploration of inertial effects. Jie and Zhu [71] show that for small Re thedrag on forward and rear particles in a pair differs and that even for Re 1in a uniform array of 7 particles the drag on the particles reduces towardsthe centre. Tokpavi et al. [135] conducted an in-depth computational studyof the (2D) flow around 2 translating cylinders in Stokes flow of a Binghamfluid, providing detailed description of the yield surfaces, drag coefficientcomputations and estimates of the yield limit.Flow past a series of cylindrical obstacles (i.e., fixed particles) has beenstudied by many authors [99, 130] and particularly in the context of modelporous media flows [7, 8, 18, 126]. In general, the resistance formulationhas been used in these studies which results in different drag coefficientsfor different particles. Limit load analysis of multiple-piles in the soil wasthe objective of a series of studies as well [50–52] which can be used forcalculating multiple-particle Yc in yield-stress fluids.Studying a suspension of yield-stress fluids numerically seems a very timeconsuming task since the numerical algorithms for these type of fluids arein any case very slow and particularly in 3D. The framework for this typeof computation (e.g. using a fictitious domain method) has been developedand tested for small numbers of particles by Yu et al. [146], but extensions tolarge numbers of particles has only been addressed for other non-Newtonianfluids by Yu and Wachs [145]. We do not know of any other studies thatreliably compute transient suspension flows of yield-stress fluids.However, yield stress suspensions have been studied theoretically/exper-imentally in a series of papers by Ovarlez and Chateau (specifically from therheological point of view). For example, Mahaut et al. [87] studied experi-50mentally the behaviour of suspensions of noncolloidal particles in yield-stressfluids. They have found that the macroscopic properties of these suspen-sions relate to the mechanical properties of the suspending fluid and theparticle volume fraction. Moreover, they showed that the elastic modulus-concentration relationship follows a Krieger-Dougherty law [79],G′(φ) = G′(0)1(1− φ/φm)2.5φm , (1.78)where φ is the volume fraction and φm = 0.57. It may not be surprisingas the Krieger-Dougherty law is usually found to fit most viscosity data ofsuspension of non-colloidal particles in Newtonian fluids as also confirmed byOvarlez et al. [103]. They also have shown that the yield stress-concentrationrelationship can be obtained simply from the elastic modulus-concentrationrelationship,τˆY (φ)τˆY (0)=√1− φ(1− φ/φm)2.5φm . (1.79)In the case of a Herschel-Bulkley suspending fluid, Chateau et al. [27], usinga theoretical framework, has shown that the properties of the suspension atthe macroscopic scale can be satisfactorily modelled as that of a Herschel-Bulkley suspending fluid with the same power-law exponent:µˆhom =τˆhomY‖ˆ˙γ‖ + Kˆhom‖ˆ˙γ‖n−1, (1.80)withτˆhomY = τˆY√(1− φ)g(φ), Kˆhom = Kˆg(φ)(g(φ)1− φ)n−1/2.This form of closure should be valid, independent of the choice of g(φ),which denotes the coefficient of proportionality, linking the macroscopic vis-cosity to the microscopic viscosity. Furthermore, by comparing these model51expressions with experimental data, they have reported that,τˆhomYτˆY=(1− φ)1/2(1− φ/φm)1.25φm , (1.81)and,KˆhomKˆ=(1− φ)(1−n)/2(1− φ/φm)1.25φm(n+1), (1.82)are well fitted to experimental measurements. This scaling is also interpretedin terms of an amplification of the microscopic shear rates with respect tothe macroscopic shear rates.Vu et al. [141] studied the rheological behaviour of isotropic bidispersesuspensions of non-colloidal particles in yield-stress fluids and observed thatat the same volume fraction, both the elastic modulus and yield stress ofbidisperse suspensions are lower than those of monodisperse suspensions.Almost on the same road but with different measurement protocol and in-strument, very recently, Dagois-Bohy et al. [33] have investigated the rheol-ogy of non-colloidal neutrally buoyant solid spheres in a yield-stress fluid inthe dense regime. Measurements for both the shear stress and the particlenormal stress have been obtained which agree well with the expression de-rived by Chateau et al. [27] but with a different coefficient. However, theyreported that when φ→ 0, that expression loses its accuracy which shows amore complex variation over the full range of volume fraction.In a more fluid mechanics/settling direction, we can also mention Ovarlezet al. [104], in which stability of coarse particles under the action of grav-ity and shear is investigated by using MRI techniques. They studied thetime evolution of the particle volume fraction during the flows in a Couettegeometry. As intuitively may be clear, they observe that shear induces sedi-mentation of the particles which are stable at rest (i.e., no shear case). Thesedimentation velocity was observed to increase with increasing shear rateand particle diameter, and to decrease with increasing yield stress of the sus-pending fluid. Moreover, the effect of volume fraction on the sedimentationvelocity has been addressed in their work.Particle transport processes in yield-stress fluids such as pipelining of52mine tailings, removal of drilling cuttings from oil wells, etc. are of cru-cial importance in industrial application. There are many studies in thetechnical literature on these subjects, but fewer that focus on fundamentalaspects. Merkak et al. [91, 92] experimentally studied the dynamics of par-ticles suspended in a viscoplastic fluid flowing along a pipe. Three differentpossibilities for the motion of particle were identified by them, as follows:(i) Particles situated in the plug zone will move with a translational ve-locity which is not necessarily the same as the plug velocity for asimple pipe (channel) in a sense that the particle itself will alter theflow. However, the translational velocity of particles are independentof their position in the plug zone. By reducing the size of the particle,they achieved larger translational velocities. No rotation or migrationwere reported for this case.(ii) Particles situated completely or partly in the sheared zone will trans-late and also rotate with velocities that are functions of yield stressand pressure drop (Oldroyd number). For small particles, migrationtoward the centreline of the pipe is recorded. Although particles mi-grate toward the plug zone, no penetration into the plug zone wasobserved. This case is sensitive to the position of the particles in thesheared zone in the sense that for those which are close to the wall nomigration has been detected.(iii) Particles which are very close to the pipe wall will adhere to the walland not move.The 3 cases above are results which have been reported for neutrally buoyantparticles. For the case in which particles are heavier than the fluid, but stillstatically stable, sedimentation was not observed in the plug zone. In thesheared zone particles settled by bypassing the plug zone and reached theirfinal position at the bottom of the pipe.531.6.7 Open problems, issues and gapsHere we highlight some gaps in the literature and motivations for the ap-proaches taken in this thesis.Variational methodsThe velocity minimization and stress maximization principles for yield stressfluids have been known since the 1950s and their application to particlesettling was pioneered by Adachi and Yoshioka [1]. However, systematicexploitation of these principles has not evolved much since that time. Thevelocity and stress field estimates of Adachi and Yoshioka [1] were crude(viewed in the light of modern computations), leading to a large uncertaintyin their solutions. Indeed the gap between lower and upper bounds of thedrag force is relatively large and grows for large Bingham numbers, whichmakes it questionable for the yield limit. There is also no clear path togenerating estimates for different shapes.Since computational methods such as the augmented Lagrangian methodare essentially variational, one could interpret numerical solutions as oneway of generating approximate velocity and stress fields, and these certainlyare accurate and useful for the yield limit, as we show later in the thesis.Nevertheless, there remains an interest in having a constructive analyticalway of generating estimates for different particles.Variational methods were used in [112] to show that in the critical limitof zero flow, the viscous dissipation converges at least one order of magnitudefaster than the plastic dissipation, which balances with the rate of work dueto the buoyancy forces. This has two consequences. First, this promptsone to consider a (potentially simplified) flow problem in which the viscousterms are ignored, but not the yield stress terms, which enters the realm ofperfect plasticity [66], as we discuss below. Secondly, though understandingthe structure of the limiting flows, we may be able to make (leading order)estimates of the plastic dissipation and rate of work. These in turn can leadto estimates of Yc.54Perfect plasticityAlthough [15, 112, 134] have all pointed to the potential advantages of usingplasticity theory and especially the characteristics/slipline method for ana-lyzing the yield limit of the particle motion in viscoplastic fluids, no in-depthsystematic attempts have been made. One of the best non-trivial examplesof using the characteristics method for calculation of the collapse load is thestudy of [114], who constructed the characteristic network around a circularpile to calculate the lateral resistance to push the pile through surround-ing soil. The more recent studies of [134] and [112] calculated plastic dragcoefficient numerically for a circular cylinder moving in a Bingham fluidand reported the same limiting drag coefficient as given by the analysis of[114], for the fully rough pile. This leads to the hope that the characteris-tics method might be more widely applicable. However, later attempts topredict the limiting drag force by the characteristics theory were not suc-cessful. In [112] the numerical calculations for long prolate ellipses differconsiderably from those that they calculated by generalizing the expressionin [114]. Moreover, Nirmalkar et al. [100] calculated drag coefficients thatare not close to the estimates of [78].There is thus a clear need to revisit the relevance of perfect plasticity tothe limiting flows of planar 2D flows around particles in yield stress fluids.This has different perspectives. (a) Does the perfectly plastic flow alwaysmatch (or approximate) the limiting viscoplastic flow, and how? (b) Arethe stress and velocity fields generated from the method of characteristicsgood as variational estimates (both for the perfectly plastic problem andthe limiting viscoplastic problem)? (c) What can the methods of perfectplasticity tell us about the shape of unyielded envelope around a particleand also pressure and stress distribution about the particle? These questionsmotivate much of chapters 2 and 3).Computational protocolsGiven the inaccuracy of e.g. Nirmalkar et al. [100] in what might appear tobe a trivial 2D computation, we believe there is a need to establish some55form of computational protocol for this type of problem. Two issues seemto be overlooked. First, Frigaard and Nouar [47] have shown that usingregularization techniques in this limit are not recommended for two reasons:(i) there is no guarantee of stress convergence; (ii) regularized flows do not“stop”. Secondly, if using problem [R] there is an additional convergenceissue, i.e. how large should B be to be close to the yield limit. These twoissues in themselves afflict many conventional computations of these flows(and yield limits), so that there are only very few numerically accurate yieldlimit studies for particles that use both problem [R] and regularization,e.g. as in [134].A supplementary issue that we believe important is the use of meshadaptivity, as we do later in the thesis. The limiting flows have stronggeometrical features which are not well represented by regular meshes. Theleading order contributions (to e.g. the plastic dissipation) often occur alongboundary layer surfaces, so again it is important that the numerical methodcan adapt to these features. Interestingly, the perfectly plastic solutions havesimilar geometric structures. As we shall see, for some flows the perfectlyplastic solutions are exact for the limiting flows and in these cases make goodanalytical test cases. Thus, although not specifically a computational thesis,we believe that the methods developed and used extensively in this thesisrepresent a reliable and accurate method for this type of computations.Multiple particlesRegarding multiple particle flows, one main drawback is in the [R] formu-lation that has been used for most of the single particle studies. Whendealing with multiple particles the resistance formulation is fundamentallylimited, e.g., flow of an array of cylinders in the resistance formulation isequivalent to flow past a series obstacles. If we want to gain some insightto suspensions of yield-stress fluids and hydrodynamic interaction of parti-cles, we need first to solve the [M] problem in these cases. The one-to-onecorrespondence between mobility and resistance formulations in Stokes flowis in general lost for multi-particle flows and only the mobility formulation56is relevant for suspensions. Moreover, although interesting semi-empiricallaws are proposed for rheology of suspensions in yield-stress fluids, we havelittle systematic development building from a fluid mechanics perspectivewhich can be founded on multiple particles dynamics. This is one of theobjectives of chapter 5.1.7 Objectives and outline of the thesisA key objective of this thesis is to develop a general way to analyze thestability/yield limit of a settling particle, using a mix of numerical andanalytical methods. Beyond that we would like to construct a general pictureof the case of having multiple particles and hydrodynamic interactions inyield-stress fluids. This thesis makes the following main contributions:(i) Analyzing the yield limit for some simple particle shapes using stan-dard numerical techniques of viscoplastic flows and also method ofcharacteristics from rigid-perfect plasticity (as just outlined).(ii) Connected with (i), we consider common features of families of parti-cles (ellipses, rectangles, diamonds) as their geometric aspect ratio χ isvaried between 0 and∞. We derive analytical approximations to theseasymptotic limits and show where the perfect plasticity estimates areaccurate.(iii) Unyielded envelope: is very important concept in studying yield limitand hence stability of particles. This envelope consists of the particleand any unyielded fluid that is trapped in contact with the particleas the yield limit is approached. It is the unyielded envelope thatis related to yield limit. The same unyielded envelope, containingmaterial of the same mean density, will have the same yield limit.However, the same unyielded envelope can cloak (or hide) particlesof different shapes. This cloaking phenomenon is explicitly studied:different particle shapes can have the same unyielded envelope aroundthem.57(iv) Unyielded envelope rule: we develop a rule that can be used to findthe unyielded envelope around 2D particles with at least one line ofsymmetry.(v) Multiple particles: still in the Stokes flow limit we begin to categorizethe many different configurations that can arise. Here we confine themotion to co-linear or inline motions of 2D circular discs.(vi) Plug regions can appear between the particles and connect them to-gether, depending on the proximity and yield number.(vii) The yielding behaviour of an assembly of particles can change signif-icantly, since the combination forms a larger (and heavier) ‘particle’.Thus, the notion of using the single particle Yc to characterize evensimple multi-particle systems is false and for suspensions this remainsa distant goal.(viii) Small particles (that cannot move alone) can be pulled/pushed bylarger particles or an assembly of particles. Large number of particlesleads to interesting chain dynamics, including breaking and reforming.Perhaps surprisingly it appears very difficult to have a moving clusterof more than 3 particles, at least in an inline configuration.1.7.1 OutlineThis thesis is composed of four main chapters. We start in the next chapter(Chapter 2) by formulating yield limit problem specifically for 2D planarflows. Three main symmetric shapes are studied, considering a wide rangeof aspect ratios: 2D ellipse, rectangle, and diamond. We show how to usemethod of characteristics to analyze yield limit and its benefits/limitations.Also we develop the numerical methodology used later in the thesis.In Chapter 3 we again consider single particles, but with more complexgeometries. We find an unyielded envelope rule which can find the shape ofthe moving object (particle+attaching unyielded regions) accurately. Know-ing the unyielded envelope around the particle and using the method of58characteristics, we will analyze the yield limit in-depth and generalize thefindings of Chapter 2 for particles with just one line of symmetry.In Chapter 4, we partly address the effect of particle orientation onstatic stability and also particle motion far from the yield limit. A simple,yet reasonable model (for at least wide range of problem parameters) is alsoproposed for this type of problems based on decomposition.In Chapter 5, we re-formulate problem [M] for the case of having morethan one particle, solve the governing equations for the case of inline motionof multiple disks, analyze the yield limit of case of two particles, and revealsome interesting effects in this case. Then we will continue with study-ing cases where we have more than two particles, considering both steadyflows and the dynamics of chains of particles. At the end we move towardcategorizing hydrodynamic interaction of particles in a yield-stress fluid.In Chapter 6 we summarize the work conducted, draw conclusions, andpresent recommendations for future research.59Chapter 2Stability of particles and theyield limitThe main focus of this chapter1 is the systematic study of the effects ofshape and aspect ratio on the limiting 2D flows of planar symmetric particlesthrough a Bingham fluid. In doing so we begin to develop a methodologythat combines high accuracy computations, variational methods and thetheory of perfect plasticity. These same 3 elements are present in Beriset al. [15], which is the seminal work in the arena of single particle dynamicsin yield stress fluids.The planar analogue of the sphere of [15] is a settling disk. The analyti-cal solution of the yield limit here comes from soil mechanics and is derivedusing perfect plasticity theory; see Randolph and Houlsby [114]. Later theapplicability of this analysis to the limiting settling disk in a yield stressfluid was confirmed in [112, 134], which employed computations of bothproblems [M] and [R], as well as variational methods. The combinationof perfect plasticity, computational and variational methods is thus natu-ral. The restriction to 2D planar geometries allows perfect plasticity to beused optimally. The geometries we consider in this chapter are convex andsymmetric: extensions from the circular disk.1The results of this chapter have appeared as: E. Chaparian, I.A. Frigaard, Yield limitanalysis of particle motion in a yield-stress fluid, J. Fluid Mech. 819 (2017) 311–351.60An outline of the present chapter is as follows. Below in Section 2.1we set out the general problem studied, introduce the variational terminol-ogy, the characteristics method, the computational tools and other basicfeatures. We start by re-examining the circular cylinder flow of [112, 134]and the characteristics solution of [114], showing the differences in the solu-tions even though the limiting plastic drag coefficients agree. This promptsexamination of three families of particles: ellipses, rectangles and diamonds,both from the perspective of the limiting viscoplastic flows (computed andestimated) and approximations from the method of characteristics; see Sec-tion 2.2 to Section 2.4. We find that the lower bound characteristics methodgenerally gives an accurate approximation to Yc, but is only exact for cer-tain specific geometries. We draw together the general features of limitingviscoplastic solutions in Section 2.5, in particular studying the common fea-tures of these particles with large and small aspect ratio χ, and of flows inthe limit Y → Yc. The chapter ends in Section 2.6 with a brief discussion.2.1 Problem statementThis chapter focuses on the motion of isolated symmetric two-dimensional(2D) particles in a viscoplastic medium. The flow is assumed to be inertialessand isothermal. The particle is denoted by X; ∂X is the boundary of theparticle; Ω and ∂Ω represent the entire domain (fluid+particle) and its outerboundary, respectively (see Figure 2.1).We align the gravitational acceleration with the positive y-direction: gˆey.Considering a circular disk as a reference geometry, we select a length-scaleLˆ such that:piLˆ2 =∫Xdxˆdyˆ = Area of X,i.e., Lˆ is the radius if X is circular. The dimensionless governing and con-stitutive equations are the same as Equation 1.19 and Equation 1.20. Re-garding the boundary conditions, in the far-field we specify,u = 0, on ∂Ω \ ∂X, (2.1)61xyv0ParticleFluiddomaingΧΩϒRigid 𝑔𝑦𝑥Ω𝑋Figure 2.1: Schematic of the particle motion.and at the particle surface (∂X) the fluid velocity is continuous with thatof the particle:u→ U , on ∂X, (2.2)but U is unknown. In this section we consider only particles that are settlingin the y-direction. Provided the particles have two orthogonal symmetryplanes, there is no torque exerted on the particle and only linear motion ofthe particle is possible. Provided that the symmetry axes orient with eythere is no drift component of velocity. Thus, we may write U = Uey.The stress at the particle surface must satisfy the following force balance:∫∂Xσ · n ds = pi1− ρey, (2.3)where n is the normal of ∂X, pointing inward to the particle (recall that pi isthe dimensionless area of the particle). Note here that the net force (per unitlength) is also in the y-direction, by consequence of the force equilibrium inthe x-direction and the symmetry of the particles.Note that since the particle velocity U is also divergence free, there isno net volumetric flux across Ω in any direction. A particular consequenceof this is: ∫Ω\X¯u · ey dA = −∫X¯U · ey dA = −pi U · ey = −piU. (2.4)62The space of admissible velocity solutions for this problem is,V = {v :∇ · v = 0, in Ω \ X¯;v = 0, on ∂Ω \ ∂X; v → V = V ey, on ∂X},where v ∈ V. The buoyancy work functional is:L(v) = piV = pi V · ey = −∫Ω\X¯v · ey dA. (2.5)The energy equation is now derived by taking the dot product of the velocityand Stokes equations, then integrating over Ω \ X¯. On using the divergencetheorem, equations (2.3) & (2.4), we finda(u,u) + Y j(u) =ρ1− ρ∫Ω\X¯u · ey dA+∫∂Xu · σ · n ds= − ρ1− ρ∫X¯U · ey dA+∫∂XU · σ · n ds= − piρ1− ρU · ey +U ·∫∂Xσ · n ds = piU = L(u).(2.6)Below (2.6) is used to define the critical yield number.The velocity minimization principle states that the solution u minimizesthe functional J (v) over v ∈ V :J (v) = 12a(v,v)+Y j(v)− ρ1− ρ∫Ω\X¯v ·ey dA−∫∂Xσij(u)njVi ds. (2.7)Note that, as V is constant (2.4) applies to the pairing (v,V ), and on using(2.3):− ρ1− ρ∫Ω\X¯v·ey dA−∫∂Xσij(u)njVi ds =piρV1− ρ−V ·∫∂Xσ(u)·n ds = −L(v),so that equivalently: J (v) = 12a(v,v) + Y j(v)− L(v).For the second variational principle, consider the space of admissible63stresses S:S ={σ′ = τ ′ − p′I :∇ · σ′ + ρ1− ρey = 0 in Ω \ X¯;∫∂Xσ′ · n ds = pi1− ρey},i.e., symmetric stress tensors satisfying the Stokes equations and the forcebalance (2.3). The solution σ maximizes over σ′ ∈ S the functional:K(σ′) =∫∂Xσ′ijnjUi ds−12∫Ω\X¯(‖τ ′‖ − Y )2+ dA. (2.8)2.1.1 Yielding of flowsAs it was mentioned an important feature of yield-stress fluids is their abilityto hold particles statically stable under certain circumstances. The ratioτˆY /τˆb is the yield number and the above argument predicts stationary flowfor sufficiently large Y . We can verify this intuition analytically, using thevariational tools.Secondly, we might re-examine the maximization (2.8). Since each ad-missible σ′ satisfies (2.3), we may also write:K(σ′) = 11− ρL(u)−12∫Ω\X¯(‖τ ′‖ − Y )2+ dA. (2.9)The first term is independent of σ′ ∈ S and the second term is negativedefinite. Any admissible stress field with ‖τ ′‖ 6 Y will therefore maximizeK. Equally if any such σ′ ∈ S can be found, then it follows that the truestress must also have ‖τ‖ 6 Y . This in turn implies that the solutionstress does not exceed the yield stress, hence that the strain rate is zeroeverywhere, and consequently that u = 0. Via the equivalence (1.13), wesee that K(σ) = J (u) = 0, in this limit. Thus, the condition for theparticle not to yield the fluid is independent of the precise type of fluid(Bingham, Casson, Herschel-Bulkley, Robertson-Stiff, etc.): any viscoplasticfluid remains preferentially static.The energy equation (2.6) can be used to define the minimal limiting64yield stress for static flow. Assuming u 6= 0, we have,a(u,u) = L(u)− Y j(u) = j(u)[L(u)j(u)− Y]6 j(u)[supv∈V, v 6=0L(v)j(v)− Y]or,Yc = supv∈V, v 6=0L(v)j(v), (2.10)with u = U = 0 if Y > Yc, [112]. As we have shown in Section 1.4.1 theviscous dissipation of the solution u will decay at least one order faster thanthe plastic dissipation. Returning to (2.6), we see that the plastic dissipationmust balance with the buoyancy work as Y → Y −c , i.e.Y j(u) ∼ L(u) = piU, (2.11)giving us directly the decay of the settling velocity as Y → Y −c .2.1.2 Mapping between problems [M] & [R]In order to have a correspondence between problems [M] and [R] we alignthe direction of particle motion in [R] problem with ey, assumed to be asymmetry axis of the particle. The mapping between problems [M] and [R]is captured by:u =YBu∗, U =YB, piBY= F ∗, (2.12)where the scaled force exerted on the particle has magnitude F ∗ and is indirection ey.One reason for introducing [R] is that it may be more convenient tocompute the solution of problem [R] (rather than [M]), and use this toevaluate Yc. In controlling relative numerical errors, the velocities in problem[R] are strictly O(1). Secondly, as argued by [112], optimality of L(v)/j(v)in (2.10) is attained by the solution u to [M], in the limit that Y → Y −c .On rescaling u with U , we arrive at the equivalent solution to [R], for B =Y/U . The functional L(v)/j(v) in (2.10) also rescales and thus Yc may be65calculated alternatively as:Yc = limY→Y −cL(u)j(u)= limY→Y −cpiUj(u)= limB→∞pij(u∗). (2.13)Although finding the solution u∗, as B → ∞, may become numericallydifficult, at least j(u∗) is an integral quantity and should be more stable tocompute than a pointwise quantity, allowing the asymptotic behaviour atlarge B to be estimated effectively from a numerical solution.A different way of evaluating the critical stopping limit, that is useful forcomparing between problems, involves the drag coefficient. Conventionally,a drag coefficient is defined by scaling the force on an object with an inertialstress scale and cross-sectional area. Here we consider inertialess flows closeto yielding. The plastic drag coefficient instead uses the yield stress in placeof an inertial stress. For the 2D flow, in either formulation:CpD =[Fˆ ∗ˆ`⊥τˆY][R]=[F ∗`⊥B][R]for problem [R],[ρˆgˆAˆpˆ`⊥τˆY][M ]=[pi`⊥Y][M ]for problem [M],(2.14)where ˆ`⊥ = `⊥Lˆ is the linear dimension of the particle perpendicular tothe flow direction. Other similar problems may be compared by scaling theparticle force Fˆ , as above with ˆ`⊥τˆY . We see from the above again thatF ∗ = piB/Y . As we increase the yield stress and approach Y → Y −c , theplastic drag coefficient approaches a critical value:CpD,c =pi`⊥Yc=[CpD][M ]Y→Y −c =[CpD][R]B→∞ . (2.15)Critical values CpD,c have been reported for various particles with simplegeometry. For example, [134] calculated CpD,c for a circular disk and [100]calculated CpD,c for a square. Both studies used the resistance formulation[R] and extrapolated to the critical value. Alternately, [112] calculated CpD,cfor 2D elliptical particles using problem [M].66and the space Qh of continuous piecewise linear functions ðP1 ÿ C0Þ. Thus Dh ¼ Th ¼ DðHhÞ is the space ofdiscontinuous piecewise linear function ðP1 ÿ Cÿ1Þ. Fig. 4 represents the mixed finite element approxima-tion. It can be shown (see [20,21]) that the discrete problem ðFV Þh admits a solution with uh and dh uniquelydetermined. This quadratic approximation of the velocity field leads to high precision results and is well-suited to exhibit fine flow patterns.3.3. Resolution of the Stokes subproblemThe finite element method leads to a discrete version of Algorithm 3.1. At step 1, we get a Stokes problem(9)–(13) discretized by the Taylor–Hood element. This gives the classical matrix problem:A BTB 0 WP ¼FG which is solved again by an Augmented Lagrangian algorithm [18].Algorithm 3.2 (Uzawa for Stokes).initialization k ¼ 0Let P 0 arbitrarily chosen.loop kP 1• step 1: Let P k being known, find W k such thatðAþ ~rBTBÞW kþ1 ¼ F þ BTGÿ BTP k:• step 2:P kþ1 :¼ P k þ ~rBW kþ1:The parameter ~r is fixed at 107, i.e. the inverse of the square root of the machine precision, in order toensure very fast convergence of this subproblem. For the computation of W kþ1, the matrix Aþ ~rBTB isfactorized once and for all, before the global iteration of Algorithm 3.1, with a LDLT method.3.4. Using quadrature formulae for the constitutive equationThe explicit computation of cnþ1 at step 2 of Algorithm 3.1 is characterized as the solution of the fol-lowing minimization problem:cnþ1 :¼ argmind2Tðgþ rÞZXkdk2 dxþ r0ZXkdkdxÿZXðrn þ 2rDðunþ1ÞÞ : ddxFig. 4. The finite element approximation.3324 N. Roquet, P. Saramito / Comput. Methods Appl. Mech. Engrg. 192 (2003) 3317–3341Figure 2.2: Finite element approximation. Reproduced from [116].2.1.3 Computational methodWe solve the present problem by finite element discretization of the aug-mented Lagrangian method. The saddle point problem is resolved usingan Uzawa algorithm. As it is stated in Section 1.3.3, assuming an initialvalue for the relaxed strain rate (γ0) and the Lagrange multiplier (whichbecomes the deviatoric stress tensor, T 0), we iterate the Uzawa steps se-quentially until the maximum of∫Ω |un+1 − un| dA,∫Ω ‖γn+1 − γn‖ dA,an∫Ω ‖γn+1 − γ˙n+1‖ dA is less than 10−7. Note that in [R] the velocitiesare of magnitude 1. The size of computational domain Ω is chosen so thatthe yielded region around the particle does not extend to the boundary, ∂Ω,and also making domain larger does not affect the results considerably.We have computed both the resistance problem [R] and the mobilityproblem [M], using the method in [112]. The majority of results presentedlater have been obtained from problem [R]. The discretization follows thatof [116] and [112] in terms of mesh adaptivity and choice of finite elementspaces. Figure 2.2 shows the elements we have used for velocity (P2 con-tinuous), stress (P1 discontinuous), relaxed strain rate (P1 discontinuous),and pressure (P1 continous). The implementation on triangular elementsand adaptivity are carried out within FreeFEM++ [62]. This formulationand computational method has been used for many years to solve 2D prob-lems. It is particularly effective for limiting problems in that the adaptivityfocuses the mesh on the yield surf ce. Benchma k results for a circular par-67YU × 10−60.1 0.11 0.12 0.13012Y[M ]c = 0.121Figure 2.3: Settling velocity of a square particle for increasing Y .ticle are addressed below. Convergence of problem [R] as B increases (i.e.,to evaluate Yc) is explored in Section 2.2.We outline typical calculations of problems [R] and [M], for a squareparticle, and illustrate mesh adaptivity. Other detailed results on conver-gence can be found in [112, 116]. To demonstrate how Yc is calculated ina [M] problem, Figure 2.3 shows the settling velocity of the square as thefunction of yield number Y . The critical yield number is shown by the redbroken line: Y[M ]c = 0.121, which corresponds to the CpD,c = 14.68. HereY[M ]c is determined simply via the criterion that the settling velocity is zerofor Y > Y[M ]c . The precision of the calculation is evidently determined bythe resolution in Y . The same mesh is used for solving the [R] problem atlarge B. Generally, we have very good agreement in the critical plastic dragcoefficients computed by solving either [M] or [R] problems (here ≈ 0.1%).The underlying mesh adaptivity and discretization is very similar to[116], who present numerous examples. To supplement these and to see theeffect of mesh adaptivity on convergence, we have solved a further classicalfluid mechanics test problem: Poiseuille flow. In Figure 2.4 the convergenceof the problem with mesh adaptation is plotted. We see little improvementafter 4-5 cycles, which is similar to that reported in [116]. Having verified68  0 5 1010−610−510−4L2L∞Figure 2.4: L2 and L∞ norm of the error versus cycles of adaptationfor Poiseuille flow0 4 86.56.51386.556.6 (a) (b)Figure 2.5: (a) j(u∗) for square case (dissipation in computationaldomain, i.e., one quarter of the physical domain) versus numberof adaptation cycles. Symbols are computed quantities and redbroken line is the limiting value (b) Part of the mesh generatedafter 8 cycles of adaptation for square particle.the improvement of the accuracy by adaptation, we now look at adaptationeffects for the present problem. Figure 2.5a shows that after nearly 5 cyclesof adaptation, the change in the calculated plastic dissipation (from whichwe compute Yc) is negligible. Nevertheless, we typically use 8 cycles ofadaptivity for the results in this study. Part of the generated mesh after 8cycles of adaptation is shown in Figure 2.5b. The number of triangles in thecomputational domain for this case is 56252.692.1.4 The yield limit and perfect plasticityFor the present discussion, the most interesting fact is that calculating ad-missible stress and velocity fields from analysis of the characteristics (orindeed any admissible field constructed by any other means) allow us tomake lower and upper bound estimates of the force on a particle translatingat unit speed within the plastic medium.In turn, these lead to estimates of the limiting plastic drag coefficientand hence Yc. Note that because we consider a 2D problem, the drag forceswe estimate below (and elsewhere in this section) are forces per unit depthof the particle in the ‘third z-direction’.• Lower bound: as outlined in (1.55) the load calculated by using anadmissible stress field is a lower bound for the actual force,[F˜ p]L =∫Γσ˜′ · n dS, (2.16)where Γ is the boundary of the region ∆ that consists of the particleplus any plugs attached to the particle surface and translating withthe same velocity. Evidently, Γ may be replaced by the boundaryof the particle (∂X). The characteristic network simply provides aconvenient method for computing an admissible stress field.• Upper bound: as outlined in (1.58) the mechanical work dissipated inany admissible velocity field v˜ is an upper bound for the actual force.In the present context of the [R] problem, i.e.,[F˜ p]U = B(∫Ω\∆¯‖˜˙γ ′‖ dV +∫Sd(v˜)|∆v˜| dSd(v˜)). (2.17)Here Sd(v˜) contains all curves that contain velocity discontinuities inΩ \ ∆¯ and ∆v˜ represents the velocity discontinuity vector. We maycompute an admissible velocity field in any way. Again we may replace∆ with X, above. In the [R] problem we are dealing with just Sv typeboundary condition. In other words, in the problem considered in70this chapter (resistance formulation) S is identical with Sv and hencethe last term on the right-hand side of Equation 1.58 vanishes. Theboundary conditions on Sv are constant: in the far-field we have zerovelocity. On the particle surface (plus attached potential rigid plugregions) we have unit velocity in the direction of motion (v˜ = 1 on ∂X).Therefore, an upper bound for the actual traction force can be foundfrom any admissible velocity field vˆ, as in Equation 2.17.If the lower and upper bounds are equal, the calculated drag force isexact. In most problems there is an uncertainty between the two boundsand this usually stems from the velocity field giving a poor estimate. Thiscan be reduced by finding a better admissible velocity field, but this becomeshard for complex geometries. A final point to acknowledge here is that theadmissible stress field that we calculate are incomplete. In constructing thecharacteristic meshes we typically find a bounded region around the particleon which the stress field (for calculating the lower bound) is resolved. Ourprimary interest is to make comparisons of the perfectly plastic flow in thisregion with that of the analogous viscoplastic flow, and in particular inthe limit of zero flow. Thus, we do not attempt to construct stress fieldsoutside of the characteristic net computed. It is simply assumed that thecomputed stress field can be coupled with such a stress field in the exterior.In cases where the perfectly plastic and limiting viscoplastic solutions appearto coincide, we comment that the viscoplastic solution has computed a stressfield with ‖τ‖ 6 B in the unyielded exterior region.2.1.5 A benchmark problem: a 2D disk particleRandolph and Houlsby [114] studied the geotechnical problem of forcinga circular pile laterally through soil, using the characteristics method toproduce analytical solutions for different adhesion factors at the soil-pile in-terface. Murff et al. [97] later found that the Randolph and Houlsby [114]solutions for partially rough piles were not exact. More recently, Martinand Randolph [88] reduced the uncertainty by proposing an alternative ad-missible velocity field. Here however, as our interest lies in comparing with71viscoplastic flow solutions (no slip), we are interested only in the fully roughsolutions of [114] which are exact, as verified by the agreement of their lowerand upper bounds. The plastic drag coefficient (= 4√2 + 2pi) computed by[114] has been verified by [134], who performed extensive computations us-ing a regularization approach to the [R] problem, finding a critical plasticdrag coefficient ≈ 11.94. Using the [M] problem [112] benchmarked theircalculations of a settling circular disk by computing a critical plastic dragcoefficient ≈ 11.94, suggesting good agreement between perfectly plastic andviscoplastic flows. However, we will show later that the two solutions (vis-coplastic problem and perfectly plastic problem) are not strictly the same.First we review the characteristics field found analytically by Randolphand Houlsby [114]; see Figure 2.6. Only a quarter of the circular geometryis presented. The following are the main features:• The maximum shear will happen on the x-axis. Hence the x-axis is astraight characteristic, QF (say α-line), and thus all the characteristicsfrom the same family will be straight (direct consequence of Henckyequations; see [66]). Indeed, Hencky equations dictate that the α-linesare straight for particles with two orthogonal lines of symmetry be-cause the symmetry line of the particle perpendicular to the directionof movement (x-axis) is a straight α-line.• The shear stress on the y-axis is zero. Therefore, the y-axis is in thedirection of principle stress and the characteristics intersect at an angle±pi/4, e.g., like AB.• The maximum shear occurs on the disk surface. Consequently, theα-lines are tangent to the disk surface, e.g., lines AB and QD. Pleasenote that this is an assumption as we are interested in no-slip conditionon the particle surface (or ‘rough pile’ as stated by Randolph andHoulsby [114]).• There is a singular point Q, which is the centre of a fan filling thespace between the last α-line tangent to the surface (QD) and thex-axis (QF ).72PABDF JHQθFigure 2.6: Characteristic network adjacent to a circle• In Figure 2.6 the dark shaded region shows the plug that is attachedto the front of the particle. In this region, both α- and β-lines arestraight: a region of constant state.Consequently, we can find different contributions of drag force with theadmissible stress field that the characteristics give us:• The shear stress component on QA:F˜1 =∫ pi40B cos θ dθ• The shear stress component on AB:F˜2 = B cos(pi4)• The normal stress component on QA:F˜3 =∫ pi40B [σ¯ + (pi + 2θ)] sin θ dθ73• The normal stress component on AB:F˜4 = B(σ¯ +3pi2)sin(pi4)where Bσ¯ designates the unknown mean stress through the equator α-line(i.e., QJF ). So far we have calculated the force acting on a quarter ofthe circle. For the total force, we repeat the same procedure to obtain theforces acting on the three remaining quarters and sum. The terms involvingBσ¯ cancel due to symmetry, whereas the other terms are added together.Therefore, the lower-bound of the critical plastic drag coefficient is,[CpD,c]L=[F˜ p]L`⊥B=4∑4i=1 F˜i2B= 2√2︸︷︷︸total shear+ 2√2 + 2pi︸ ︷︷ ︸total normal= 4√2 + 2pi.(2.18)A closer look into the normal contribution shows that ‘2pi’ is coming fromthe effect of stress singularity in point Q. In other words, it is the result ofchange of state of stress in centred fan DQF , while the ‘2√2’ is the result ofnatural change of normal stress in subsequent characteristic lines from QDto AB.We may find the upper bound by an admissible velocity field. Thevelocity field associated with the characteristics is governed by Geiringerequations. We find u˜α = 0 and u˜β = const. along the β-lines. The velocityboundary condition on the particle surface defines u˜β. Along the group ofβ-lines emanating from AB, u˜β = 1/√2 (note the particle velocity is 1),and for the β-lines emanating from QA, u˜β = sin θ, where θ is shown inFigure 2.6.Consequently, we can find different contributions of drag force with theadmissible velocity filed that the characteristics give us:• Work dissipation along QA:∆W1 = B∫ pi40cos θ dθ74• Work dissipation along AB:∆W2 = B sin(pi4)• Work dissipation along BD:∆W3 = B sin(pi4)∫ pi40(1 +pi4− θ)dθ• Work dissipation along DF :∆W4 = B sin(pi4)∫ pi40(1 +pi4)dθ3• Work dissipation in zone ABDFJHA:∆W5 = B∫ 3pi40dθ3√2• Work dissipation in zone AHQA:∆W6 = B∫ pi40∫ θ10[(θ1 − θ2) cos θ1 − sin θ1] dθ2 dθ1• Work dissipation in zone QHJQ:∆W7 = B∫ pi40∫ pi20(θ1 cos θ1 − sin θ1) dθ3 dθ1where θ1 and θ2 are just two different θ values that we chose to make theintegration simpler. Moreover, θ3 is the angle that the green arrow pointermakes with equator QF . Please note that A is the point on the circle surfacewhere θ = pi4 , i.e., the very beginning point of the rigid zone attached to theparticle surface; H is the point where the β-line emanating from A passesthrough QD; J is the point where this β-line intersect the equator QF . Itis clear that β-lines in the zone QHDFJQ are all arcs of circles. Putting allwork dissipations together and also take into account the three remaining75quarters of the domain, we will end up with the upper bound of the criticalplastic drag coefficient equal to 4√2 + 2pi which is strictly the same asthe lower bound. So it means that the plastic problem solution gives usCpD,c = 4√2 + 2pi ≈ 11.94. It is interesting that the viscoplastic problemalso give us same number but we now will show that the two solutions arenot exactly the same.Figure 2.7 plots the shear stress τ∗xy computed from the [R] problem atB = 104 (left) and that from the characteristics analysis τ˜xy (right). Bothshear stresses have been normalized with B. Firstly, we see that the envelopeof the characteristics does not agree fully with the outer yield surface of theviscoplastic computation. Secondly, we see that the shear stresses are differ-ent. The characteristics solution has a single rigid plug at the front, whereasthe viscoplastic solution also has a second plug along the side of the cylin-der. The frontal plugs also differ slightly in size. Figure 2.8 compares thevelocity along the x-axis. The perfectly plastic velocity has a discontinuityin the velocity at both the cylinder surface and at the outer characteris-tic surface. The viscoplastic fluid has no discontinuities. A narrow viscousboundary layer is found close to the cylinder, then a rigid plug, followed byyielding and slow deformation into a (near) constant downwards velocity,then a further viscous layer at the outside of the flow envelope. Given thedisparities in both stress and velocity fields it is perhaps surprising that thecomputational comparisons of the limiting plastic drag coefficients in [134]and [112] can be so close to the perfectly plastic values.Further investigation (not shown) reveals that the outer viscous layerdoes decrease very slowly in width as B → ∞, as does the sheared zonealong the y-axis at the front of the particle, (also they appear to decreasein width at the same rate with B). The equatorial velocities in Figure 2.8eventually match, although the viscoplastic solutions remain continuous andadopt a boundary layer profile. The rigid plug at the side of the particlein the left panel of Figure 2.7 remains as B → ∞. The reader is referredto [134] for a detailed quantitative description of such features. It is thusinteresting that although B = 104 is quite adequate for convergence of thelimiting plastic drag coefficients, (as we see below) other features of velocity76Figure 2.7: Normalized shear stresses: left half from computation ofthe [R] problem with B = 104, i.e. τ∗xy/B; right half from thecharacteristics prediction τ˜xy/B. Rigid regions are plotted gray.x − 1uy−0.5 0 0.5 1 1.5 2 2.5−0.500.51  Numerical computationVelocity in the upper bound mechanismFigure 2.8: Computed velocity in the y-direction ([R] problem withB = 104), measured with distance r from the cylinder surfacealong the x-axis.77Ellipse Rectangle Diamondˆ`⊥ 2aˆe 2aˆr 2aˆdˆ`|| 2bˆe 2bˆr 2bˆdLˆ√aˆebˆe 2√aˆr bˆr/pi√2aˆdbˆd/piˆ`||/ˆ`⊥ χ χ χ`⊥ 2χ−1/2 = 2ae√piχ−1/2 = 2ar√2piχ−1/2 = 2ad`|| 2χ1/2 = 2be√piχ1/2 = 2br√2piχ1/2 = 2bdTable 2.1: Dimensional and dimensionless parameters for the 3 ge-ometries. 2       al= blla 𝑎𝑎𝑏𝑙2 𝑎𝑒 𝑏𝑒= l 2 𝑎𝑟 = l2 𝑎𝑑 = l2 𝑏𝑒 = l2 𝑏𝑟 = l2 𝑏𝑑= lχ = 1.6χ = 0.625Figure 2.9: Dimensional geometries considered.and stress require significantly larger B.To examine if this methodology is useful or not, we will consider moregeneral particle shapes, i.e., three other symmetric geometries: ellipse, rect-angle and diamond. For each particle geometry, `⊥ = ˆ`⊥/Lˆ and `|| = ˆ`||/Lˆrepresent the dimensionless lengths of the particle in the x- and y-directions,respectively. The aspect ratio is always denoted χ = `||/`⊥ and the dimen-sionless area is pi; see Table 2.1 and Figure 2.9.2.2 EllipsePutz and Frigaard [112] computed viscoplastic flows around a settling ellipse78(c)(a) (b)Figure 2.10: Characteristic network around an ellipse: (a) χ = 10;(b) zoom of frontal plug region for χ = 10; (c) χ = 0.2.and approximated the limiting plastic drag coefficient by generalizing theexpression of [114] for a circle. However, the proposed approximation waspoor for χ > 1, the main problem being that the characteristic networkwas not constructed. The characteristic network may be constructed bygeneralizing the statements made about the circle. Figure 2.10 shows thecharacteristics for χ = 10 & 0.2. It can be seen that for χ < 1 the rigid plugat the front is much larger than for χ > 1. In the limiting case χ → 0 theellipse approaches a thin plate moving in the direction of its thickness andthe plug region approaches a right angle triangle with height equals to theplate half-length.79Lower bound calculationsFrom the characteristic network we are able to calculate the lower boundof the plastic drag coefficient as follows; see Figure 2.11 for the geometricalparameters.• Shear stress contribution on AB:F˜1−1 = χ−1/2B cos θ∗1 (2.19)• Normal stress contribution on AB:F˜2−1 = χ−1/2B(σ¯ +3pi2)cos θ∗1 (2.20)• Shear stress contribution on AKSQ:F˜3−1 =∫ θ∗10B sin ζ√χ−1 sin2 θ1 + χ cos2 θ1 dθ1 (2.21)• Normal stress contribution on AKSQ:F˜4−1 =∫ θ∗10B [σ¯ + 2 (pi − ζ)] cos ζ√χ−1 sin2 θ1 + χ cos2 θ1 dθ1(2.22)where, θ1 = tan−1 [χ−1 tan θ] and θ∗1 = tan−1 χ. The unknown mean stressthrough the QNJF is designated by Bσ¯.The lower bound plastic drag coefficient is:[CpD,c]L= (2 + 3pi) cos θ∗1+ 2∫ θ∗10{χ cos θ1 +[pi + 2 cot−1(χ cot θ1)]sin θ1}dθ1.(2.23)80𝜋/4𝜁𝜃1𝐴𝐵𝐶𝐷𝐻𝐿𝑀𝐼𝐸𝐹 𝐽 𝑁𝜃2𝑄 𝑃𝐺𝐾𝑆 𝑢0𝑎𝑒𝑏𝑒𝜃𝜃∗(a) (b)𝑏𝑒𝑎𝑒Figure 2.11: Geometrical parameters for lower and upper bound cal-culationsUpper bound calculationsWe also construct velocity solutions on the characteristic network, whichlead to an upper bound for the force and plastic drag coefficient. First,we find the velocity field associated with the characteristic network usingGeiringer equations by relying on the characteristics network constructedin previously. For the entire plastic region u˜α = 0 (note that along thestraight α-lines, dφ = 0) and u˜β is constant along each β-line, determinedfrom its value along Γ. Please note that for the viscoplastic problem no slipconditions are imposed on the particle surface, but for the perfectly plasticslip is allowed. However, the normal component of velocity is continuousas material cannot penetrate the particle. Additionally, the velocity fieldshould be solenoidal. We assume here that the tangential component of theperfectly plastic velocity is zero (i.e., this case is associated with adhesionat the pile surface equal to the soil cohesion, α = 1, in the soil mechanicscontext, e.g., [114]). Therefore, where β is the angle of β-line with the x-axis on the Γ (see Figure 2.11). Five different plastic regions have beenidentified by [114] around a circular pile, which can be observed here again.The whole mechanical work dissipation in a quarter of the plastic zone canbe calculated by seven terms:81• Velocity discontinuity on interface AB:∆W1 = χ−1/2B cos θ∗1 (2.24)• Velocity discontinuity on interface BCD:∆W2 = B sin(pi4)∫ θ∗10(√2χ−1/2 cos θ∗1 +∫ θ∗1θ1Θ(t) dt)Θ−2(θ1)dθ1(2.25)• Velocity discontinuity on interface DEF :∆W3 = B sin(pi4)∫ pi/20(√2χ−1/2 cos θ∗1 +∫ θ∗10Θ(t) dt)dθ2 (2.26)• Work dissipation in region BDEFJIHA:∆W4 =3pi4Bχ−1/2 cos θ∗1 (2.27)• Velocity discontinuity on surface AKSQ:∆W5 = B∫ θ∗10sin ζ Θ(θ1) dθ1 (2.28)• Work dissipation in region AGHLQK:∆W6 = B∫ θ∗10∫ θ10Θ−2(θ′1)[χ1/2 cos θ1 Θ−3(θ1)∫ θ1θ′1Θ(t) dt− χ−1/2 sin θ1]dθ1 dθ′1 (2.29)82• Work dissipation in region HIJNQL:∆W7 = B∫ θ∗10∫ pi/20[χ1/2 cos θ1 Θ−3(θ1)∫ θ10Θ(t) dt− χ−1/2 sin θ1]dθ2 dθ1 (2.30)where Θ(·) =√χ−1 sin2(·) + χ cos2(·). Note that θ and θ′ are ∠QPK and∠QPS, respectively. The total dissipation in one quarter of the flow will be∆W =∑7i=1 ∆Wi, and the plastic drag coefficient reads,[CpD,c]U=4∆W`⊥B. (2.31)On comparing with the lower bound, we find a discrepancy for χ 1, (i.e.,more than 200%), while for χ < 1, the lower and upper bounds are veryclose. To reduce the uncertainty associated with large χ, we devise a secondplastic mechanism for the upper bound calculation.The proposed mechanism can be defined by three geometrical optimiza-tion parameters: θˆ, ψ, and δ, shown in Figure 2.12. It consists of a rigidblock AIJQK which goes through a pure rotation about O with angularvelocity,ω =1` sinψ(χ cot θˆ1 − cotψ) , (2.32)where, ` is the length of segment OA (= be sin θˆ1/ sinψ). The interestingfeature of this angular velocity is that it makes the motion compatible withthe motion of the particle (i.e., no material penetration will occur on theparticle surface), as can be verified for a typical point K on the particlesurface. All the material in region BDEFJIA has a uniform velocity sin δ:ABD is a centred fan and DEFJIA is a fan shear zone. Thus, the workdissipated consists of six terms:83 𝜃ψ𝑎𝑒𝑄 𝑂𝐾𝐴𝐵𝐷 𝑢0𝑏𝑒𝜃3𝛿𝐸𝐹 𝐽𝐼𝑟𝑃 𝑢0lFigure 2.12: Schematic of mechanism II.• Velocity discontinuity on interface AB:∆W1 = χ−1/2B cos θˆ1 cot δ (2.33)• Velocity discontinuity on interface BD:∆W2 = χ−1/2B cos θˆ1(δ − ψ + pi2)(2.34)• Velocity discontinuity on interface DEF :∆W3 = B sin δ(`+χ−1/2 cos θˆ1sin δ)ψ (2.35)• Velocity jump on AIJ :∆W4 = B ` ψ∣∣∣∣sin δ + 1cosψ − χ sinψ cot θˆ1∣∣∣∣ (2.36)84• Velocity discontinuity on surface AKQ:∆W5 = B∫ θˆ10(sin ζ − 1cosψ − χ sinψ cot θˆ1)r`sin(ζ+θ3) Θ(θ1) dθ1(2.37)• Work dissipation in region BDEFJIA:∆W6 =3pi4χ−1/2B cos θˆ1 (2.38)The optimized upper bound associated with this mechanism is much closerto the lower bound for large aspect ratios.Comparison with viscoplastic solutionsIn addition to the lower bound and two different mechanisms which proposedfor upper bound of the plastic drag coefficient, we have also computed thelimiting viscoplastic solutions, using both [R] and [M] problems, for a widerange of χ.Convergence of the calculation of Yc using (Equation 2.13) and the solu-tion of the [R] problem at large B is illustrated in Figure 2.13 for 3 differentχ: for B > 104 no significant changes in plastic drag coefficient occur butthe computations become more difficult.Calculations of Yc using the [M] problem are direct, but involve iteratingon Y until the flow stops. The agreement between Yc using [R] or [M] iswithin 1% for all our results illustrated, and is often better as discussedpreviously for a specific example in Section 2.1.3. Figure 2.14 compares thebounds of the plastic drag coefficient with those obtained numerically fromthe viscoplastic flow. The [R] problem data are obtained at B = 104. Quitegood agreement can be observed between the lower bound plastic solutionand the computed coefficients from the viscoplastic problem, although theydo not match exactly (except at χ = 1). The upper bound plastic solutionsare less accurate. Some example computations of the viscoplastic flow areshown in Figure 2.15 and other detailed examples are shown in [112]. Theseshow regions of constant speed for χ 6 1 and a large rigid plug region at the85Bpij(u∗)101 102 103 104 105 1060.060.06120.0614Bpij(u∗)101 102 103 104 105 1060.1240.1316Bpij(u∗)101 102 103 104 105 1060.150.1690.181(b)(a)(c)Figure 2.13: Dependency of plastic dissipation on the Bingham num-ber for ellipse: (a) χ = 10; (b) χ = 1; (c) χ = 0.2. TheBlue symbols are data from the [R] problem. Red discontinu-ous lines are the characteristics predictions and black ones arethe asymptotic values of the blue symbols for large Binghamnumbers.sides of the particle, for χ > 1. The yield surfaces are marked in white 2.2.3 RectangleMotion of a 2D square in a viscoplastic fluid was studied by Nirmalkar et al.[100]. Here we consider more general case of 2D rectangle with differentaspect ratios. Moreover, to the best of our knowledge, there is no priorstudy concerning the characteristic network around a rectangular particle,although [78] estimate an upper bound for the mechanical work dissipatedin the plastic region around a square cross-section pile by suggesting a (non-2For the plasticity theorists, note that yield surfaces in the viscoplastic fluids context(as in Figure 2.15 and elsewhere in this study) are simply the level set where the deviatoricstress equals the yield stress, and not a curve in stress space.86  10−1 100 101 1020100200300400500600700 Lower bound (slip-line)Upper bound (slip-line)Upper bound (mech. II)[R] problem[M] problemCpD,cχFigure 2.14: Critical plastic drag coefficient of 2D elliptical particlesFigure 2.15: Velocity magnitude colour map, computed from problem[R] at B = 104, (the white lines are the yield surfaces): (a)χ = 10, (b) χ = 1, (c) χ = 0.2.87𝐵𝐴𝐸𝐹𝑄𝑏𝑟𝐷2𝜆𝐸𝐹𝐴𝑄𝐷𝐵𝑎𝑟𝑏𝑟(a) (b)𝑎𝑟Figure 2.16: (a) schematic of characteristic network adjacent to arectangle, (b) geometrical parameters for calculation of upperboundadmissible) velocity field.lower bound calculationsThe characteristic network for the rectangle is illustrated in Figure 2.16.Again the y-axis bisects the right angle at point B. The points A and Q ofthe ellipse network (e.g., Figure 2.6) have shrunk to the singular point A atthe corner of the rectangle. Here again the x-axis would be an α-line. Thus,a region of two orthogonal families of parallel straight lines will be formed(AEFQ) along the side of the rectangle. The lower bound for the plasticdrag force for this case consists of the following contributions:• Shear stress contribution on AB:F˜1−1 =√piχ−1/2B/2. (2.39)• Shear stress contribution on AQ:F˜2−1 =√piχ1/2B/2. (2.40)88• Normal stress contribution on AB:F˜3−1 =√piχ−1/2B(σ¯ +3pi2)/2. (2.41)Summing over 4 quarters of the rectangle (cancelling the mean normal stresscontributions σ¯ and noting `⊥ =√pi/χ), leads us to,[CpD,c]L=[F˜ p]L`⊥B= 2 + 3pi + 2χ. (2.42)Upper bound calculationsWe may also calculate an upper bound for the plastic drag coefficient basedon the general mechanism illustrated in Figure 2.16b. This mechanism isdefined by a single geometrical optimization parameter, λ. The upper boundbased on the characteristic network then can simply be obtained by settingλ = pi/4. Different contributions will be:• Velocity discontinuity on line AB:∆W1 =√piχ−1/2/2B cotλ (2.43)• Velocity discontinuity on line BDE:∆W2 =√piχ−1/2/2B(pi2+ λ)(2.44)• Work dissipation in area BDEA:∆W3 =√piχ−1/2/2B(pi2+ λ)(2.45)• Velocity discontinuity on line EF :∆W4 =√piχ1/2/2B sinλ (2.46)89• Velocity discontinuity on line AQ:∆W5 =√piχ1/2/2B (1 + sinλ) (2.47)The upper bound is obtained by repeating and summing over the 4 quartersof the domain:[CpD,c]U= 2 cotλ+ (2pi + 4λ) + χ (2 + 4 sinλ) . (2.48)As can be observed, the upper bound associated with the characteristicnetwork (λ = pi/4) will converge to the lower bound as χ → 0, but has alarge discrepancy as χ→∞, ([CpD,c]U − [CpD,c]L = 2√2χ). In principle, thisdiscrepancy can be reduced for large χ by selecting λ optimally. The result ofthis alternative velocity field is represented as mechanism II in Figure 2.17.Comparison with viscoplastic solutionsWe have also calculated the limiting viscoplastic solutions, using both prob-lems [M] and [R] (again at B = 104). The computed CpD,c are again close tothe perfect plasticity lower bound; see Figure 2.17. Some example computa-tions of the velocity magnitude colormaps and the yield surfaces are shownin Figure 2.18. The pi/4 angle between the yield surface of the attached plugand the direction of motion is clearly visible. Despite the good agreementwith the limiting drag coefficients we observe a very noticeable differencebetween the shape of outer yield surface and the outermost β-line in theplastic solutions; see Figure 2.16a. This is particularly evident in the caseof large χ. The boundary layer between the particle and the plug appearsto elongate with large χ, extending along the sides of the rectangle. Weexplore this boundary layer and the limiting flows further in Section 2.5.3.It may be of interest to look at the square case in more detail. Here wecompare the present and other known results for the square case. The criticalyield number has been shown to be Y[M ]c = 0.121, which corresponds to theCpD,c = 14.68 in Figure 2.3. Known critical plastic drag coefficients for thesquare particle are summarized in Table 2.2. With respect to viscoplastic90  100 101 1020100200300400500Lower bound (slip-line)Upper bound (slip-line)Upper bound (mech. II)[R] problem[M] problemχCpD,cFigure 2.17: Critical plastic drag coefficients for a rectangular parti-cle.Figure 2.18: Velocity magnitude colour map, computed from problem[R] at B = 104, (the white lines are the yield surfaces): (a)χ = 10; (b) χ = 1; (c) χ = 0.2.computations, the only results we know are those of [100], which we seediffer significantly from ours. We note also that in [100] the calculated CpD,cfor a circular particle also do not match CpD,c ≈ 11.94, as have been verifiedby [112, 134], and also in the present study. We believe the root cause ofthe discrepancy is that the regularization method that [100] have used isnot ideally suited to determining such limits without extensive attentionto mesh refinement, convergence etc., e.g., as in [134]. Regarding perfect91CpD,cNirmalkar et al. [100] 14.91[R] problem - this study 14.70[M] problem - this study 14.68Lower bound - this study 13.42Upper bound (mech. I) - this study 16.25Upper bound (mech. II) - this study 15.84Knappett et al. [78] 19.94Table 2.2: Critical plastic drag coefficients for a square particleplasticity estimates, we note that our bounds are improvements on that of[78], although there is still a significant discrepancy between our lower andupper bounds. It is apparent that our computed viscoplastic CpD,c for thesquare lies nicely between the calculated lower and upper bounds.2.4 DiamondThe diamond shaped particle is interesting chiefly because of the behaviourof the plug attached to the front of the particle. For χ < 1 the attached plugextends in front (and behind) of the entire diamond, to form an angle pi/4with the y-axis, so that the particle and plug together form a square rotatedby pi/4. However, the story is different for χ > 1: no attached frontal plugregion will be formed. Instead two centred fans, with radius equal to thelength of the side, cover the entire plastic region. Figure 2.19 shows thecharacteristic network for 3 diamonds with different aspect ratios χ.The lower and upper bounds of the plastic drag force can now be deter-mined easily by re-examining the terms associated to the centred fan in thecharacteristic network of the rectangle. The most interesting point is thatthe upper and lower bounds (calculated based on the characteristic network)are identical for this case, which means that the calculated plastic drag force92(c)(a)(b)Figure 2.19: Characteristic network around the diamond: (a) χ = 10;(b) χ = 1; (c) χ = 0.2.(and coefficient) is exact:CpD,c ={2χ+ 4(pi − tan−1 χ) χ > 1,2 + 3pi χ < 1.(2.49)This result has been known for some time in the plasticity community,e.g. derived also by Poulos and Davis [109] who examined the effect of aspectratio on lateral resistance of a rough pile (with rhombus cross section). Fig-ure 2.20 compares (2.49) with the limiting plastic drag coefficient computedfrom the viscoplastic problem, showing very good agreement. Figure 2.21presents 3 example computations for the diamond-shaped particle. Theouter yield surfaces in both the characteristics analysis and the computedviscoplastic solution are very similar. The excellent agreement is largelydue to there being no ‘side’ region to the particle. In the case of the ellipseand rectangle (at large χ) the tip and sides of the particle are different, witha boundary layer emerging along the sides. The diamond has only a long tip.93  10−1 100 101 102050100150200250Slipline analysis[R] problem[M] problemχCpD,cFigure 2.20: Critical plastic drag coefficient of diamondFigure 2.21: Velocity magnitude contours around the diamond: (a)χ = 10; (b) χ = 1; (c) χ = 0.2.2.5 General features of limiting viscoplasticsolutionsSo far we have seen that the lower bound perfectly plastic solutions appear togive a very reasonable approximation to the limiting plastic drag coefficientscomputed for the viscoplastic problem. This is both interesting and in somesense remarkable. Although there are cases for which the solutions are veryclose or exact, e.g., the diamond, others cases such as the rectangle havecompletely different yielded envelopes around the particle. Intermediate94cases such as the ellipse contain special geometries (the circle) for whichplastic and viscoplastic agree, although the stress fields are not totally thesame.The upper bound plastic velocity fields are much less effective as esti-mates for the limiting viscoplastic solutions. One reason for this is thatthe plastic test functions may be discontinuous whereas the viscoplastic so-lutions vary continuously due to viscous effects, albeit often approachingdiscontinuity in thin boundary layers. In many cases the transition occursin (relatively) thin layers of (relatively) high strain rate, which contributesignificantly to the plastic dissipation in the limit of zero flow. The otherdifference in the upper bound velocity fields is that the boundary conditionson the tangential component is not specified at a solid surface.In addition, we have presented computations of viscoplastic problem ofthese geometric families of particles, at various aspect ratios, close to thestatic limit. First, in understanding the generic behaviour of this yieldingprocess, two geometric trends are apparent:• as χ→ 0 the flow geometries all appear very similar.• for large χ we generate boundary layers in the static limit and thereseems to be some degree of commonality between different geometries.We explore these limits below. The second observation is that althoughperfectly plastic flows may on occasion predict the limit Yc accurately, theycontain no information on the convergence behaviour (with B or Y −c − Y ).Perfectly plastic flow solutions contain B only as a scaling parameter: thestress and velocity fields otherwise depend only on the particle shape, i.e., χin each geometry. Thus, here we concentrate on the viscoplastic solutions.2.5.1 The limit χ→ 0The limit χ → 0 gives the greatest agreement between drag coefficientscomputed from the viscoplastic flow and the characteristics analysis. Thestructure of the solution that we have seen in our viscoplastic computationsconsists of (up to) five regions as χ → 0. Referring to one quarter of theflow domain, these regions are:95(i) A rigid plug attached to the front (and rear) of the particle, of width∼ `⊥/2 as χ → 0, and forming a triangular region fore-aft of theparticle, making an angle pi/4 with the y-axis.(ii) A rotating zone that is centred at the ‘corner’ of the rigid plug, whereit contacts the particle. The fluid rotates on circular streamlines aboutthis point and appears to have approximately constant speed in theazimuthal direction. The radius of this region is ∼ `⊥/√2 and theazimuthal speed is ∼ 1/√2.(iii) A thin shear layer between the attached rigid plug and the rotatingzone. The length of this boundary layer is `⊥/√2. Across the layer thenormal component of velocity is continuous and the tangential dropfrom 1/√2 to 0.(iv) An outer boundary layer, surrounding the rotating zone, across whichthe azimuthal velocity drops from 1/√2 to 0. The circumferentiallength of this boundary layer is 3pi`⊥/(4√2). We have not studied thethickness, but it does not appear to grow as χ→ 0.(v) A ‘side region’ that connects the rotating regions fore and aft. Thisregion has width ∼ `⊥/2 and height ∼ χ`|| (ellipse) or ∼ `|| (rectangle).This region is absent in the case of the diamond shape.We can estimate the contributions to j(u∗) from each of the above re-gions as follows: (i) 0; (ii) ∼ 3pi`⊥/2; (iii) ∼ 2`⊥; (iv) ∼ 3pi`⊥/2; (v) ∼ χ`||/2(ellipse) or ∼ `||/2 (rectangle). Combining these we find:CpD,c = limB→∞j(u∗)`⊥∼ 2 + 3pi,for all three geometries as χ → 0, which agrees with the characteristicsestimates at leading order.In order to estimate convergence with respect to χ we would need toknow the next order of terms in each of the above regions, which is notapparent. Figure 2.22 plots the critical plastic drag coefficient for different96  0 0.25 0.5 0.75 1112 + 3pi1315Ellipse (slip-line)Rectangle (slip-line)Diamond (slip-line)Ellipse ([R] problem)Rectangle ([R] problem)Diamond ([R] problem)χCpD,cFigure 2.22: Critical plastic drag coefficient for χ → 0, computedwith B = 104.χ, as χ → 0, computed with B = 104. It is rather interesting that thediamond shape shows no variation with χ for χ < 1. As the diamond has no‘sides’ and only the tip region, the main role of χ is to determine the widthof the particle, from which we have seen there results an attached fore-aftplug. The unyielded plug is thus indistinguishable from the particle and wesee no variation with χ.By the same analogy, the parts of the ellipse and rectangle that lie withinthe fore-aft angular plug regions as χ→ 0 should not have any influence onthe flow convergence with χ. This suggest that χ dependence comes onlyfrom the side region: denoted (v) above. For the ellipse the surface tangentthat is at pi/4 to the y-axis meets the surface aty =`||2√χ21 + χ2∼ χ`||2.97The contribution from this side region to j(u∗) is thus ∼ χ`|| and to CpD,cis ∼ χ`||/`⊥ ∼ χ2. For the rectangle, we have instead the contributionto CpD,c is ∼ `||/`⊥ ∼ χ. We see that these estimates do agree with the(approximately) quadratic and linear variations in Figure 2.22, for ellipseand rectangle respectively.Regarding B-dependency, generally this must be examined numerically.However, for the diamond we may progress via a re-scaling argument asfollows. Defining xˇ = 2x/`⊥ we rescale the lengths while keeping velocitiesunchanged in problem [R]. If we also re-scale: τˇ∗ij = (`⊥/2)τ∗ij , pˇ∗ = (`⊥/2)p∗,the Stokes equations are satisfied by the re-scaled stress tensor, which ischaracterized by a single parameter:Bˇ =`⊥2B. (2.50)The re-mapped diamond intersects the xˇ-axis at ±1 and the yˇ-axis at ±χ.For any fixed χ < 1 the limit B →∞ is the same as Bˇ →∞.We have already seen that as Bˇ →∞ the solution will have a rigid plugattached to the diamond that intersects the yˇ-axis at ±1, and then regions(ii)-(iv), as described above. The contributions to j(uˇ∗) from regions (ii)-(iv) are 3pi, 4 and 3pi, respectively. However, since now uˇ∗ depends only onBˇ, we may postulate convergencej(uˇ∗) ∼ 4 + 6pi + 2ABˇ−t, as Bˇ →∞, (2.51)for constants A and t > 0. On rescaling lengths: j(u∗) = `⊥2 j(uˇ∗) ∼`⊥(2 + 3pi +ABˇ−t), as Bˇ →∞. From this we deduce:CpD,c = limB→∞j(u∗)`⊥∼ 2 + 3pi +O(B−t). (2.52)The exponent t can be deduced economically from computation of uˇ∗ asBˇ →∞. We have found t ≈ 0.70 from fitting a power law expression to ourdata.982.5.2 A general frameworkWe would like to understand the limit Y → Y −c . The rescaling leading to(2.12) clarifies that Y → Y −c and B → ∞ are the same limit, but does notactually determine the relationship between the two parameters. As well asclarifying this relationship, we would like to be able to determine the decayrates. The only general results we have are the decay bounds (1.29) and(1.30) for problem [M], as Y → Y −c .To this end, let us suppose that the particle velocity U ∼ b(χ)(1−Y/Yc)ν ,for some positive exponent ν = ν(χ). The bound (1.29) implies that ν > 1and (2.11) gives us j(u) ∼ b(χ)(1 − Y/Yc)νpi/Yc. The rescaling of (2.12)then tells us thatB =YU∼ Yc(χ)b(χ)(1− Y/Yc)−ν , (2.53)which couples the limits Y → Y −c and B →∞. To find ν we must estimatethe limiting solutions.Suppose we compute problem [R] at large B and substitute the velocityu∗ into (2.13), i.e., we approximate Yc using the limiting values of j(u∗). Ifthe contributions to j(u∗) come from the entire flow domain, it seems clearthat determination of ν is best carried out numerically. This occurs primarilyfor χ ∼ O(1). The best studied example is the circular disk. Looking backat Figure 2.8 it becomes evident that j(u∗) will have contributions from atleast 3 regions:(i) The boundary layer close to the particle,(ii) The outer shear layer,(iii) The zone of near-constant azimuthal velocity.The plug region converges to a fixed size as B → ∞ [134], and so eachof these regions continue to contribute at leading order. Tokpavi et al.[134] find that the boundary layer thickness between the small rigid plugon the disk equator and the particle scales like B−0.52 and the convergenceof their plastic drag coefficient to its limiting value scales like B−0.55, (i.e.,ν ≈ 1.82). Our results (see Figure 2.13b) also have ν ≈ 2, consistent with99[134]. Putz and Frigaard [112] calculate the limiting exponents for ellipticalparticles at χ = 0.5, 1, 2, as ν = 1.37 ± 0.23, 1.20 ± 0.26, 1.84 ± 0.25,respectively. The convergence rate for χ = 1 does not agree with our presentcomputations, nor those of [134], although the values for ν = 0.5 & 2 areconsistent with our calculations. Evidently, calculating each such χ ∼ O(1)geometry is laborious. The small χ limit offers some simplification. Theprocedure outlined for the diamond, leading to the estimate ν ≈ 1.43 forall χ < 1 (t ≈ 0.70), should also be applicable to the rectangle and ellipsefor sufficiently small χ (as the side regions vanish). Large χ also potentiallyallows for simplification (see Section 2.5.3 below).Lastly, note that the separation of geometric dependency from the decayin the solution as Y → Y −c , leading to (2.53), is justified by numerous exam-ples. Classical 1D flows such as Poiseuille flow (pipe or planar) have ν = 2.This also seems to be the case for more complex flows along uniform ducts,e.g., Frigaard and Scherzer [48] find ν = 2 for buoyancy driven two-fluid ex-change flows in circular pipes, with either concentric or stratified interfaces.These flows all have a single shear stress active along the bounding surfaceor at an interface, which seems to lead to ν = 2. For 2D flows along unevenchannels, Roustaei et al. [122] find both ν ≈ 2 and ν ≈ 1.5, depending on thechannel shape, with the latter relating to limiting flows with an extensionalstress component.2.5.3 Long particlesThe only way in which we might make general deductions about the limitY → Y −c is if the viscoplastic solutions develop distinct geometric forms,in which predictable regions of the flow make the dominant contribution toj(u∗). This might be true in the case of boundary layers, which occur inat least two situations. Firstly, for the ellipse and rectangle at large χ, wehave seen that there is a sharp transition from the particle velocity to thevelocity of the large plug regions along the sides of the particle, that arein rigid rotation. Secondly, for the diamond the particle transitions into afan-like structure over a relatively thin layer; see Figure 2.21a. We consider100these 2 cases below. Note that in both cases the length of the boundarylayer is expected to scale with `||.Ellipse and rectangleAt large χ, the general solution structure is as illustrated in Figure 2.18a.Along the sides of the particle there is an inner boundary layer (width δi),then the rigid plug and finally an outer boundary layer (width δo). The rigidplug has width `p and rotates with angular velocity ωp. While the rigid plugdoes not contribute to j(u∗) the outer boundary layer does. Equally, thereis another contribution from near the tip of the particle, where the fluidpushed out from in front of the plug and dragged upwards in the boundarylayer turns and moves back downwards into the plug and outer boundarylayer. Thus, we may write j(u∗) = ji(u∗) + jo(u∗) + jt(u∗), denoting thecontributions from the 3 regions.Writing u∗ = (u∗, v∗), we may evaluate the no-flux condition (2.4) alongthe x-axis for problem [R]:0 =`⊥2+∫ `⊥/2+δi`⊥/2v∗(x, 0) dx+∫ `⊥/2+δi+`p`⊥/2+δiv∗(x, 0) dx+∫ `⊥/2+δi+`p+δo`⊥/2+δi+`pv∗(x, 0) dx= O(χ−1/2) +O(δi)−O(ωp`2p)−O(ωp`pδo). (2.54)In the limit of χ→∞, our computations of the viscoplastic solution suggestthat ωp ∼ χ−1 and `p ∼ χ1/2 (see Figure 2.23). This suggests that asχ→∞, the second and third terms on the right-hand-side should balance:δi ∼ O(1). In other words, to leading order, the mass flux in the yieldedboundary layer dragged upwards by the plate, will turn back within the tipregion into the rigidly rotating plug.We may also evaluate ji(u∗), jo(u∗), and jt(u∗) directly. Figure 2.24presents the variation in these contributions at large χ, evaluated numeri-cally at B = 104 using problem [R]. The inner boundary layer contributionji(u∗) ∼ χ0.5 is clearly dominant. The outer boundary layer contribution101  250 500 100010−410−3  250 500 1000101102(a)(b)110.51ℓpωpχχFigure 2.23: Geometric behaviour at large χ, computed for the rect-angular particle at B = 104 using problem [R]: (a) ωp; (b)`p: (symbols from computation and solid lines fitted, with theindicated slopes).jo(u∗) is approximately constant (very weak function of χ) and the dissipa-tion in the tip region decays jt(u∗) ∼ χ−0.3.We now consider fixed large χ and variations with (1 − Y/Yc) (equiva-lently B), starting with the inner boundary layer. The velocity u∗ decaysfrom ∼ 1 to ∼ O(1− Y/Yc) across the width δi and has a length ∼ `|| alongthe particle. It follows that the contribution to j(u∗) is ∼ `||. This must becompatible with problem [M]. On rescaling of u∗ we find:pib(χ)Yc(χ)(1− Y/Yc)ν ∼ j(u) = Uj(u∗) ∼ b(χ)`||(1− Y/Yc)ν , (2.55)suggesting Yc(χ) ∼ 1/`|| ∼ χ−1/2. We might explicitly take the leading orderterm as corresponding to the drop in velocity along the full length of theparticle:j(u∗) ∼ ji(u∗) ∼ 2`|| +O(χ−1/2) +O(1− Y/Yc), (2.56)102250 500 100050100150250 500 10002.052.15250 500 10001.52(a)(b)(c)1110.50.030.3χji(u∗)jo(u∗)jt(u∗)χχFigure 2.24: Contributions to the plastic dissipation at large χ, com-puted for the rectangular particle at B = 104 using problem[R]: (a) ji(u∗); (b) jo(u∗); (c) jt(u∗): (symbols from compu-tation and solid lines fitted, with the indicated slopes).accounting for geometric effects and small non-zero velocity at the edge ofthe boundary layer. This results explicitly in Yc(χ) ∼ pi/(2`||) at leadingorder.This asymptotic behaviour is entirely compatible with our earlier results.The critical plastic drag coefficient is:CpD,c =pi`⊥Yc= limB→∞j(u∗)`⊥. (2.57)In every example computed (and those estimated from the lower boundcharacteristics solutions) we have CpD,c ∼ 2χ at large χ. This coincidesprecisely with the leading order contribution j(u∗) ∼ 2`|| and with Yc(χ) ∼pi/(2`||).The boundary layer contribution to the viscous dissipation a(u∗,u∗) is103∼ `||/δi, which translates to,a(u,u) = U2a(u∗,u∗) ∼ b2(χ)`||(1− Y/Yc)2νδi. (2.58)Our analysis of (2.54) suggests that: δi ∼ (1−Y/Yc)k, as χ→∞. The bound(1.29) implies k 6 2(ν − 1), whereas (1.30) is more restrictive, implyingk 6 ν − 1, (note also ν > 1). For the narrowest possible boundary layer:k = ν−1 and the widest possible k = 0. However, numerical evidence is thatk > 0. In terms of B we have: δi ∼ (Yc/b)k/νB−k/ν . As we have determinedthat δi is independent of χ as χ→∞ we deduce that:b(χ) ∼ Yc(χ) ∼ χ−1/2, as χ→∞. (2.59)We may even conclude further that:j(u) ∼ `||YcB∼ B−1, (2.60)a(u,u) ∼ `||Y1/νcb1/ν−2B−2+k/ν ∼ χ−1/2B−2+k/ν , (2.61)B ∼ (1− Y/Yc)−ν . (2.62)Figure 2.25 shows the values ν calculated by fitting our computations, allat χ = 100, for the 3 geometries. We observe different convergence rates ν,with the smaller exponents coming from the rectangle.So far this is non-conclusive. We may try to estimate the exponents ν andk directly from physical grounds or simpler assumptions. A simple balance ofthe buoyancy force with a viscous shear stress in the boundary layer suggestsδi ∼√`||YcB−1/2, i.e., k = ν/2. For the flat plate limit we might followthe boundary layer analysis of [102], who derived k/ν = 1/3, meaning hereν = 3/2 for the narrowest boundary layer. Recently however, Balmforthet al. [9] have shown that Oldroyd’s analysis and scaling is erroneous for theflat plate, but does apply to thin free shear layers. Following their analysisthe boundary layer close to a flat plate has k/ν = 1/2, to which our results104B1 − YYc  102 10310−2.510−1.5EllipseRectangleDiamondfitted line (m=-0.622)fitted line (m=-0.756)fitted line (m=-0.623)Figure 2.25: Convergence at large B for the 3 geometries at χ = 100:symbols are computes and lines are power law fits with expo-nent m (the slope of the fitted lines in log-log). This givesν ≈ 1.61, 1.32, 1.60 for ellipse, rectangle and diamond, re-spectively.should converge as χ → ∞. It may be however, that for χ = 100, we arestill far from the flat plate limit χ→∞.Lets now examine the next largest contribution to Yc, i.e. jo(u∗). Aswell as the χ-dependency considered earlier, both ωp and `p may vary withB, sayωp ∝ B−q, `p ∝ Bs. (2.63a, b)To maintain the leading order balance in (2.54) we require: q = 2s+k/ν. Wemay now estimate the contribution of the outer boundary layer in problem[R] to the plastic dissipation, say jo(u∗). Across the outer boundary layerthe velocity drops form ∼ ωp`p to zero, along the length ∼ `||, i.e.,jo(u∗) ∼ ωp`p`|| ∼ χ0B−s−k/ν . (2.64)105In terms of the limit B →∞ the dependency of `p observed computationallyappears very weak, so that we might infer s ≈ 0. In other words, the extentof the yielded region in the limit of no flow depends on χ but not significantlyon B. We conclude that jo(u∗)/j(u∗)→ 0, both as B →∞ and as χ→∞.Finally, it is interesting to reflect physically on why for large χ the vis-coplastic model and perfectly plastic model differ, in terms of outer yieldsurface and velocity. In the absence of viscosity in the rigid-plastic model,the plate does not drag the material as it is moving and hence no rigidrotation needs to be formed. This may also explain why in the viscoplas-tic solution we get a significantly smaller plastic drag coefficient for therectangle than the upper bound based on the proposed velocity fields inSection 2.3. The dissipation in the tip region is in fact smaller in the plas-tic solution than that in the viscoplastic solution, (decreasing like χ−1/2compared to χ−0.3). However, the upper bound plastic solution has leadingorder term: ∼ (1+√2)√pi2 χ1/2, whereas the viscoplastic solution has leadingorder contribution√pi2 χ1/2.DiamondThe flow around a long diamond (χ  1) shares many characteristics withthe small χ flow of Section 2.5.1, but is simpler in having only three regionscontributing to j(u∗). Referring to one quarter of the flow domain, thesethree regions are:(i) A rotating zone that is centred at the ‘corner’ of the diamond onthe x-axis. The fluid rotates on circular streamlines about this pointand appears to have approximately constant speed in the azimuthaldirection. The radius of this region is ∼ `||/(2 cosα) and the azimuthalspeed is ∼ sinα. Here α is the angle that the tip of the diamond makeswith the y-axis, i.e., we have α = tan−1(χ−1). Please note in this caseα < pi/4.(ii) A thin shear layer between the particle and the rotating zone. Thelength of this boundary layer is ∼ `||/(2 cosα). Across this layer the106normal component of velocity is continuous and the tangential velocitydrops from cosα to 0.(iii) An outer boundary layer, surrounding the rotating zone, across whichthe azimuthal velocity drops from ∼ sinα to 0. The circumferentiallength of this boundary layer is `||(pi/2 + α)/(2 cosα).We can estimate the contributions to j(u∗) from each of the above regionsas follows:(i) ∼ 2`||(pi/2 + α) tanα,(ii) ∼ 2`||,(iii) ∼ 2`||(pi/2 + α) tanα.Combining these we find:CpD,c = limB→∞[j(u∗)/`⊥] ∼2`||[1 + (pi + 2α) tanα]`⊥∼ 2χ+ 4(pi − tan−1 χ), (2.65)as χ → ∞. Note that the leading order contribution has come from theboundary layer (ii) as χ → ∞. This expression agrees with that of thecharacteristics analysis.2.6 Summary and discussionIn this chapter we have studied the relationship between perfect plasticand limiting viscoplastic flows, in the context of flows around buoyant 2Dsymmetric particles in Stokes regime. Three different families of shapedparticles have been considered: ellipse, rectangle and diamond, allowing afairly general study of effects of aspect ratio χ.For the viscoplastic flow we have computed the limiting flows using ei-ther a mobility [M] or resistance [R] formulation and evaluated the criticalplastic drag coefficient from these solutions. Calculations were performedusing an established finite element discretization with adaptive meshing, im-plementing an augmented Lagrangian approach. Problem [M] allows a more107direct approach to the yield limit Yc, but problem [R] is slightly easier com-putationally. Both formulations give consistent results, within numericalaccuracy.The critical plastic drag coefficients CpD,c(= pi/(`⊥Yc)) were comparedwith estimates from the characteristics analysis (perfect plasticity). Bothlower and upper bound estimates are generated. Some shapes have an uncer-tainty between lower and upper bounds for all χ, e.g., the rectangle. Othershave a small discrepancy for most χ, but coincide at particular values, e.g.,the ellipse. Finally, some agree for all χ, e.g., the diamond, and so the cal-culated plastic drag coefficient are exact in the plastic problem. In somecases we have derived multiple test velocity solutions, based on different as-sumed mechanisms, and improved the upper bound estimates to shrink theuncertainty.In terms of comparison with the limiting viscoplastic flows, the lowerbound estimates always appear to give a very reasonable approximation toCpD,c, even though the envelopes of yielded fluid about the particle may differsignificantly from the viscoplastic flows in some cases, e.g., the rectangle.Our formulation fixes the buoyancy force (i.e., the area of the 2D particlesis identical regardless of shape). The two extreme cases of large and smallχ correspond to an infinite plate moving longitudinally or laterally throughthe quiescent Bingham fluid. As χ → 0 the lower bound perfectly plasticlimit and the viscoplastic limit coincide. The limit is also identical for allshapes considered: CpD,c ∼ 2+3pi. The plastic dissipation in this limit comesfrom order 1 contributions originating from 3 different zones: two boundarylayers and a zone rotating at constant angular speed.Convergence to the above limit, with respect to χ appears to be dom-inated by the sides of the particle. At small χ the flow is dominated bythe large rigid plugs fore and aft of the particle, which make an angle pi/4with the y-axis. Thus, those changes in particle geometry that lie withinthe rigid plugs do not appear to affect CpD,c. Thus, the diamond shape forχ < 1 shows no χ-dependency. The ellipse and rectangle converge to theflat plate limit quadratically and linearly, respectively, as χ→ 0.Convergence with respect to rheological effects ((1− Y/Yc) or B) terms108does not feature in the characteristics analysis. For the viscoplastic flows, ingeneral this dependency must be evaluated computationally at each χ < 1.For the diamond shape the picture is slightly different as the limiting flowsmay all be mapped to the same problem, of a settling unit square (rotatedthrough pi/4). Here we have found convergence to CpD,c proceeds as B0.7 or(1− Y/Yc)1.43, from computing the mapped problem.At large χ the dominant contribution to the dissipation in the limitingflow comes from a boundary layer along the sides of the particle. This leadsagain to a universal limit: CpD,c ∼ 2χ (or equivalently Yc ∼ pi/(2`||)), inagreement with the lower bound characteristics analysis at leading order inχ. The leading order mass flow balances the fluid dragged upwards in theboundary layer with that flowing downwards in the large rotating plug.A general analysis of convergence to this limit with respect to B or(1 − Y/Yc), results in the settling velocity U ∼ b(χ)(1 − Y/Yc)ν , whereb(χ) ∼ Yc(χ) ∼ χ−1/2 as χ → ∞. The exponent ν ∈ [1, 2], depends onthe boundary layer thickness (in terms of B we converge as B−ν). Weinterpret ν = 2 as corresponding to a simple one-dimensional shear flow inthe boundary layer and ν = 1 implies that the boundary layer thickness isindependent on (1−Y/Yc). Finding ν requires computation and the resultsshow different rates for the 3 geometries studied, e.g., ν = 1.61, 1.32, 1.60were found for ellipse, rectangle and diamond, respectively, at χ = 100.In principle each geometry should converge to the same flat-plate flow asχ→∞ and this limiting flow has been studied recently by Balmforth et al.[9]. The different exponents suggest that still for relatively large χ we arefar from this limit.The boundary layer analysis and solution structure at large χ gives in-teresting insights relevant to the recent experimental observations of Boujlelet al. [20], who studied flow around a dragging flat plate in a yield-stressfluid (pastes). Closer examination of the visualized velocity field in theirpaper (see Figures 2.26 and 2.27) reveals that there is a rigid rotation ofsmall angular velocity adjacent to their boundary layer. Boujlel et al. [20],however, interpret this part of their flow as a constant velocity region: theyneglect the small slope (ωp) that the velocity profile has in this region. This109motion of the plate through the fluid and the mass conservation (assuming that the fluid isincompressible). However, the simple upward displacement of a fluid volume equal to theplate volume moving downward cannot explain such a value for DV=V. Indeed, in thatcase we would expect DV=V  el=2LD, i.e., about 0.2%. The larger value that we foundresults from the downward displacement of a significant volume of fluid around the movingplate. If we roughly assume that the average velocity of this (boundary) layer is half theplate velocity we find that its total thickness (including both sides of the plate) should be ofFIG. 5. Velocity field averaged over a plate displacement of 1 cm (Sec. III C) for V ¼ 1mm:sÿ1 as directlyobtained from PIV measurements for two different positions of the plate with regards to the window of observa-tion: (left) plate tip at 3 cm above the window bottom, (b) plate tip at 10 cm below the window bottom. The av-erage plate position is represented by the grey area.FIG. 6. Effective velocity profile (in the frame of the container) along the plate direction during the penetrationthrough a Carbopol gel (0.5%) at V ¼ 1mm=s at x ¼ 6 cm. The dotted line corresponds to v ¼ ÿ0:03mm=s.The plate is situated at y ¼ 0 and the container wall at y ¼ 7:4 cm.1095BOUNDARY LAYER IN YIELD STRESS FLUIDSFigur 2.26: Veloci y field av raged over a pla e displacement dir ctlyobtained from PIV measurements for two different positionsof the plate with regards to the window of observation: (left)plate tip at 3 cm above the window bottom, (b) plate tip at10 cm below the window bottom. Reproduced from [20].is inconsistent with the zero velocity condition far from the plate (whetherthe low velocity region is interpreted as a constant velocity rigid plug or asa rigid rotation): the outer boundary layer is also neglected in the analy-sis of [20]. The perceived structure of the flow is however evident in thecomments they made: “this slight motion obviously finds its origin in themotion of the plate through the fluid and mass conservation · · · however,the simple displacement of a fluid volume equal to the plate volume cannotexplain such a value · · · the larger value that we found results from the dis-placement of a significant volume of fluid around the moving plate.” Thisperceived structure is wrong, at least in the limit χ → ∞, although theobservation regarding the plate volume is in agreement. As we have notedearlier the leading order mass balance is between the fluid dragged upwardsin the boundary layer (that remains of O(1) as χ → ∞) and the rotatingplug (of extent `p ∼ χ and angular speed ωp ∼ χ−0.5).To summarize, we h ve found that the characteristics method doe notin general produce the same stress fields as in the limiting viscoplastic flows,110motion of the plate through the fluid and the mass conservation (assuming that the fluid isincompressible). However, the simple upward displacement of a fluid volume equal to theplate volume moving downward cannot explain such a value for DV=V. Indeed, in thatcase we would expect DV=V  el=2LD, i.e., about 0.2%. The larger value that we foundresults from the downward displacement of a significant volume of fluid around the movingplate. If we roughly assume that the average velocity of this (boundary) layer is half theplate velocity we find that its total thickness (including both sides of the plate) should be ofFIG. 5. Velocity field averaged over a plate displacement of 1 cm (Sec. III C) for V ¼ 1mm:sÿ1 as directlyobtained from PIV measurements for two different positions of the plate with regards to the window of observa-tion: (left) plate tip at 3 cm above the window bottom, (b) plate tip at 10 cm below the window bottom. The av-erage plate position is represented by the grey area.FIG. 6. Effective velocity profile (in the frame of the container) along the plate direction during the penetrationthrough a Carbopol gel (0.5%) at V ¼ 1mm=s at x ¼ 6 cm. The dotted line corresponds to v ¼ ÿ0:03mm=s.The plate is situated at y ¼ 0 and the container wall at y ¼ 7:4 cm.1095BOUNDARY LAYER IN YIELD STRESS FLUIDSFigure 2.27: Effective velocity profile (in the frame of the container)along the plate direction during the penetration through a Car-bopol gel. Reproduced from [20].although the lower bound solutions are generally a reasonable estimate andin exceptional cases agree exactly.Both small and large aspect ratio particles lead to universal leadingorder CpD,c, with speed of convergence depending on particle shape. Theflow structure is similar in these limits, being dominated at small χ by thefore-aft rigid plugs and associated rotating regions, and for large χ by thelongitudinal boundary layers.At intermediate χ away from these limits the yielded flow envelope isfinite, but different flow regions are significant in contributing to the plasticdissipation. This makes analytic determination of the limiting flows difficultand computation is the main tool.In terms of applicability of our results, we should firstly remind the readerthat the particles considered are 2D. The restriction to planar flows comesprincipally from the analysis of the plasticity problem via the characteristicsmethod. Other restrictions are in the symmetry of the particles, ensuringthat motion is in the y-direction, and evidently the range of shapes studied111has limits. Nevertheless, the general qualitative picture of variations withχ may also hold for symmetric 3D particles and at least we have a baselinefrom which to look at such studies.In terms of applicability to complex industrial flows, such as drill cuttingstransport, we remain quite distant due to the 3D and irregular shape of thecuttings. For fractionation applications such as in [86], in a lab setting withwell controlled particle size/shape distributions, the qualitative results thatwe have derived may have application.112Chapter 3Cloaking and unyieldedenvelope ruleAs introduced in chapter 2, single particles in a yield stress fluid move onlywhen the driving body force exceeds a critical value depending on the yieldstress, particle shape and orientation. We have explored a range of sym-metric 2D particles and find that the critical limit Yc is not unique. Forexample, we have seen that each geometric family of the previous chapterbehaves analogously as χ→ 0. Very revealing in this particular example isthat the settling particles each “trap” two large wedges of fluid (symmet-rically fore and aft) which move with the particle. The flow around thisunyielded envelope (meaning particle and trapped fluid) is identical for the3 geometries as χ→ 0.Since the particle and fluid within the envelope move together, it isapparent that the unyielded envelope rather than the particle shape is im-portant in determining Yc, (equivalently the critical plastic drag coefficient).Using the critical plastic drag coefficient CpD,c, places the focus on the forcesexerted on the particle. As discussed earlier in Section 1.6, questions re-garding the shape of unyielded envelope around a particle, the magnitudeand line of action of the limiting particle force, have been identified as im-portant since the 1960s [19], although not clearly answered. In this chapterwe approach these questions directly.113An outline of the present chapter is as follows 1. Below in Section 3.1we explain the problem studied, discuss the methodology and numericalmethod, leaning heavily on the previous chapter to avoid repetition. Theresults are in 3 sections. In section 3.2.1 we illustrate vividly the cloak-ing phenomenon, namely that quite different particle shapes may have thesame unyielded envelope surrounding them, i.e. the particle shape can behidden or cloaked within the unyielded envelope. We explain some of theconsequences. Section 3.2.2 shows how to construct the unyielded enve-lope for a symmetric particle (with possibly non-convex shape), and thatthis again leads to effective lower bound solutions using the methods ofperfect plasticity. In section 3.2.3 this method is generalized to particleswith only left-right symmetry, showing how a characteristics network canbe constructed between mismatched fore and aft particle shapes (and yieldenvelopes). This is illustrated with a number of representative examples,all of which result in a good estimate of the critical limit from the perfectplasticity lower bound. The chapter ends with a brief discussion.3.1 Problem statementIn this chapter, again we are dealing with same equations as before andtwo classic formulations [R] and [M]: we focus on the motion of isolated2D particles in an infinite region of yield-stress fluid. We are specificallyinterested in the yield limit, i.e., when the particle is just able to move,and hence consider only inertialess flows. The particle is again denotedby X, ∂X is the boundary of the particle, Ω represents the entire domain(fluid+particle) and ∂Ω is its outer boundary (see Figure 3.1).For the sake of conciseness, derivation of the dimensionless equations isnot repeated. However, there are two main differences between the presentproblem and the problem considered in the previous chapter:(i) Here the only restriction for the particle shape is that it should haveone symmetry line (say y-axis) parallel to the direction of motion: for1The results of this chapter have appeared as: E. Chaparian, I.A. Frigaard, Cloaking:Particles in a yield-stress fluid, J. non-Newton. Fluid Mech. 243 (2017) 47–55.114xyv0ParticleFluiddomain𝑔ΧΩϒRigidYield-stressParticleFigure 3.1: Schematic of the problem considered.the [R] problem, u∗ on ∂X is U∗ = U∗ey and for the [M] problem,the gravitational acceleration is aligned with the positive y-direction(gˆ = gˆ ey).(ii) With respect to Lˆ, in this chapter we do not use, Lˆ =(Aˆppi) 12, becauseof quite irregular and complex particle shapes that we will consider inthis chapter. Instead, we provide the dimensionless size of the consid-ered particles in Figure A.1.With respect to boundary conditions, the velocity u(∗) vanishes in thefar-field for either [R] or [M] problems. Moreover, in the [M] problem, thevelocity is continuous at the particle surface (no-slip) and the stress at theparticle surface must satisfy the following force balance:∫∂Xσ · n ds = Ap1− ρey, (3.1)where n is the normal of ∂X, pointing inward to the particle and Ap is thedimensionless area of the particle.As it has been shown in the previous chapter, solving either [R] or [M]problems is equivalent for moving particles and either can be used for study-ing the limit of no flow. The connection is made via the critical plastic drag115coefficient CpD,c:CpD,c = [CpD][M ]Y→Y −c = [CpD][R]B→∞. (3.2)Conventionally, the plastic drag coefficient is defined as: CpD = Fˆ3D/(Aˆ⊥τˆY )with Fˆ3D the net force on the particle and Aˆ⊥ representing the the frontalarea of the particle, perpendicular to the direction of motion. Here however,as we work in 2D it is implicit that the geometries have an infinite lengthin the 3rd dimension. Thus, we write CpD = Fˆ /(ˆ`⊥τˆY ), where ˆ`⊥ is thefrontal length of the particle and Fˆ is the net force per unit length in the3rd dimension. Indeed, when we discuss “force” below, it is implicit that thismeans force per unit length. The plastic drag coefficient may be evaluatedfrom the solution of [M] or [R]. In the case of [M], in dimensionless termswe have simply CpD = Ap/(`⊥Y ) and in the case of [R], we must calculatethe dimensionless force from the solution.The numerical method for the viscoplastic problem is again the aug-mented Lagrangian method which converts the velocity minimization func-tion to a saddle point problem by removing the non-differentiability of plasticdissipation term. Also, here we will use the same convergence criterion asthe previous chapter. The majority of results presented later have howeverbeen obtained from problem [R] with large B. In general, B = 104 appearsadequate for relatively regular particles as here.3.2 ResultsResults of this study relate to three problem areas: the cloaking phenomenonand its consequences; finding the unyielded envelope for symmetric parti-cles and use of the characteristics (slipline) method for approximating CpD,c;extension of the same method to particles without fore-aft symmetry.3.2.1 Cloaking and its consequencesIn the previous chapter, we studied 2D particles with two orthogonal sym-metry lines (ellipses, rectangles and diamonds), for different aspect ratios,χ. Here χ = `||/`⊥, with `|| representing the particle length in the settling116Figure 3.2: Speed contours and yield surfaces (white): (a) Ellipse, (b)‘Batman’.direction. Although the limiting flows at large χ were somewhat differentbetween the geometries, for small χ the limiting flows all approached an iden-tical flow, characterized only by χ, and leading to a universal CpD,c = 2 + 3pias χ→ 0 (see Figure 2.22). What is interesting about these flows is that ineach case the (increasingly wide) particle was encased within an unyieldedenvelope of fluid. Where the shape of the unyielded envelope containingthe particle was identical between two flows, the same limiting CpD,c wasobtained for different particles.To clarify, we show two examples. Each of Figs. 3.2 & 3.3 show particleswith different geometries, but which have the same unyielded envelopes asthe yield limit is approached. The colourmaps indicate velocity magnitudeand the white lines in these figures are the yield surfaces. In each case wesee that the unyielded region essentially cloaks the shape of particle inside.The consequence of cloaking is that identical CpD,c are obtained. In themobility formulation we have (dimensionally):CpD =Fˆˆ`⊥τˆY=(ρˆp − ρˆf )Aˆpgˆˆ`⊥τˆY.In both the case of the ‘Batman’ and triangle examples (Figs. 3.2 & 3.3) wehave ensured that the perpendicular lengths ˆ`⊥ are the same between leftand right figures. Thus, comparing left and right in each case suggests that117Figure 3.3: Speed contours and yield surfaces (white): (a) Tiltedsquare, (b) Triangle.the limiting flows can identify (ρˆp − ρˆf )Aˆp only. The area of the ‘Batman’is clearly less than that of the ellipse, but can achieve the same CpD,c withlarger density difference (ρˆp − ρˆf ). More concisely, in the triangle example,the same CpD,c results if (ρˆp− ρˆf ) in Figure 3.3b is twice that in Figure 3.3a.These simple observations have consequences for fractionation processesas discussed in Section 1.1.1. In such processes one may also vary gˆ (inter-preted more generally as acceleration, e.g., in a centrifuge), or indeed τˆY .Evidently, particle orientation may affect ˆ`⊥ as well. We may also comparethe shapes of the unyielded regions between Figures 3.2 & 3.3. Althoughnot identical (due to the rounded corner of the ellipse/Batman shapes), thelimiting flows are quite similar and one might expect that CpD,c will be close.Here we computed CpD,c ≈ 11.54 for the shapes in Fig. 3.2, compared toCpD,c ≈ 11.46 for the shapes in Fig. 3.3 (a difference of < 1%). Therefore,practically speaking, knowledge of CpD,c without any a priori information onthe particle shape and orientation will limit the precision of any fractionationprocesses.3.2.2 Finding the unyielded envelope: two lines ofsymmetryThe examples shown in the previous section exhibit cloaking effects within anunyielded envelope that is similar in shape to those found in previous chapter118Figure 3.4: Speed contour: (a) Rounded-end rectangle (a = 1, b =1.5), (b) ‘Dumbbell’ (a = 1, b = 1.5, b′ = 0.5). See Figure A.1.for symmetric particles with small aspect ratio. However, other shapes ofunyielded envelopes can also be found depending on the particle shape,as illustrated in Figure 3.4. For any such unyielded envelope the cloakingeffects are similar. The interest in the unyielded envelope is therefore two-fold. First, for a given particle shape, what is the shape of the unyieldedenvelope (and hence we may understand potential cloaked shapes)? Second,for a given unyielded envelope the hope is that it will be relatively easy tobuild an admissible stress tensor using the theory of perfect plasticity (e.g.,[23]) and the associated characteristic (slipline) network. This procedureleads directly to a lower-bound estimate for CpD,c, which has generally beenfound to also be a good approximation to the limiting viscoplastic behaviour(see our results in previous chapter or in [25]).We start with particles containing two orthogonal lines of symmetry, oneof which is aligned with the direction of motion. This alignment is chosento avoid consideration of particle drift in Stokes flow. We thus consideronly one quadrant of the flow regime, due to symmetry. After conductinga large number of computations, we devised the following descriptive rulethat appears to characterize the unyielded envelope.Unyielded envelope rule: One quarter of the moving object (unyieldedenvelope plus contained particle) is a convex shape (∆) bounded by the twolines of symmetry and by a surface Γ. The boundary Γ is smooth, except119𝐴𝐾𝑆𝑄𝐹 𝑃𝐷𝐶𝐵𝐸∆𝜂2𝜂1𝛤𝑃(𝑎) (𝑏)Figure 3.5: Schematic of quarter of the unyielded envelope (particle+ attached plugs): (a) quarter of the bounding surface Γ, illus-trating the angles η1 and η2; (b) quarter of the slipline (charac-teristic) network generated from the unyielded envelope.possibly at a finite number of points. Γ intercepts the y and x-axes at anglesη1 6 pi/4 and η2 6 pi/2, respectively, as illustrated in Figure 3.5. The lengthmes(Γ) is minimal amongst the set of all such admissible boundaries. Thematerial trapped between the bounding surface Γ, and the particle surfacemoves as a plug region.Figure 3.5a shows the descriptive geometry of the above rule. In simpleterms, Γ consists of straight line segments (AB and SK in Figure 3.5b)and possibly segments of the particle surface (between points A and K,also between S and Q in Figure 3.5b), which represent the places whereΓ coincides with the particle surface. In the case illustrated Γ starts frompoint B at the vertical symmetry line with an angle pi/4, and hits the particlesurface (point A) where the tangent to the surface makes an angle pi/4 withthe x-direction. Then it adheres to the particle surface up to point K wherethe tangent to the particle surface is the same as the tangent line at pointS. Then it again follows the particle surface between points S and Q.To make things more clear regarding mes(Γ), 3 different computations120Figure 3.6: Speed contour for 3 different particles close to yielding.It should be noted that only a quarter of the whole domain ispresented here.are shown in Figure 3.6. The attached plugs in front of the particles arethe same in all 3 cases: Γ starts from the vertical axis, intercepting at angleη1 = pi/4, and touches the particle surface where the tangent to the surfacealso makes pi/4 angle with the y-axis. Then Γ follows the particle surface fora short interval. However, we can see that the plugs attached to the ‘sides’ ofthe particles are quite different in each case. In other words, the boundary Γconsists of straight line segments and other segments that simply follow thecurves of the particle surface. Indeed, Γ is the evolute of the straight-linecharacteristic (slipline) family (say α-lines); see [23].The three particle geometries of Figure 3.6 are described parametricallyin Figure A.1, using 2 constant parameters a and b. For each case, having121Fig. 3.6a Fig. 3.6b Fig. 3.6cParameters a = 1, b = 0 a = 1, b = 2 a = 1, b = 5[CpD,c]L11.68 12.91 14.61CpD,c 11.68 12.92 14.62[CpD,c]U11.68 20.93 29.68Table 3.1: Computed critical plastic drag coefficients compared tolower and upper bounds, for each of Figure 3.6.found bounding curve Γ of the unyielded envelope, we may construct thecharacteristic network and hence compute an admissible perfectly plasticstress field, which provides a lower bound[CpD,c]L6 CpD,c and also a velocitytest solution that can be used to give an upper bound[CpD,c]U> CpD,c.Details of the upper bounds calculations for this family of particles canbe found in Appendix B. Results of this calculation, and of comparingwith CpD,c computed directly from the viscoplastic problem, are shown inTable 3.1.Evidently, the lower bound stress fields give an excellent approximationto CpD,c. The lower bound computation gives the following expressions:[CpD,c]L=√3 + 2√2(1√2+3pi2√2)+ pi(1− 32√2)+3√22,[CpD,c]L= 3 + 2√2 + 2pi +(3√25+ 2−√2)tan−1(13),and,[CpD,c]L= 5 + 2√2 + 2pi +(5√213+ 2−√2)tan−1(15),for Figure 3.6a-c, respectively. The upper bound agrees identically with thelower bound for Figure 3.6a, but is otherwise a poor approximation for themore elongated particles.122Note that although for some cases already shown (e.g., Figure 3.2b andFigure 3.3b) the particle itself does not have two orthogonal symmetry lines,the unyielded envelope does, i.e., the conditions of this section apply whenthe unyielded envelope is symmetric.3.2.3 Particles with only left-right symmetryWe now consider two examples in this section to show the construction ofthe characteristics network for more complex particles with just one lineof symmetry (left-right symmetry about the y-axis). The extension of thismethod to more general shapes is then fairly straightforward. The simplestcase is a ‘kite’: two isosceles triangles stuck to each other at their edges.If we just consider two isolated isosceles triangles, the network for each ofthese triangles will be a simple centred fan (see Figure 3.7a). However, itis obvious this network is not of any interest for finding a test stress tensoror velocity fields. However, the black characteristics are common betweenthe two networks. So our idea to find a non-trivial characteristic network isto keep the black characteristics in Figure 3.7a as initial characteristics fora numerical construction. The numerical scheme we implement for findingsuch characteristics networks is outlined in Appendix C.In Figure 3.7b the computed network is presented. The blue line isthe outermost β-line in the black region of Figure 3.7a, from which theconstruction is launched. Beyond this last β-line, the α-lines will not bestraight any more. So we continue finding the α-lines until the red α-lineis achieved (i.e., the last α-line which is initiated in the centred fan). Fromnow onward, all the β-lines should be initiated from the limiting line (whichis the particle surface here), because characteristic lines cannot be continuedbeyond the particle surface. For more details on limiting lines see [111]. Theconstruction will be continued up to the point where the last α- and β-lineshit each other at front of the particle. For more complicated shapes themethod is the same. For example, combining a semicircular domain with anisosceles triangle, we may consider a parachute shape (see Figure 3.8).To demonstrate that this construction leads to a characteristic network123(a) (b)Figure 3.7: Characteristic network adjacent to a ‘kite’ (see Figure A.1,a = b′ = 1, b = 2): (a) two separate centred fans; (b) calculatednetwork ([CpD,c]L= 11.85). Only half of the domain is shown.that produces a lower bound stress field, figures 3.9 and 3.10 show a com-parison of the characteristics network (and corresponding[CpD,c]L) with thecomputed limiting viscoplastic solutions, for both a kite and a parachuteshape. The lower bounds estimates in both cases give an excellent approx-imation. There is also a strong agreement between the shape of yieldedenvelope in the viscoplastic computation and the outermost characteristicof the network.3.3 ConclusionsThis chapter has three main contributions. Firstly, we have identified thecloaking phenomenon and importance of the unyielded envelope around aparticle. The force resulting from stresses evaluated along the boundaryof the unyielded envelope balances with the body force in the zero flowlimit and it is therefore the shape of the unyielded envelope that is the keycharacteristic of this limit: i.e. a signature of CpD,c or Yc. The unyielded124  (b)(a)Figure 3.8: Characteristic network adjacent to a ‘parachute’(seeFigure A.1d, a = b = 1): (a) two separate networks; (b)calculated network ([CpD,c]L= 11.70). Only half of the domainis shown.envelope, however, can be the same for different particles, as we have shownconvincingly: we may cloak (or hide) the particle shape within the unyieldedenvelope, in the zero flow limit. This development provides a direct answerto the long-lasting question of how we can calculate the CpD,c or Yc of aparticle and basically where does the yield stress of the fluid acts over theparticle surface.Secondly, we have explored the consequences of cloaking for using yieldstress fluids as fractionating agents, e.g., [86]. In the first instance thisnegates the idea that the yield stress can be used as a precise selector ofparticle size/shape. However, this is perhaps too harsh a judgement. Inmany practical situations one would be dealing with a particle size and ori-entation distribution within a suspension, which anyway is likely to mollifyany selection. Most other fractionation procedures, such as screening orviscous settling also do not provide a sharp cut-off in effectiveness. Indeed125E. Chaparian, I.A. Frigaard / Journal of Non-Newtonian Fluid Mechanics 243 (2017) 47–55 53 Fig. 9. Elongated ‘Kite’ (see Fig. A.11 e, a = b ′ = 1 , b = 5 ): (a) Slipline network ( [C p D,c ]L = 14 . 79 ); (b) speed contour computed from the [R] problem ( C p D,c = 14 . 79 ). Fig. 10. Elongated ‘parachute’ (see Fig. A.11 d, a = 1 , b = 4 ): (a) Slipline network ( [C p D,c ]L = 13 . 87 ); (b) speed contour from the [R] problem ( C p D,c = 13 . 97 ). Acknowledgement We greatly appreciate financial support of Natural Sciences and Engineering Research Council of Canada via their strategic projects programme (grant number STPGP 447180-13 ). Appendix A. Particle geometries A parametric description of the particle geometries considered earlier is shown in Fig. A.11 . Fig. A.11. Schematic of particle geometries. Figure 3.9: Elongated ‘Kite’ (see Figure A.1e, a = b′ = 1, b = 5):(a) Slipline network ([CpD,c]L= 14.79); (b) speed contour com-puted from the [R] problem (CpD,c = 14.79).viscous drag coefficients are similarly non-unique. Using a yield-stress fluidstill provides a different (rheological) dimension with which to control frac-tionation and it is the mechanical balance captured in Yc (or CpD,c) that isimportant here.Thirdly, we have extended the characteristics-based methods from per-fect plasticity to a class of 2D particles with left-right symmetry. We havefirst shown how to construct the unyielded envelope around particles. Thenwe have clarified how to construct the characteristic network and finallyhave demonstrated that the lower bound estimates give a very good approx-imation to CpD,c.Considering future perspectives, the next step in considering single par-ticles in planar 2D flows is to study orientation effects and drift, i.e., whenthe particle symmetry axis is not aligned with the body forc (objectiveof the next chapter). Eventually, it would be interesting to explore parti-cle geometries with no symmetries, when rotational motion must also be126E. Chaparian, I.A. Frigaard / Journal of Non-Newtonian Fluid Mechanics 243 (2017) 47–55 53 Fig. 9. Elongated ‘Kite’ (see Fig. A.11 e, a = b ′ = 1 , b = 5 ): (a) Slipline network ( [C p D,c ]L = 14 . 79 ); (b) speed contour computed from the [R] problem ( C p D,c = 14 . 79 ). Fig. 10. Elongated ‘parachute’ (see Fig. A.11 d, a = 1 , b = 4 ): (a) Slipline network ( [C p D,c ]L = 13 . 87 ); (b) speed contour from the [R] problem ( C p D,c = 13 . 97 ). Acknowledgement We greatly appreciate financial support of Natural Sciences and Engineering Research Council of Canada via their strategic projects programme (grant number STPGP 447180-13 ). Appendix A. Particle geometries A parametric description of the particle geometries considered earlier is shown in Fig. A.11 . Fig. A.11. Schematic of particle geometries. Figure 3.10: Elongated ‘parachute’ (see Fig. A.1d, a = 1, b = 4): (a)Slipline network ([CpD,c]L= 13.87); (b) speed contour fromthe [R] problem (CpD,c = 13.97).accounted for in the zero-flow limit.Whether the characteristics methodology can be easily used for thesegeometries remains to be seen. Although constructive, the method requiressome physical insight in order to build the network. Indeed, this is a draw-back of this method, compared to e.g., the viscoplastic computations whichinvolve standardized FEM tools. The latter can be readily applied to morecomplex flows and it would be of value to have a numerical method that isalso widely applicable. Another direction of current interest concerns flowswith multiple particles, where again the characteristics methodology mayhave problems. We turn to this in Chapter 5.Lastly, real particles are 3D and eventually we aim to develop under-standing of yielding in these circumstances. Here the plasticity methodsare not used as easily (even when axisymmetric) and there are few knownresults beyond that of [15] for the sphere.127Chapter 4Effect of particle orientationAs stated at the outset, in general we expect the yield limit to dependon particle size, density, shape and orientation. In the present chapter weconsider the effect of particle orientation on sedimentation and the yieldlimit. We know that when particles settle in the (Newtonian) Stokes regime,particles can drift without any rotation or re-orientation [57]. For particleswith sufficient symmetry, it is possible to calculate the drift angle fromthe eigenvalues of the resistance matrix via a simple force balance. Settlingproblems have been investigated by many scholars specifically in the limit ofhaving slender bodies, e.g., for a viscous suspending fluid case [11, 31, 32] andalso for a viscoelastic fluid [42, 81]. In the case of yield-stress fluids, Wachsand Frigaard [142] addressed the stability of 2D rectangles with aspect ratio2 with different orientations, but only briefly as part of a test problem intheir study.The complexity of the equations governing the problem of particle sedi-mentation with orientation in a yield-stress fluid makes any analytical studycomplicated. However, we may use the same tools as in the previous chap-ters. This short chapter presents only preliminary results, where we havetried to find simple expressions for the yield limit and for particle settling.We leave a more in-depth analysis to future studies, as it needs on its own,an extensive study.The outline of the present chapter is as follows. We briefly define the128problem considered in Section 4.1. The stability of particles with orientationis considered in Section 4.2: a model is proposed and also the validity of thismodel far from the yield limit is investigated. The chapter is closed with abrief summary.4.1 Problem statementThe equations governing this problem are the same as equations (1.19) and(1.20). With respect to the particle geometry we consider particles with twoorthogonal axes of symmetry. Most of our computations are for rectangularparticles, with a few additional ones for 2D elliptical particles to confirm anydiscussion we provide. A schematic of the problem is sketched in Figure 4.1.The orientation of the particle is designated by φ and α shows the steadydrift angle.𝜙𝛼𝑦𝑥𝑈𝑝ො𝑔Figure 4.1: Schematic of the problem considered.Regarding the length scale, we choose Lˆ consistent with chapters 2 & 5and also [25], in order to make any potential comparison convenient. There-fore,Lˆ =(Aˆppi) 12= 2(aˆbˆpi) 12,for the rectangle, where Aˆp is the dimensional particle area (4aˆbˆ). The129aspect ratio of the rectangle is again designated by χ = bˆaˆ =ba . It should benoted that in the present study we only consider particles with χ > 1, sinceany problem with χ′ < 1 and φ′ can be considered as a problem with χ = 1χ′and φ = φ′ + pi2 . In other words,P(χ, φ, Y ) ≡ P( 1χ, φ+pi2, Y ),where P denotes the problem defined by equations (1.19) and (1.20).4.2 ResultsFigures 4.2 and 4.3 show example computations for χ = 2 and χ = 5, re-spectively. As well as the velocity field and yield surfaces, as before, weillustrate the direction of motion of the particles. We observe some fea-tures as previously, e.g. attached angular plugs of fluid, some symmetry ofsolutions, some large rotating regions of unyielded fluid. There are also so-lutions such as Figs. 4.2b and 4.3b, for which the velocity contours appearto spiral. As can be seen in these examples, fully quantifying the solutionsof the problem P (χ, φ, Y ), as functions of the three parameters (χ, φ, Y )will be a non-trivial task. We therefore first put effort to finding the effectof orientation of particles on the static stability or yield limit.4.2.1 Yield limitThe present problem, P (χ, φ, Y ), may be decomposed (from a force per-spective) into two sub-problems as follows (see Figure 4.4),(i) PI (χ, 0, Y‖): a [M] problem of a particle with aspect ratio χ, and yieldnumber Y‖ = Ycosφ in which the particle length is in the direction of‘gravity’ (or in other word we consider only the component of gravityin the y′ direction).(ii) PII ( 1χ , 0, Y⊥) = PII (χ, pi2 , Y⊥): a [M] problem of a particle with aspectratio 1χ , and yield number Y⊥ =Ysinφ in which the particle widthmoves in the direction of ‘gravity’ (or in other word we consider only130      0 2 4 6x 10−40 5 10x 10−40 2 4 6x 10−4(a) (b) (c)Figure 4.2: Velocity contours for χ = 2: (a) φ = 30, Y = 0.122, (b)φ = 60, Y = 0.095, (c) φ = 75, Y = 0.0925. The blue arrowshows the steady direction of motion of the particle and whitelines are yield surfaces.      (a) (b) (c)0 1 2 3 4x 10−40 1 2 3x 10−30 2 4x 10−3Figure 4.3: Velocity contours for χ = 5: (a) φ = 30, Y = 0.13, (b)φ = 60, Y = 0.075, (c) φ = 75, Y = 0.06. The blue arrow showsthe steady direction of motion of the particle and white linesare yield surfaces.the component of gravity in the x′ direction).It is worth mentioning that problem P (χ, φ, Y ) of course is not a linearproblem from the velocity perspecive. However, by this simple decomposi-tion, we will later show that we can get a good estimate of certain aspects ofthe problem P (χ, φ, Y ). Indeed, this finds its root in this fact that in manycases (based on different values of χ, φ, and Y ) the solution to the eitherproblems PI or PII is zero: the particle is static in one of the problems PI131𝜙𝛼𝑈𝑝ො𝑔ො𝑔sin𝜙= +ො𝑔cos𝜙𝑌 =Ƹ𝜏𝑌∆ො𝜌 ො𝑔 ෠𝐿𝑌 =Ƹ𝜏𝑌∆ ො𝜌 ො𝑔cos𝜙 ෠𝐿=𝑌cos𝜙 ||𝑌 =Ƹ𝜏𝑌∆ ො𝜌 ො𝑔sin𝜙 ෠𝐿=𝑌sin𝜙( I ) ( II )Figure 4.4: Schematic of the decomposition.or PII.For instance, as we go to large χ limit, the critical yield number forproblem (I) is much higher than the critical yield number of problem (II).We will show these two critical values by Y‖,c and Y⊥,c, respectively. Asit has been shown in Chapter 2, the critical plastic drag coefficient of PIIasymptotes to 2 + 3pi for 1χ → 0. However, regarding PI, the leading-orderof critical plastic drag coefficient is√pi2 χ12 when χ→∞. It suggests,Y‖,c = 2, Y⊥,c =√pi2 + 3pi√1χ, for χ→∞. (4.1)SinceY‖Y⊥ = tanφ, it means that when χ→∞, for φ < tan−1(2(2+3pi)√pi√χ),the problem PI is really controlling the motion of the particle, i.e. we expectthat α ∼ φ and there is almost no motion in the x′−direction. There willbe only a narrow range of φ ≈ pi2 , for which problem PI is comparable to PIIand consequently there is no motion in y′−direction (α ≈ 0).This analysis is not restricted to case the χ→∞. For finite χ cases weshall claim that:132• If Y‖,c cosφ Y⊥,c sinφ then Yc ≈ Y⊥,c sinφ.• Otherwise if Y‖,c cosφ Y⊥,c sinφ then Yc ≈ Y‖,c cosφ.We denote by φc this critical value of φ at which the P problem switchesfrom being dominated by either PI or PII. Hence,{Yc ≈ Y⊥,c sinφ if φ 6 φc,Yc ≈ Y‖,c cosφ if φ > φc.(4.2)To show this, we have plotted the Yc for χ = 1, 2, 15, 50 in Figs.4.5 to 4.8, respectively. In these figures the red and blue continuous linesshow expression (4.2) and the black asterisks are the present computationalresults. As we expect and as is clear in these figures, for large χ, expression(4.2) is more accurate because of the dramatic difference between Y‖,c andY⊥,c.0 15 30 45 60 75 90?00.030.060.090.120.15Y cFigure 4.5: Yc versus φ for a rectangular particle of χ = 1.In a further attempt to approximate Yc over a range of orientations, wefirst note that PI and PII are problems that correspond to stress fields thatscale as Y/ cosφ and Y/ sinφ, respectively. Thus, we might propose a more1330 15 30 45 60 75 90?00.030.060.090.120.15Y cFigure 4.6: Yc versus φ for a rectangular particle of χ = 2. The blackcircles are results computed by Wachs and Frigaard [142].0 15 30 45 60 75 90?00.030.060.090.120.15Y cFigure 4.7: Yc versus φ for a rectangular particle of χ = 15.general trignometric interpolation of form,Yc = Y‖,c cosn φ+ Y⊥,c sinn φ, (4.3)1340 15 30 45 60 75 90?00.020.040.060.080.10.12Y cFigure 4.8: Yc versus φ for a rectangular particle of χ = 50.and use the numerical data to fit n. We have found that n = 9/4 gives anexcellent fit to the intermediate range, for a wide range of aspect ratios. Thedotted-discontinuous brown lines in Figs. 4.5 to 4.8 represents expression(4.3) for n = 9/4. Therefore, we may propose expression,Yc = max(Y‖,c cosφ, Y‖,c cos94 φ+ Y⊥,c sin94 φ), (4.4)which works for entire range of χ and φ.To check that expression (4.4) is more universal, (i.e. not only fits therectangular particle shapes), we computed Yc for a range of 2D ellipses aswell. Figs. 4.9 and 4.10 provide the data for these cases: 2D ellipses withaspect ratio 2 and 15, respectively. As can be seen, expression (4.4) alsopredicts Yc for 2D ellipses quite effectively. We have no physical rationalefor the n = 9/4 exponent.4.2.2 Settling velocityHaving seen that the decomposition method has been effective in predictingYc for oriented particles, it is natural to explore a similar method for the set-tling velocity, with Y far from the yield limit. Our methodology is as follows.1350 15 30 45 60 75 9000.030.060.090.120.150.18φYcFigure 4.9: Yc versus φ for a 2D ellipse with χ = 2.0 15 30 45 60 75 9000.030.060.090.120.15φYcFigure 4.10: Yc versus φ for a 2D ellipse with χ = 15.For given Y , at fixed χ and φ; then we solve PI(χ, 0, Y‖) and PII( 1χ , 0, Y⊥);and compare the velocity of the particle in the actual problem, Up(P), with√Up(PI)2 + Up(PII)2. We then repeat this comparison for a wide range ofY .This comparison is made for χ = 2 & φ = 30 and χ = 50 & φ = 15 inFigs. 4.11 and 4.12, respectively. In these figures the bottom horizontal axisshows Y/ cosφ and the top one shows Y‖. Vertical broken lines represent136critical yield numbers: red for Y‖,c, blue for Y⊥,c, and black for Yc. Hence, weexpect non-zero solutions for all of problems P, PI, and PII in the region tothe left of all three broken lines. As we move to the right either of problemsPI or PII can give us a zero solution as we cross the yield limit. Thus, inFigure 4.11 we have 4 different regions:(i) To the left of the vertical blue broken line the velocity of particle inall three problems are non-zero.(ii) To the right of the vertical blue broken line and the left of the thevertical red broken line problem PII is static, but problems P and PIgive us non-zero solutions.(iii) To the right of the vertical red broken line and the left of the thevertical black broken line, both problems PI and PII are zero, butthe particle is moving in the actual problem P. This zone can beconsidered as the uncertainty between Yc and Y‖,c for the case (χ =2, φ = 30) which is also sensible from Figure 4.6.(iv) To the right of the vertical black broken all three problems are static.The circles and continuous black lines in the figures below show the ve-locity of particle in problem P. The asterix and broken line represent√Up(PI)2 + Up(PII)2: purple (red+blue) when both Up(PI) and Up(PII)are non-zero; red when Up(PII) is zero; and blue when Up(PI) is zero. Forthe two particular cases in Figs. 4.11 and 4.12, there is a nice agreementbetween the actual velocity of the particle and the velocity approximation(using this decomposition method). Even far from Yc the approximation isgood. It should be mentioned that we did not plot Y⊥,c in Figure 4.12 as itis far to the left of Y‖,c and Yc which are in good agreement and cannot bedistinguished in Figure 4.12.This approximation does however loses accuracy for a particle withsmaller χ (see e.g. Fig. 4.11) and also as φ → φc. As an example of thelatter see Figure 4.13. These cases are where the decomposition methodpredicts poor results in terms of velocity of the particle.1370.035 0.055 0.075 0.095 0.115 0.135 0.155Y= cos?00.020.040.060.080.10.12U p(P)0.035 0.055 0.075 0.095 0.115 0.135 0.155Yk00.020.040.060.080.10.12pU p(PI)2+U p(PII)2Figure 4.11: Velocity of the particle versus yield number (χ = 2, φ =30).0.02 0.04 0.06 0.08 0.1 0.12Y= cos?00.020.040.060.080.10.12U p(P)0.02 0.04 0.06 0.08 0.1 0.12Yk00.020.040.060.080.10.12U p(PI)Figure 4.12: Velocity of the particle versus yield number (χ = 50, φ =15).4.3 SummaryIn this chapter, we first considered the effect of particle orientation on staticstability, i.e. Yc. Our main tool to attack this problem is computation. As1380.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12Y= sin?00.010.020.030.040.050.060.07U p(P)0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12Y?00.010.020.030.040.050.060.07pU p(PI)2+U p(PII)2Figure 4.13: Velocity of the particle versus yield number (χ = 2, φ =60).to the theoretical modelling of this problem, we proposed a very simplemodel in which the actual problem is decomposed into two sub-problems:motion of the particle in the orthogonal directions of its length and widthwith yield numbers modified by the gravitational direction. This methodworks very well for predicting Yc for particles with large aspect ratios whileit loses its sharpness as χ → 1. In generalizing this decomposition methodby interpolating between the two end values: φ = 0, pi/2, we have fitted apower law expression in cosφ and sinφ that is able to predict Yc very wellfor all aspect ratios and orientations. We have also checked the accuracy ofthe expression (4.4) for 2D ellipse particles as well and again nice agreementhas been observed.Regarding the settling velocity of the particle, this decomposition methodagain works well for large aspect ratio particles. For particles with small as-pect ratios, it still can predict convincingly results as long as φ is not closeto φc. When χ→ 1 and φ→ φc, the only reliable tool seems to be compu-tation.139Chapter 5Inline motion of particles andhydrodynamic interactionHaving considered single particles in the earlier chapters, we now move tomulti-particle systems. As reviewed in Section 1.6.6 there are a number ofknowledge gaps where multiple particles and yield stress fluids are concernedand this is an active area of research, e.g. in measuring, modelling andcomputing yield stress suspension behaviours. In line with the direction ofprevious chapters we try to study these flows systematically and in a waythat extends our qualitative understanding of these flows. Avoiding the fullcomplexity, we confine particle motions studied to flows in one directionand collinear, i.e. inline motion along the axis of the falling particles. Weconsider only circular disks.A number of different problems seem interesting and are approachedbelow. Mostly we use direct computation, although we also explore theeffectiveness of the characteristics method of perfect plasticity, with partialsuccess. Some of the problems considered are as follows.• Computational methods: although both problems [R] and [M] canbe formulated the correspondence between the problems is not clear.Solving problem [R] is a straightforward extension of the previousmethods, but has little meaning for a settling problem. Thus, here140we extend our methods for problem [M] to multi-particles.• We have seen that single particles settle within an finite yielded region.This leads to natural questions about whether multiple particles suffi-ciently far apart will settle in their own yielded envelopes completelyindependently or whether the stress fields influence particles beyondtheir yielded region• The effects of multiple particles on the yield number is not known. Wemight expect that particles close together form a larger (and heavier)‘particle’ (or cluster) and light therefore be harder to resist motion.• For problem [R], depending on proximity, it is known that particlescan be bridged together by plug regions. Whether such bridges oc-cur naturally for problem [M] can appear between the particles andconnect them together is not known, nor is the dependence of this onproximity and yield number.• In multi-particle systems the general situation will be that the particlesdo not move at the same velocity. Thus, in understanding the evolutionof a multi-particle flow the relative positions of particles changes, eventhough the flow at any instance may be given by a Stokes flow problem.This means that a dynamic aspect to the flow must be considered.An outline of the present chapter is as follows.1 Below in Section 3.1We start with a brief mathematical definition of the present problem andsome motivating results with two particles, to launch the main study inthe next two sections. In Section 5.2 we will consider steady (Stokesian)motion of mono-size particles N = 2, 3 and what can be said in generalabout N > 3. Section 5.3 uses the Stokesian results to study dynamics ofparticle chains, focusing first at N = 3 and then generalizing. We addressgeneral categorization of hydrodynamic interaction of particles in yield-stressfluids in Section 5.4, where the main distinctions between flows come from1The results of this chapter will appear as: E. Chaparian, A. Wachs, I.A. Frigaard,Inline motion and hydrodynamic interaction of 2D particles in a viscoplastic fluid, Physicsof Fluids, accepted for publication and to appear, 2018.141X1X2X3XNBinghamfluidgˆ⌦Figure 5.1: Schematic of the problem considered.a consideration of whether the velocity and/or stress fields are localised ornot. The chapter closes with a brief summary section.5.1 Inline particle motionWe study the inline inertialess motion of heavy two-dimensional (2D) par-ticles (circular disks) settling within a Bingham fluid, i.e., 2D Stokes flowaround particles arranged as illustrated in Figure 5.1. Dimensionless mo-mentum and continuity equations are:∇ · σ + ρ11− ρ1ey = 0, ∇ · u = 0 in Ω \ X¯, (5.1)where X =⋃ni=1Xi and ρ1 = ρˆf/ρˆ1 < 1 (ρˆ1 is the density of the disks andρˆf the fluid density). We index the particles as in Figure 5.1.The particles are all of the same density and the problem is posed fromthe mobility perspective in which buoyancy is the driving force for mo-tion. Thus, all stresses are scaled with the buoyancy stress of the firstparticle: (ρˆ1 − ρˆf )gˆRˆ1, where gˆ is the gravitational acceleration and Rˆ1is the radius of the first disk (also used as the length-scale). The dimen-142sionless constitutive equation of the Bingham fluid is the same as Equa-tion 1.20 where Y represents the balance of yield stress to the buoyancystress: Y = τˆY /[(ρˆ1 − ρˆf )gˆRˆ1].The radius of the i-th disk is Rˆi, and when made dimensionless: χi =Rˆi/Rˆ1. The inter-particle distance is `i, which is equal to the distancebetween the centres of i-th and (i + 1)-th disks. In the previous chapters[24, 25], we have considered a single particle in R2 and have focused on thecritical Y at which motion stops, e.g., for a single disk we find Y ∗c = 0.1316,agreeing with previous reported results [114, 134]. In moving to a sequenceof particles, the i-th particle senses a yield number,Yi =τˆY(ρˆi − ρˆf )gˆRˆi=τˆY(ρˆ1 − ρˆf )gˆRˆi= Y1 · 1χi. (5.2)Please note that Y1 = Y . One intuitive notion is that particle ith will notmove if Yi > Y ∗c , meaning no motion of the i-th particle when Y > χiY ∗c .Thus, smaller particles should become static at lower Y . This is indeed foundto be the case, but Yi > Y ∗c is the correct criterion only as the inter-particledistance `i becomes sufficiently large. A second interesting possibility arisesat small `i: here two adjacent particles may combine so that the criterionY > Y ∗c is insufficient to stop motion of the largest particle. These are thequestions we explore.In this chapter, we will use same methodology as previous chapters (i.e.,augmented Lagrangian method and method of characteristics) to investigatethe present problem.5.1.1 Example results for two particles (χ2 < 1)To gain intuition, we start with example flows involving 2 particles, andtake χ2 < 1 without loss of generality (i.e., reversing the direction of theflow would place the smaller particle in front and rescaling lengths with thesmaller particle radius would give χ2 > 1). To illustrate the type of flowfound, Figure 5.2a shows the speed colormap for Y = 0.11, χ2 = 1/2, `1 = 4.We observe firstly that, as for single particles, all of the motion is confined143  2 2.5 3 3.5 4 4.5 500.0040.0080.0120.016ℓ1V  −V1−V20123456x 10−3(a)(b)Figure 5.2: Example resutls for Y = 0.11, `1 = 4, χ2 = 1/2: (a) Speedcolourmap and yield surfaces (white); (b) settling velocities ofthe two particles.within a yielded envelope, i.e., the stresses due to the buoyant particlesdecay below the yield stress at a finite distance from the particles. Withinthis envelope the flow field around the larger particle is characteristic of asingle particle, with fore-aft unyielded caps and lateral rigid plugs. However,further out the flow is displaced in a wider region, moving slowly from underthe particle around to the back where particle 2 is located. The stress fieldsof the two particles clearly affect each other: the fore-aft symmetry of theflow around particle 1 is only disturbed marginally close to the particle,whereas particle 2 shows no fore-aft symmetry.The settling speed of the smaller particle 2 in Figure 5.2a is less thanhalf that of particle 1, but it is still mobile although Y > χ2Y ∗c here. As wevary `1 the particle velocities vary, as shown in Figure 5.2b. Two interestingfeatures emerge from this plot: first, the velocities coincide below some144        (a) (b) (c) (d)(e) (f) (g) (h)0 0.012 0 0.009 0 0.0075 0 0.0068Figure 5.3: Speed colormap and yield surfaces (white) for an assem-bly of two particles. Top panels show the flow before break-ing the plug bridge and bottom panels are associated withslightly increase the inter-particle distance, which results in theplug breaking: Y1 = 0.11. (a,e) χ2 =12 , (b,f) χ2 =13 , (c,g)χ2 =14 , (d,h) χ2 =15 ; `1 = 2.55, 1.85, 1.62, and 1.5.critical `1; second, the smaller particle becomes static above a second critical`1. This suggests that interesting qualitative transitions are taking place inthe flows around the particles. To explore further rate, we fix Y = 0.11 < Y ∗cand compute flows for a range of different χ2 in Figure 5.3.145The top row of Figure 5.3 shows particles with χ2 = 1/2, 1/3, 1/4, 1/5,positioned at separation distances `1 = 2.55, 1.85, 1.62 and 1.5, respectivelyfor Figure 5.3a–d. We observe that the two particles are connected by abridge of unyielded fluid, which is the reason for settling at the same speedin Figure 5.2b. We have focused the images close to the particles. Furtheraway the fluid moves only within a finite yielded envelope surrounding theenvelope of both particles, e.g., similar to Figure 5.2a. Intuitively, thisplug bridge might have been expected. The key point here however is thatthe small particles, which cannot move alone, can be pulled/pushed by thenearby larger particles. As we have seen, motion of the smaller particle isnot restricted to having a plug bridge between particles, but does requireproximity. In Figure 5.3e–h the inter-particle distance has been marginallyincreased from that in the top row: the plug bridge is now broken andthe particles move at different speeds, i.e., this captures the first transitionevident in Figure 5.2b.As `1 is increased further beyond this critical breaking distance the set-tling speed of both particles is reduced, with the smaller particle of coursemoving slower. Note however that the stress field of the smaller particlealso influences the larger, so that the larger particle speed also decreaseswith separation. Eventually, since we have Y > χ2Y∗c the smaller particlebecomes stationary at large enough `1, while the larger continues to move.Through exhaustive computation, these two critical distances can be calcu-lated and the flow regimes are as plotted in Figure 5.4.Approximating Yc for two disksAs observed above, the flows depend significantly on (χ2, `1) at each Y .Nevertheless, we expect that for any fixed (χ2, `1) the flow will stop forsufficiently large Y > Yc. Here we use the method of characteristic toapproximate the critical yield number, Yc. Following Chapter 3 and [24], wefirst find the plugs attached to the particle surface(s) by using the ‘unyieldedenvelope rule’, and then compute the sliplines to calculate the normal stressand shear stress contributions on the unyielded envelope which surrounds the1461 3 5 7 90.10.30.50.70.9ℓ1χ2MovingseparatelySmaller diskstaticbridgePlugFigure 5.4: Critical distances and flow regimes for different χ2 and`1: Y = 0.11. The discontinuous blue line represents where thediscs make contact. Blue crosses indicate points at which theflow has been computed and characterized.particle. Since the two particles in close proximity are connected by a plugbridge (forming a ‘single’ heavier and larger particle), we apply the sameidea; see Figure 5.5. Details of our calculations are given in Appendix D.The lower bound of the critical drag coefficient will be:[CpD,c]L= 2√2 (χ2 + 1) + `1 + 2pi + 2(1− χ2)2`1. (5.3)Note that if χ2 = 0 (i.e., Rˆ2 = 0) and `1 =√2 then CpD,c = 4√2+2pi ≈ 11.94,which is the known value for a single disk. Knowing the lower bound of thecritical drag coefficient, an estimation of the critical yield number then canbe calculated as:[Yc]U =As`⊥[CpD,c]L=As2[CpD,c]L=τˆY(ρˆs − ρˆf )gˆRˆ1, (5.4)where As is the total area of the solid particles in Figure 5.5, i.e., As =147ABCDEFGFigure 5.5: Schematic of the unyielded envelope around assembly oftwo particles (the red discontinuous line shows the symmetryline).pi(1 + χ22). Hence, we find:[Yc]U =pi(1 + χ22)4√2 (χ2 + 1) + 2`1 + 4pi + 4(1−χ2)2`1. (5.5)Note that this expression is valid only when [Yc]U > Y∗c . Otherwise, if[Yc]U 6 Y ∗c , this suggests that the smaller particle has no effect and theflow is then expected only to occur around the first particle, i.e., Yc = Y∗c .Figure 5.6 shows the comparison of expression (5.5) with computed results.For small χ2 the slipline approximation is clearly very good, but accuracydegrades at larger χ2. As χ2 → 1 the rigid region, contained between theparticles and inside the line DE, is rectangular supplemented by the roundedends of the two particles. Although there should be a slipline solution thatcorresponds to the limiting viscoplastic problem, Chaparian and Frigaard[25] has shown that finding the characteristics in this way is not very accuratefor this type of geometry. Indeed, deducing/constructing the correspondingslipline solution is not a trivial task.148ℓ1Y1.5 2.5 3.5 4.5 5.5Y ∗c0.140.150.16ℓ1Y1.25 1.75 2.25 2.75Y ∗c0.1350.13750.14(b)(a)Figure 5.6: Comparison of Yc for assembly of two disks: (a) χ2 = 1/2,(b) χ2 = 1/4. Blue line shows the slipline theory prediction(Equation 5.5) and red symbols are numerical results.5.2 Results for uniform disksWe now fix χi = 1 for the remainder of our study in this chapter. This allowsus to focus on the effects of inter-particle separation on steady Stokes flowsand eventually on the dynamics. Also, setting χi = 1 introduces a knownbaseline for motion, i.e., for large enough `i all particles become isolated andstatic at the same Yc.5.2.1 N = 2We start with two disks. Different scenarios can happen based on the yieldnumber and the inter-particle distance. It is intuitively clear that if particlesare far away from each other and the yield number is less than Y ∗c , then bothare moving in their own yielded envelope and are separate from each other(see Figure 5.7a). If we now take Y > Y ∗c at the same `1, the particles maystop e.g., Figure 5.7b, or may continue to move. In the latter case it is the149stress field from each particle that influences the other particle, contributingto yield the fluid and allow motion.Due to symmetry, in all cases the settling speeds of the two particles areidentical. As in the previous section, as we decrease the distance betweenmobile particles sufficiently, a single yield envelope will form: eventuallyunyielded fluid bridges connect the particles. This sequence is illustrated inFigure 5.7a, c, e, for Y = 0.12 < Y ∗c , and in Figure 5.7b, d, f, for Y = 0.14 >Y ∗c . In the latter case, although the isolated disks are static the bridged disksremain mobile. If we increase Y > 0.15, then the intermediate regime inwhich we have two separate moving particles in the same yield envelope,will disappear. Only two possibilities remain: two connected moving disksor two static disks for small and large inter-particle distance, respectively.Figure 5.8 shows the different regimes in the (`1, Y )-plane. The colormaprepresents particle velocities. Also shown in this figure are the data fromother researchers. The limit between 2 connected and 2 stationary particlesis in excellent agreement with the extensive computational study by Tokpaviet al. [135], which also describes quantitatively other features such as sizesof plug regions. Experimental results of [74] are also shown and are slightlyhigher than the computed values. Interestingly, these experimental valuesagree reasonably with the Yc values predicted by expression (5.5). Thedeviation of the experimental values from our ideal yield stress fluid resultscould have many reasons, e.g., slip, elasto-plastic yielding, etc.5.2.2 N = 3For N = 3 the 3-disk system in R2 is characterised by Y and two inter-particle separations: (`1, `2). At any fixed Y we may compute the particlevelocities Vi at each (`1, `2) and hence characterize the possible flows. Bysymmetry, we have both V2(`1, `2) = V2(`2, `1) and V3(`1, `2) = V1(`2, `1).Let us first consider any fixed Y < Y ∗c , for which no particles may bestationary. If `1 = `2 = 2, then the 3 particles touch. We may identify thefollowing regimes for `i > 2.• Regime (I): By continuity, within a neighbourhood of `1 = `2 = 2150    (a) (b)0 0.5 1 1.5 2x 10−30 0.5 1 1.5 2x 10−3    (c) (d)0 2 4 6 8x 10−40 0.003 0.006 0.009 0.012    (e) (f)0 0.005 0.01 0.015 0.020 0.01 0.02 0.03 0.04Figure 5.7: Speed colourmap and yield surfaces (white lines). `1 = 10:(a) Y = 0.12, (b) Y = 0.14; `1 = 6.5: (c) Y1 = 0.12, (d)Y1 = 0.14; `1 = 3: (e) Y1 = 0.12, (f) Y1 = 0.14. Please notethat 0.12 < Y ∗c = 0.1316 < 0.14.151ℓ1Y  2 4 6 8 10 12 14 160.080.1Y ∗c0.160.180.20.220.24−0.2−0.15−0.1−0.05Movingdisks in a sameenvelopeSeparatemoving disksConnectedStationarydisksFigure 5.8: Different flow regimes in the (`1, Y )-plane for two disksof the same size (χ2 = 1). The colormap indicates the settlingspeed of the disks. The solid blue line marks the border betweentwo separate moving particles and two static particles regimes,while the dotted blue line just shows Y ∗c . Broken black lineis Yc computed by Tokpavi et al. [135]. Broken green line isthe Yc calculated by expression (5.5). Diamond symbols areexperimental measurements by Jossic and Magnin [74].we can expect that the 3 particles remain connected by an unyieldedbridge.• Regime (II): Suppose `1 = `2 and we increase the separation distancesequally. At some distance the bridges to both particles will break, andall 3 particles move individually, yet within a single yielded envelope.We expect that particle 2nd settles the fastest. For the `1 > `2 closeto `1 = `2 we call this regime (IIa) and its symmetric reflection about`1 = `2 is regime (IIb).• Regime (III): Again moving away from regime (I), but keeping say`2 ∼ 2 and increasing `1 we might expect to first break the bridgebetween particles 1 and 2, while that between 2 and 3 remains intact.We call this regime (IIIa) and the symmetric reflection about `1 = `2is regime (IIIb), where 1 & 2 remain bridged and particle 3 movesseparately.152• Regime (IV): lies at larger separations, where all 3 particles moveindividually, but 2 of the particles share the same yielded envelope.Again (IVa) refers to `1 > `2 and (IVb) refers to `1 < `2.• Regime (V): where all 3 particles move individually, each within theirown yielded envelope.Secondly, we consider Y > Y ∗c for which sufficiently distant particles mightbe static. Now:• Regimes (I), (II) & (III) are as described for Y < Y ∗c .• Regimes (IV) & (V) may or may not exist, depending on Y .• Regime (VI): denotes where two particles are moving within a singleyielded envelope and the other is static.• Regime (VII): denotes where two particles are moving within theirown yielded envelopes and the other is static.• Regime (VIII): denote where particle 2nd is mobile and the other 2particles are static.• Regime (IX): denotes where all 3 particles are static.Each of regimes (VI)-(IX) will have parameter regimes (a & b) reflectedsymmetrically about `1 = `2. The precise configuration of these regimesrequires computation to evaluate, for any fixed Y . Although regime bound-aries may be uncertain, we expect that each Vi varies smoothly with (`1, `2),so that precise determination of each regime is mainly of academic interest.Depending on Y not all of the regimes may exist, e.g., at large enough Y onlyregime (IX) will exist, Y may be large enough to require that two particlesinteract in order to yield the fluid, etc. Other than extensive computation,if either `1 or `2 is sufficiently large we might expect the closest 2 particlesto behave analogously to a 2 particle system.As an example, we illustrate the flows for Y = 0.15 in Figure 5.9. Wefirst show examples of regimes (VI), (III) and (II) in Figure 5.9a–c, respec-tively. Figure 5.9d shows the computed regime boundaries. It is interesting153    2 4 6 8 10 12 142468101214ℓ1ℓ2  0 0.001 0.002 0.0030 0.008 0.016 0.024 0 0.0017 0.0034 0.00512 2.05 2.1 2.1522.052.12.15ℓ1ℓ2(b) (c)(a)(d) (e)(Ia,b)(IIIa)(IIIb)(IIIb)(IIb)(IIa)(IIb)(IIIa)(VIIa)(VIIb)(IIa)(IXb)(IXa)Figure 5.9: Triple disks map and examples (Y = 0.15): (a) `1 =10.5, `2 = 2.1 (region (VII)), (b) `1 = 9.75, `2 = 4.9 (region(III)), (c) `1 = 6.5, `2 = 6.5 (region (II)), (d) Different regionsin `1 − `2 map, (e) Zoom of the panel (d) around `1 ∼ `2 ∼ 2.Computational points are not marked in this figure in order toavoid cluttering it.to note that regime (I) is extremely small and is localised about `1 = `2 ' 2.For `1 6= `2 one of the bridges tends to break, see e.g., Figure 5.10a. Thislimit appears difficult to compute precisely; perhaps because the numericalmethod is targeted at identifying yielded/unyielded fluid on an individualelement, rather using this information to construct a yield surface. Depend-ing on Y we may certainly compute 3 connected particles at `1 6= `2, seee.g., Figure 5.10b. Thus, we consider that the boundary between regimes(II) and (III) (green circles in Figure 5.9d) in fact approaches the regime (I)boundary for `1 > `2 as `1 → 2, and perhaps asymptotically `1 → 2 at theregime (I)-(II)-(III) triple point where the boundaries intersect.154    (a) (b)0 0.02 0.04 0.06 0.08 0 5 10x 10−4Figure 5.10: Regime (I) flows: (a) Y = 0.15 `1 = 2.065, `2 = 2.05; (b)Y = 0.25, `1 = 2.01, `2 = 2.07.5.2.3 N > 3For N > 3, it appears that the number of possible flow regimes increases.In principle, each particle may be static or may settle. A settling particlemay settle in its own yielded envelope or may share a yielded envelope withone or more neighbours. Particles that share the same envelope may moveindividually or may be connected via a bridge of unyielded fluid. If oneis to study these flows in a systematic way, two questions appear to be ofinterest. First, given an N -particle system, is there a minimal `N−1 abovewhich the N -th particle is static? For such `N−1, to what extent may weconsider the behaviour of the (reduced) N−1-particle system independentlyof the N -th particle? Second, for fixed Y , what is the minimal `i such that,for separations below this distance, all the particles form a single connectedchain? We have seen that connected particles settle the fastest, so thisappears to be a relevant question for bounding the settling speed.The first of these questions we address later in this chapter. The second155      (a) (b) (c)0 0.113 0.226 0 0.021 0.042 0 0.021 0.042Figure 5.11: Five disks `1 = `2 = 2.1: (a) Y = 0.15, (b) Y = 0.25,(c) Y = 0.35.of these questions, surprisingly we found was not always possible. Fig-ure 5.11 shows the flow field around N = 5 disks with `i = 2.1 as Y isincreased. For the relatively small `i considered, it seems that initially nobridges are formed. Then, no bridge is formed between the outermost parti-cles and the inner trio of particles (which have bridged). At sufficiently largeY all the particles stop. Evidently, as we have observed before, the centralparticles feel the largest magnitude stress fields from the other particles andhence move the fastest, here as a trio. The same result was found for N = 4,with the central pair of particles bridging but leaving the outer two particlesunconnected.It is unclear whether or not there can be any bridged inline combinationsfor N > 3 with `i > 2. Figure 5.12 shows the velocity along the y-axis fortwo successive values of Y as we approach the limit of zero motion. In bothcases the central 3 disks are bridged. We calculate the ratio V5/V3 of theouter disk velocity to those of the central 3. We see that V5/V3 < 1 andthat this ratio in fact seems to decrease as Y → Yc, which suggests that asY → Y −c the outer disks do not bridge at all.5.3 Dynamics of motionThe previous section computes the regimes and velocity of different inlinedisk configurations. The computed disk velocities may be integrated forward1560 5 10 15 20−0.05−0.04−0.03−0.02−0.010yV  Y1 = 0.32V5V3= 0.8120V5V3= 0.94843 4 5Y1 = 0.25Figure 5.12: Velocity at the centerline versus y-coordinate which isfixed at the centre of the middle particle. The filled grey boxesshow inside of particles and numbers on that are index of theparticles.in time to provide new relative locations and hence track particle dynamicsin the Stokesian regime. We have observed that the central disk(s) in anyconfiguration move the fastest. For Y > Y ∗c the consequence of this is thatthe uppermost particles are left behind the central particles, with `N−1 even-tually becoming large enough that VN = 0 and the uppermost particle is leftbehind. The central particles also catch up the lower particles, potentiallybridging together with those below. However, as we have seen above, havingmore than 3 particles bridged does not seem possible. Thus, at long timesit seems that any initial N -particle system will degenerate into a 3-particlesystem, comprising the lowest 3 particles. For this reason we study the caseN = 3 in detail.5.3.1 N = 3For 3-particles in R2, the particle speeds depend only on the separationdistances between particles and not on the explicit positions. Thus, the 3-157disk system is governed by the following autonomous system of differentialequations:d`1dt= dV1(`1, `2) = V1(`1, `2)− V2(`1, `2), (5.6)d`2dt= dV2(`1, `2) = V2(`1, `2)− V3(`1, `2). (5.7)Due to the symmetry of V1 to V3, we have that dV1(`1, `2) = −dV2(`2, `1).The velocities V1 to V3 are evaluated numerically at regular intervals in(`1, `2) and used to construct the functions dV1(`1, `2) and dV2(`1, `2) on theRHS of equations (5.6) and (5.7). The function dV1(`1, `2) is shown in thecolormap of Figure 5.13. These functions are then used to integrate equa-tions (5.6) & (5.7) and `1 and `2 as functions of time, i.e., the solutions.The set of all solutions to equations (5.6) and (5.7) can be represented asthe phase paths of the system: lines with slope dV2(`1, `2)/dV1(`1, `2) inthe (`1, `2)-plane. These are sketched in Figure 5.13. We observe that thephase paths are symmetric about `1 = `2. A consequence is that any initialconfiguration of (`1, `2) = (`1,0, `2,0) evolves towards a final configuration(`1, `2) = (`2,0, `1,0) after an orbit time T (`1,0, `2,0). With reference to Fig-ure 5.13, the orbit times are shortest when `2,0 ≈ 2 and are longest (infinite)as regime (IX) with 3 static particles is approached.The positions of the disk centres, say y1(t), y2(t), and y3(t), can also becomputed during any phase orbit. The cases in which initially the first diskis static and second and third disks are connected is of most interest (i.e.,regime (III) of Figure 5.9). Figure 5.14a shows the case of `1,0 = 12 and`2,0 = 3. In panels (b-e) we see how the three particles move and changetheir positions during the trajectory. The position of each of these figuresis marked on a phase path of Figure 5.13 using the same symbols. Theparticle positions versus ‘time’ are shown in Figure 5.14f. As can be seen,the first particle (blue) is static for the first stages until the top two, whichare initially falling with the same velocity, get close enough. The red particlevelocity then decreases as the lower two particles approach and accelerate.The plug bridge between the top two particles breaks. All three particlesmove, but the second particle catches up to the lowest particle and eventually158ℓ1ℓ2  2 4 6 8 10 1224681012−0.025−0.02−0.015−0.01−0.0053 4 5051015ℓ2T×10−3Figure 5.13: Phase paths of the system (5.6) and (5.7) with Y1 =0.15. Phase paths are superimposed upon the colourmap ofdV1(`1, `2) map, Y1 = 0.15. The small panel in the right toprepresents T × 10−3 versus `2 where T is the orbit ‘time’ foreach path. Please note that in all paths the initial conditionis `1 = 12.a plug bridge forms between the lower two disks. These leave behind theupper disk, which eventually becomes static. Note that the final separationof `2 is the initial separation of `1, due to symmetry, i.e., this position definesthe end time of the orbit. However, although the phase paths are symmetricthe disk positions in Figure 5.14f have all shifted downwards by a length `s.5.3.2 Chain dynamics for N > 3In principle we may consider any chain of N particles in R2. In place ofequations (5.6) and (5.7), now we have the following N − 1 dimensional159y−25−20−15−10−50510152025t0 700 1400(b)(a) (c) (e)(d) (f)Figure 5.14: Different snapshots of three particle dynamics. In allpanels (a-f) y-axis shows the y coordinate. Asterisks in panel(f) mark the ‘times’ of panels (a-e). Also same symbols areused in Figure 5.13 to mark the time-lapses of panels (a-e) in(`1, `2) domain.autonomous differential equation system:d`kdt= dVk(`) = Vk(`)− Vk+1(`), k = 1, 2, ...N − 1, (5.8)where ` = (`1, `2, ...., `N−1). Constructing dV(`) via computation becomesprohibitively expensive and we have little to say about the general system.However, motivated by the previous N = 3 example (Figure 5.14) weask if there is an extension of this type of motion to N > 3? More explicitly,we ask if we can find an initial configuration such that motion is transferredfrom the pair (N,N −1) to (N −1, N −2), .... to (2, 1), in each case leavingthe previous particle static.Let us consider a chain of N > 3 particles, with N−1 of particles initiallypositioned at uniform separations `initial such that they are static. The lastparticle is then inserted between the uppermost 2 particles: (i) sufficientlyclose to the top particle that this pair begins to move; (ii) sufficiently distant160from the next particle that it remains initially static. We see that this isthe case for the initial N = 3 spacing in Figure 5.14f. At the outset wesee that `initial must be larger than that for the 3 particle system, since wehave observed that the configuration shifts down by a distance `s. As a firstguess we should simply add `s to the initial 3-particle separation.This appears to work, in that we have computed Stokes flow solutionsfor some 4 particle systems that indicate that the particle motion would betransferred down the chain. However, we have not constructed the functionsdV(`) fully nor computed the dynamics for N = 4 or higher. Although itappears that we may extend to larger N , the question arises of whetherthere is a `initial that allows this type of chain dynamics as N → ∞. Twoconsiderations are:(i) Does `s increase with N (and then converge as N → ∞, as would benecessary)?(ii) does the stress field from the additional (initially static) N -th particleaffect the motion of the other N−1 particles. if `initial is large enough?This is the same question as asked earlier and concerns the stress field.5.4 Locality of the stressMany of the computed solutions illustrated earlier are strongly localisedabout the particle. However, we note that the localization refers to thevelocity solutions and not necessarily the stress field. Certainly we knowthat the stress field is less localised than the velocity field as many of ourexamples have shown motion for Y > Y ∗c , due to proximity of particles.Stress effects are also evident for Y < Yc. For example, at N = 2, theminimal distance required to ensure that two disks are separate and movingin their own yielded envelopes (e.g., Figure 5.7a) is much larger than twicethe maximum height of the yielded envelope of a single disk. For example, forY = 0.11, the yield surface of a single disk reaches approximately y ≈ 2.81(see Figure 5.15) while the critical distance to have two separate yieldedenvelopes at Y = 0.11 is `1 ' 11 (see Figure 5.8).161  012345x 10−32.806Figure 5.15: Speed contour around a single disk, Y = 0.11.Thus, in proximity we see that even when adjacent particles move intheir own yielded envelopes, their stress fields contribute to flows within theother envelopes. On the other hand, as the separation distance increasesthis effect does diminish and we can question if there is a finite “cut-off”separation above which there is no effect on the other particle. This wouldbe satisfied if the deviatoric stress fields around a particle falls to zero at afinite distance, i.e., in this case we could construct a multi-particle chain bypiecing together single particle solutions.The problem with the above approach for any flow modelled using theBingham model (or its variants, e.g., Herschel-Bulkley, etc.) is that thestress field is indeterminate below the yield stress. While computationalmethods such as the augmented Lagrangian method used here computean admissible stress field in unyielded regions, it is not unique. Thus, weare unable to predict how the stresses will be distributed and decay withdistance from a particle in unyielded situations. Note that any such decayprediction of the stress field must only be a property of the Stokes equations,independent of constitutive law. For example, in Newtonian fluids many ofthe analytical results available come from fundamental solutions of the linearelliptic PDE’s that result from the linear constitutive law inserted into the162  00.020.040.060.080.10.120.14  00.020.040.060.080.10.120.140.16(b)(a)Figure 5.16: Second invariant of stress: (a) Y = 0.12 < Y ∗c , (a) Y =0.15 > Y ∗c .Stokes equations.To illustrate, in Figure 5.16 we have plotted ‖τ‖ for a single disk moving(Y = 0.12 < Y ∗c ) and static (Y = 0.15 > Y ∗c ). The fluid is unyielded outsidethe white line in Figure 5.16a and everywhere in Figure 5.16b. The stressdistribution (scaled with Y ) is similar in both unyielded regions, adoptinga dramatic bowtie pattern.163This pattern may be a consequence of the boundary conditions imposedin the far-field. No conditions are imposed on the stress field (non-essential inthe finite element formulation). The velocity is set to zero at the boundariesof the computational domain, as we know that the velocity field is localisedaround the particle. This far-field condition is equivalent (for the velocity) toimposing no-slip on stationary walls. However, rigid walls are not equivalentto far-field decay for the stress problem. Figure 5.16 was calculated bysetting ∂Ω at x, y = ±30. Extending ∂Ω to x, y = ±60 produces the samevelocity solutions but different stress fields: the bowtie region extends out tothe side walls (essentially stretched). This illustrates nicely that (as notedpreviously) these are only admissible stress fields and generally there is nounique correct stress field below the yield stress.Another point of interest in Figure 5.16 is the ‘triangular’ for-aft regionsin which ‖τ‖ = Y , i.e., these particular boundary conditions do not pro-duce a stress field that decays to zero at a finite distance from the particle.Instead the stress field is reminiscent of a perfectly plastic stress field. In-deed, the 45 degree angle separating these regions from the bowtie is alsoevident in the far-field stress field constructed by Randolph and Houlsby[114]. In conclusion, if there is a building block type of y-periodic singleparticle solution, that can be used to piece together chains of static inlineparticles, it must be found using different boundary conditions. This type ofsolution would be relevant to the question of building static arrays of disks(2D suspensions), understanding their stability and yielding properties.We may also approach these problems from the perspective of movingparticles. Once particles are moving within their own yielded envelopes, theonly communication between the particles is via the stress field. It seemsreasonable therefore to examine the velocity field and at what separationdistance particle velocities become identical, as this indicates that the stressfield has no influence.To illustrate, we set Y = 0.12 and compute solutions for N = 3. Threeseparation distances are plotted in Figure 5.17a-c: `1 = `2 = 20, 14, 11,respectively. Figure 5.17d plots the velocity along the y-axis for these 3cases. It appears we may divide the hydrodynamic interactions of particles164into three main categories, as follows.• When particles are far away and have no sense of each other, e.g.,Figure 5.17a. Particles move not only in their own yielded envelopebut also the same speed, which is the same speed as a single disksettling at the same Y . We see the particle velocities are identical inFigure 5.17d for `1 = `2 = 20.• When particles are close enough that their stress fields interact butstill the velocity field is localised in a yielded envelope around eachparticle, e.g., Figure 5.17b. Visually this may be hard to detect thedifference, but since the stress field influences the particles, we seedifferent particle velocities, e.g., Figure 5.17d shows that the centralparticle settles fastest.• When particles are much closer they interact not only via the stressfield but also by their velocity fields: the velocity field is no longerlocalised and a single yielded envelope encapsulate particles, e.g., Fig-ure 5.17c.Although the above characterization is confirmed only by example, i.e., wehave no theoretical results, it does suggest that the influence of the stress islimited in the case when particles move in their own yielded envelopes. ForNewtonian disks in 2D flows, ‖τ‖ ∼ 1/r as r → ∞, and Tanner [133] hasused the same decay rate to infer that the 2D Stokes paradox is eliminatedfor shear-thinning fluids. However, here we see no effect on the velocityfields of adjacent particles with `1 = `2 = 20, which suggests a much fasterdecay.5.5 SummaryThis chapter looked at inline distributions of settling particles in a Binghamfluid. Despite the simplification of restricting motion to one direction, thedynamics of these multi-particle systems remains complex.For paired particles we have shown that there are essentially 3 types ofbehaviour: (i) the particles move together in a yielded envelope; (ii) the165  00.00050.00100.0015  00.0010.0020.0030.004  00.00050.0010.00150.0020 5 10 15 20 25−5−4−3−2−10x 10−3(b)(a)(c) (d)Figure 5.17: Speed contours (Y = 0.12): (a) `1 = `2 = 20, (b)`1 = `2 = 14, (c) `1 = `2 = 11. Panel (d) shows velocity onthe centerline versus y-coordinate which is fixed at the centerof the middle disk. In this panel the red line corresponds to(a), blue discontinuous line to (b), and the black dashed-dottedline to (c).particles move separately in their own envelope; (iii) the particles are static.In the case that the particle radii are different, intermediate between (ii) &(iii) the smaller particle can remain static. Our study of 2 particles of differ-ent radii has also revealed some inaccuracies in using the slipline method toestimate Yc for multi-particle systems, leaving us with computation as theonly reliable tool.Setting χi = 1 we have succeeded to characterize the systems for N = 2& 3 particles, in terms of the flow regimes and particle settling velocity.For N = 2 the particles always move at the same speed. For N = 3 the166main characteristic behaviour (when not static) is that the middle particlemoves the fastest and hence eventually distances itself from the uppermostparticle. Thereafter the two particles settle inline as a pair. To some extentthis form of chain dynamics seems to extend to N > 3.Another interesting feature revealed is the apparent difficulty of havingmore that 3 particles settling together, joined by a bridge of unyielded fluid.It seems that for N > 3 the inner pair (or triple) moves the fastest andbreaks the fluid bridges to the outer particles. Even as Y → Y −c for fixed `iit seems that all the particle velocities Vi → 0, but with ratios Vi/Vi+1 6= 1.This result was quite unexpected: our initial intuition was that a bridgedN -particle chain would be a natural occurrence.Lastly, for large `i the interesting questions relate to whether the parti-cles move independently. We have shown that this is the case: for spacingsbeyond a certain critical `i settling velocities of adjacent particles are identi-cal and the particles move within their own yielded envelopes. This suggestsa stress decay rate faster than would be expected with viscous fluids.167Chapter 6Summary and conclusionsIn this thesis we have studied particle motion and sedimentation in yield-stress fluids. The methodology has been both computational and analytical.In the yield limit or static stability analysis we used plasticity theory to buildour physical insight of flow yielding. However, far from the yield limit it isimpossible to rely only on plasticity theory and so we need the full solutionto viscoplastic flow problems, which are only available computationally ingeneral. Thus, we have developed and relied on computational tools formuch of the thesis.In the present chapter we first summarize the specific results and insightsof each chapter (Section 6.1). In making the novel contributions of thisthesis we have progressed from the initially identified knowledge gaps to thepresent point, and of course there remain untouched problems and challengeson the horizon. Therefore, we try to develop the bigger contextual pictureof the thesis study. We look back at the research motivations, identify thelimitations of our study and discuss possible improvements in Section 6.2.Finally, the thesis closes with our suggestions for future research directionsin this area.1686.1 Results and contributions from theindividual chapters6.1.1 Stability of particles and yield limit (Chapter 2)(i) Studying yield limit or stability of particles in a yield-stress fluids ispossible either by looking at a [R] or a [M] problem. In a [R] problemwe focus on the limit B → ∞, and in a [M] problem at Y → Y −c .Defining a critical plastic drag coefficient makes the analysis muchsimpler as it links both formulations through the drag force.(ii) To the best of our knowledge, this was a first study of its kind in thecontext of particles in a yield-stress fluid in which we have tried tosystematically construct slipline networks around a particle. Thesesliplines are the characteristics of the hyperbolic governing equationsin 2D flows of rigid-perfectly plastic material. We have shown that thismethodology is useful in analyzing the particles yield limit. Indeed,these sliplines give us an admissible stress field and we can use it tofind the lower bound of the critical plastic drag coefficient.(iii) Revisiting the flow about a 2D circle, we have shown that althoughthe lower bound estimation of the slipline network is perfect for thisspecific case, the stress field of the solution to the viscoplastic problemis not strictly the same as the one for plasticity problem.(iv) Although it is true that yield surfaces of the plasticity problem andalso stress fields are not the same as ones of the visoplastic problem,after considering 3 families of 2D particle shapes (ellipse, rectangle,and diamond) with wide ranges of aspect ratio, we have shown thatthe slipline analysis is useful in being: (a) constructive, at least inthe present context, and (b) giving a very reasonable estimate of thecritical plastic drag coefficient.(v) Postulating an admissible velocity field, we are able to find the upperbound of the critical plastic drag coefficient as well, i.e.from plasticity.169However, it is less interesting to compare with the stress fields becauseimproving these velocity fields to get a more reliable estimation (closeto the lower bound to shrink the uncertainty gap) is not a trivial task.One of the issues is that in the absence of viscosity in the rigid-perfectplasticity problem, the flow can slip over the particle surface. Thus, weend up with the open question of setting a velocity boundary conditionon the particle surface in the plastic case.(vi) We have fixed the buoyancy force (i.e., the area of the 2D particlesis identical regardless of shape) in our analysis. Hence, the two ex-treme cases of large and small χ correspond to an infinite plate mov-ing longitudinally or laterally through the quiescent Bingham fluid.As χ→ 0 the lower bound of perfectly plastic limit and the viscoplas-tic limit coincide. The limit is also identical for all shapes considered:CpD,c ∼ 2+3pi. The plastic dissipation in this limit comes from order 1contributions originating from 3 different zones: two boundary layersand a zone rotating at constant angular speed. However, convergenceto the above limit, with respect to χ appears to be dominated by thesides of the particle. At small χ the flow is dominated by the largerigid plugs fore and aft of the particle, which make an angle pi/4 withthe y-axis. Thus, those changes in particle geometry that lie within therigid plugs do not appear to affect CpD,c. In particular the diamondshape for χ < 1 shows no χ-dependency. The ellipse and rectangleconverge to the flat plate limit quadratically and linearly, respectively,as χ→ 0.(vii) Slipline analysis cannot describe the convergence with respect to rheo-logical parameters, (1− Y/Yc) or B. In general this dependency mustbe evaluated computationally for the viscoplastic flows at each χ < 1.For the diamond shape the picture is slightly different as the limit-ing flows may all be mapped to the same problem of a settling unitsquare (rotated through pi/4). Here we have found convergence toCpD,c proceeds as B0.7 or (1− Y/Yc)1.43, from computing the mappedproblem.170(viii) At large χ the dominant contribution to the dissipation in the lim-iting flow comes from a boundary layer along the sides of the parti-cle. This leads again to a universal limit: CpD,c ∼ 2χ (or equivalentlyYc ∼ pi/(2`||)), in agreement with the lower bound analysis at leadingorder in χ. The leading order mass flow balances the fluid dragged up-wards in the boundary layer with that flowing downwards in the largerotating plug. That is why the outer yield surfaces about rectanglesare different in this limit: in the plasticity problem the material whichis pushed from the tip of the particle surface slips over the particlesurface and moves around to the rear. In the viscoplastic problem, itis dragged upwards in the boundary layer by the particle side surfacesand then flows back downwards in the large rotating plugs. Writingthe leading order mass balance and evaluating the terms using ourcomputational results, shows that the fluid dragged upwards in theboundary layer remains of O(1) as χ → ∞, while the rotating plughas extent `p ∼ χ and angular speed ωp ∼ χ−0.5, as χ→∞.(ix) A general analysis of convergence to this limit suggests the settlingvelocity U ∼ b(χ)(1−Y/Yc)ν , where b(χ) ∼ Yc(χ) ∼ χ−1/2 as χ→∞.The exponent ν ∈ [1, 2], depends on the boundary layer thickness (interms of B we converge as B−ν). We interpret ν = 2 as correspondingto a simple one-dimensional shear flow in the boundary layer and ν = 1implies that the boundary layer thickness is independent on (1−Y/Yc).Finding ν requires computation and the results show different ratesfor the 3 geometries studied, e.g., ν = 1.61, 1.32, 1.60 were found forellipse, rectangle and diamond, respectively, at χ = 100. In principleeach geometry should converge to the same flat-plate flow as χ→∞.The different exponents suggest that for relatively large χ we are stillfar from this limit.(x) At intermediate χ away from these limits, the yielded flow envelope isfinite, but different flow regions are significant in contributing to theplastic dissipation. This makes analytic determination of the limitingflows difficult and computation is the main tool.1716.1.2 Cloaking and unyielded envelope rule (Chapter 3)(i) Finding the unyielded envelope which encapsulates the particle is veryimportant in finding the critical plastic drag coefficient. This ques-tion dates back to the very first studies on this topic but has neverfound a complete answer, i.e. what is the shape and how to find it?After performing a large number of computations for different particleshapes, we have found a general rule, using that we are able to findthe limiting unyielded envelope about the particle.(ii) Through finding this unyielded envelope rule, we have identified thecloaking phenomenon. Indeed, we realized that particles with differentshapes could have same unyielded envelopes. Consequently, particleswith different shapes can have same critical plastic drag coefficient.(iii) We have extended the characteristics/slipline methods from perfectplasticity to a class of 2D particles with left-right symmetry. We havefirst shown how to construct the unyielded envelope around particles(using unyielded envelope rule). Then clarified how to construct thecharacteristic network: we should divide the particle shapes into sub-parts from the point of having maximum width; then we construct theslipline network for each part individually; find the mutual parts; andthen construct the remaining parts.(iv) We have demonstrated that the lower bound plasticity estimates givea very good approximation to CpD,c, for more complex particle shapesthat in (Chapter 2), with one line of symmetry.(v) We have also postulated/constructed admissible velocity fields to findthe upper bounds of CpD,c, but as stated before, it is less clear howwe can improve these types of estimate, e.g., ambiguities in boundaryconditions. Hence, it is more reasonable to rely on numerical compu-tations of the viscoplastic problem and lower bound predictions usingslipline network.1726.1.3 Effect of particle orientation (Chapter 4)(i) Particle orientation with respect to body force direction (e.g., gravity)can change the yielding behaviour.(ii) Considering 2D rectangular particles, we have addressed the effect oforientation on yield limit or static stability of particles.(iii) A model has been proposed to capture the yielding behaviour of parti-cles with orientation based on decomposing the force into two compo-nents: one in the direction of the length of the particle and the otherone in the width direction. This model works very well for particleswith large aspect ratios.(iv) For particles with small aspect ratios, the model is modified slightlyto capture the Yc precisely.(v) We have shown that this model also works far from the yield limit(predicts the velocity of the particle) for wide range of aspect ratiosand orientations.6.1.4 Inline motion of particles and hydrodynamicinteraction (Chapter 5)(i) We have looked at inline distributions of settling particles in a Bing-ham fluid because despite the simplification of restricting motion toone direction, the dynamics of these multi-particle systems remainscomplex. The main aim was to again explore the characteristics ofyielding and motion in the Stokes flow limit, but the longer term ob-jective is to build understanding of settling in yield stress suspensions,such as used in fractionation applications and other industrial sus-pensions, e.g., concrete. Although some multi-particle studies can befound in viscoplastic fluids literature, an important difference of thepresent study is that we use the mobility and not the resistance formu-lation. When dealing with multiple particles the resistance formulation173is fundamentally limited, e.g., flow of an array of cylinders in the re-sistance formulation is equivalent to flow past a series of cylindricalobstacles and are particularly of interest in the context of model porousmedia flows, not suspensions. In general, the resistance formulationresults in different drag coefficients for different particles. Hence, using[M] formulations is a key factor because if they are free to move in theflow, over time, these configurations change and the velocities change:the one-to-one correspondence between mobility and resistance formu-lations in Stokes flow is in general lost for multi-particle flows and onlythe mobility formulation is relevant for suspensions.(ii) Implementing a [M] problem for two disks with different radii, we havefound that plug regions can appear between the particles based on theyield number and proximity of the particles. So the two particlesare moving with the same velocity and this can change the yieldingbehaviour as we now are dealing with a “larger” and “heavier” particle.(iii) Small particles, which cannot move alone, can be pulled/pushed byneighbour larger particles.(iv) This picture remains the same when we have particles with the samesize. Indeed, formation of an assembly of particles is possible againbased on yield number and proximity. However, it is less clear if wecan get a chain of more than three particles because even by increasingthe yield number and keeping the inter-particle distances constant, thevelocity ratio of the particles increase. It seems that for N > 3 theinner pair (or triple) moves the fastest and breaks the fluid bridges tothe outer particles. Our initial intuition was that a bridged N -particlechain would be a natural occurrence.(v) These findings are important in the context of stability of suspensionsof yield-stress fluids, as most models that include yielding base theiranalysis on a single particle. It seems that this results in an underes-timation of Yc.174(iv) Setting χi = 1 we have succeeded to characterize the systems forN = 2& 3 particles, in terms of the flow regimes and particle settling velocity.For N = 2 the particles always move at the same speed. For N = 3the main characteristic behaviour (when not static) is that the middleparticle moves the fastest.(v) Consequently, we have found that eventually the middle particle dis-tances itself from the uppermost particle. Thereafter the two particlessettle inline as a pair. To some extent, this form of chain dynamicsseems to extend to N > 3 and we can see this kind of breaking-reforming patterns.(vi) For large inter-particle distance cases, the interesting questions relateto whether the particles move independently. We have shown thatthis is the case: for spacings beyond a certain critical inter-particledistance, settling velocities of adjacent particles are identical and theparticles move within their own yielded envelopes. This suggests astress decay rate faster than would be expected with viscous fluids.(vii) Particles can affect their neighbours motion even if they do not sharetheir yielded envelopes with each other. In other words, the veloc-ity field could be localised about the particles, but still stress fieldsof individual particles can interact which results in different settlingvelocities.(viii) Characterizing stress decay in simple yield-stress fluid models, e.g.,Bingham, is almost not possible since the computed stress field farfrom the particles (when ‖τ‖ < Y ) is only an admissible stress fieldand generally there is no unique correct stress field below the yieldstress.6.2 Thesis limitations and future directionsInevitably, the methodology adopted for the present study includes somelimitations. In what follows, we address these limitations and propose dif-175ferent future research directions.6.2.1 Rheological idealizationsLimitationsOne limitation with our approach is in the adoption of a ‘simple’ yield-stressfluid: the Bingham model. This model is the simplest model that describesyield-stress fluids and retains the key feature of plasticity, i.e., the finiteresistance to motion—yield stress. Although some very interesting featuresof yield-stress fluids flows can be captured by this model (e.g., yield limitand stability of particles), the simplicity of the constitutive model restrictsus from finding and explaining complex phenomena like the presence ofa negative-wake in Figure 1.6. The issue of using more complex thixo-elasto-visco-plastic models is the large number of parameters that they have,e.g., the model proposed by Dimitriou and McKinley [37] has 9 physicalmaterial parameters which makes the analysis extensively challenging. Alarge number of similar models have proposed recently in the community ofrheology.Although most of these models are valuable from the rheological point ofview, still some validation need to be done in the context of fluid mechanicsbefore widespread usage of these models. Otherwise, using a range of thixo-elasto-visco-plastic models would be time consuming and exhausting. In thiscontext this thesis could be considered as only a first step towards producingmore general insight into particle motion/sedimentation in real yield-stressfluids.Future directionsAs just discussed, we may say that the next step could be using more com-plex yield-stress models: models with either thixotropy or elasticity. Al-though the general features of the flows in the present context may be hold,there are some interesting features that could be different such as the hydro-dynamic interaction of particles where elastic stresses arise below the yield176stress. Thixotropy could reveal some interesting features and could affectthe results, e.g., those in Chapter 5.6.2.2 Computational challengesLimitationsHere we just studied the 2D problems in the context of particle motion inyield-stress fluids. One reason is that solving plasticity problems analyti-cally with the von Mises yield criterion even in axisymmetric cases is not atrivial task although a few studies can be found in axisymmetric cases withTresca’s yield criterion [66]. The other reason is that 3D computations arefar slower than the 2D computations and this will manifest itself more inslow algorithms such as augmented Lagrangian schemes. Even with regu-larized techniques, there are few known results beyond that of [15] for thesphere.Future directionsVery recently Treskatis et al. [137] have proposed a new scheme based on thedual problem to accelerate augmented Lagrangian method for visocplasticflows. Saramito [124] has also proposed a new technique to increase theconvergence speed of augmented Lagrangian method. Potentially one canimplement these schemes to increase efficiency and investigate more complexaxisymmetric or 3D flows.6.2.3 Applicability to industrial flowsLimitationsIn terms of applicability of our results, we should again remind the readerthat the particles considered are 2D. The restriction to planar flows comesprincipally from the analysis of the plasticity problem via the characteristicsmethod. Other restrictions are in the symmetry of the particles, ensuringthat motion is in the y-direction. Nevertheless, the general qualitative pic-177ture of variations with χ in Chapter 2 may also hold for symmetric 3Dparticles and at least we have a baseline from which to look at such studies.In terms of applicability to complex industrial flows, such as drill cuttingstransport, we remain quite distant due to the 3D and irregular shape of thecuttings. For fractionation applications such as in [86], in a lab setting withwell controlled particle size/shape distributions, the qualitative results thatwe have derived may have application. Although the cloaking phenomenon(Chapter 3) negates the idea that the yield stress can be used as a preciseselector of particle size/shape, in many practical situations one would bedealing with a particle size and orientation distribution within a suspension,which anyway is likely to mollify selection. Other fractionation procedures,such as screening or viscous settling also do not provide a sharp cut-offin effectiveness. Indeed viscous drag coefficients are similarly non-unique.Using a yield-stress fluid still provides a different (rheological) dimensionwith which to control fractionation and it is the mechanical balance capturedin Yc (or CpD,c) that is important here.Future directionsConsidering future perspectives, it would be interesting to explore particlegeometries with no symmetries, for which rotational motion must also beaccounted for in the zero-flow limit. Also increasing number of particles inmore general settings to get information about suspensions of yield-stressfluids is another interesting avenue to go along. Finally, studying orientationeffects and drift (Chapter 4) with much more depth could be interesting.178Bibliography[1] K. Adachi and N. Yoshioka. On creeping flow of a visco-plastic fluidpast a circular cylinder. Chem. Eng. Sci., 28(1):215–226, 1973. →pages 43, 44, 54[2] F. Ahonguio, L. Jossic, and A. Magnin. Motion and stability of conesin a yield stress fluid. 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A fictitious domain method forparticulate flows with heat transfer. J. Comput. Phys., 217(2):424–452, 2006. → pages 50191[147] T. Zisis and E. Mitsoulis. Viscoplastic flow around a cylinder keptbetween parallel plates. J. Non-Newtonian Fluid Mech., 105(1):1–20,2002. → pages 46192Appendix AParticle geometriesconsidered in Chapter 3A parametric description of the particle geometries considered earlier inChapter 3 is shown in Figure A.1.𝑏𝑎𝑎𝑎𝑏𝑎𝑏𝑏′(c) (d) (e)𝑏𝑎(a)𝑏′𝑎(b)𝑏Figure A.1: Schematic of particle geometries.193Appendix BDetails of upper-boundcalculation in Chapter 3We illustrate the procedure for the upper bound critical plastic drag coeffi-cient calculation by considering the earlier example of Figure 3.6b here. Amore detailed derivation for other particle geometries can be found in [25].Note that here we assume that the particle velocity is 1 (which makes nodifference in the drag coefficient since finally the total force should be scaledby the particle velocity).We identify the following sources of dissipation in the velocity field (seeFigure B.1):• Velocity discontinuity on interface AB:∆W1 =B√2∫ pi4θ∗(1 +pi4− θ2)dθ2 (B.1)• Velocity discontinuity on interface BC:∆W2 =B√2∫ θ∗0(1 +pi4+√10− θ3)dθ3 (B.2)194D H K N PLMJGBAFEOTCISQFigure B.1: Deformation mechanism.• Velocity discontinuity on interface CD:∆W3 =B pi2√2(1 +pi4+√10)(B.3)• Velocity discontinuity on interface AE:∆W4 = B/√2 (B.4)• Velocity discontinuity on interface EI:∆W5 = B∫ pi4θ∗cos θ1 dθ1 (B.5)• Velocity discontinuity on interface IL:∆W6 = B√10 cos θ∗ (B.6)195• Velocity discontinuity on interface LP :∆W7 = B∫ θ∗0cos θ3 dθ3 (B.7)• Work dissipation in region ABCDHGFE:∆W8 =3pi4√2B (B.8)• Work dissipation in region EFI:∆W9 = B∣∣∣∣∣∫ pi4θ∗∫ θ10[(θ1 − θ2) cos θ1 − sin θ1] dθ1 dθ2∣∣∣∣∣ (B.9)• Work dissipation in region FGJI:∆W10 = B∣∣∣∣∣∫ θ∗0∫ pi4θ∗[(θ1 +√10− θ3) cos θ1 − sin θ1]dθ1 dθ3∣∣∣∣∣(B.10)• Work dissipation in region GHKJ :∆W11 = B∣∣∣∣∣∫ pi4θ∗∫ pi20[(θ1 +√10) cos θ1 − sin θ1]dθ1 dθ5∣∣∣∣∣ (B.11)• Work dissipation in region IJKNML:∆W12 =√10 B(pi2+ θ∗)(B.12)• Work dissipation in region LMP :∆W13 = B∣∣∣∣∣∫ θ∗0∫ θ10[(θ1 − θ3) cos θ1 − sin θ1] dθ1 dθ3∣∣∣∣∣ (B.13)196• Work dissipation in region MNP :∆W14 = B∣∣∣∣∣∫ θ∗0∫ pi20(θ4 cos θ4 − sin θ4) dθ4 dθ5∣∣∣∣∣ (B.14)where θ∗ = tan−1 (1/3). The angles θ1,2, θ3,4, and θ5 are measured clockwisefrom SQ, PT , and DP , respectively. Hence,[CpD,c]U=4∑14i=1 ∆Wi4aB≈ 20.93 (B.15)197Appendix CDetails of slipline fieldcalculations in Chapter 3As mentioned in section 3.2.3, an initial centred fan with angle pi/4 −tan−1(b/a)− tan−1(b′/a) and radius min(√a2 + b2,√a2 + b′2) is consideredin the corner of the ‘kite’. The α-lines are the ‘spokes’ of this fan and thecircular arcs are designated as β-lines. Therefore, the construction will bestart from the last circular arc (the blue line in Fig. C.1) and will be con-tinued. Here we use a numerical scheme proposed by Dubash et al. [40]. Tolaunch the numerical scheme, a finite number of nodes are considered on thelast β-line of the centred fan (say N) and also an assumed pressure on thenode (1,1). The node (1,2) then can be found by solving,x(1,2) − x(2,1)y(2,1) − y(1,2)= tan[12(φ(1,2) + φ(2,1))], (C.1)p˜(1,2) + 2φ(1,2) = p˜(2,1) + 2φ(2,1), (C.2)for y(1,2) and p(1,2) given that φ(1,2) = −pi/4 and x(1,2) = 0.Finding node (1,2), we can calculate the parameters at the next nodesof the same β-line by solving,x(i,2) − x(i+1,1)y(i+1,1) − y(i,2)= tan[12(φ(i,2) + φ(i+1,1))], (C.3)198b'a(1,1)(N,1)(1,2)(1,M)(2,2)(3,1)(2,1)(a)(N,2)(N-1,2)(N,M)b(b)Figure C.1: Schematic of the slipline network: (a) aft of the ‘kite’,(b) fore of the ‘kite’.x(i,2) − x(i−1,2)y(i−1,2) − y(i,2)= tan[12(φ(i,2) + φ(i−1,2))− pi2], (C.4)p˜(i,2) + 2φ(i,2) = p˜(i,1) + 2φ(i,1), (C.5)p˜(i,2) − 2φ(i,2) = p˜(i−1,2) − 2φ(i−1,2), (C.6)for x(i,2), z(i,2), φ(i,2), and p˜(i,2). We proceed until finding the red slipline(see Figure C.1 b). Indeed, this is the last α-line emanating from the initialcentre fan. Knowing that the particle surface is a limiting line (see [111] formore details), we can proceed with the network construction until touchingthe particle surface. This means that we should start by finding node (N, 2)using nodes (N, 1) and (N − 1, 2):x(N,2) − x(N−1,2)x(N−1,2) − x(N,2)= tan[12(φ(N,2) + φ(N−1,2))− pi2], (C.7)p˜(N,2) − 2φ(N,2) = p˜(N−1,2) − 2φ(N−1,2). (C.8)199Then we can find the whole next α-line (nodes (N − 1, 3), (N − 2, 4),· · · , (N −M + 2,M)). We proceed by finding node (N, 3) and then repeatthe entire steps to find the whole characteristics network. Iterating on M iscontinued up to the point that the node (N,M) hits the front of the particle.200Appendix DDetails of lower-boundcalculations in Chapter 5Details of how we can find the unyielded envelope around two connecteddisks at the yield limit is the same as having a single particle since theyare connected and moving with the same velocity. Hence, here we use the‘unyielded envelope rule’ of Chapter 3 and [24] to discover Figure 5.5: i.e.,FG and AB are tangents to the particle surfaces and make angle pi/4 withy-axis; ED is also tangent to the both particles. We designate the anglebetween ED and y-direction by θ∗ and its length by `∗. Please note thatif χ2 = 1, then θ∗ = 0 and `∗ = `1. So different contributions of the shearand normal stresses on the unyielded envelope could be found by using themethod of characteristic which is discussed generally before in the presentcontext. Different contributions are as follows:• Shear on AB:F˜1 = R1B cos(pi4)(D.1)• Shear on BC:F˜2 =∫ pi40R1B cos θ dθ (D.2)201• Shear on CD:F˜3 =∫ θ∗0R1B cos θ dθ (D.3)• Shear on DE:F˜4 = `∗B cos θ∗ (D.4)• Shear on EF :F˜5 =∫ pi4θ∗R2B cos θ dθ (D.5)• Shear on FG:F˜6 = R2B cos(pi4)(D.6)• Normal on AB:F˜7 = R1B(σ¯ − 3pi2)sin(pi4)(D.7)• Normal on BC:F˜8 =∫ pi40B [σ¯ − (pi + 2θ1)]R1 sin θ1 dθ1 (D.8)• Normal on CD:F˜9 =∫ θ∗0B [σ¯ + (pi + 2θ1)]R1 sin θ1 dθ1 (D.9)• Normal on DE:F˜10 = B [σ¯ + (pi + 2θ∗)] `∗ sin θ∗ (D.10)• Normal on EF :F˜11 =∫ pi4θ∗B [σ¯ + (pi + 2θ1)]R2 sin θ1 dθ1 (D.11)202• Normal on FG:F˜12 = R2B(σ¯ +3pi2)sin(pi4)(D.12)where the unknown mean stress on the horizontal line passing through C isdesignated by Bσ¯. Therefore, the lower bound of the drag coefficient couldbe calculated by gathering different contributions as,[CpD,c]L=2∑12i=1 F˜i`⊥ B=∑12i=1 F˜iB. (D.13)203

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