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Creeping flow around particles in a Bingham fluid. (English) Zbl 1274.76059
Summary: We revisit the variational setting of the creeping flow of Bingham fluid about a particle. We investigate two problems: the resistance and the mobility problem, which arise in the context of sedimentation. We present results and the general framework for uniqueness, symmetry and reversibility properties of solutions, and clarify the relation between resistance and mobility problems. We also consider general properties of the static stability limit (or load limit). In the second part of the paper we apply insights gained from the first part to the computation of 2D exterior flow around an infinite cylinder with elliptical cross-section. We study the static stability limit and find that the limiting flow solution approaches the perfectly plastic solid solution.

MSC:
76A05 Non-Newtonian fluids
76T20 Suspensions
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[1] Andres, U. T.: Equilibrium and motion of spheres in a viscoplastic liquid, Sov. phys. Dokl. (U.S.A.) 5, 723 (1961) · Zbl 0108.38504
[2] Du Plessis, M.; Ansley, R.: Settling parameters in solids pipelining, J. pipeline div. (ASCE) 93, 1 (1967)
[3] Ansley, R. W.; Smith, T. N.: Motion of spherical particles in a Bingham plastic, Aiche J. 13, 1193 (1967)
[4] Whitmore, R.: Drag forces in Bingham plastics, , 472 (1969)
[5] W. Prager, On slow visco-plastic flow, in: Studies in Mathematics and Mechanics presented to Richard von Mises, Studies in Mathematics and Mechanics, Academic Press Inc., Publishers, 1954, pp. 208 – 216.
[6] Mosolov, P. P.; Miasnikov, V. P.: Variational methods in the theory of the fluidity of a viscous plastic medium, J. appl. Math. mech. 29, No. 3, 545-577 (1965) · Zbl 0168.45505
[7] Mosolov, P. P.; Miasnikov, V. P.: On stagnant flow regions of a viscous-plastic medium in pipes, J. appl. Math. mech. 30, No. 4, 841-854 (1966) · Zbl 0168.45601
[8] Mosolov, P. P.; Miasnikov, V. P.: Asymptotic theory of rigid plastic shells, J. appl. Math. mech. 41, No. 3, 550-566 (1977) · Zbl 0395.73032
[9] Mosolov, P. P.: On certain mathematical questions of the theory of incompressible viscoplastic media, J. appl. Math. mech. 42, No. 4, 787-799 (1978) · Zbl 0438.73025
[10] Duvaut, G.; Lions, J. L.: Inequalities in mechanics and physics, (1976) · Zbl 0331.35002
[11] Glowinski, R.: Numerical methods for nonlinear variational problems, (1983) · Zbl 0612.76033
[12] Adachi, K.; Yoshioka, N.: On creeping flow of a viscoplastic fluid past a circular cylinder, Chem. eng. Sci. 28, 215-226 (1973)
[13] Yoshioka, N.; Adachi, K.: On variational principles for a non-Newtonian fluid, J. chem. Eng. jpn. 4, 217-220 (1971)
[14] Yoshioka, N.; Adachi, K.: Applications of the extremum principles for non-Newtonian fluids, J. chem. Eng. jpn. 4, 221-226 (1971)
[15] Beris, A. N.; Tsamopoulos, J. A.; Armstrong, R. C.; Brown, R.: Creeping motion of a sphere through a Bingham plastic, J. fluid mech. 158, 219-244 (1985) · Zbl 0581.76010
[16] Blackery, J.; Mitsoulis, E.: Creeping motion of a sphere in tubes filled with a Bingham plastic material, J. non-Newton fluid mech. 70, 59-77 (1997)
[17] Beaulne, M.; Mitsoulis, E.: Creeping motion of a sphere in tubes filled with Herschel-Bulkley fluids, J. non-Newton fluid mech. 72, No. 1, 55-71 (1997)
[18] Liu, B. T.; Muller, S. J.; Denn, M. M.: Convergence of a regularization method for creeping flow of a Bingham material about a rigid sphere, J. non-Newton fluid mech. 102, 179-191 (2002) · Zbl 1082.76577
[19] De Besses, B. Deglo; Magnin, A.; Jay, P.: Sphere drag in a viscoplastic fluid, Aiche J. 50, No. 10, 2627-2629 (2004)
[20] Zisis, T. H.; Mitsoulis, E.: Viscoplastic flow around a cylinder kept between parallel plates, J. non-Newton fluid mech. 105, 1-20 (2002) · Zbl 1006.76507
[21] Mosolov, P. P.; Miasnikov, V. P.: Boundary layer in the problem of longitudinal motion of a cylinder in a viscoplastic medium, J. appl. Math. mech. 38, No. 4, 634-644 (1974)
[22] Roquet, N.; Saramito, P.: An adaptive finite element method for Bingham fluid flows around a cylinder, Comput. methods appl. Mech. eng. 192, No. 31 – 32, 3317-3341 (2003) · Zbl 1054.76053
[23] De Besses, B. Deglo; Magnin, A.; Jay, P.: Viscoplastic flow around a cylinder in an infinite medium, J. non-Newton fluid mech. 115, 27-49 (2003) · Zbl 1052.76005
[24] Mitsoulis, E.: On creeping drag flow of a viscoplastic fluid past a circular cylinder: wall effects, Chem. eng. Sci. 59, 789-800 (2004)
[25] Tokpavi, D. L.; Magnin, A.; Jay, P.: Very slow flow of Bingham viscoplastic fluid around a circular cylinder, J. non-Newton fluid mech. 154, 65-76 (2008) · Zbl 1273.76028
[26] Randolph, M. F.; Houlsby, G. T.: The limiting pressure on a circular pile loaded laterally in cohesive soil, Geotechnique 34, No. 4, 613-623 (1984)
[27] Liu, B. T.; Muller, S. J.; Denn, M.: Interactions of two rigid spheres translating collinearly in creeping flow in a Bingham material, J. non-Newton fluid mech. 113, 49-67 (2003) · Zbl 1065.76519
[28] Tokpavi, D. L.; Jay, P.; Magnin, A.: Interaction between two circular cylinders in slow flow of Bingham viscoplastic fluid, J. non-Newton fluid mech. 157, No. 3, 175-187 (2009) · Zbl 1274.76073
[29] Jie, P.; Zhu, K. -Q.: Drag force of interacting coaxial spheres in viscoplastic fluids, J. non-Newton fluid mech. 135, 83-91 (2006) · Zbl 1195.76028
[30] Dean, E. J.; Glowinski, R.; Pan, T. W.: A fictitious domain method for the numerical simulation of particulate flow for Bingham visco-plastic, Numerical methods for scientific computing: variational problems and applications, 11-19 (2003)
[31] Wachs, A.; Yue, Z.: A fictitious domain method for dynamic simulation of particle sedimentation in Bingham fluids, J. non-Newton fluid mech. 145, No. 2 – 3, 78-91 (2007) · Zbl 1195.76411
[32] Atapattu, D. D.; Chhabra, R. P.; Uhlherr, P. H. T.: Wall effect for spheres falling at small Reynolds number in a viscoplastic medium, J. non-Newton fluid mech. 38, No. 1, 31-42 (1990)
[33] Atapattu, D. D.; Chhabra, R. P.; Uhlherr, P. H. T.: Creeping sphere motion in Herschel-bulkey fluids: flow field and drag, J. non-Newton fluid mech. 59, 245-265 (1995)
[34] Hariharaputhiran, M.; Subramanian, R. S.; Campell, G. A.; Chhabra, R. P.: Settling of spheres in a viscoplastic fluid, J. non-Newton fluid mech. 79, 87-97 (1998) · Zbl 0973.76513
[35] Jossic, J.; Magnin, A.: Drag and stability of objects in a yield stress fluid, Aiche J. 47, 2666-2672 (2001)
[36] Merkak, O.; Jossic, L.; Magnin, A.: Spheres and interactions between spheres moving at very low velocities in a yield stress fluid, J. non-Newton fluid mech. 133, 99-108 (2006) · Zbl 1195.76037
[37] Tabuteau, H.; Coussot, P.; De Bruyn, J. R.: Drag force on a sphere in steady motion through a yield-stress fluid, J. rheol. 51, No. 1, 125-137 (2007)
[38] Tabuteau, H.; Sikorski, D.; De Bruyn, J. R.: Shear waves and shocks in soft solids, Phys. rev. E (3) 75, No. 1, 012201 (2007)
[39] Putz, A. M. V.; Burghelea, T. I.; Frigaard, I. A.; Martinez, D. M.: Settling of an isolated spherical particle in a yield stress shear thinning fluid, Phys. fluids 20, No. 3, 033102 (2008) · Zbl 1182.76615
[40] Chafe, N. P.; De Bruyn, J. R.: Drag and relaxation in a bentonite Clay suspension, J. non-Newton fluid mech. 131, 44-52 (2005)
[41] B. Gueslin, L. Talini, B. Herzhaft, Y. Peysson, C. Allain, Flow induced by a sphere settling in an aging yield-stress fluid, Phys. Fluids 18 (2006).
[42] B. Gueslin, L. Talini, B. Herzhaft, Y. Peysson, C. Allain, Aggregation behavior of two spheres falling through an aging fluid, Phys. Rev. E (3) 74 (2006).
[43] Oldroyd, J. G.: A rational formulation of the equations of plastic flow for a Bingham solid, Proc. Cambridge philos. Soc. 43, No. 1, 100-105 (1947) · Zbl 0029.32702
[44] Oldroyd, J. G.: Two-dimensional plastic flow of a Bingham solid. A plastic boundary-layer theory for slow motion, Proc. Cambridge philos. Soc. 43, No. 3, 383-395 (1947) · Zbl 0029.32703
[45] Aris, R.: Vectors, tensors and the basic equations of fluid mechanics, (1962) · Zbl 0123.41502
[46] I. Ekeland, R. Temam, Convex Analysis and Variational Problems, no. 28 in Classics in Applied Mathematics, SIAM, 1999. · Zbl 0281.49001
[47] Dubash, N.; Frigaard, I. A.: Conditions for static bubbles in viscoplastic fluids, Phys. fluids 16, No. 12, 4319-4330 (2004) · Zbl 1187.76131
[48] Dubash, N.; Frigaard, I. A.: Propagation and stopping of air bubbles in carbopol solutions, J. non-Newton fluid mech. 142, No. 1 – 3, 123-134 (2007) · Zbl 1143.76325
[49] Hill, R.: The mathematical theory of plasticity, Oxford classic texts, (1950) · Zbl 0041.10802
[50] Temam, R.; Strang, G.: Functions of bounded deformation, Arch. rational mech. Anal., 7-21 (1980) · Zbl 0472.73031
[51] Tsamopoulos, J.; Dimakopoulos, Y.; Chatzidai, N.; Karapetsas, G.; Pavlidis, M.: Steady bubble rise and deformation in Newtonian and viscoplastic fluids and conditions for bubble entrapment, J. fluid mech. 601, 123-164 (2008) · Zbl 1151.76602
[52] Glowinski, R.; Pan, T. W.; Hesla, T. I.; Joseph, D. D.; Periaux, J.: A distributed Lagrange multiplier/fictitious domain method for the simulation of flow around moving rigid bodies: application to particulate flow, Comput. methods appl. Mech. eng. 184, 241-267 (2000) · Zbl 0970.76057
[53] Hu, H. H.; Patankar, N. A.; Zhu, M. Y.: Direct numerical simulations of fluid-solid systems using the arbitrary Lagrangian – Eulerian technique, J. comput. Phys., 427-462 (2001) · Zbl 1047.76571
[54] Glowinski, R.; Le Tallec, P.: Augmented Lagrangian and operator-splitting methods in nonlinear mechanics, studies in applied mathematics, (1989) · Zbl 0698.73001
[55] R. Glowinski, J.L. Lions, R. Tremolieres, Analyse Numérique des inéquations variationelles, vol. 2, Dunod, Paris, 1976. · Zbl 0358.65091
[56] Roquet, N.; Saramito, P.: Stick-slip transition capturing by using an adaptive finite element method, M2AN math. Model. numer. Anal. 38, No. 2, 249-260 (2004) · Zbl 1130.76368
[57] A.M.V. Putz, I.A. Frigaard, D.M. Martinez, On the lubrication paradox and the use of regularisation methods for lubrication flows, J. Non-Newton. Fluid Mech. (available online 8 July 2009). · Zbl 1274.76196
[58] Saramito, P.; Roquet, N.: An adaptive finite element method for viscoplastic fluid flow in pipes, Comput. methods appl. Mech. eng. 190, No. 40 – 41, 5391-5412 (2001) · Zbl 1002.76071
[59] H. Borouchaki, L. George, P.F. Hecht, P. Laug, E. Saltel, Delaunay mesh generation governed by metric specifications. I. Algorithms, Finite Elem. Anal. Des. 25 (1 – 2) (1997) 61 – 83 (adaptive meshing, Part 1). · Zbl 0897.65076
[60] Castro-Díaz, M. J.; Hecht, F.; Mohammadi, B.; Pironneau, O.: Anisotropic unstructured mesh adaption for flow simulations, Int. J. Numer. methods fluids 25, No. 4, 475-491 (1997) · Zbl 0902.76057
[61] Faxen, O. H.: Forces exerted on a rigid cylinder in a viscous fluid between two parallel fixed planes, Proc. R. Swed. acad. Eng. sci. 187, 1-13 (1946)
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