On discretization in time in simulations of particulate flows

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June 2011, 15(4): 935-956. doi: 10.3934/dcdsb.2011.15.935

On discretization in time in simulations of particulate flows

Matthieu Hillairet ^1, , Alexei Lozinski ^2, and Marcela Szopos ^3,

1.	Ceremade, UMR CNRS 7534, Université Dauphine, Place du Maréchal De Lattre De Tassigny, F-75016 Paris, France
2.	Institut de Mathématiques de Toulouse, UMR 5219, Université de Toulouse et CNRS, 118 route de Narbonne, F-31062 Toulouse Cedex 9, France
3.	Institut de Recherche Mathématique Avancée, UMR 7501, Université de Strasbourg et CNRS, 7 rue René Descartes, F-67084 Strasbourg Cedex, France

Received January 2010 Revised July 2010 Published March 2011

Abstract
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We propose a time discretization scheme for a class of ordinary differential equations arising in simulations of fluid/particle flows. The scheme is intended to work robustly in the lubrication regime when the distance between two particles immersed in the fluid or between a particle and the wall tends to zero. The idea consists in introducing a small threshold for the particle-wall distance below which the real trajectory of the particle is replaced by an approximated one where the distance is kept equal to the threshold value. The error of this approximation is estimated both theoretically and by numerical experiments. Our time marching scheme can be easily incorporated into a full simulation method where the velocity of the fluid is obtained by a numerical solution to Stokes or Navier-Stokes equations. We also provide a derivation of the asymptotic expansion for the lubrication force (used in our numerical experiments) acting on a disk immersed in a Newtonian fluid and approaching the wall. The method of this derivation is new and can be easily adapted to other cases.

Keywords: near contact, Particulate flow, lubrication force..

Mathematics Subject Classification: Primary: 65Z05, 76T99; Secondary: 65L0.

Citation: Matthieu Hillairet, Alexei Lozinski, Marcela Szopos. On discretization in time in simulations of particulate flows. Discrete and Continuous Dynamical Systems - B, 2011, 15 (4) : 935-956. doi: 10.3934/dcdsb.2011.15.935

References:

[1]	J. F. Brady and G. Bossis, Stokesian dynamics, Annual Review of Fluid Mechanics, 20 (1988), 111-157. doi: oi:10.1146/annurev.fl.20.010188.000551.
[2]	J. Butcher, "Numerical Methods for Ordinary Differential Equations," John Wiley & Sons Ltd., Chichester, 2003. doi: doi:10.1002/0470868279.
[3]	M. Cooley and M. O'Neill, On the slow motion generated in a viscous fluid by the approach of a sphere to a plane wall or stationary sphere, Mathematika, 16 (1969), 37-49. doi: doi:10.1112/S0025579300004599.
[4]	S. L. Dance and M. R. Maxey, Incorporation of lubrication effects into the force-coupling method for particulate two-phase flow, Journal of Computational Physics, 189 (2003), 212-238. doi: doi:10.1016/S0021-9991(03)00209-2.
[5]	W. R. Dean and M. E. O'Neill, A slow motion of viscous liquid caused by the rotation of a solid sphere, Mathematika, 10 (1963), 13-24. doi: doi:10.1112/S0025579300003314.
[6]	G. P. Galdi, "An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. I, Linearized Steady Problems," vol. 38 of Springer Tracts in Natural Philosophy, Springer-Verlag, New York, 1994.
[7]	D. Gérard-Varet and M. Hillairet, Regularity issues in the problem of fluid structure interaction, Arch. Ration. Mech. Anal., 195 (2010), 375-407. doi: doi:10.1007/s00205-008-0202-9.
[8]	R. Glowinski, T.-W. Pan, T. I. Hesla and D. D. Joseph, A distributed lagrange multiplier/fictitious domain method for particulate flows, Int. J. Multiphase Flow, 24 (1999), 755-794. doi: doi:10.1016/S0301-9322(98)00048-2.
[9]	F. Hecht, O. Pironneau, A. L. Hyaric and K. Ohtsuka, Freefem++, ver. 3.7, http://www.freefem.org/ff++, (2009).
[10]	M. Hillairet, Lack of collision between solid bodies in a 2D incompressible viscous flow, Comm. Partial Differential Equations, 32 (2007), 1345-1371. doi: doi:10.1080/03605300601088740.
[11]	M. Hillairet and T. Takahashi, Collisions in three-dimensional fluid structure interaction problems, SIAM J. Math. Anal., 40 (2009), 2451-2477. doi: doi:10.1137/080716074.
[12]	M. S. Ingber, A. Mammoli, P. Vorobieff, T. McCollam and A. Graham, Experimental and numerical analysis of irreversibilities among particles suspended in a couette device, Journal of Rheology, 50 (2006), 99-114. doi: doi:10.1122/1.2169806.
[13]	N. Lecocq, R. Anthore, B. Cichocki, P. Szymczak and F. Feuillebois, Drag force on a sphere moving towards a corrugated wall, J. Fluid Mech., 513 (2004), 247-264. doi: doi:10.1017/S0022112004009942.
[14]	N. Lecocq, F. Feuillebois, N. Anthore, R. Anthore, F. Bostel and C. Petipas, Precise measurement of particle-wall hydrodynamic interactions at low reynolds number using laser interferometry, Phys. Fluids A, 5 (1993), 3-12. doi: doi:10.1063/1.858787.
[15]	A. Lefebvre, Numerical simulation of gluey particles, M2AN Math. Model. Numer. Anal., 43 (2009), 53-80. doi: doi:10.1051/m2an/2008042.
[16]	A. Lozinski and M. Romerio, Motion of gas bubbles, considered as massless bodies, affording deformations within a prescribed family of shapes, in an incompressible fluid under the action of gravitation and surface tension, M3AS Math. Mod. Meth. Appl. Sci., 17 (2007), 1445-1478. doi: doi:10.1142/S0218202507002340.
[17]	A. A. Mammoli, The treatment of lubrication forces in boundary integral equations, Royal Society of London Proceedings Series A, 462 (2006), 855-881. doi: doi:10.1098/rspa.2005.1600.
[18]	B. Maury, A gluey particle model, ESAIM: Proceedings, 18 (2007), 133-142. doi: doi:10.1051/proc:071811.
[19]	M. O'Neill, A slow motion of viscous liquid caused by a slowly moving solid sphere, Mathematika, 11 (1964), 67-74.
[20]	M. O'Neill and K. Stewartson, On the slow motion of a sphere parallel to a nearby plane wall, J. Fluid Mech., 27 (1967), 705-724. doi: doi:10.1017/S0022112067002551.
[21]	L. Pasol, M. Chaoui, S. Yahiaoui and F. Feuillebois, Analytical solutions for a spherical particle near a wall in axisymmetrical polynomial creeping flows, Phys. Fluids, 17 (2005), 1-13. doi: doi:10.1063/1.1955272.
[22]	F. Qi, N. Phan-Tien and X. J. Fan, Effective moduli of particulate solids: The completed double layer boundary element method with lubrication approximation, Zeitschrift Angewandte Mathematik und Physik, 51 (2000), 92-113. doi: doi:10.1007/PL00001509.
[23]	A. Sierou and J. F. Brady, Accelerated Stokesian dynamics simulations, Journal of Fluid Mechanics, 448 (2001), 115-146. doi: doi:10.1017/S0022112001005912.
[24]	J. Smart and D. Leighton, Measurements of the hydrodynamic roughness of non colloidal spheres, Phys. Fluids A, 1 (1989), 52-60. doi: doi:10.1063/1.857523.
[25]	O. Vinogradova and G. Yakubov, Surface roughness and hydrodynamic boundary conditions, Phys. Rev. E, 73 (2006), p. 045302(R). doi: doi:10.1103/PhysRevE.73.045302.

show all references

References:

[1]	J. F. Brady and G. Bossis, Stokesian dynamics, Annual Review of Fluid Mechanics, 20 (1988), 111-157. doi: oi:10.1146/annurev.fl.20.010188.000551.
[2]	J. Butcher, "Numerical Methods for Ordinary Differential Equations," John Wiley & Sons Ltd., Chichester, 2003. doi: doi:10.1002/0470868279.
[3]	M. Cooley and M. O'Neill, On the slow motion generated in a viscous fluid by the approach of a sphere to a plane wall or stationary sphere, Mathematika, 16 (1969), 37-49. doi: doi:10.1112/S0025579300004599.
[4]	S. L. Dance and M. R. Maxey, Incorporation of lubrication effects into the force-coupling method for particulate two-phase flow, Journal of Computational Physics, 189 (2003), 212-238. doi: doi:10.1016/S0021-9991(03)00209-2.
[5]	W. R. Dean and M. E. O'Neill, A slow motion of viscous liquid caused by the rotation of a solid sphere, Mathematika, 10 (1963), 13-24. doi: doi:10.1112/S0025579300003314.
[6]	G. P. Galdi, "An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. I, Linearized Steady Problems," vol. 38 of Springer Tracts in Natural Philosophy, Springer-Verlag, New York, 1994.
[7]	D. Gérard-Varet and M. Hillairet, Regularity issues in the problem of fluid structure interaction, Arch. Ration. Mech. Anal., 195 (2010), 375-407. doi: doi:10.1007/s00205-008-0202-9.
[8]	R. Glowinski, T.-W. Pan, T. I. Hesla and D. D. Joseph, A distributed lagrange multiplier/fictitious domain method for particulate flows, Int. J. Multiphase Flow, 24 (1999), 755-794. doi: doi:10.1016/S0301-9322(98)00048-2.
[9]	F. Hecht, O. Pironneau, A. L. Hyaric and K. Ohtsuka, Freefem++, ver. 3.7, http://www.freefem.org/ff++, (2009).
[10]	M. Hillairet, Lack of collision between solid bodies in a 2D incompressible viscous flow, Comm. Partial Differential Equations, 32 (2007), 1345-1371. doi: doi:10.1080/03605300601088740.
[11]	M. Hillairet and T. Takahashi, Collisions in three-dimensional fluid structure interaction problems, SIAM J. Math. Anal., 40 (2009), 2451-2477. doi: doi:10.1137/080716074.
[12]	M. S. Ingber, A. Mammoli, P. Vorobieff, T. McCollam and A. Graham, Experimental and numerical analysis of irreversibilities among particles suspended in a couette device, Journal of Rheology, 50 (2006), 99-114. doi: doi:10.1122/1.2169806.
[13]	N. Lecocq, R. Anthore, B. Cichocki, P. Szymczak and F. Feuillebois, Drag force on a sphere moving towards a corrugated wall, J. Fluid Mech., 513 (2004), 247-264. doi: doi:10.1017/S0022112004009942.
[14]	N. Lecocq, F. Feuillebois, N. Anthore, R. Anthore, F. Bostel and C. Petipas, Precise measurement of particle-wall hydrodynamic interactions at low reynolds number using laser interferometry, Phys. Fluids A, 5 (1993), 3-12. doi: doi:10.1063/1.858787.
[15]	A. Lefebvre, Numerical simulation of gluey particles, M2AN Math. Model. Numer. Anal., 43 (2009), 53-80. doi: doi:10.1051/m2an/2008042.
[16]	A. Lozinski and M. Romerio, Motion of gas bubbles, considered as massless bodies, affording deformations within a prescribed family of shapes, in an incompressible fluid under the action of gravitation and surface tension, M3AS Math. Mod. Meth. Appl. Sci., 17 (2007), 1445-1478. doi: doi:10.1142/S0218202507002340.
[17]	A. A. Mammoli, The treatment of lubrication forces in boundary integral equations, Royal Society of London Proceedings Series A, 462 (2006), 855-881. doi: doi:10.1098/rspa.2005.1600.
[18]	B. Maury, A gluey particle model, ESAIM: Proceedings, 18 (2007), 133-142. doi: doi:10.1051/proc:071811.
[19]	M. O'Neill, A slow motion of viscous liquid caused by a slowly moving solid sphere, Mathematika, 11 (1964), 67-74.
[20]	M. O'Neill and K. Stewartson, On the slow motion of a sphere parallel to a nearby plane wall, J. Fluid Mech., 27 (1967), 705-724. doi: doi:10.1017/S0022112067002551.
[21]	L. Pasol, M. Chaoui, S. Yahiaoui and F. Feuillebois, Analytical solutions for a spherical particle near a wall in axisymmetrical polynomial creeping flows, Phys. Fluids, 17 (2005), 1-13. doi: doi:10.1063/1.1955272.
[22]	F. Qi, N. Phan-Tien and X. J. Fan, Effective moduli of particulate solids: The completed double layer boundary element method with lubrication approximation, Zeitschrift Angewandte Mathematik und Physik, 51 (2000), 92-113. doi: doi:10.1007/PL00001509.
[23]	A. Sierou and J. F. Brady, Accelerated Stokesian dynamics simulations, Journal of Fluid Mechanics, 448 (2001), 115-146. doi: doi:10.1017/S0022112001005912.
[24]	J. Smart and D. Leighton, Measurements of the hydrodynamic roughness of non colloidal spheres, Phys. Fluids A, 1 (1989), 52-60. doi: doi:10.1063/1.857523.
[25]	O. Vinogradova and G. Yakubov, Surface roughness and hydrodynamic boundary conditions, Phys. Rev. E, 73 (2006), p. 045302(R). doi: doi:10.1103/PhysRevE.73.045302.

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