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

On discretization in time in simulations of particulate flows

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

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.
Citation: Matthieu Hillairet, Alexei Lozinski, Marcela Szopos. On discretization in time in simulations of particulate flows. Discrete & 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.  doi: oi:10.1146/annurev.fl.20.010188.000551.  Google Scholar

[2]

J. Butcher, "Numerical Methods for Ordinary Differential Equations,", John Wiley & Sons Ltd., (2003).  doi: doi:10.1002/0470868279.  Google Scholar

[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.  doi: doi:10.1112/S0025579300004599.  Google Scholar

[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.  doi: doi:10.1016/S0021-9991(03)00209-2.  Google Scholar

[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.  doi: doi:10.1112/S0025579300003314.  Google Scholar

[6]

G. P. Galdi, "An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. I, Linearized Steady Problems,", vol. \textbf{38} of Springer Tracts in Natural Philosophy, 38 (1994).   Google Scholar

[7]

D. Gérard-Varet and M. Hillairet, Regularity issues in the problem of fluid structure interaction,, Arch. Ration. Mech. Anal., 195 (2010), 375.  doi: doi:10.1007/s00205-008-0202-9.  Google Scholar

[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.  doi: doi:10.1016/S0301-9322(98)00048-2.  Google Scholar

[9]

F. Hecht, O. Pironneau, A. L. Hyaric and K. Ohtsuka, Freefem++, ver. 3.7,, \url{http://www.freefem.org/ff++}, (2009).   Google Scholar

[10]

M. Hillairet, Lack of collision between solid bodies in a 2D incompressible viscous flow,, Comm. Partial Differential Equations, 32 (2007), 1345.  doi: doi:10.1080/03605300601088740.  Google Scholar

[11]

M. Hillairet and T. Takahashi, Collisions in three-dimensional fluid structure interaction problems,, SIAM J. Math. Anal., 40 (2009), 2451.  doi: doi:10.1137/080716074.  Google Scholar

[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.  doi: doi:10.1122/1.2169806.  Google Scholar

[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.  doi: doi:10.1017/S0022112004009942.  Google Scholar

[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.  doi: doi:10.1063/1.858787.  Google Scholar

[15]

A. Lefebvre, Numerical simulation of gluey particles,, M2AN Math. Model. Numer. Anal., 43 (2009), 53.  doi: doi:10.1051/m2an/2008042.  Google Scholar

[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.  doi: doi:10.1142/S0218202507002340.  Google Scholar

[17]

A. A. Mammoli, The treatment of lubrication forces in boundary integral equations,, Royal Society of London Proceedings Series A, 462 (2006), 855.  doi: doi:10.1098/rspa.2005.1600.  Google Scholar

[18]

B. Maury, A gluey particle model,, ESAIM: Proceedings, 18 (2007), 133.  doi: doi:10.1051/proc:071811.  Google Scholar

[19]

M. O'Neill, A slow motion of viscous liquid caused by a slowly moving solid sphere,, Mathematika, 11 (1964), 67.   Google Scholar

[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.  doi: doi:10.1017/S0022112067002551.  Google Scholar

[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.  doi: doi:10.1063/1.1955272.  Google Scholar

[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.  doi: doi:10.1007/PL00001509.  Google Scholar

[23]

A. Sierou and J. F. Brady, Accelerated Stokesian dynamics simulations,, Journal of Fluid Mechanics, 448 (2001), 115.  doi: doi:10.1017/S0022112001005912.  Google Scholar

[24]

J. Smart and D. Leighton, Measurements of the hydrodynamic roughness of non colloidal spheres,, Phys. Fluids A, 1 (1989), 52.  doi: doi:10.1063/1.857523.  Google Scholar

[25]

O. Vinogradova and G. Yakubov, Surface roughness and hydrodynamic boundary conditions,, Phys. Rev. E, 73 (2006).  doi: doi:10.1103/PhysRevE.73.045302.  Google Scholar

show all references

References:
[1]

J. F. Brady and G. Bossis, Stokesian dynamics,, Annual Review of Fluid Mechanics, 20 (1988), 111.  doi: oi:10.1146/annurev.fl.20.010188.000551.  Google Scholar

[2]

J. Butcher, "Numerical Methods for Ordinary Differential Equations,", John Wiley & Sons Ltd., (2003).  doi: doi:10.1002/0470868279.  Google Scholar

[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.  doi: doi:10.1112/S0025579300004599.  Google Scholar

[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.  doi: doi:10.1016/S0021-9991(03)00209-2.  Google Scholar

[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.  doi: doi:10.1112/S0025579300003314.  Google Scholar

[6]

G. P. Galdi, "An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. I, Linearized Steady Problems,", vol. \textbf{38} of Springer Tracts in Natural Philosophy, 38 (1994).   Google Scholar

[7]

D. Gérard-Varet and M. Hillairet, Regularity issues in the problem of fluid structure interaction,, Arch. Ration. Mech. Anal., 195 (2010), 375.  doi: doi:10.1007/s00205-008-0202-9.  Google Scholar

[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.  doi: doi:10.1016/S0301-9322(98)00048-2.  Google Scholar

[9]

F. Hecht, O. Pironneau, A. L. Hyaric and K. Ohtsuka, Freefem++, ver. 3.7,, \url{http://www.freefem.org/ff++}, (2009).   Google Scholar

[10]

M. Hillairet, Lack of collision between solid bodies in a 2D incompressible viscous flow,, Comm. Partial Differential Equations, 32 (2007), 1345.  doi: doi:10.1080/03605300601088740.  Google Scholar

[11]

M. Hillairet and T. Takahashi, Collisions in three-dimensional fluid structure interaction problems,, SIAM J. Math. Anal., 40 (2009), 2451.  doi: doi:10.1137/080716074.  Google Scholar

[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.  doi: doi:10.1122/1.2169806.  Google Scholar

[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.  doi: doi:10.1017/S0022112004009942.  Google Scholar

[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.  doi: doi:10.1063/1.858787.  Google Scholar

[15]

A. Lefebvre, Numerical simulation of gluey particles,, M2AN Math. Model. Numer. Anal., 43 (2009), 53.  doi: doi:10.1051/m2an/2008042.  Google Scholar

[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.  doi: doi:10.1142/S0218202507002340.  Google Scholar

[17]

A. A. Mammoli, The treatment of lubrication forces in boundary integral equations,, Royal Society of London Proceedings Series A, 462 (2006), 855.  doi: doi:10.1098/rspa.2005.1600.  Google Scholar

[18]

B. Maury, A gluey particle model,, ESAIM: Proceedings, 18 (2007), 133.  doi: doi:10.1051/proc:071811.  Google Scholar

[19]

M. O'Neill, A slow motion of viscous liquid caused by a slowly moving solid sphere,, Mathematika, 11 (1964), 67.   Google Scholar

[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.  doi: doi:10.1017/S0022112067002551.  Google Scholar

[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.  doi: doi:10.1063/1.1955272.  Google Scholar

[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.  doi: doi:10.1007/PL00001509.  Google Scholar

[23]

A. Sierou and J. F. Brady, Accelerated Stokesian dynamics simulations,, Journal of Fluid Mechanics, 448 (2001), 115.  doi: doi:10.1017/S0022112001005912.  Google Scholar

[24]

J. Smart and D. Leighton, Measurements of the hydrodynamic roughness of non colloidal spheres,, Phys. Fluids A, 1 (1989), 52.  doi: doi:10.1063/1.857523.  Google Scholar

[25]

O. Vinogradova and G. Yakubov, Surface roughness and hydrodynamic boundary conditions,, Phys. Rev. E, 73 (2006).  doi: doi:10.1103/PhysRevE.73.045302.  Google Scholar

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