Numerical simulation of gluey particles
ESAIM: Modélisation mathématique et analyse numérique, Tome 43 (2009) no. 1, pp. 53-80.

We propose here a model and a numerical scheme to compute the motion of rigid particles interacting through the lubrication force. In the case of a particle approaching a plane, we propose an algorithm and prove its convergence towards the solutions to the gluey particle model described in [B. Maury, ESAIM: Proceedings 18 (2007) 133-142]. We propose a multi-particle version of this gluey model which is based on the projection of the velocities onto a set of admissible velocities. Then, we describe a multi-particle algorithm for the simulation of such systems and present numerical results.

DOI : 10.1051/m2an/2008042
Classification : 65L20, 74F10, 76T20
Mots-clés : fluid/particle systems, fluid/solid interaction, lubrication force, contacts, Stokes fluid 
@article{M2AN_2009__43_1_53_0,
     author = {Lefebvre, Aline},
     title = {Numerical simulation of gluey particles},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {53--80},
     publisher = {EDP-Sciences},
     volume = {43},
     number = {1},
     year = {2009},
     doi = {10.1051/m2an/2008042},
     mrnumber = {2494794},
     zbl = {1163.76056},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an/2008042/}
}
TY  - JOUR
AU  - Lefebvre, Aline
TI  - Numerical simulation of gluey particles
JO  - ESAIM: Modélisation mathématique et analyse numérique
PY  - 2009
SP  - 53
EP  - 80
VL  - 43
IS  - 1
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an/2008042/
DO  - 10.1051/m2an/2008042
LA  - en
ID  - M2AN_2009__43_1_53_0
ER  - 
%0 Journal Article
%A Lefebvre, Aline
%T Numerical simulation of gluey particles
%J ESAIM: Modélisation mathématique et analyse numérique
%D 2009
%P 53-80
%V 43
%N 1
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an/2008042/
%R 10.1051/m2an/2008042
%G en
%F M2AN_2009__43_1_53_0
Lefebvre, Aline. Numerical simulation of gluey particles. ESAIM: Modélisation mathématique et analyse numérique, Tome 43 (2009) no. 1, pp. 53-80. doi : 10.1051/m2an/2008042. http://www.numdam.org/articles/10.1051/m2an/2008042/

[1] Y. Achdou, O. Pironneau and F. Valentin, Effective boundary conditions for laminar flows over periodic rough boundaries. J. Comp. Phys. 147 (1998) 187-218. | MR | Zbl

[2] Y. Assou, D. Joyeux, A. Azouni and F. Feuillebois, Mesure par interférométrie laser du mouvement d'une particule proche d'une paroi. J. Phys. III 1 (1991) 315-330.

[3] L. Bocquet and J.-L. Barrat, Hydrodynamic boundary conditions, correlation functions, and Kubo relations for confined fluids. Phys. Rev. E 49 (1994) 3079-3092.

[4] J.F. Brady and G. Bossis, Stokesian dynamics. Ann. Rev. Fluid Mech. 20 (1988) 111-157.

[5] D. Bresh and V. Milisic, High order multi-scale wall-laws, part I: The periodic case. Quat. Appl. Math. (to appear) ArXiv:math/0611083v2.

[6] R.G. Cox, The motion of suspended particles almost in contact. Int. J. Multiphase Flow 1 (1974) 343-371. | Zbl

[7] R.G. Cox and H. Brenner, The slow motion of a sphere through a viscous fluid towards a plane surface - II - Small gap width, including inertial effects. Chem. Engng. Sci. 22 (1967) 1753-1777.

[8] S.L. Dance and M.R. Maxey, Incorporation of lubrication effects into the force-coupling method for particulate two-phase flow. J. Comp. Phys. 189 (2003) 212-238. | MR | Zbl

[9] B. Desjardin and M.J. Esteban, Existence of weak solutions for the motion of rigid bodies in a viscous fluid. Arch. Ration. Mech. Anal. 146 (1999) 59-71. | MR | Zbl

[10] A. Einstein, A new method of determining molecular dimensions. Ann. Phys. Leipsig 19 (1906) 289-306. | JFM

[11] A. Einstein, Correction to my work: a new determination of molecular dimensions. Ann. Phys. Leipsig 34 (1911) 591-592. | JFM

[12] E. Feireisl, On the motion of rigid bodies in a viscous incompressible fluid. J. Evol. Equ. 3 (2003) 419-441. | MR | Zbl

[13] R. Glowinski, T.-W. Pan, T.I. Heslaand and D.D. Joseph, A distributed Lagrange multiplier/fictitious domain method for particulate flows. Int. J. Multiphase Flow 25 (1999) 755-794. | Zbl

[14] M. Hillairet, Lack of collision between solid bodies in a 2D constant-density incompressible viscous flow. Comm. Partial Diff. Eq. 32 (2007) 1345-1371. | MR | Zbl

[15] H.H. Hu, Direct simulation of flows of solid-liquid mixtures. Int. J. Multiphase Flow 22 (1996) 335-352. | Zbl

[16] A.A. Johnson and T.E. Tezduyar, Simulation of multiple spheres falling in a liquid-filled tube. Comput. Methods Appl. Mech. Engrg. 134 (1996) 351-373. | MR | Zbl

[17] S. Labbé, J. Laminie and V. Louvet, CSiMoon. Calcul scientifique, méthodologie orientée objet et environnement: de l'analyse mathématique à la programmation. Technical report RT 2001-01, Laboratoire de Mathématiques, Université Paris-Sud, France (2004).

[18] 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.

[19] N. Lecoq, 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. | Zbl

[20] A. Lefebvre, Fluid-Particle simulations with FreeFem++, in ESAIM: Proceedings 18, J.-F. Gerbeau and S. Labbé Eds. (2007) 120-132. | MR | Zbl

[21] A. Lefebvre, Simulation numérique d'écoulements fluide/particules. Ph.D. thesis, Université Paris-Sud XI, Orsay, France (Nov. 2007).

[22] B. Maury, A many-body lubrication model. C.R. Acad. Sci. Paris 325 (1997) 1053-1058. | MR | Zbl

[23] B. Maury, Direct simulation of 2D fluid-particle flows in biperiodic domains. J. Comp. Phys. 156 (1999) 325-351. | MR | Zbl

[24] B. Maury, A time-stepping scheme for inelastic collisions. Numer. Math. 102 (2006) 649-679. | MR | Zbl

[25] B. Maury, A gluey particle model, in ESAIM: Proceedings 18, J.-F. Gerbeau and S. Labbé Eds. (2007) 133-142. | MR | Zbl

[26] S. Nasseri, N. Phan-Thien and X.J. Fan, Lubrication approximation in completed double layer boundary element method. Comput. Mech. 26 (2000) 388-397. | Zbl

[27] N.A. Patankar, P. Singh, D.D. Joseph, R. Glowinski and T.-W. Pan, A new formulations for the distributed Lagrange multiplier/fictitious domain method for particulate flows. Int. J. Multiphase Flow 26 (2000) 1509-1524. | Zbl

[28] S. Richardson, A model for the boundary condition of a porous material. Part 2. J. Fluid Mech. 49 (1971) 327-336. | Zbl

[29] J.A. San Matín, V. Starovoitov and M. Tucsnak, Global weak solutions for the two-dimensional motion of several rigid bodies in an incompressible viscous fluid. Arch. Ration. Mech. Anal. 161 (2002) 113-147. | MR | Zbl

[30] P. Singh, T.I. Hesla and D.D. Joseph, Distributed Lagrange multiplier method for particulate flows with collisions. Int. J. Multiphase Flow 29 (2003) 495-509. | Zbl

[31] J.R. Smart and D.T. Leighton, Measurement of the hydrodynamic roughness of non colloidal spheres. Phys. Fluids A 1 (1989) 52.

[32] D.E. Stewart, Rigid-body dynamics with friction and impact. SIAM Rev. 42 (2000) 3-39. | MR | Zbl

[33] T. Takahashi, Analysis of strong solutions for the equations modeling the motion of a rigid-fluid system in a bounded domain. Adv. Differential Equations 8 (2003) 1499-1532. | MR | Zbl

[34] T. Takahashi, Existence of strong solutions for the problem of a rigid-fluid system. C.R. Math. Acad. Sci. Paris 336 (2003) 453-458. | MR | Zbl

[35] G.I. Taylor, A model for the boundary condition of a porous material. Part 1. J. Fluid Mech. 49 (1971) 319-326. | Zbl

[36] O.I. Vinogradova and G.E. Yacubov, Surface roughness and hydrodynamic boundary conditions. Phys. Rev. E 73 (2006) 045302(R).

[37] D. Wan and S. Turek, Direct numerical simulation of particulate flow via multigrid FEM techniques and the fictitious boundary method. Int. J. Numer. Meth. Fluids 51 (2006) 531-566. | MR | Zbl

Cité par Sources :