Hostname: page-component-745bb68f8f-s22k5 Total loading time: 0 Render date: 2025-01-11T07:51:28.699Z Has data issue: false hasContentIssue false
  • English
  • Français

Numerical simulation of gluey particles

Published online by Cambridge University Press:  16 October 2008

Aline Lefebvre
Aline Lefebvre*
Affiliation:
Laboratoire de Mathématiques, Université Paris-Sud, 91405 Orsay Cedex, France. [email protected]
Get access

Abstract

We propose here a model and a numerical scheme to compute the motionof rigid particles interacting through the lubrication force. In thecase of a particle approaching a plane, we propose an algorithm andprove its convergence towards the solutions to the gluey particle modeldescribed in [B. Maury, ESAIM: Proceedings18 (2007)133–142]. We propose a multi-particle version ofthis gluey model which is based on the projection of the velocitiesonto a set of admissible velocities. Then, we describe a multi-particle algorithmfor the simulation of such systems and present numerical results.

Keywords

Fluid/particle systemsfluid/solid interactionlubrication forcecontactsStokes fluid.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 43 , Issue 1 , January 2009 , pp. 53 - 80
Copyright
© EDP Sciences, SMAI, 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Achdou, Y., Pironneau, O. and Valentin, F., Effective boundary conditions for laminar flows over periodic rough boundaries. J. Comp. Phys. 147 (1998) 187–218. CrossRef
Assou, Y., Joyeux, D., Azouni, A. and Feuillebois, F., Mesure par interférométrie laser du mouvement d'une particule proche d'une paroi. J. Phys. III 1 (1991) 315–330.
Bocquet, L. and Barrat, J.-L., Hydrodynamic boundary conditions, correlation functions, and Kubo relations for confined fluids. Phys. Rev. E 49 (1994) 3079–3092. CrossRef
Brady, J.F. and Bossis, G., Stokesian dynamics. Ann. Rev. Fluid Mech. 20 (1988) 111–157. CrossRef
D. Bresh and V. Milisic, High order multi-scale wall-laws, part I: The periodic case. Quat. Appl. Math. (to appear) ArXiv:math/0611083v2.
Cox, R.G., The motion of suspended particles almost in contact. Int. J. Multiphase Flow 1 (1974) 343–371. CrossRef
Cox, R.G. and Brenner, H., 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. CrossRef
Dance, S.L. and Maxey, M.R., Incorporation of lubrication effects into the force-coupling method for particulate two-phase flow. J. Comp. Phys. 189 (2003) 212–238. CrossRef
Desjardin, B. and Esteban, M.J., Existence of weak solutions for the motion of rigid bodies in a viscous fluid. Arch. Ration. Mech. Anal. 146 (1999) 59–71. CrossRef
Einstein, A., A new method of determining molecular dimensions. Ann. Phys. Leipsig 19 (1906) 289–306. CrossRef
Einstein, A., Correction to my work: a new determination of molecular dimensions. Ann. Phys. Leipsig 34 (1911) 591–592. CrossRef
Feireisl, E., On the motion of rigid bodies in a viscous incompressible fluid. J. Evol. Equ. 3 (2003) 419–441. CrossRef
Glowinski, R., Pan, T.-W., Heslaand, T.I. and Joseph, D.D., A distributed Lagrange multiplier/fictitious domain method for particulate flows. Int. J. Multiphase Flow 25 (1999) 755–794. CrossRef
Hillairet, M., Lack of collision between solid bodies in a 2D constant-density incompressible viscous flow. Comm. Partial Diff. Eq. 32 (2007) 1345–1371. CrossRef
Direct, H.H. Hu simulation of flows of solid-liquid mixtures. Int. J. Multiphase Flow 22 (1996) 335–352.
Johnson, A.A. and Tezduyar, T.E., Simulation of multiple spheres falling in a liquid-filled tube. Comput. Methods Appl. Mech. Engrg. 134 (1996) 351–373. CrossRef
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).
Lecocq, N., Feuillebois, F., Anthore, N., Anthore, R., Bostel, F. and Petipas, C., Precise measurement of particle-wall hydrodynamic interactions at low Reynolds number using laser interferometry. Phys. Fluids A 5 (1993) 3–12. CrossRef
Lecoq, N., Anthore, R., Cichocki, B., Szymczak, P. and Feuillebois, F., Drag force on a sphere moving towards a corrugated wall. J. Fluid Mech. 513 (2004) 247–264. CrossRef
A. Lefebvre, Fluid-Particle simulations with FreeFem++, in ESAIM: Proceedings 18, J.-F. Gerbeau and S. Labbé Eds. (2007) 120–132.
A. Lefebvre, Simulation numérique d'écoulements fluide/particules. Ph.D. thesis, Université Paris-Sud XI, Orsay, France (Nov. 2007).
Maury, B., A many-body lubrication model. C.R. Acad. Sci. Paris 325 (1997) 1053–1058. CrossRef
Maury, B., Direct simulation of 2D fluid-particle flows in biperiodic domains. J. Comp. Phys. 156 (1999) 325–351. CrossRef
Maury, B., A time-stepping scheme for inelastic collisions. Numer. Math. 102 (2006) 649–679. CrossRef
B. Maury, A gluey particle model, in ESAIM: Proceedings 18, J.-F. Gerbeau and S. Labbé Eds. (2007) 133–142.
Nasseri, S., Phan-Thien, N. and Fan, X.J., Lubrication approximation in completed double layer boundary element method. Comput. Mech. 26 (2000) 388–397. CrossRef
Patankar, N.A., Singh, P., Joseph, D.D., Glowinski, R. and Pan, T.-W., A new formulations for the distributed Lagrange multiplier/fictitious domain method for particulate flows. Int. J. Multiphase Flow 26 (2000) 1509–1524. CrossRef
Richardson, S., A model for the boundary condition of a porous material. Part 2. J. Fluid Mech. 49 (1971) 327–336. CrossRef
San Matín, J.A., Starovoitov, V. and Tucsnak, M., 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.
Singh, P., Hesla, T.I. and Joseph, D.D., Distributed Lagrange multiplier method for particulate flows with collisions. Int. J. Multiphase Flow 29 (2003) 495–509. CrossRef
Smart, J.R. and Leighton, D.T., Measurement of the hydrodynamic roughness of non colloidal spheres. Phys. Fluids A 1 (1989) 52. CrossRef
Stewart, D.E., Rigid-body dynamics with friction and impact. SIAM Rev. 42 (2000) 3–39. CrossRef
Takahashi, T., 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.
Takahashi, T., Existence of strong solutions for the problem of a rigid-fluid system. C.R. Math. Acad. Sci. Paris 336 (2003) 453–458. CrossRef
Taylor, G.I., A model for the boundary condition of a porous material. Part 1. J. Fluid Mech. 49 (1971) 319–326. CrossRef
Vinogradova, O.I. and Yacubov, G.E., Surface roughness and hydrodynamic boundary conditions. Phys. Rev. E 73 (2006) 045302(R). CrossRef
Wan, D. and Turek, S., Direct numerical simulation of particulate flow via multigrid FEM techniques and the fictitious boundary method. Int. J. Numer. Meth. Fluids 51 (2006) 531–566. CrossRef