Skip to main content Accessibility help

An accurate method to include lubrication forces in numerical simulations of dense Stokesian suspensions

  • A. Lefebvre-Lepot (a1), B. Merlet (a2) and T. N. Nguyen (a2)


We address the problem of computing the hydrodynamic forces and torques among $N$ solid spherical particles moving with given rotational and translational velocities in Stokes flow. We consider the original fluid–particle model without introducing new hypotheses or models. Our method includes the singular lubrication interactions which may occur when some particles come close to one another. The main new feature is that short-range interactions are propagated to the whole flow, including accurately the many-body lubrication interactions. The method builds on a pre-existing fluid solver and is flexible with respect to the choice of this solver. The error is the error generated by the fluid solver when computing non-singular flows (i.e. with negligible short-range interactions). Therefore, only a small number of degrees of freedom are required and we obtain very accurate simulations within a reasonable computational cost. Our method is closely related to a method proposed by Sangani & Mo (Phys. Fluids, vol. 6, 1994, pp. 1653–1662) but, in contrast with the latter, it does not require parameter tuning. We compare our method with the Stokesian dynamics of Durlofsky et al. (J. Fluid Mech., vol. 180, 1987, pp. 21–49) and show the higher accuracy of the former (both by analysis and by numerical experiments).


Corresponding author

Email address for correspondence:


Hide All
Alouges, F., DeSimone, A., Heltai, L., Lefebvre, A. & Merlet, B. 2013 Optimally swimming Stokesian robots. Discrete Contin. Dyn. Syst. Ser. B 18 (5), 11891215.
Alouges, F., DeSimone, A. & Lefebvre, A. 2008 Optimal strokes for low Reynolds number swimmers: an example. J. Nonlinear Sci. 18 (3), 277302.
Brady, J. F. & Bossis, G. 1988 Stokesian dynamics. Annu. Rev. Fluid Mech. 20, 111157.
Cichocki, B., Ekiel-Jezewska, M. L. & Wajnryb, E. 1999 Lubrication corrections for three-particle contribution to short-time self diffusion coefficients in colloidal dispersions. J. Chem. Phys. 111 (7), 32653273.
Cichocki, B., Felderhof, B. U., Hinsen, K., Wajnryb, E. & Blawzdziewicz, J. 1994 Friction and mobility of many spheres in Stokes flow. J. Chem. Phys. 100, 37803790.
Cox, R. G. 1974 The motion of suspended particles almost in contact. Intl J. Multiphase Flow 1, 343371.
Dance, S. L. & Maxey, M. R. 2003 Incorporation of lubrication effects into the force-coupling method for particulate two-phase flow. J. Comput. Phys. 189, 212238.
Douglas, S. M., Bachelet, I. & Church, G. M. 2012 A logic-gated nanorobot for targeted transport of molecular payloads. Science 335, 831834.
Durlofsky, L., Brady, J. F. & Bossis, G. 1987 Dynamic simulation of hydrodynamically interacting particles. J. Fluid Mech. 180, 2149.
Jeffrey, D. J. & Onishi, Y. 1984 The forces and couples acting on two nearly touching spheres in low-Reynolds-number flow. Z. Angew. Math. Phys. 35, 634641.
Kim, S. & Karrila, S. J. 1991 Microhydrodynamics: Principles and Selected Applications. Butterworth-Heinemann.
Ladd, A. J. C. 1988 Hydrodynamic interactions in a suspension of spherical particles. J. Chem. Phys. 88, 50515063.
Ladd, A. J. C. 1994a Numerical simulations of particulate suspensions via a discretized Boltzmann equation. Part 1. Theoretical foundation. J. Fluid Mech. 271, 285309.
Ladd, A. J. C. 1994b Numerical simulations of particulate suspensions via a discretized Boltzmann equation. Part 2. Numerical results. J. Fluid Mech. 271, 311339.
Ladd, A. J. C. 2002 Lubrication corrections for lattice-Boltzmann simulations of particle suspensions. Phys. Rev. E 66, 046708.
Lefebvre-Lepot, A. & Merlet, B. 2009 A Stokesian submarine. In CEMRACS 2008—Modelling and Numerical Simulation of Complex Fluids, ESAIM Proceedings, vol. 28, pp. 150161. EDP Sci.
Najafi, A. & Golestanian, R. 2004 Simple swimmer at low Reynolds number: three linked spheres. Phys. Rev. E 69 (6), 062901.
Patankar, N. A., Singh, P., Joseph, D. D., Glowinski, R. & Pan, T.-W. 2000 A new formulations for the distributed Lagrange multiplier/fictitious domain method for particulate flows. Intl J. Multiphase Flow 26, 15091524.
Pozrikidis, C. 1992 Boundary Integral and Singularity Methods for Linearized Viscous Flow. Cambridge University Press.
Sangani, A. S. & Mo, G. 1994 Inclusion of lubrication forces in dynamic simulations. Phys. Fluids 6, 16531662.
Yeo, K. & Maxey, M. R. 2010 Simulation of concentrated suspensions using the force-coupling method. J. Comput. Phys. 229, 24012421.
MathJax is a JavaScript display engine for mathematics. For more information see

JFM classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed