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

Published online by Cambridge University Press:  25 March 2015

A. Lefebvre-Lepot ,
B. Merlet  and
T. N. Nguyen
A. Lefebvre-Lepot
Affiliation:
CNRS – Ecole Polytechnique/CMAP, route de Saclay, 91128 Palaiseau CEDEX, France
B. Merlet*
Affiliation:
Ecole Polytechnique/CMAP, route de Saclay, 91128 Palaiseau CEDEX, France
T. N. Nguyen
Affiliation:
Ecole Polytechnique/CMAP, route de Saclay, 91128 Palaiseau CEDEX, France
*
†Email address for correspondence: [email protected]
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Abstract

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).

JFM classification

Low-Reynolds-number flows: Boundary integral methods Low-Reynolds-number flows: Lubrication theory Low-Reynolds-number flows: Stokesian dynamics
Type
Papers
Information
Journal of Fluid Mechanics , Volume 769 , 25 April 2015 , pp. 369 - 386
Copyright
© 2015 Cambridge University Press 

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References

Alouges, F., DeSimone, A., Heltai, L., Lefebvre, A. & Merlet, B. 2013 Optimally swimming Stokesian robots. Discrete Contin. Dyn. Syst. Ser. B 18 (5), 1189–1215.Google Scholar
Alouges, F., DeSimone, A. & Lefebvre, A. 2008 Optimal strokes for low Reynolds number swimmers: an example. J. Nonlinear Sci. 18 (3), 277–302.CrossRefGoogle Scholar
Brady, J. F. & Bossis, G. 1988 Stokesian dynamics. Annu. Rev. Fluid Mech. 20, 111–157.Google Scholar
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), 3265–3273.Google Scholar
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, 3780–3790.Google Scholar
Cox, R. G. 1974 The motion of suspended particles almost in contact. Intl J. Multiphase Flow 1, 343–371.Google Scholar
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, 212–238.Google Scholar
Douglas, S. M., Bachelet, I. & Church, G. M. 2012 A logic-gated nanorobot for targeted transport of molecular payloads. Science 335, 831–834.Google Scholar
Durlofsky, L., Brady, J. F. & Bossis, G. 1987 Dynamic simulation of hydrodynamically interacting particles. J. Fluid Mech. 180, 21–49.Google Scholar
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, 634–641.CrossRefGoogle Scholar
Kim, S. & Karrila, S. J. 1991 Microhydrodynamics: Principles and Selected Applications. Butterworth-Heinemann.Google Scholar
Ladd, A. J. C. 1988 Hydrodynamic interactions in a suspension of spherical particles. J. Chem. Phys. 88, 5051–5063.Google Scholar
Ladd, A. J. C. 1994a Numerical simulations of particulate suspensions via a discretized Boltzmann equation. Part 1. Theoretical foundation. J. Fluid Mech. 271, 285–309.CrossRefGoogle Scholar
Ladd, A. J. C. 1994b Numerical simulations of particulate suspensions via a discretized Boltzmann equation. Part 2. Numerical results. J. Fluid Mech. 271, 311–339.CrossRefGoogle Scholar
Ladd, A. J. C. 2002 Lubrication corrections for lattice-Boltzmann simulations of particle suspensions. Phys. Rev. E 66, 046708.Google Scholar
Lefebvre-Lepot, A. & Merlet, B. 2009 A Stokesian submarine. In CEMRACS 2008—Modelling and Numerical Simulation of Complex Fluids, ESAIM Proceedings, vol. 28, pp. 150–161. EDP Sci.Google Scholar
Najafi, A. & Golestanian, R. 2004 Simple swimmer at low Reynolds number: three linked spheres. Phys. Rev. E 69 (6), 062901.CrossRefGoogle ScholarPubMed
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, 1509–1524.CrossRefGoogle Scholar
Pozrikidis, C. 1992 Boundary Integral and Singularity Methods for Linearized Viscous Flow. Cambridge University Press.CrossRefGoogle Scholar
Sangani, A. S. & Mo, G. 1994 Inclusion of lubrication forces in dynamic simulations. Phys. Fluids 6, 1653–1662.Google Scholar
Yeo, K. & Maxey, M. R. 2010 Simulation of concentrated suspensions using the force-coupling method. J. Comput. Phys. 229, 2401–2421.Google Scholar