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Computation of the drag force on a sphere close to a wall

The roughness issue

Published online by Cambridge University Press:  15 March 2012

David Gérard-Varet  and
Matthieu Hillairet
David Gérard-Varet
Affiliation:
Institut de Mathématiques de Jussieu, 175 rue du Chevaleret, 75013 Paris, France. [email protected]
Matthieu Hillairet
Affiliation:
Ceremade, Place du Maréchal de Lattre de Tassigny, 75775 Paris Cedex 16, France; [email protected]
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Abstract

We consider the effect of surface roughness on solid-solid contact in a Stokes flow. Various models for the roughness are considered, and a unified methodology is given to derive the corresponding asymptotics of the drag force in the close-contact limit. In this way, we recover and clarify the various expressions that can be found in previous studies.

Keywords

Fluid mechanicsStokes equationsdragroughnesshomogenizationNavier boundary condition
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 46 , Issue 5 , September 2012 , pp. 1201 - 1224
Copyright
© EDP Sciences, SMAI, 2012

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References

Achdou, Y., Pironneau, O. and Valentin, F., Effective boundary conditions for laminar flows over periodic rough boundaries. J. Comput. Phys. 147 (1998) 187–218. Google Scholar
Barnocky, G. and Davis, R. H., The influence of pressure-dependent density and viscosity on the elastohydrodynamic collision and rebound of two spheres. J. Fluid Mech. 209 (1989) 501–519. Google Scholar
Basson, A. and Gérard-Varet, D., Wall laws for fluid flows at a boundary with random roughness. Comm. Pure Appl. Math. 61 (2008) 941–987. Google Scholar
Bocquet, L. and Barrat, J., Flow boundary conditions from nano-to micro-scales. Soft Matt. 3 (2007) 985–693. Google Scholar
Brenner, H. and Cox, R.G., The resistance to a particle of arbitrary shape in translational motion at small Reynolds numbers. J. Fluid Mech. 17 (1963) 561–595. Google Scholar
Bresch, D., Desjardins, B. and Gérard-Varet, D., On compressible Navier-Stokes equations with density dependent viscosities in bounded domains. J. Math. Pures Appl. 87 (2007) 227–235. Google Scholar
Cooley, M. and O’Neill, M., 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) 37–49. Google Scholar
Davis, R.H., Zhao, Y., Galvin, K.P. and Wilson, H.J., Solid-solid contacts due to surface roughness and their effects on suspension behaviour. Philos. Transat. Ser. A Math. Phys. Eng. Sci. 361 (2003) 871–894. Google ScholarPubMed
Davis, R.H., Serayssol, J. and Hinch, E., The elastohydrodynamic collision of two spheres. J. Fluid Mech. 163 (2006) 045302. Google Scholar
Gérard-Varet, D., The Navier wall law at a boundary with random roughness. Commun. Math. Phys. 286 (2009) 81–110. Google Scholar
Gérard-Varet, D. and Hillairet, M., Regularity issues in the problem of fluid structure interaction. Arch. Rational Mech. Anal. 195 (2010) 375–407. Google Scholar
Hillairet, M., Lack of collision between solid bodies in a 2D incompressible viscous flow. Commun. Partial Differ. Equ. 32 (2007) 1345–1371. Google Scholar
Hocking, L., The effect of slip on the motion of a sphere close to a wall and of two adjacent sheres. J. Eng. Mech. 7 (1973) 207–221. Google Scholar
Jäger, W. and Mikelić, A., Couette flows over a rough boundary and drag reduction. Commun. Math. Phys. 232 (2003) 429–455. Google Scholar
Kamrin, K., Bazant, M. and Stine, H., Effective slip boundary conditions for arbitrary periodic surfaces: the surface mobility tensor. Phys. Rev. Lett. 102 (2009). Google Scholar
Kunert, C., Harting, J. and Vinogradova, O., Random roughness hydrodynamic boundary conditions. Phys. Rev. Lett. 105 (2010) 016001. Google ScholarPubMed
E. Lauga, M. Brenner and H. Stone, Microfluidics: The no-slip boundary condition (2007).
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. Google Scholar
Lefebvre, A., Numerical simulation of gluey particles. ESAIM: M2AN 43 (2009) 53–80. Google Scholar
Luchini, P., Asymptotic analysis of laminar boundary-layer flow over finely grooved surfaces. Eur. J. Mech. B, Fluids 14 (1995) 169–195. Google Scholar
O’Neill, M., A slow motion of viscous liquid caused by a slowly moving solid sphere. Mathematika 11 (1964) 67–74. Google Scholar
O’Neill, M. and Stewartson, K., On the slow motion of a sphere parallel to a nearby plane wall. J. Fluid Mech. 27 (1967) 705–724. Google Scholar
Smart, J. and Leighton, D., Measurement of the hydrodynamic surface roughness of noncolloidal spheres. Phys. Fluids 1 (1989) 52–60. Google Scholar
Vinogradova, O. and Yakubov, G., Surface roughness and hydrodynamic boundary conditions. Phys. Rev. E 73 (1986) 479–487. Google ScholarPubMed