Computation of the drag force on a sphere close to a wall
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 46 (2012) no. 5, pp. 1201-1224.

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.

DOI : 10.1051/m2an/2012001
Classification : 35Q35, 35Q30, 74F10
Mots clés : fluid mechanics, Stokes equations, drag, roughness, homogenization, Navier boundary condition
@article{M2AN_2012__46_5_1201_0,
     author = {G\'erard-Varet, David and Hillairet, Matthieu},
     title = {Computation of the drag force on a sphere close to a wall},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {1201--1224},
     publisher = {EDP-Sciences},
     volume = {46},
     number = {5},
     year = {2012},
     doi = {10.1051/m2an/2012001},
     mrnumber = {2916378},
     zbl = {1267.76020},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/m2an/2012001/}
}
TY  - JOUR
AU  - Gérard-Varet, David
AU  - Hillairet, Matthieu
TI  - Computation of the drag force on a sphere close to a wall
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2012
SP  - 1201
EP  - 1224
VL  - 46
IS  - 5
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/m2an/2012001/
DO  - 10.1051/m2an/2012001
LA  - en
ID  - M2AN_2012__46_5_1201_0
ER  - 
%0 Journal Article
%A Gérard-Varet, David
%A Hillairet, Matthieu
%T Computation of the drag force on a sphere close to a wall
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2012
%P 1201-1224
%V 46
%N 5
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/m2an/2012001/
%R 10.1051/m2an/2012001
%G en
%F M2AN_2012__46_5_1201_0
Gérard-Varet, David; Hillairet, Matthieu. Computation of the drag force on a sphere close to a wall. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 46 (2012) no. 5, pp. 1201-1224. doi : 10.1051/m2an/2012001. http://archive.numdam.org/articles/10.1051/m2an/2012001/

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

[2] G. Barnocky and R. H. Davis, The influence of pressure-dependent density and viscosity on the elastohydrodynamic collision and rebound of two spheres. J. Fluid Mech. 209 (1989) 501-519.

[3] A. Basson and D. Gérard-Varet, Wall laws for fluid flows at a boundary with random roughness. Comm. Pure Appl. Math. 61 (2008) 941-987. | MR | Zbl

[4] L. Bocquet and J. Barrat, Flow boundary conditions from nano-to micro-scales. Soft Matt. 3 (2007) 985-693.

[5] H. Brenner and R.G. Cox, The resistance to a particle of arbitrary shape in translational motion at small Reynolds numbers. J. Fluid Mech. 17 (1963) 561-595. | MR | Zbl

[6] D. Bresch, B. Desjardins and D. Gérard-Varet, On compressible Navier-Stokes equations with density dependent viscosities in bounded domains. J. Math. Pures Appl. 87 (2007) 227-235. | MR | Zbl

[7] M. Cooley and M. O'Neill, 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. | Zbl

[8] R.H. Davis, Y. Zhao, K.P. Galvin and H.J. Wilson, 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. | MR | Zbl

[9] R.H. Davis, J. Serayssol and E. Hinch, The elastohydrodynamic collision of two spheres. J. Fluid Mech. 163 (2006) 045302.

[10] D. Gérard-Varet, The Navier wall law at a boundary with random roughness. Commun. Math. Phys. 286 (2009) 81-110. | MR | Zbl

[11] D. Gérard-Varet and M. Hillairet, Regularity issues in the problem of fluid structure interaction. Arch. Rational Mech. Anal. 195 (2010) 375-407. | MR | Zbl

[12] M. Hillairet, Lack of collision between solid bodies in a 2D incompressible viscous flow. Commun. Partial Differ. Equ. 32 (2007) 1345-1371. | MR | Zbl

[13] L. Hocking, 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. | Zbl

[14] W. Jäger and A. Mikelić, Couette flows over a rough boundary and drag reduction. Commun. Math. Phys. 232 (2003) 429-455. | Zbl

[15] K. Kamrin, M. Bazant and H. Stine, Effective slip boundary conditions for arbitrary periodic surfaces: the surface mobility tensor. Phys. Rev. Lett. 102 (2009). | Zbl

[16] C. Kunert, J. Harting and O. Vinogradova, Random roughness hydrodynamic boundary conditions. Phys. Rev. Lett. 105 (2010) 016001.

[17] E. Lauga, M. Brenner and H. Stone, Microfluidics: The no-slip boundary condition (2007).

[18] 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

[19] A. Lefebvre, Numerical simulation of gluey particles. ESAIM: M2AN 43 (2009) 53-80. | Numdam | MR | Zbl

[20] P. Luchini, Asymptotic analysis of laminar boundary-layer flow over finely grooved surfaces. Eur. J. Mech. B, Fluids 14 (1995) 169-195. | MR | Zbl

[21] M. O'Neill, A slow motion of viscous liquid caused by a slowly moving solid sphere. Mathematika 11 (1964) 67-74. | MR | Zbl

[22] M. O'Neill and K. Stewartson, On the slow motion of a sphere parallel to a nearby plane wall. J. Fluid Mech. 27 (1967) 705-724. | MR | Zbl

[23] J. Smart and D. Leighton, Measurement of the hydrodynamic surface roughness of noncolloidal spheres. Phys. Fluids 1 (1989) 52-60.

[24] O. Vinogradova and G. Yakubov, Surface roughness and hydrodynamic boundary conditions. Phys. Rev. E 73 (1986) 479-487.

Cité par Sources :