Effective boundary condition at a rough surface starting from a slip condition

Abstract We consider the homogenization of the Navier–Stokes equation, set in a channel with a rough boundary, of small amplitude and wavelength ϵ . It was shown recently that, for any non-degenerate roughness pattern, and for any reasonable condition imposed at the rough boundary, the homogenized boundary condition in the limit e = 0 is always no-slip. We give in this paper error estimates for this homogenized no-slip condition, and provide a more accurate effective boundary condition, of Navier type. Our result extends those obtained in Basson and Gerard-Varet (2008) [6] and Gerard-Varet and Masmoudi (2010) [13] , in which the special case of a Dirichlet condition at the rough boundary was examined.

[1]  Dorin Bucur,et al.  On the asymptotic limit of the Navier–Stokes system on domains with rough boundaries , 2008 .

[2]  Christophe Ybert,et al.  Achieving large slip with superhydrophobic surfaces: Scaling laws for generic geometries , 2007 .

[3]  Didier Bresch,et al.  Effect of rugosity on a flow governed by stationary Navier-Stokes equations , 2001 .

[4]  D. Gérard-Varet,et al.  Wall laws for fluid flows at a boundary with random roughness , 2006, math/0606768.

[5]  J. Lions,et al.  Les inéquations en mécanique et en physique , 1973 .

[6]  Frédéric Valentin,et al.  Constructing wall laws with Domain Decomposition or asymptotic expansion techniques , 1998 .

[7]  V. Solonnikov,et al.  Determination of the solutions of boundary value problems for stationary stokes and Navier-Stokes equations having an unbounded Dirichlet integral , 1983 .

[8]  N. Masmoudi,et al.  Relevance of the Slip Condition for Fluid Flows Near an Irregular Boundary , 2010 .

[9]  Juan Casado-Díaz,et al.  Why viscous fluids adhere to rugose walls: A mathematical explanation , 2003 .

[10]  Howard A. Stone,et al.  Microfluidics: The no-slip boundary condition , 2005, cond-mat/0501557.

[11]  Alain Bourgeat,et al.  Stochastic two-scale convergence in the mean and applications. , 1994 .

[12]  P. Luchini.  Asymptotic analysis of laminar boundary-layer flow over finely grooved surfaces , 1995 .

[13]  Frédéric Valentin,et al.  Effective Boundary Conditions for Laminar Flows over Periodic Rough Boundaries , 1998 .

[14]  Willi Jäger,et al.  On the Roughness-Induced Effective Boundary Conditions for an Incompressible Viscous Flow , 2001 .

[15]  D. Gérard-Varet.  The Navier Wall Law at a Boundary with Random Roughness , 2007, 0711.3610.

[16]  Willi Jäger,et al.  Couette Flows over a Rough Boundary and Drag Reduction , 2001 .

[17]  J. Cooper.  SINGULAR INTEGRALS AND DIFFERENTIABILITY PROPERTIES OF FUNCTIONS , 1973 .

[18]  C. Tropea,et al.  Microfluidics: The no-slip boundary condition , 2005, cond-mat/0501557.

[19]  C. Villani,et al.  On a variant of Korn's inequality arising in statistical mechanics , 2002 .

[20]  J. Nitsche.  On Korn's second inequality , 1981 .

[21]  V. Zhikov,et al.  Homogenization of Differential Operators and Integral Functionals , 1994 .