Computation of the effective slip of rough hydrophobic surfaces via homogenization

Matthieu Bonnivard (1) , Anne-Laure Dalibard (2) , David Gérard-Varet (3)

1 LJLL - Laboratoire Jacques-Louis Lions
2 DMA - Département de Mathématiques et Applications - ENS Paris
3 IMJ - Institut de Mathématiques de Jussieu

We present a quantitative analysis of the effect of rough hydrophobic surfaces on viscous newtonian flows. We use a model introduced by Ybert and coauthors in which the rough surface is replaced by a flat plane with alternating small areas of slip and no-slip. We investigate the averaged slip generated at the boundary, depending on the ratio between these areas. This problem reduces to the homogenization of a non-local system, involving the Dirichlet to Neumann map of the Stokes operator, in a domain with small holes. Pondering on works by Allaire, we compute accurate scaling laws of the averaged slip for various types of roughness (riblets, patches). Numerical computations complete and confirm the analysis.

Domaines: Equations aux dérivées partielles [math.AP]

Matthieu Bonnivard, Anne-Laure Dalibard, David Gérard-Varet. Computation of the effective slip of rough hydrophobic surfaces via homogenization. Mathematical Models and Methods in Applied Sciences, 2014, 24 (11), pp.2259-2285. ⟨hal-00777283v2⟩