Effective boundary conditions at a rough wall: a high-order homogenization approach

Effective boundary conditions, correct to third order in a small parameter $$\epsilon$$ , are derived by homogenization theory for the motion of an incompressible fluid over a rough wall with periodic micro-indentations. The length scale of the indentations is l, and $$\epsilon = l/L \ll 1$$ , with L a characteristic length of the macroscopic problem. A multiple scale expansion of the variables allows to recover, at leading order, the usual Navier slip condition. At next order the slip velocity includes a term arising from the streamwise pressure gradient; furthermore, a transpiration velocity $${\mathcal {O}}(\epsilon ^{2})$$ appears at the fictitious wall where the effective boundary conditions are enforced. Additional terms appear at third order in both wall-tangent and wall-normal components of the velocity. The application of the effective conditions to a macroscopic problem is carried out for the Hiemenz stagnation point flow over a rough wall, highlighting the differences among the exact results and those obtained using conditions of different asymptotic orders