Random homogenization for a fluid flow through a membrane

Abstract

Effective boundary conditions for the flow of a viscous fluid across randomly leaky permeable membranes are studied. Threshold leak conditions of subgradient type, introduced by Fujita [Res. Inst. Math. Sci. Kokyuroku 888 (1994), 199–216, J. Comp. Appl. Math. 149(1) (2002), 56–79], are considered on the randomly distributed solid part of the membrane. The effective conditions are of subgradient type with an effective yield limit, in the case of a densely distributed solid part, or of Navier slip type, in the critical case; in the dilute case the tangential slip cancels, whereas the normal velocity and stress are continuous. We thus extend our results [Complex Variables and Elliptic Equations 57(2–4) (2012), 437–453], from the periodic permeable membrane, to a randomly permeable membrane model.