Random Homogenization for the Stokes Flow through a Leaky Membrane

Maris, Razvan Florian

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Random Homogenization for the Stokes Flow through a Leaky Membrane

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We study a random homogenization problem concerning the flow of a viscous fluid through a permeable membrane with a highly oscillatory geometry and nonlinear boundary condition on it. Along an interface we consider a periodic distribution of small permeable obstacles with a random geometry. Leak boundary conditions of threshold type are considered on the obstacle part of the membrane: the normal velocity of the fluid is zero until the jump of the normal component of the stress acting on it reaches a certain limit, and then the fluid may pass freely. The problem is studied first in the deterministic case, and then in the random case, for which assumptions on the randomness of the solid obstacles are needed in order to obtain a limiting behaviour. The description of the obstacles is given in terms of a random set-valued variable defined on a probability space and a dynamical system acting on it. Effective boundary conditions for the fluid are derived, and these depend on the relative size of the obstacles. We establish two major cases, in one of them we obtain an effective permeability across the membrane and in the critical case a slip boundary condition of Navier type. If the dynamical system is assumed to be ergodic, the limiting behaviour of the fluid is deterministic. The approach is based on the Mosco convergence, which also allows us to pass from the stationary case to the time dependent case via the convergence of the associated semigroups.

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