Effective interfacial conditions for the Stokes flow of a fluid on periodically rough surfaces

This work is concerned with the Stokes flow of a fluid through the channel between two parallel solid walls, of which one is smooth and homogeneous and the other is periodically rough and heterogeneous. The main purpose of this work is to homogenize the resulting periodically rough solid/fluid interface so as to replace it by an equivalent smooth homogeneous solid/fluid interface characterized by appropriate effective slip lengths. To achieve this objective, a semi-analytic approach is elaborated by carrying out the Fourier expansions of the velocity and pressure fields in the plane normal to the thickness direction of the channel and by resorting to the point collocation method. In comparison with the relevant approaches reported in the literature, the present one has the advantage of being valid for the general case where the distance between the two parallel walls is arbitrary. With the help of the elaborated approach, the effective slip lengths are determined for a variety of periodically rough surface microstructures including, for example, longitudinal grooves, transverse grooves, chessboard texture and the Hashin-Shtrikman's nested circles. The results obtained by the proposed semi-analytical approach are compared systematically with the ones provided by the finite element method and discussed in the light of some available bounds or analytical results. It follows from these comparisons and discussions that our semi-analytical approach is particularly efficient and accurate.