A reciprocal theorem for boundary-driven channel flows

In a variety of physical situations, a bulk viscous flow is induced by a distribution of surface velocities, for example, in diffusiophoresis (as a result of chemical gradients) and above carpets of cilia (as a result of biological activity). When such boundary-driven flows are used to pump fluids, the primary quantity of interest is the induced flow rate. In this letter, we propose a method, based on the reciprocal theorem of Stokes flows, to compute the net flow rate for arbitrary flow distribution and periodic pump geometry using solely stress information from a dual Poiseuille-like problem. After deriving the general result, we apply it to straight channels of triangular, elliptic, and rectangular geometries and quantify the relationship between bulk motion and surface forcing. (C) 2015 AIP Publishing LLC.