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.