A reciprocal theorem for boundary-driven channel flows

TL;DR

This paper introduces a reciprocal theorem-based method to efficiently compute flow rates in boundary-driven channel flows, applicable to various geometries and surface velocity distributions, with potential applications in biological and chemical systems.

Contribution

It presents a novel reciprocal theorem approach to determine flow rates in boundary-driven flows, simplifying calculations for complex geometries and surface velocity patterns.

Findings

Method accurately computes flow rates in different channel geometries.

Flow rate depends on surface forcing and channel shape.

Applicable to biological and chemical flow systems.

Abstract

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.