Reciprocal theorem for calculating the flow rate of oscillatory channel flows

Abstract

We demonstrate the use of the Lorentz reciprocal theorem in obtaining corrections to the steady flow rate due to flow oscillations in rigid channels. Starting from the unsteady Stokes equations, we derive the suitable reciprocity relation, assuming all quantities can be expressed as time-harmonic phasors. The auxiliary problem is the steady Hagen--Poiseuille flow solution, from which the reciprocal theorem allows us to calculate the first-order correction in the Womersley number to the steady flow rate in a straight rigid channel. We also consider nonuniform channels, specifically with variable height in the flow-wise direction, in which case the flow rate correction provides the leading-order effect of the interplay between the oscillations of the fluid flow and the given shape of the channel.