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We demonstrate the use of the Lorentz reciprocal theorem in obtainingcorrections to the flow rate for oscillatory flow in channels. We start fromthe unsteady Stokes equations and derive the suitable reciprocity relations forboth rigid and deformable channels, assuming all quantities can be expressed astime-harmonic phasors. For the case of a rigid channel, the auxiliary problemis a steady flow solution, and the reciprocal theorem allows us to calculatethe first-order correction in the Womersley number to the steady flow rate. Forthe case of a deformable channel, we account for the fluid--structureinteraction using domain perturbation under the assumption of a smallcompliance number. Using the oscillatory flow solution in a rigid channel asthe auxiliary problem, we obtain the first-order correction in the compliancenumber to the rigid channel flow rate. This correction, validated againstnumerical simulations, provides the leading-order effect of the interplaybetween the oscillations of the fluid flow and the compliance of the channelfor arbitrary values of the Womersley number.

Reciprocal theorems for calculating the flow rate of oscillatory channel flows