The Lorentz reciprocal theorem applied to oscillatory flows in rigid and deformable channels

We demonstrate the use of the Lorentz reciprocal theorem in obtaining corrections to the flow rate--pressure drop relation for oscillatory flow in channels. We start from the unsteady Stokes equations and derive the suitable reciprocity relations for both rigid and deformable channels, assuming all quantities can be expressed as time-harmonic phasors. For the case of a rigid channel, the auxiliary problem is a Poiseuille flow, and the reciprocal theorem allows us to calculate the first-order correction to the pressure drop in the Womersley number. For the case of a deformable channel, we account for the fluid-structure interaction using a domain perturbation approach under the assumption of a small compliance number. Using the Womersley flow solution in a rigid channel as an auxiliary problem, we obtain the first correction to the flow rate-pressure drop relations due to compliance of the channel.