Joint use of fluidic and elastic reciprocal theorems to find the flow rate–pressure drop relation for deformable microchannels

Viscous flows through compliant conduits apply forces at the fluid-solid interface, leading to deformation of the conduit's cross-section, which affects the flow rate–pressure drop relation. Conventionally, calculating this relation requires solving the two-way coupled elastohydrodynamic problem of flow and deformation. Instead, we employ reciprocal theorems for Stokes flow and linear elasticity to derive a closed-form expression for the flow rate-pressure drop relation in deformable channels, using a domain perturbation expansion that that only requires the fluid flow solution and the elastic deformation due to the fluid stress distribution in an undeformed channel. Thus, we bypass solving the coupled fluid-structure-interaction problem. Unlike previous results, our approach shows a trade-off between the compliance of the deformable wall and the drag due to rigid sidewalls on the pressure drop. We find that compliance decreases the pressure drop, while drag due to sidewalls increases it.