  Viscous flows through configurations manufactured from soft materials apply
both pressure and shear stress at the solid-liquid interface, leading to
deformation of the cross-section, which affects the flow rate-pressure drop
relation. Conventionally, calculating this flow rate-pressure drop relation
requires solving the complete elastohydrodynamic problem, which couples the
fluid flow and elastic deformation. In this work, we use the reciprocal
theorems for Stokes flow and linear elasticity to derive a closed-form
expression for the flow rate-pressure drop relation in deformable channels,
bypassing the detailed calculation of the solution to the
fluid-structure-interaction problem. For small deformations (under a domain
perturbation scheme), our theory provides the leading-order effect, of the
interplay between the fluid stresses and the compliance of the channel, on the
flow rate-pressure drop relation. Our approach uses solely the fluid flow
solution and the elastic deformation due to the fluid stress distribution in an
undeformed channel, eliminating the need to solve the coupled
elastohydrodynamic problem. Unlike previous theoretical studies that neglected
the presence of lateral sidewalls and considered shallow geometries of
effectively infinite width, our approach allows to determine the influence of
confining sidewalls on the flow rate-pressure drop relation. For the
flow-rate-controlled situation and the plate-bending theory for the elastic
deformation, we show a trade-off between the effect of compliance of the
deforming top wall and the drag due to sidewalls on the pressure drop. While
increased compliance decreases the pressure drop, the effect of the sidewalls
increases it. Our theoretical framework may provide insight into existing
experimental data and pave the way for the design of novel optimized soft
microfluidic configurations of different cross-sectional shapes.
