Interplay between complex fluid rheology and wall compliance affects the hydrodynamic resistance of deformable channels

Viscous flows through configurations manufactured from soft materials apply pressure and shear stress at the solid-liquid interface, leading to deformation. The resulting fluid-structure interaction affects the relationship between the pressure drop and the flow rate or the hydrodynamic resistance, which is the ratio between the two. While the hydrodynamic resistance in deformable configurations has been extensively studied for Newtonian fluids, it remains largely unexplored for non-Newtonian fluids even at low Reynolds numbers. In this work, we present a theoretical framework for calculating the hydrodynamic resistance of complex fluids in deformable axisymmetric channels. Our theory applies to a wide class of shear-thinning and viscoelastic constitutive models in the weakly non-Newtonian limit and provides the leading-order effect of the interplay between complex fluid rheology and wall compliance on the hydrodynamic resistance, bypassing the detailed calculations of the non-Newtonian fluid-structure-interaction problem. We illustrate our approach for a viscoelastic Oldroyd-B fluid and a shear-thinning Carreau fluid, highlighting the physical mechanisms underlying the response for each non-Newtonian fluid.