Interplay between complex fluid rheology and wall compliance in the flow resistance of deformable axisymmetric configurations

Abstract

Viscous flows through configurations fabricated from soft materials exert stresses at the solid–liquid interface, leading to a coupling between the flow field and the elastic deformation. The resulting fluid–structure interaction affects the relationship between the pressure drop $\Delta p$ and the flow rate $q$, or the corresponding flow resistance $\Delta p/q$. While the flow resistance in deformable configurations has been extensively studied for Newtonian fluids, it remains largely unexplored for non-Newtonian fluids even at low Reynolds numbers. We analyze the steady low-Reynolds-number fluid–structure interaction between the flow of a non-Newtonian fluid and a deformable tube. We present a theoretical framework for calculating the leading-order effect of the complex fluid rheology and wall compliance on the flow resistance, which holds for a wide class of non-Newtonian constitutive models. For the weakly non-Newtonian limit, our theory provides the first-order non-Newtonian correction for the flow resistance solely using the known Newtonian solution for a deformable tube, bypassing the detailed calculations of the non-Newtonian fluid–structure-interaction problem. We illustrate our approach for a weakly viscoelastic Oldroyd-B fluid and a weakly shear-thinning Carreau fluid. In particular, we show analytically that both the viscoelasticity and shear thinning of the fluid and the compliance of the deformable tube decrease the flow resistance in the weakly non-Newtonian limit and identify the physical mechanisms governing this reduction.