Viscous flows in soft-walled microfluidic and biological systems induce deformation due to interfacial stresses, significantly altering the pressure drop-flow rate relationship. While this interaction has been extensively studied for Newtonian and different types of non-Newtonian fluids, the case of inhomogeneous fluids remains largely unexplored. In this talk, we analytically derive flow rate-pressure drop relations for fluids with spatially varying viscosity in deformable rectangular channels and axisymmetric tubes. Utilizing a combination of the reciprocal theorem and perturbation expansions, our results reveal that fluid-structure interactions at the boundaries generally reduce the pressure drop by increasing the cross-sectional area, irrespective of the viscosity distribution. However, the form of viscosity inhomogeneity leads to different results, including the possibility of "turning" the pressure drop reduction via the inhomogeneous viscosity. This finding provides insights into the development of more accurate microfluidic platforms for mimicking biological and physiological flows, wherein the inhomogeneity may arise from a cell-free layer near the vessel wall due to the Fahraeus-Lindqvist effect.
