Flow rate--pressure drop relations for shear-thinning fluids in deformable configurations: theory and experiments

We provide an experimental framework to measure the flow rate--pressure drop relation for Newtonian and shear-thinning fluids in two common deformable configurations: (i) a rectangular channel and (ii) an axisymmetric tube. Using the Carreau model to describe the shear-dependent viscosity, we identify the key dimensionless rheological number, $Cu$, which characterizes shear thinning, and we show that our experiments lie within the power-law regime of shear rates. To rationalize the experimental data, we derive the flow rate--pressure drop relation taking into account the two-way-coupled fluid--structure interaction between the flow and its compliant confining boundaries. We thus identify the second key dimensionless number, $\alpha$, which characterizes the compliance of the conduit. We then compare the theoretical flow rate--pressure drop relation to our experimental measurements, finding excellent agreement between the two. We further contrast our results for shear-thinning and Newtonian fluids to highlight the influence of $Cu$ on the flow rate--pressure drop relation. Finally, we delineate four distinct physical regimes of flow and deformation by mapping our experimental flow rate--pressure drop data for Newtonian and shear-thinning fluids into a $Cu-\alpha$ plane.