  In a wide range of applications, microfluidic channels are implemented in
soft substrates. In such configurations, where fluidic inertia and
compressibility are negligible, the propagation of fluids in channels is
governed by a balance between fluid viscosity and elasticity of the surrounding
solid. The viscous-elastic interactions between elastic substrates and
non-Newtonian fluids are particularly of interest due to the dependence of
viscosity on the state of the system. In this work, we study the
fluid-structure interaction dynamics between an incompressible non-Newtonian
fluid and a slender linearly elastic cylinder under the creeping flow regime.
Considering power-law fluids and applying the thin shell approximation for the
elastic cylinder, we obtain a non-homogeneous p-Laplacian equation governing
the viscous-elastic dynamics. We present exact solutions for the pressure and
deformation fields for various initial and boundary conditions for both
shear-thinning and shear-thickening fluids. We show that in contrast to Stokes'
problem where a compactly supported front is obtained for shear-thickening
fluids, here the role of viscosity is inversed and such fronts are obtained for
shear-thinning fluids. Furthermore, we demonstrate that for the case of a step
in inlet pressure, the propagation rate of the front has a $t^{\frac{n}{n+1}}$
dependence on time ($t$), suggesting the ability to indirectly measure the
power-law index ($n$) of shear-thinning liquids through measurements of elastic
deformation.
