# Viscous-elastic dynamics of power-law fluids within an elastic cylinder

## TL;DR

This paper investigates the complex interaction between non-Newtonian power-law fluids and elastic cylinders in microfluidic channels, deriving exact solutions and revealing how viscosity influences fluid front propagation, with implications for measuring fluid properties.

## Contribution

It introduces a novel analytical framework for viscous-elastic dynamics of power-law fluids in elastic cylinders, including exact solutions and insights into front propagation behavior.

## Key findings

- Exact solutions for pressure and deformation fields for shear-thinning and shear-thickening fluids.
- Inversion of viscosity role compared to Stokes' problem, with shear-thinning fluids exhibiting front propagation.
- The front propagation rate scales as t^{n/(n+1)}, enabling indirect measurement of the fluid's power-law index.

## Abstract

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.