Analysis of a two-fluid Taylor-Couette flow with one non-Newtonian fluid

TL;DR

This paper investigates the dynamics of a two-fluid Taylor-Couette flow with a non-Newtonian inner fluid, deriving evolution equations, analyzing different regimes, and proving existence and shape convergence of solutions.

Contribution

It introduces a mathematical model for two-fluid Taylor-Couette flow with a non-Newtonian fluid and analyzes interface behavior under various dominant effects.

Findings

Solutions behave like Newtonian flow when surface tension and shear stresses are comparable.

Local existence of solutions is proven for shear-thinning and shear-thickening fluids.

Interfaces close to a circle converge to a circle in finite time under dominant surface tension.

Abstract

We study the dynamic behaviour of two viscous fluid films confined between two concentric cylinders rotating at a small relative velocity. It is assumed that the fluids are immiscible and that the volume of the outer fluid film is large compared to the volume of the inner one. Moreover, while the outer fluid is considered to have constant viscosity, the rheological behaviour of the inner thin film is determined by a strain-dependent power-law. Starting from a Navier--Stokes system, we formally derive evolution equations for the interface separating the two fluids. Two competing effects drive the dynamics of the interface, namely, the surface tension and the shear stresses induced by the rotation of the cylinders. When the two effects are comparable, the solutions behave, for large times, as in the Newtonian regime. We also study the regime in which the surface tension effects dominate the stresses induced by the rotation of the cylinders. In this case, we prove local existence of positive weak solutions both for shear-thinning and shear-thickening fluids. In the latter case, we show that interfaces which are initially close to a circle converge to a circle in finite time and keep that shape for later times.