Mode-coupling approach to non-Newtonian Hele-Shaw flow

TL;DR

This paper develops a mode-coupling analytical approach to study the early nonlinear evolution of fluid interfaces in non-Newtonian Hele-Shaw flows, revealing how shear-dependent viscosity affects pattern formation and interface morphology.

Contribution

It introduces a differential equation for interface modes considering vorticity and mode-coupling in non-Newtonian fluids, extending traditional models.

Findings

Mode-coupling influences tip-splitting and side branching.

Vorticity from modified Darcy's law affects interface evolution.

Non-Newtonian patterns depend on shear-thinning or thickening behavior.

Abstract

The Saffman-Taylor viscous fingering problem is investigated for the displacement of a non-Newtonian fluid by a Newtonian one in a radial Hele-Shaw cell. We execute a mode-coupling approach to the problem and examine the morphology of the fluid-fluid interface in the weak shear limit. A differential equation describing the early nonlinear evolution of the interface modes is derived in detail. Owing to vorticity arising from our modified Darcy's law, we introduce a vector potential for the velocity in contrast to the conventional scalar potential. Our analytical results address how mode-coupling dynamics relates to tip-splitting and side branching in both shear thinning and shear thickening cases. The development of non-Newtonian interfacial patterns in rectangular Hele-Shaw cells is also analyzed.