Viscosity-Stratified Flow in a Hele-Shaw Cell

TL;DR

This paper develops mathematical models to analyze viscosity-stratified flows in Hele-Shaw cells, including numerical simulations and a new kinematic-wave model that captures viscous fingering and layer interactions.

Contribution

It introduces a hierarchical modeling framework and a novel kinematic-wave model accounting for inter-layer friction in viscosity-stratified flows.

Findings

The models accurately predict viscous finger velocity and thickness.

Numerical simulations illustrate the effects of inertia and friction.

The kinematic-wave model aligns well with two-dimensional calculations.

Abstract

A hierarchy of mathematical models describing viscosity-stratified flow in a Hele-Shaw cell is constructed. Numerical modelling of jet flow and development of viscous fingers with the influence of inertia and friction is carried out. One-dimensional multi-layer flows are studied. In the framework of three-layer flow the interpretation of the Saffman--Taylor instability is given. A kinematic-wave model of viscous fingering taking into account friction between the fluid layers is proposed. Comparison with calculations on the basis of two-dimensional equations shows that this model allows to determine the velocity of propagation and the thickness of the viscous fingers.