Fingering instability of self-similar radial flow of miscible fluids in a Hele-Shaw cell

TL;DR

This paper theoretically analyzes the linear stability of radial miscible fluid flow in a Hele-Shaw cell, revealing conditions under which viscous fingering occurs based on viscosity ratios and providing detailed decay rates of disturbances.

Contribution

It introduces a theoretical framework for the stability of self-similar radial flow in Hele-Shaw cells, identifying critical viscosity ratios for fingering instability and analyzing disturbance decay rates.

Findings

Flow is unstable to fingering if viscosity ratio > 1.5.

Flow is stable if viscosity ratio < 1.5.

Disturbance decay rates depend on viscosity ratio and are characterized mathematically.

Abstract

The linear stability of miscible displacement for radial source flow at infinite P\'eclet number in a Hele-Shaw cell is calculated theoretically. The axisymmetric self-similar flow is shown to be unstable to viscous fingering if the viscosity ratio $m$ between ambient and injected fluids exceeds $3\over2$ and to be stable if $m<{3\over2}$. If $1<m<{3\over2}$ small disturbances decay at rates between $t^{-3/4}$ and $t^{-1}$ relative to the $t^{1/2}$ radius of the axisymmetric base-state similarity solution; if $m<1$ they decay faster than $t^{-1}$. Asymptotic analysis confirms these results and gives physical insight into various features of the numerically determined relationship between the growth rate and the azimuthal wavenumber and viscosity ratio.