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

Abstract

The linear stability of miscible displacement for radial source flow at infinite Péclet 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 $\frac{3}{2}$ and to be stable if $m<\frac{3}{2}$. If $1<m<\frac{3}{2}$ 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.

References in corpus (3)

• Fingering versus stability in the limit of zero interfacial tension
• Compression-driven viscous fingering in a radial Hele-Shaw cell
• Delayed onset and the transition to late time growth in viscous fingering