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

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 $3/2$ , and to be stable if $m\lt {3/2}$ . If $1\lt m\lt {3/2}$ , then small disturbances decay at rates between $t^{-3/4}$ and $t^{-1}$ (the exact range depending on $m$ ) relative to the $t^{1/2}$ radius of the axisymmetric base-state similarity solution; if $m\lt 1$ , then 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.