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 $\frac{3}{2}$ and to be stable if $m<\frac{3}{2}$. If $1<m<\frac{3}{2}$ small disturbances decay at rates between $t^{-3\mathrm{/}4}$ and $t^{-1}$ relative to the $t^{1\mathrm{/}2}$ 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.