This thesis examines the possible instability of radially spreading interfaces to the formation of fingers that break the axial symmetry. A well-known example of this occurs when a less viscous fluid displaces a more viscous immiscible fluid either in a porous medium or in a Hele-Shaw cell, which is commonly referred to as the Saffman–Taylor instability.

There are three related problems studied in this thesis: a single-layer viscous gravity current spreading from a point source over a rigid surface, radial spreading of an intrusion displacing miscible fluid in a Hele-Shaw cell, and finally, a viscous gravity current spreading from a constant-flux point source over a uniform layer of ambient fluid with equal density but different viscosity.

For single-layer viscous gravity currents with constant volumes, an analytical solution is available, which is known to be stable. By means of a numerical linear stability analysis, it is shown here that more general currents, with volumes growing as power laws in time, are stable as well. For currents with constant influx, considering a small shift in temporal origin yields the least stable axisymmetric perturbation mode. This analytic solution is generalised, first to non-axisymmetric perturbations, and then to more general power-law influxes. The derived growth rate confirms theoretically the stability of this least stable mode. Further perturbation modes are found numerically, exploiting a scaling-invariance symmetry of the governing equations, and using a change of independent variable to mitigate the singular nature of the nose. Finally, the stability of a general moving front within the framework of lubrication theory is established by considering the asymptotic limit of large azimuthal wavenumber.

Miscible intrusions in a Hele-Shaw cell with negligible diffusion are known to form flat frontal shocks for a sufficiently viscous ambient fluid. Experiments and theoretical work suggest that these fronts become unstable, similar to the Saffman–Taylor instability. However, no formal stability analysis has been done thus far. This thesis caries out this linear stability analysis, showing both that intrusions without a shock are stable and that intrusions with a shock are unstable. An asymptotic analysis of large azimuthal wavenumber shows that the model based on lubrication theory predicts rapidly growing perturbations in this limit. Therefore, the full three-dimensional Stokes equations would be required to predict a most unstable wavenumber. Analytic solutions for the general nonlinear evolution of the intrusion are found in the cases of axisymmetric perturbations and of equal-viscosity fluids.

Finally, a viscous gravity current spreading from a constant-flux point source over a uniform layer of ambient fluid is examined for the case of equal-density fluids. This case is identified as a singular limit in which the evolution equation for the interface becomes hyperbolic instead of parabolic. As a consequence, vertical shocks are predicted to form at the front of the intruding current for a sufficiently viscous ambient fluid layer, similar to the shocks found in Hele-Shaw flows. Reintroduction of a small density difference yields an Oleinik entropy condition, which predicts a unique shock height for the self-similar base state. The subsequent linear stability analysis reveals many similarities to Hele-Shaw flows, in particular the singular nature of large azimuthal wavenumbers. Experimental data obtained by others, compares very well overall to predictions of the theory. Finally, the cases of a single-layer current and of a Hele-Shaw intrusion are established as formal asymptotic limits of this two-layer current for large and small influxes, respectively.