Self-similar gravity currents in porous media: Linear stability of the Barenblatt–Pattle solution revisited

We study the linear stability properties of the Barenblatt–Pattle (B–P) self-similar solutions of the porous medium equation which models flow including viscous and porous media gravity currents. Grundy and McLaughlin [R.E. Grundy, R. McLaughlin, Eigenvalues of the Barenblatt–Pattle similarity solution in nonlinear diffusion, Proc. Roy. Soc. London Ser. A 383 (1982) 89–100] have shown that, in both planar and axisymmetric geometries, the B–P solutions are linearly stable to symmetric perturbations. Using a new technique that eliminates singularities in the linear stability analysis, we extend their result and establish that the axisymmetric B–P solution is linearly stable to asymmetric perturbations. This suggests that the axisymmetric B–P solution provides the intermediate asymptotics of gravity currents that evolve from a wide range of initial distributions including those that are not axisymmetric. We use the connection between the perturbation eigenfunctions and the symmetry transformations of the B–P solution to demonstrate that the leading order rate of decay of the perturbations can be maximised by redefining the volume, time and space variables. We show that, in general, radially symmetric perturbations decay faster than asymmetric perturbations of equal amplitude. These theoretical predictions are confirmed by numerical results.