Known similarity solutions of the shallow-water equations representing the motion of constant-volume gravity currents are studied in both plane and axisymmetric geometries. It is found that these solutions are linearly stable to small correspondingly symmetric perturbations and that they constitute the large-time limits of the solutions of the initial-value problem. Furthermore, the analysis reveals that the similarity solution is approached in an oscillatory manner. Two initial-value problems are solved numerically using finite differences and in each case the approach to the similarity solution is compared with the analytic predictions.

SUMMARYThe paper considers the large-time behaviour of positive solutions of the equationwith -oo < x < oo and t ^ 0, for pulse-type initial data. In suitably scaled variables this equation models the one-dimensional flow of a solute through a porous medium with the solute undergoing absorption by the solid matrix of the medium. With the total mass both absorbed and in solution invariant, it is shown that the asymptotic solution depends crucially on the value of p. For p > 2 the solution approaches the symmetric solution of the linear heat equation centred on x = t, while for p = 2 this becomes asymmetric due to the effect of nonlinearity. For 1 < p < 2 convection dominates at large time and the solution approaches the form of an asymmetric pulse moving at unit speed along the positive x-axis. Diffusion effects are confined to regions near the leading and trailing edges of the pulse. For 0 < p < 1 the pulse is still convection-dominated but it no longer moves under a simple translation. Again diffusion only becomes important near the leading and trailing edges. It is shown that for p ^ 2 the asymptotic solutions are uniformly valid in x but for 0 < p < 2 convection-dominated outer solutions have to be supplemented by diffusion boundary layers. In this latter case uniformly-valid composite solutions can be constructed. Finally our asymptotic analyses for the various values of p are compared with numerical solutions of the initial-value problem.

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