Evolution of self-similarity, and other properties of waiting-time solutions of the porous medium equation: the case of viscous gravity currents

Abstract

The one-dimensional porous medium equation ht= (hmhx)x (m>0) admits waiting-time solutions, whose front remains motionless during a finite time interval tw before starting to move. We consider a family of initial value problems, and investigate the asymptotics, close to the front and near start-up, which we expect to be self-similar. We obtain numerical solutions for viscous gravity currents (m=3) and power-law initial conditions (h∝xp, h is proportional to the thickness of the fluid, x is the distance to the front). We find that: (a) if p<2/3 the front starts moving immediately, (b) if p=2/3 the front remains motionless during a finite time, (c) if p>2/3 one obtains waiting-time solutions in which a moving corner layer (a small interval Δx in which hx varies strongly) appears behind the front; the front starts moving when it is overrun by the corner layer. The corner layer strengthens (Δx reduces and the variation of hx increases) as it approaches the front. Our initial conditions produce waiting-time solutions whose front starts moving with nonzero velocity. We determine tw(p) and study the motion of the corner layer and the front, as well as other properties of the solutions. We compare the results with the theoretical upper and lower bounds of tw. We investigate the asymptotics of the numerical solutions for p>2/3, close to the corner layer and the front, and near start-up. To represent this asymptotics various kinds of similarity solutions are available, that can be classified according to the self-similarity exponent δ. We find that only two types (called L and A) are relevant. The L solutions correspond to 1<δ<13/10, and have an infinite series of corner layers that accumulate at the front. The part of these solutions behind the first corner layer of the series represents the asymptotics of the numerical solutions in a domain that excludes the region between the corner layer and the front, for a time interval excluding the neighbourhood of start-up. The A solutions have δ⩽1, and represent the evolution of the strong corner layer that is arriving at the front. The numerical evidence shows that the constant front velocity solution (type A with δ=1) describes the asymptotics close to, and including start-up, so that the motion of the corner layer joins smoothly with that of the front.