Asymptotic regimes of ridge and rift formation in a thin viscous sheet model

Abstract

We numerically and theoretically investigate the evolution of the ridges and rifts produced by the convergent and divergent motions of two substrates over which an initially uniform layer of a Newtonian liquid rests. We put particular emphasis on the various asymptotic self-similar and quasi-self-similar regimes that occur in these processes. During the growth of a ridge, two self-similar stages occur; the first takes place in the initial linear phase, and the second is obtained for a large time. Initially, the width and the height of the ridge increase as t 1/2 . For a very large time, the width grows as t 3/4 , while the height increases as t 1/4 . On the other hand, in the process of formation of a rift, there are three self-similar asymptotics. The initial linear phase is similar to that for ridges. The second stage corresponds to the separation of the current in two parts, leaving a dry region in between. Last, for a very large t, each of the two parts in which the current has separated approaches the self-similar viscous dam break solution.

Self–similar asymptotics in convergent viscous gravity currents of non–Newtonian liquids

“…The analogy between the present results and those derived for a Newtonian liquid [6] can be traced to the fact that the constitutive relation (1) introduces a single-dimensional parameter (A) into the problem, as in the case of a Newtonian liquid (the viscosity coefficient). In both instances this dimensional parameter can be scaled out by an appropriate definition of the dependent variable, and thus does not appear in the final governing equations.…”

“…In the beginning the profile h(x, t) looks like the self-similar solution (6)(7). Later on h(x, t) tends to the solution (10)(11) as t → ∞.…”

“…In figure 7 we compare the values of h w for n = 3 from the numerical solution with those given by the self-similar and the quasi-self-similar solutions. We can improve the description far from the crest if we follow an approach similar to that described in [6]. To this end, we look for a solution that satisfies F (∞) = t −n/n 3 (to ensure that h(∞) = 1), sacrificing the condition at ξ = 0.…”

“…This substrate is divided in two parts, that for t > 0 are pushed one against the other [6]. This convergent motion drags the liquid, and produces a ridge (see figure 1).…”

Self–similar asymptotics in non–symmetrical convergent viscous gravity currents

“…The basic aspects of the process of mountain building and of the models developed to study it are discussed in the introduction of [1] and we shall not repeat them here. Here we extend our recent work [3] to include the effects of an asymmetric motion of the substrate. To this purpose we investigate a simple model that consists of a uniform layer of a Newtonian liquid resting over a horizontal substrate divided in two parts, that for t > 0 are pushed one against the other with different velocities.…”