Self-similar viscous gravity currents: phase-plane formalism

Abstract

A theoretical model for the spreading of viscous gravity currents over a rigid horizontal surface is derived, based on a lubrication theory approximation. The complete family of self-similar solutions of the governing equations is investigated by means of a phase-plane formalism developed in analogy to that of gas dynamics. The currents are represented by integral curves in the plane of two phase variables, Z and V, which are related to the depth and the average horizontal velocity of the fluid. Each integral curve corresponds to a certain self-similar viscous gravity current satisfying a particular set of initial and/or boundary conditions, and is obtained by solving a first-order ordinary differential equation of the form dV/dZ = f(Z, V), where f is a rational function. All conceivable self-similar currents can thus be obtained. A detailed analysis of the properties of the integral curves is presented, and asymptotic formulae describing the behaviour of the physical quantities near the singularities of the phase plane corresponding to sources, sinks, and current fronts are given. The derivation of self-similar solutions from the formalism is illustrated by several examples which include, in addition to the similarity flows studied by other authors, many other novel ones such as the extension to viscous flows of the classical problem of the breaking of a dam, the flows over plates with borders, as well as others. A self-similar solution of the second kind describing the axisymmetric collapse of a current towards the origin is obtained. The scaling laws for these flows are derived. Steady flows and progressive wave solutions are also studied and their connection to self-similar flows is discussed. The mathematical analogy between viscous gravity currents and other physical phenomena such as nonlinear heat conduction, nonlinear diffusion, and ground water motion is commented on.

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

“…For very large t, h tends to the solution of the viscous dam break problem, 13 as seen by an observer moving with velocity −U 0 . This solution is self-similar and in this reference frame has the form h = f͑xЈ / f ͱ t͒, where xЈ = x − U 0 t and f = −0.492¯.…”

“…13 In the previous expression, we have not yet specified h w . The logical choice is expression ͑20͒, which describes very well the time behavior at the crest, but this improvement is not sufficient because the solution so obtained does not satisfy yet the boundary condition at x =0 ͑except in the limit t → ϱ͒.…”

Fluid drainage from the edge of a porous reservoir

“…Typically, the latter refers to injection of supercritical CO 2 into saline aquifers, where it is expected that, following injection, the lower-density CO 2 will migrate over more dense interstitial fluid. In many situations, these gravity-driven flows occur in long narrow geometries and can be described by similarity solutions (Gratton & Minotti 1990;Huppert & Woods 1995), including allowance for power-law injection scenarios (Huppert & Woods 1995;Lyle et al 2005). Specific application of these ideas to CO 2 sequestration include a study of front propagation due to a constant injection rate in a confined porous medium (Nordbotten & Celia 2006b), analysis of † Email address for correspondence: hastone@princeton.edu…”

Elements of Similarity and Singularity

“…(21) is determined by the mass flux Qn through a section of unit area in the flow direction, of the porous medium: v = (Q/ρ)n. The system (21) and (22) is closed with the equation of state p = func(ρ), which for instance in the case of a gas is of the form p/p 0 = (ρ/ρ 0 ) γ , with γ the ratio of specific heats (ρ 0 , p 0 are density and pressure reference constants, respectively). Using the equation of state in (21) …”

“…There is an extensive literature on the self-similar solutions of (19), see for example the works by Gratton et al 21 , 10 , 22 , where cases of second kind similarity, arising for convergent currents, are also treated.…”