Interfacial dynamics in transport-limited dissolution

Abstract

Various model problems of "transport-limited dissolution" in two dimensions are analyzed using time-dependent conformal maps. For diffusion-limited dissolution (reverse Laplacian growth), several exact solutions are discussed for the smoothing of corrugated surfaces, including the continuous analogs of "internal diffusion-limited aggregation" and "diffusion-limited erosion". A class of non-Laplacian, transport-limited dissolution processes are also considered, which raise the general question of when and where a finite solid will disappear. In a case of dissolution by advectiondiffusion, a tilted ellipse maintains its shape during collapse, as its center of mass drifts obliquely away from the background fluid flow, but other initial shapes have more complicated dynamics.The analysis of interfacial dynamics using conformal maps ("Loewner chains") is a classical subject, which is finding unexpected applications in physics [1,2,3]. For dynamics controlled by Laplacian fields, there is a vast literature on continuous models of viscous fingering [4,5], and stochastic models of diffusion-limited aggregation (DLA) [3,6] and fractal curves in critical phenomena [1,2]. Conformal-map dynamics has also been formulated for a class of non-Laplacian growth phenomena of both types [7], driven by conformally invariant transport processes [8]. For growth limited by advection-diffusion in a potential flow, the connection between continuous and stochastic growth patterns has been elucidated [9], and the continuous dynamics has also been studied in cases of freezing in flowing liquids [10,11,12,13].In all of these examples, the moving interface separates a "solid" region, where singularities in the map reside, a "fluid" region, where the driving transport processes occur and the map is univalent. (In viscous fingering, these are the inviscid and viscous fluid regions, respectively.) Most attention has been paid to problems of "transportlimited growth", where the solid region grows into the fluid region, since the dynamics is unstable and typically leads to cusp singularities in finite time [14,15,16] (without surface tension [4]). Here, we consider various time-reversed problems of "transport-limited dissolution" (TLD), where the solid recedes from the fluid region, e.g. driven by advection-diffusion in a potential flow. These are stable processes, so we focus on continuous dynamics, without surface tension.Stochastic diffusion-limited dissolution (DLD), sometimes called "diffusion-limited erosion" (DLE) or "anti-DLA", has been simulated by allowing random walkers in the fluid to annihilate particles of the solid upon contact [17,19,20,21], rather than aggregating as in DLA [18]. Outward radial DLE on a lattice, or "internal DLA" (IDLA), where the random walkers start at the origin and cause a fluid cavity to grow in an infinite solid, has also been studied by mathematicians, who proved that the asymptotic shape is a sphere in any dimension [22].We begin our analysis by summarizing some exact solutions for continuous DLD, which cou...

Asymmetric collapse by dissolution or melting in a uniform flow

“…The results for Figs. 2(a) and 2(b) match those that were previously studied analytically [44]. For the case of no flow where B = 0 where the system is mathematically equivalent to (time-reversed) Laplacian growth, and bubble contraction in a porous medium, it is known that the ellipse is the most generic self-similar shape [61-63].…”

“…The dissolution process is therefore described entirely in terms of a and q 3 , and could therefore be determined analytically using Eqs. (24) and (27), as considered in previous work [29,44,64]. Since q 0 remains at zero, the collapse point is at the origin.…”

Electrokinetic Control of Viscous Fingering

“…This phenomenon illustrates the rich physics of "Vector Laplacian Growth" (VLG), a general mathematical model of interfacial dynamics driven by the gradient of a vector-valued potential function through a generalized mobility tensor. The "one-sided" VLG model (with field gradients only on one side of the interface) is known to be unstable, leading to fractal patterns, during growth [7] and stable, resulting in smooth collapse, during retreat [20,21]. Our theory shows that stable growth is also possible, if field gradients exist on both sides of the interface.…”

Steadily Translating Parabolic Dissolution Fingers

“…The conformal invariance of these processes provides an effective way of solving these problems [10]. In the context of growth processes, these techniques have been used to track the evolution of the interface in solidification and melting under the action of a potential flow [34,19,33,14,13,3,5]. However in these works the growth has been taking place in external flows (with the fluid outside of the growing object), in contrast to the present case where the medium is porous, and the driving flow goes through the growing finger and then into the undissolved medium.…”