Asymmetric collapse by dissolution or melting in a uniform flow

Abstract

An advection-diffusion-limited dissolution model of an object being eroded by a two-dimensional potential flow is presented. By taking advantage of the conformal invariance of the model, a numerical method is introduced that tracks the evolution of the object boundary in terms of a time-dependent Laurent series. Simulations of a variety of dissolving objects are shown, which shrink and collapse to a single point in finite time. The simulations reveal a surprising exact relationship, whereby the collapse point is the root of a non-analytic function given in terms of the flow velocity and the Laurent series coefficients describing the initial shape. This result is subsequently derived using residue calculus. The structure of the non-analytic function is examined for three different test cases, and a practical approach to determine the collapse point using a generalized Newton-Raphson root-finding algorithm is outlined. These examples also illustrate the possibility that the model breaks down in finite time prior to complete collapse, due to a topological singularity, as the dissolving boundary overlaps itself rather than breaking up into multiple domains (analogous to droplet pinch-off in fluid mechanics). The model raises fundamental mathematical questions about broken symmetries in finite-time singularities of both continuous and stochastic dynamical systems.

Mineralogical and transport controls on the evolution of porous media texture using direct numerical simulation

“…Thus, as the grain dissolves, its center of mass of the grain moves downgradient and this tendency increases at faster flow rates. This is in agreement with the results of Rycroft and Bazant [], who used an analytical solution to solve for advection‐diffusion‐limited dissolution of a cylindrical object being eroded by two‐dimensional potential flow. However, the main difference is that in Rycroft and Bazant [] the cylindrical grain preserves its shape as it evolves.…”

“…This is in agreement with the results of Rycroft and Bazant [], who used an analytical solution to solve for advection‐diffusion‐limited dissolution of a cylindrical object being eroded by two‐dimensional potential flow. However, the main difference is that in Rycroft and Bazant [] the cylindrical grain preserves its shape as it evolves. At relatively fast flow rates (small numbers), the grain evolves to an oblong shape.…”

“…As a result, mineral grain surfaces in close proximity may be subject to different reactive regimes due to pore‐scale variations of the flow velocity; areas of the grain surface thus contribute little to the overall dissolution rate in the case where slow flow allows for closer‐to‐equilibrium conditions locally [ Molins et al ., ; De Baere et al ., ]. As a result, these nonuniform rates at mineral surfaces may result in nonuniform evolution of mineral grains [ Rycroft and Bazant , ], thus affecting the texture of porous media.…”

A boundary-integral framework to simulate viscous erosion of a porous medium

“…Recent work in the high-Reynolds-number regime has highlighted the importance of a shape-flow feedback that occurs during erosion [33,42,45,46,64] and the related processes of dissolution and melting [14,15,21,34,36,40,65]. In these cases, the fluid alters the morphology of immersed structures, which in turn modifies the surrounding flow field, and so on.…”

Moving Boundary Problems for Quasi-Steady Conduction Limited Melting

“…As mentioned in the Introduction, the governing equations (2.2e) with (2.2b)-(2.2d) are also relevant for the problem of a bubble that is forced to contract in a saturated medium, where the fluid flow is governed by Darcy's law [12,28,45], as well as the two-dimensional analogue for Hele-Shaw flow [15,14,42]. These equations also arise in other moving boundary problems, for example the small Péclet number limit of advection-diffusion-limited dissolution/melting models [6,27,32,53,57], for which it is also of interest to track the moving boundary and predict its shape and location (the collapse point [53]) close to the extinction time; other closely related advection-diffusion-like moving boundary problems in potential flow have similar governing equations in the small Péclet number limit [4,7].…”