Interfacial dynamics in transport-limited dissolution

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 is also considered, which raises the general question of when and where a finite solid will disappear. In a case of dissolution by advection-diffusion, 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.

Figure 1

The diffusion-limited dissolution of a corrugated surface from left to right, for $h\left(0\right)=0$, $a\left(0\right)=0.3$, $t=0,0.25,0.50,\dots ,2.50$ in Eq. (2).

Figure 2

Outward radial diffusion-limited dissolution driven by a point sink at the origin for an initial four-lobed perturbation of a circle. This exact solution is given by Eqs. (7, 8, 9) with $N=5$, $c=0.15$, and $t=0,0.2,0.4,\dots ,1.0$.

Figure 3

Inward radial diffusion-limited dissolution driven by a sink at $\infty$ for a four-pointed shape. This exact solution is given by Eqs. (11, 12) with $N=3$, $c=0.3$, and $t=0,0.04,0.08,\dots ,0.36,0.3649$; the collapse occurs at $t_c=0.365$.

Figure 4

Inward advection-diffusion-limited dissolution of a tilted elliptical solid in a uniform background flow from left to right ($c=0.5i$, $b=0.8$, $t=0,0.04,0.08,\dots ,0.36,0.3749$). The major axis of the ellipse remains oriented at $\pi ∕4$, while its center of mass moves at an angle of $\pi ∕6$ relative to the background flow. At time $t_c=0.375$, the solid disappears at the point $a_0\left(t_c\right)=0.4+0.2i$.