Morphological Attractors in Natural Convective Dissolution

Recent experiments demonstrate how a soluble body placed in a fluid spontaneously forms a dissolution pinnacle—a slender, upward pointing shape that resembles naturally occurring karst pinnacles found in stone forests. This unique shape results from the interplay between interface motion and the natural convective flows driven by the descent of relatively heavy solute. Previous investigations suggest these structures to be associated with shock formation in the underlying evolution equations, with the regularizing Gibbs-Thomson effect required for finite tip curvature. Here, we find a class of exact solutions that act as attractors for the shape dynamics in two and three dimensions. Intriguingly, the solutions exhibit large but finite tip curvature without any regularization, and they agree remarkably well with experimental measurements. The relationship between the dimensions of the initial shape and the final state of dissolution may offer a principle for estimating the age and environmental conditions of geological structures.

Figure 1

Dissolution-induced sharpening. (a) Limestone structures form the stone forests of Borneo (Grant Dixon). (b)–(c) Dissolution of lab-scale planar and axisymmetric objects unveils the sharpening process; images from the same set of experiments reported in Ref. [23]. The observed noise in the curvature measurements results from surface impurities, like bubbles, affecting image tracking. The final radius of curvature at the tip was measured to be $60\mu \mathrm{m}$.

Figure 2

Simulating dissolution-induced sharpening. (a) Evolution of the initial shape $\theta =\mathrm{arccot}s/\ell$ in two dimensions. (b) Zooming in near the apex illustrates the strong sharpening effect. (c) Model schematic. (d) Profiles of the tangent-angle $\theta (s,t)$ show a steep gradient develop near the apex, $s=0$, consistent with (e) a tip curvature that increases by 5 orders of magnitude. (f) Characteristic curves show contours of constant $\theta$, with the physical trajectories shown in (a) with red.

Figure 3

Convergence towards the equilibrium morphology. (a) Left: Overlaying interfaces in Fig. 2 shows a common shape to emerge at late times. Right: Zooming-in near the apex further reveals the convergence towards an equilibrium. (b) The rescaled radius of curvature $R/R_0$ tends to the exact distribution predicted by Eq. (8). (c) Choosing initial shapes near the equilibrium can lead to sharpening or blunting, as predicted by Eq. (10). Inset: physical shape evolution of the case $\gamma =5/3$ reveals straight characteristic paths.

Figure 4

Comparison with laboratory experiments shown in Fig. 1. (a) When overlaid, the profiles measured from the planar (2D) and axisymmetric (3D) experiments are seen to converge to the shape predicted by Eq. (8). Plotting the experimentally measured surface coordinates on a log-scale confirms the far-field prediction Eq. (11).