Self-sculpting of a dissolvable body due to gravitational convection

Natural sculpting processes such as erosion or dissolution often yield universal shapes that bear no imprint or memory of the initial conditions. Here we conduct laboratory experiments aimed at assessing the shape dynamics and role of memory for the simple case of a dissolvable boundary immersed in a fluid. Though no external flow is imposed, dissolution and consequent density differences lead to gravitational convective flows that in turn strongly affect local dissolving rates and shape changes, and we identify two distinct behaviors. A flat boundary dissolving from its lower surface tends to retain its overall shape (an example of near perfect memory) while bearing small-scale pits that reflect complex near-body flows. A boundary dissolving from its upper surface tends to erase its initial shape and form an upward spike structure that sharpens indefinitely. We propose an explanation for these different outcomes based on observations of the coupled shape dynamics, concentration fields, and flows.

Figure 1

Shape evolution of a dissolving body. (a) Side-view photograph of candy body (initially a sphere) after dissolution in water for 70 min. (b) Measured profile shape displayed every 10 min. The upper surface remains smooth while the undersurface becomes pitted and dissolves several times faster.

Figure 2

Shadowgraph images of a dissolution within water at $t=0,30,60$, 90, and 120 min. The intensity variations beneath the body correspond to a turbulent descending jet of solute-laden fluid. The dark horizontal stripe near the midheight of the body corresponds to the interface of a sharp density stratification. The scale bar is 1 cm.

Figure 3

Tank size has little affect on shape dynamics. Images of dissolving body for (a) tank volume 180 ${\mathrm{cm}}^3$ and (b) tank volume $150\times {10}^3{\mathrm{cm}}^3$. The stratification interface formed in the smaller tank of (a) causes an optical distortion (dashed lines and arrow) and in reality the bodies are not noticeably different.

Figure 4

Interpretation of the buoyancy-driven flow around a dissolving body. Dense solute-laden fluid descends as an attached boundary-layer flow on the upper surface and a highly separated flow over the lower surface. Accumulation of dense mixed fluid in the lower portion yields a sharp stratification. The downward flows induced near the body are matched by broader, slower return flows.

Figure 5

Shape preservation of dissolving wedges. (a) Sketch of initial configuration: cross-sectional shape evolution at 10-min intervals for bodies with initial base angle (b) ${\theta }_0={60}^{\circ }$ and (c) ${\theta }_0={100}^{\circ }$.

Figure 6

Small-scale roughness on the underside of a partially dissolved wedge. The wedge has dissolved for 10 min and the scale bar is 1 cm.

Figure 7

Shape memory in dissolving wedges. (a) Evolution of the base angle for wedges of differing initial angles. (b) Angle of the wedge normalized by the initial angle as a function of the change in area, normalized by the initial area.

Figure 8

Dissolution velocity on the lower side is constant with space and time. (a) Dissolution velocity as a function of arc length around a triangular wedge, averaged over time. (b) Average dissolution velocity on the lower side of a triangular wedge as a function of the change in area, normalized by the initial area.

Figure 9

Mean dissolution rate on the lower side varies little with base angle $\theta$.

Figure 10

Quasi-2D and 3D axisymmetric geometries. Horizontal cylinder of (a) circular, (b) triangular, and (c) rectangular cross sections. Analogous 3D solids of revolution include (d) a sphere, (e) a cone, and (f) an upright cylinder.

Figure 11

Comparing shape evolution of quasi-2D and 3D bodies. The initial shapes correspond to those of Fig. 10, with (a)–(c) 2D cylindrical geometries and (d)–(f) 3D solids of revolution. The wedge and cone have ${60}^{\circ }$ base angle. In all cases, cross-sectional profiles are shown every 10 min and the scale is 1 cm.

Figure 12

Millimeter-scale plumes descend from a quasi-1D geometry. Shadowgraph images of wire coated in candy, whose dissolution in water results in fine-scale plumes descending under gravity: (a) startup of flow and (b) long-term behavior. The scale bar is 1 cm.

Figure 13

Shape evolution of a tall axisymmetric body dissolving within water. (a) Photographs shown at 60-min intervals. (b) Sketch of the initial geometry. (c) Extracted cross sections shown every 60 min and at the same scale as images in (a). (d) Zoomed-in and rescaled profiles of the tip region, with all contours shifted to place the tip at the origin.

Figure 14

Sharpened initial shape. (a) Shape profiles shown at 60-min intervals. (b) Zoomed-in, rescaled, and shifted profile shapes to focus on the evolution of the tip. The inset shows a sketch of the initial geometry.

Figure 15

Tip curvature continuously increases at later times. The angle $\theta$ of the surface with respect to gravity is plotted vs arc length $s$ near the tip for (a) an initially blunt body and (b) a sharpened body. (c) Time evolution of the principal curvature $\kappa (s=0)$ at the tip for initially blunt (blue) and sharpened (orange) bodies.