The convective Stefan problem: Transitional shapes under natural convection

Fluids sculpt many of the shapes we see in the world around us, from melting ice cubes to "stone forests" of limestone rock spires. We present a new mathematical model describing the shape evolution of a body that dissolves or melts under gravitationally stable buoyancy-driven convection, driven by thermal or solutal transfer at the solid-fluid interface. For high Schmidt number, the system is reduced to a single integro-differential equation for the shape evolution. Focusing on the case of an initially conic or wedge-shaped body, we derive complete predictions for the underlying self-similar shapes, intrinsic scales and descent rates that apply to bodies that melt or dissolve in a quiescent ambient fluid. The theoretical predictions show excellent agreement with the results of a new series of laboratory experiments.