Shaping of dissolving and melting bodies under natural convection

The coupling between buoyancy-driven fluid flow and the evolution of a boundary resulting from dissolution or melting presents a rich variety of free-boundary problems, from the melting of an ice cube or iceberg to the sculping of rock spires. We address the problem of the shapes of solids left to melt or dissolve in an ambient fluid driven by stable natural convection along its surface. The theory forms a convective form of a Stefan problem in which the evolution is controlled by a two-way coupling between the shape of the body and stable convection along its surface. Beginning with the boundary-layer equations of free convection, we develop a new model describing the evolution of such bodies in two-dimensional or axisymmetric geometries and analyse it using numerical solution and analytical methods. We find that different initial conditions are found to lead to different fundamental shapes and descent rates. However, we prove that the evolving shape near the tip always forms a parabola. For steeply inclined bodies, there is a more intricate double-decked asymptotic structure to the tip shape comprising a broad 4/3-power intermediate near-tip region connected to a deeper parabola at the finest scale. While this forms a universal structure describing the shape of a sufficiently steep dissolving tip, the shape of the body as a whole (the "outer region") remains dependent on the initial condition for all times. The model results apply universally for any given relationship between density, viscosity, diffusivity and concentration, including two-component convection. A surprising result is that, depending on the initial shape, the tip can either sharpen or blunt with time - this is despite the surface smoothing at all other positions. A new series of laboratory experiments involving the dissolution of cones of sugar candy in water are found to collapse systematically onto our theoretically predicted shapes and descent rates with no adjustable parameters.