Free-convection-assisted shape-preserving dissolution or etching of a rotationally symmetric body

A rotationally symmetric body which is totally immersed in a liquid that contains reactants which dissolve the solid is considered. The body is in a vertical position, which means that the heavier liquid close to the body moves downwards under the action of the force of gravity. This free-convective motion creates a concentration field around the body, with the concentration gradient at the surface being proportional to the speed of dissolution or etch rate. A special body shape is sought for which the dissolution process results in a succession of body shapes which are all the same, that is, the etching process is shape-preserving. It turns out that a fully analytical solution describing the body shape can be obtained. The speed at which the body moves downwards as a result of etching is proportional to -1/4, where is the radius of curvature at the body apex. This analytical solution is ideally suited as a benchmark against which numerical codes can be tested.