The convective Stefan problem: shaping under natural convection

Abstract

Abstract What is the shape formed by a body that is melting or dissolving into an ambient fluid? We present a theoretical analysis of the dynamics of melting or dissolving bodies in the common situation where the transfer of heat or solute at the surface creates a thin thermal or solutal convective boundary layer along its surface. By conducting a general analysis of a mathematical model describing the shape evolution of such bodies (Pegler & Davies Wykes, J. Fluid Mech. , vol. 900, 2020, A35), we reveal new phenomena relating to the emergence of fundamental similarity solutions, asymptotic transitions, tip structure and the conditions for the development of sharp versus blunted tips. A universal regime diagram is developed showing asymptotic transitions between two different classes of similarity solutions. With

$t$

time, the tip of initially rectangular bodies is found to descend as

$t^{4/3}$

at early times, but transitions to the considerable faster power of

$t^4$

at long times, for example. Surprisingly, the tips of certain shapes, including initially rectangular bodies, sharpen continuously, whilst those of others, including initially conic bodies, blunt for all times. For the former case, the tip curvature grows rapidly as

$t^{12}$

, forming a needle-like shape. More general initial shapes can produce multiple transitions between sharpening and blunting. These results provide foundational understanding of buoyancy-driven fluid sculpting that underlies numerous natural and industrial applications.