Diapiric penetration with melting

We consider a series of problems in which a solid or an immiscible fluid body melts its way through a solid medium. Both internal heat generation and viscous heating are considered. We hypothesize that the melted medium is confined to a thin squeeze film on the upstream portion of the body and that the lubrication approximation is valid in this squeeze film. We find that thermal runaway, the unbounded conversion of potential into thermal energy, will occur for spherical bodies with a radius greater than a critical value. For immisscible fluid bodies we find an infinite family of allowed shapes for a fixed volume of fluid. Thin bodies are more efficient at melting their way through a medium than blunt bodies.