Solving coupled Stefan-flow problems using Immersed Boundary Smooth Extension

Natural convection accompanies many Stefan problems such as the dissolution and melting of solid objects in fluid. Gravity driven flows are responsible for many land-forming processes, for example the formation of Karst landscapes and the "stone forests" of China and Madagascar. Smaller structures, like iceberg "scallops", are also a consequence of natural convection. Recent experimental results show that even much simpler cases of dissolution or melting of initially spherical solids can lead to very nontrivial pattern formation. In this talk, we will present a numerical study of fluid-coupled Stefan problems, based on a high-order Immersed Boundary Smooth Extension method for evolving the boundaries of soluble solids immersed in a fluid. The method yields solutions with high regularity across boundaries, which allows us to evolve the geometry with high order of accuracy. An efficient spectral method reduces the computational cost and allows for high resolution. We demonstrate the efficacy of our approach with examples of melting and dissolution that produce high Grashof number flows.