Moving boundary problems are found in many contexts including coastlines and the seabed on the large scale to atherosclerosis and blood clotting on the micro scale. The coupling between the fluid flow and evolving solid bodies is an important consideration when modelling these problems. We have run numerical simulations for the melting front of ice for both interior and exterior flows; classical examples of the Stefan problem. We also modelled erosion and colloidal deposition problems which involved other constitutive equations. The terminal shape of an eroding cylinder in cross flow was a rounded triangular profile pointed upstream, which agreed well with experimental results. All of these problems require careful attention to deformation of the mesh at the interface, and we have developed a node shuffle algorithm for preserving mesh quality at these boundaries.