Transient heat transfer, fluid flow and phase change in a weldpool with a deformed free surface are modelled using a two-dimensional coordinate transformation. The theoretical approach makes it possible to accurately model high Marangoni number flows even when the free surface is not horizontal. The method is used to examine the role of surface deformation and surface tension forces in controlling weld penetration during cathode spot welding of steel and aluminum. The calculated flow fields, temperature distributions and weldpool shapes are consistent with experimental findings, and are in reasonable agreement with previously published modelling efforts.