In this paper we study a phase change problem for non-isothermal incompressible viscous flows. The underlying continuum is modelled as a viscous Newtonian fluid where the change of phase is either encoded in the viscosity itself, or in the Brinkman–Boussinesq approximation where the solidification process influences the drag directly. We address these and other modelling assumptions and their consequences in the simulation of differentially heated cavity flows of diverse type. A second order finite element method for the primal formulation of the problem in terms of velocity, temperature, and pressure is constructed, and we provide conditions for its stability. We finally present several numerical tests in 2D and 3D, corroborating the accuracy of the numerical scheme as well as illustrating key properties of the model.