In this thesis a variety of problems are considered, the first of which is associated with the welding process. Experiments indicate that the shape of a weld pool is influenced by convection in the liquid metal. In recent years it has been shown that this convection is crucially affected by the Marangoni (or thermocapillary) force. Recently Craine and Belgrove ([30], [7]) have developed a two-dimensional, axisymmetric model which includes the Marangoni force on the free surface of a semi-infinite region of liquid steel when a point source of current and heat is incident on the free surface. An asymptotic solution to this problem is obtained in this thesis, and the surface tension gradient with respect to temperature, dj/dT, a parameter which is crucial to the magnitude of the Marangoni force, is found to affect every coefficient in the leading and first order asymptotic expansions. In various theoretical and experimental models purely poloidal flow bifurcates to a rotating flow. To investigate this possibility for our flow a linear stability analysis is performed on a numerically obtained poloidal solution for the flow and temperature distribution in a hemisphere (a model first derived in [7]). For the azimuthal stability mode m = 0 the equation governing the linear stability of the rotating motion is found to decouple from the corresponding poloidal equations. The poloidal and azimuthal stability equations both become unstable at different critical currents dependent on the sign and magnitude of d^/dT. An investigation of the eigenvectors indicates the onset of instability near to the point source. For the upper modes instability occurs only when m = 1 and in a very small region of parameter space. In the second part of this thesis a freezing sphere problem with flow is used to compare a sharp interface Stefan model and a diffuse interface phase-field model. Firstly a Stefan model that includes a disparity between the density of the solid and liquid phases is derived and solved numerically. This model is compared with a recent phase-field model with flow, derived by Anderson et al. in [2]. In this thesis the one-dimensional isotropic version of Anderson's model is obtained in spherical polar coordinates and using certain simplifications when the dimensionless thickness of the interface £5 is vanishingly small a leading order asymptotic expression reproduces the Stefan model with flow. The phase-field model is subsequently modified and solved numerically, and the results are compared with the sharp interface model. Close agreement is observed between these models when es < 0.01.