A finite element model has been developed for the computation of melting/solidifying process under the combined action of buoyancy and surface tension forces. Validated on the square cavity benchmark of Gobin and Le Quéré (Bertrand et al. [1], Gobin and Le Quéré [2]), the numerical model is used to extend this previous analysis to the free surface case where surface tension can drive the flow (capillary flow). A comparison of the results obtained for three types of boundary conditions applied at the top of the melting pool is performed. It shows that in the studied case of tin where the thermal Bond number is moderated (Bo=200), the flow is still mainly dominated by buoyancy effect as long as the melted pool is deep enough like in the square cavity case of the above mentioned benchmark.