This work presents an analysis of the melting process of a pure substance enclosed in a rectangular cavity heated from one of the verticals walls. The solid phase of the phase-change material is maintained isothermal at the fusion temperature. Consequently, the process is controlled by thermal natural convection in the liquid phase. The density gradients result from the temperature field only. The flow is laminar and the analysis is two-dimensional. The fluid is Newtonian. The Boussinesq hypothesis is assumed, that is the fluid density is considered to be constant, except in the driving term of the momentum equation. Various approximations are proposed for this term to describe the density behavior with the temperature. The proposed approximations depend on the substance and on the temperature range. Materials submitted to the melting phenomena with different Prandtl numbers are numerically investigated using a Finite Volume Method. The numerical simulation is based on the hypothesis of the quasi-stationary processes. So, from a certain position of the melting front, the natural convection equations in a fixed cavity are solved. The shape of the solid-liquid interface creates some difficulties. An algebraic method of transformation of non-orthogonal coordinates is used to map the irregular physical space occupied by the liquid into a rectangular computational space. The resulting local heat transfer at the interface is then used to determine the position of the melting front at the end of the time step. Next, the convective flow at the new liquid cavity is calculated. The procedure is repeated until the substance is almost completely melted. Special attention is given to the case of water, where a maximum density is attained at a temperature close to 4 degrees Celsius. The anomalous behavior of the water density can strongly influence the heat transfer during the melting process of ice. The lack of works in this field have motivated an experimental investigation. The particular interest of ice melting is based on the fact that the maximum density occurs over a wide range of temperature frequently found in nature and many technological applications. The experiments performed allow temperature measurements and photographic observations of the flow and the melting front motion. The results show that the non-linear density variation in the buoyancy term affects the heat transfer by natural convection, the interface motion and the flow structure.