The problem considered is that of the melting of a phase change material, initially at its saturation temperature, which is enclosed within a spherical shell whose surface temperature is suddenly raised to some fixed value. The density of the solid is assumed to exceed the density of the liquid, the implication being that the solid will continually drop toward the bottom of the shell as melting progresses. This bulk motion of the solid generates a flow field within the liquid, which gives rise to shear and pressure forces that must balance the weight of the solid. The energy equation written for the liquid region is solved in conjunction with an interface heat balance equation, while the velocity field is calculated approximately by assuming a parabolic profile for the polar component of velocity and neglecting the radial component. After making a suitable variable' transformation, the energy equation and the interface equation are expressed in finite difference form. The energy equation is solved by the alternating direction method, while the interface equation is solved using a modified Newton-Raphson procedure. Experimental evidence confirms the dropping solid hypothesis, and has permitted a limited quantitative verification of the mathematical model.