The problem of the thermofluid mechanics of the steady squeezing of a viscous fluid between a hot wall and a surface of the phase-change material is studied theoretically. The partial differential system consisting of the Navier-Stokes equation, the energy equation and the boundary conditions is found to be reduced to an ordinary differential system depending on only one coordinate in the direction normal to the surface of the hot wall. This ordinary differential system is solved by using the approximate and numerical methods. The problem is governed by two nondimensional parameters, that is, the Stefan number and the Prandtl number. It is found that the surface of the phase-change material melted by a flat wall with a higher temperature than the melting temperature is always left flat. The relationship between the thickness of the liquid layer, the melting rate and the force acting on the interface is obtained, as well as the distributions of the velocity, the pressure and the temperature in the liquid layer. If the inertia term in the Navier-Stokes equation is disregarded, the ordinary differential system is solved analytically. Its solution is obtained in the closed form. It is found to be very useful in many practical cases.