We study a solid and a liquid coexisting at a plane interface in the nonequilibrium steady-state (NESS) condition on a hydrothermodynamic level. Both solid and liquid are treated as continuums. From the equations of mass-, momentum-, and energy-conservation we obtain relations for the velocity of the interface, the net mass- flow velocity, and thermodynamic variables such as temperature, pressure, and density. The second law of thermodynamics gives a criterion for the possibility of the NESS processes. It is shown that the NESS interface motion without extraction or addition of latent heat of fusion is possible.