On the instability of the melting film over a moving hot plate
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The flow of a viscous film emerging over a plate moving through a linear elastic solid in the case when the plate temperature exceeds the melting temperature of the solid is studied. An approximate model for the melting film flow and the solid temperature is stated in the case of known dependencies of the kinematic viscosity and thermal conductivity of the melt on temperature. In the case of constant kinematic viscosity and thermal conductivity, the model describing steady flows was formulated in the work of G. G. Chernyi, "The motion of a plate in a melting solid medium," J. Appl. Math. Mech. 55, 287 (1991), where a self-similar solution was analytically found for relatively thin films. In this paper, the self-similar solution is numerically calculated in a general case, and it is demonstrated that solutions of the initial value problem for steady flows tend to that solution but do not reach it except a specific case of the inlet flow rate. Linear stability analysis shows that steady flows are unstable in regard to perturbations whose velocities are close to the liquid velocity at the solid-liquid interface.