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.

中文翻译：

---

关于移动热板上熔膜的不稳定性

研究了在板温度超过固体熔化温度的情况下，在穿过线弹性固体的板上出现的粘性膜的流动。在已知熔体运动粘度和热导率与温度相关的情况下，给出了熔膜流动和固体温度的近似模型。在运动粘度和热导率恒定的情况下，描述稳定流动的模型是在 GG Chernyi 的工作中制定的，“板在熔化固体介质中的运动”，J. Appl。数学。机械。55, 287 (1991)，其中分析发现了相对较薄的薄膜的自相似解。本文对一般情况下的自相似解进行数值计算，并且证明了稳态流的初值问题的解趋于该解，但除了入口流速的特定情况外，不会达到该解。线性稳定性分析表明，对于固液界面处速度接近液体速度的扰动，稳态流动是不稳定的。