Linear stability analysis of a solidification process with convection in a bounded region of space

Abstract

The morphological/dynamic instability of crystallization process in a bounded region in the presence of intense convection in liquid is studied. The paper considers a linear theory of morphological instability with a flat solid-liquid interface on the example of molten metal and magma. The mathematical model includes heat transfer equations and convective type boundary conditions at the interface. The equations for perturbations of the temperature field and interfacial boundary are found, allowing to obtain the dispersion relation. Its analysis has shown the existence of morphological instability of the flat interfacial boundary for a wide range of wavenumbers. Dynamic perturbations (perturbations of the quasi-stationary crystallization velocity) were also analyzed and two solutions for the perturbation frequency were obtained. One of them is stable and the other one is unstable. The system selects one of them depending on the action of convection. The result of morphological and dynamic instability is the appearance of a two-phase region in front of a flat solid-liquid interface. Therefore, the paper also considers the dynamic instability of stationary crystallization with a two-phase region replaced by a discontinuity surface. In this case, the dynamic instability was also found for a wide range of crystallization velocities.