Linear Stability Analysis of Solidification in a Bounded Region in the Presence of Convection in a Liquid

Abstract—The morphological/dynamic instability of directional solidification in a bounded region in the presence of convection in a melt is subjected to linear analysis. A linear theory, which takes into account convection in a liquid, has been developed for the morphological instability of stationary directional solidification with a flat solid–liquid interface. Equations for temperature-field and interface perturbations are written, the solution to these equations is constructed, and a dispersion relation is derived. Its analysis shows the existence of morphological instability in a wide wavenumber range and at various parameters of the solidifying system. This instability is caused by the perturbations that enter the solid–liquid interface from a cooled wall through the solid phase. In addition, the flat solid–liquid interface is shown to be unstable to dynamic perturbations, i.e., zero-wavenumber perturbations (steady-state solidification rate perturbations). There is also a branch of a stable solution for dynamic perturbations. The solidifying system can choose one of these branches (unstable or stable) depending on the action of convection. The result of morphological and dynamic instabilities is the appearance of a mushy zone in front of the flat solid–liquid interface. Therefore, the next step in the work was to analyze the dynamic instability of stationary solidification with the mushy zone. This zone is replaced by a discontinuity surface located between purely solid and liquid phases. An analysis of equations for the perturbation amplitudes revealed dynamic instability over a wide solidification rate range. This instability is caused by perturbations in the solid material near the cooled wall and extends up to the interface. When a certain solidification rate is reached, solutions bifurcate, which leads to the coexistence of both unstable and stable branches. As earlier, the system chooses one of them depending on the influence of convection. In general, the solidifying system can be morphologically/dynamically unstable to small perturbations, which arrive at the interface due to heat- and mass-exchange equipment fluctuations (e.g., temperature fluctuations at the cooled boundary).