Solid–liquid interface stability in solidification of a binary mixture under conductive transport and convective flow

A linear instability analysis of the planar solid–liquid interface propagating into a binary liquid is revisited for a steady-state mode of crystallization. The model statement includes convective and conductive transport of heat and mass in bulk phases together with these transfer contributions at the solid–liquid interface. Following our analysis [D. V. Alexandrov and P. K. Galenko, “The Mullins-Sekerka theory: 60 years of morphological stability,” J. Appl. Phys. 136 (2024) 055103], it is shown that the directional solidification with the convective and conductive transport also becomes possible only if the finite distance h of the solidification front from the cooling unit (cold boundary) exists and is taken into account in the formal analysis. If the cooling unit is removed from the interface to the spatially infinite distance (as accepted in many previous works), the directional solidification stops. The obtained dispersion relation for the system with the conductive and convective transport takes into account the existence of perturbations appearing from the cooling unit, solid–liquid interface, and bulk liquid. Therefore, the range of instability essentially depends on the distance h and friction velocity of the flow, which characterizes the liquid convection and convective contributions of heat and mass fluxes. Special cases of bounded and unbounded solidification domains that may affect the front instability are investigated. It is shown that within the bounded domain perturbations from convective flow or temperature fluctuations affect the solid–liquid interface more strongly than in the unbounded domain. This effect leads to a broader range of wavenumbers that provide the front instability.