Theory of Wavelike Interface Formation during Liquid Solidification with Allowance for Convective–Conductive Heat Transfer

Abstract—The evolution of the morphological perturbations that form at the solid–liquid interface in the presence of a convective–conductive heat transfer mechanism is studied. A mathematical model is formulated for convective–conductive heat transfer; it takes into account small sinusoidal deviations of the interface shape from plane geometry. An analytical solution describing the undisturbed ground state of a solidifying material has been found. The evolution of small sinusoidal morphological interface perturbations can cause an unstable solidification mode, which was studied using a linear analysis of morphological stability. Equations describing the evolution of small morphological interface and temperature perturbations are derived by expanding boundary conditions into Taylor series on the surface between liquid and solid phases. An analysis of the equations derived for interface and temperature perturbations allowed us to determine a dispersion relation, which is the relation between the perturbation frequency and wavenumber. At a zero perturbation frequency, the dispersion relation determines a neutral stability curve. Model calculations are performed for two systems (metal, magma) in the presence of a liquid phase flow along a solidified material. The calculations performed show that the perturbation frequency changes its sign with a wavenumber. This confirms the existence of two different solidification modes with a stable flat interfacial boundary of molten and solidified materials and an unstable wavelike interface. The maximum morphological perturbation frequency, which determines the most dangerous rapidly growing perturbations, has been found. The developed theory has a limiting transition to the previously constructed theory of a purely convective heat transfer mechanism in the liquid phase of a system.