Self-consistent theory of dendritic growth with convection

Abstract

A theory is described which treats dentritic growth with forced convection in the melt as a free boundary problem. This approach yields self-consistent solutions for the rate of propagation of an isothermal interface and the temperature and velocity fields surrounding it. The solutions to the free boundary problem reduce to Ivantsov's pure conduction solution when the Reynolds number is zero. It is shown that the Oseen, Stokes and potential flow approximation of the Navier-Stokes equations for convection in a subcooled melt yield exact solutions for the shape preserving growth of a parabolic dendrite. The results given here, and our previous results, yield an infinite family of solutions. Therefore microscopic solvability theory is used to separate the growth velocity, V G, and tip radius, R, for a fixed subcooling, Δ T. The theoretical results which include convection and the solvability relationship agree well with the experimental data of Huang and Glicksman and with the results of Ananth and Gill who used the experimental data directly with the convection theory to separate V G and R. In contrast, Ivantsov's theory, which neglects convection, yields large errors when compared with these experiments, which are correlated well by 2αd 0 V GR 2.1 = 0.03 . This implies that the solvability relationship, 2αd 0 V GR 2 = 60 ϵ 7 4 , in contrast to the heat transfer calculation, is not affected profoundly by convection in the melt in the range of the subcooling used in the experiments. However, the value of V G R 2 decreases slightly, for a fixed degree of anisotropy, ϵ, as the subcooling gets smaller and convection increases in intensity. Forced flow experiments in which flow velocities are orders of magnitude larger than those in natural convection, would help to clarify the effect of convection on interfacial stability.