Curved Solidification Front Dynamics in Melts with Convection

Abstract—A convective boundary integral equation has been obtained to describe nonisothermal solidification from a binary melt/solution. The convective boundary integral is derived in three-dimensional and two-dimensional cases and is verified for fixed surface shapes, namely, a paraboloid of revolution, an elliptical paraboloid, and a parabolic cylinder. These surface shapes correspond to needle and lamellar dendrites growing in a liquid-phase flow uniformly flowing onto a crystal. The convective boundary integral equation for dendrites having the shape of a paraboloid of revolution and a parabolic cylinder is shown to give exactly the same dependence of supercooling on the Peclet, Reynolds, and Prandtl numbers as the direct solution to a boundary-value differential problem. A thermal-concentration boundary integral has been verified at various impurity concentrations in a liquid. The results are numerically compared, since the analytical forms of the solutions are different. Another approach to verification is to reduce the new integral equation to the well-known solutions by limiting transitions. The convective boundary thermal-concentration equation is shown to transform into a convection-free equation at the liquid flow velocity tending to zero. The convective boundary integral is calculated for the growth of a parabolic dendrite in an incident ideal liquid flow. The dependences of the supercooling at the dendritic surface on the Peclet number, which were constructed for convection in an ideal liquid and a viscous liquid in the Oseen approximation, nearly coincide for metals and metal alloys but differ sharply for organic materials and aqueous solutions. A parameter that determines the need to take viscosity into account is found. This parameter is the Prandtl number, which has an order of 10^–2 for metals and 10¹ for aqueous solutions. The Prandtl number allows us to compare the following two different heat transfer mechanisms: a diffusion mechanism and energy transfer via viscous friction. In metals, the Prandtl number is small due to a low viscosity and a high thermal conductivity, and the diffusion mechanism of heat transfer prevails. Therefore, a much simpler ideal liquid model can be used instead of a viscous liquid model can be sued for metallic alloys.