Directional solidification with heat losses

We consider the upward directional solidification of a dilute binary alloy in a tall, thin, two-dimensional slot. Heat losses from the sidewalls produce curved isotherms, interface deformations, and melt convection. We use an asymptotic approach which relates the small heat loss to the thinness of the solidification cell and solve for the resulting melt convection, interface deflection, and the distribution of heat and solute. The model is compared to the numerical results of Chang and Brown which exclude the effects of solute buoyancy and thermodynamic equilibrium at the interface. We compute the radial segregation at the interface (consistent with both interface shapes and temperature fields) and find that the radial segregation is independent of interfacial shapes; one cannot calculate the radial segregation from only a knowledge of the interface shape.