The limiting behavior of the gradient function for an isolated paraboloid growing in a positive temperature gradient

Abstract

The limiting behavior of the gradient function, N g( p), for an isolated paraboloid growing in a positive temperature gradient, is re-examined here, using an asymptotic expansion for large p. It is shown that N g( p)<0 for large values of the chemical Peclet number, p. Also, as shown by Laxmanan, it appears that N g( p)→−1 as p→∞, instead of zero, as suggested by Trivedi. With N g( p)→−1, instead of zero, the model for directional solidification will lead to a number of unacceptable conclusions near the plane front limit, i.e. as the tip radius r t→∞ at any finite growth rate, R, close to the constitutional or the absolute stability limits. For example, if N g( p)→−1, the solute buildup at the tip of the paraboloid becomes negative, as p→∞, even for the case of when the partition ratio k<1.