Asymptotic Solution of Dendritic Growth in External Flow (I): The Case of Rapid Growth U ≫ U∞
Part of the Advances in Mechanics and Mathematics book series (AMMA, volume 7)
In the following two chapters, we are going to investigate the effect of enforced uniform flow on dendritic growth. During the past several years, this subject has been studied by a number of authors, such as Ananth and Gill; Ben Amar, Bouillou and Pelce; Saville and Beaghton, Xu and Yu, etc., numerically or analytically. The problem, however, was not well resolved. To describe the flow field induced by dendritic growth in the external flow, Ananth and Gill used an Oseen model solution of the uniform flow past a paraboloid (Ananth and Gill, 1989, 1991). Xu made the first attempt to derive an asymptotic expansion solution for the free boundary problem, in terms of the Navier-Stokes model of fluid dynamics for the case of the Prandtl number Pr → ∞ (Xu, 1994a). Neither Ananth and Gill’s solution, nor Xu’s solution yields the correct stream function approaching that of the given uniformly flow in the up-stream far-field. Therefore, the problem needs to be reconsidered with the Navier-Stokes model of fluid dynamics and the fully justified mathematical formulation. We shall separately discuss the two limiting cases:
The case of rapid growth or, weak external flow: U ∞/U ≪ 1 with Pr = O(1);
The case of large Prandtl number: Pr ≫ 1 with U ∞/U = O(1).
KeywordsPrandtl Number Stream Function Asymptotic Solution Interface Condition Dendritic Growth
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© Springer Science+Business Media Dordrecht 2003