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Volume 1,     Number 2,     Spring 1993

 

GLOBAL INSTABILITY AND PATTERN FORMATION
IN DENDRITIC SOLIDIFICATION OF
A DILUTE BINARY MIXTURE
JIAN-JUN XU

Abstract. This work deals with the global instability mechanisms of solidification from a dilute binary alloy. The results obtained in the present paper show that the interfacial wave theory previously developed for dendritic solidification from a pure substance can be extended in a very similar fashion to binary alloy systems. In a binary alloy system, there exists also a special simple turning point. Due to the presence of this turning point, a binary alloy system also permits a discrete set of unstable Global Trapped Wave (GTW) modes, which describe the characteristics of waves trapped in a region near the tip of the dendrite. We obtained uniformly valid asymptotic expansion solutions and quantum conditions for the corresponding eigenvalues of these global modes. The self-sustaining GTW-mode in a binary alloy system is calculated, and this explains the origin and persistence of a dendritic micro-structure in solidification expressed in terms of the Global Neutral Stability (GNS)-condition. We also calculated the critical values of the stability parameter ε* versus various physical parameters. This critical value ε* selects the tip-velocity of the dendrite.

 

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