Morphological stability in the presence of fluid flow in the melt
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INTRODUCTION
THE occurrence of
fluid flow in the melt during solidification has received considerable attention; see, e.g., the reviews by Hurle, ~ Carruthers, 2 and Pimputkar and Ostrach 3 It is well known that convection can have a significant effect on the structure and properties of the resulting solid. In particular, the morphological stability 4'5'6 of the crystal-melt interface may be altered considerably by the presence of fluid flow. For example, Delves has shown 5 that a forced flow parallel to the interface helps stabilize the interface against a Mullins-Sekerka instability, 6 while Coriell, et al. 7 show that thermosolutal instabilities in the melt may interact with the morphological instability in a complicated fashion and may, in fact, result in segregation at solute levels much lower than those predicted on the basis of morphological stability theory alone. A surprising effect of convection on an otherwise stable solid-liquid interface has recently been investigated experimentally by Fang, Glicksman, and Mickalonis) '9'~~ They studied a pure sample of succinonitrile (SCN) with a vertical, stationary cylindrical crystal-melt interface in a radial symmetric temperature field that increases into the liquid. Although according to morphological stability theory the cylindrical interface should be stable in the absence of convection, they found that the buoyancy-driven flow induced by the temperature gradient in the melt will under certain conditions produce a slowly rotating helical interface, accompanied by a more complex time-dependent flow. The critical Grashof number for the onset of this instability 1~ is an order of magnitude lower than the critical Grashof numbers associated with hydrodynamic instabilities in rigidwalled containers. ~2,~3 G.B. McFADDEN, Mathematical Analysis Division, Center for Applied Mathematics; S. R. CORIELL, Metallurgy Division, Center for Materials Science; and R. E BOISVERT, Scientific Computing Division, Center for Applied Mathematics, are with the National Bureau of Standards, Gaithersburg, MD 20899. M.E. GLICKSMAN and Q.T. FANG are with the Materials Engineering Department, Rensselaer Polytechnic Institute, Troy, NY 12181. This paper is based on a presentation made at the symposium "Fluid Flow at Solid-Liquid Interfaces" held at the fall meeting of the TMS-AIME in Philadelphia, PA on October 5, 1983 under the TMS-AIME Solidification Committee. METALLURGICALTRANSACTIONS A
Reference 14 contains a detailed description of the experiment and a comparison of experimental results with theoretical predictions of a linear stability analysis. A more detailed theoretical study for the related case of a planar geometry has also been performed, ~5 showing the dependence of the critical Grashof number on the Prandtl number of the melt. In this paper the experimental results are reviewed briefly, and some new theoretical results concerning the nature of the crystal-melt instability are presented. It is shown that the instability may be viewed either as a large alteration of a hy
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