Effects of flow on morphological stability during directional solidification
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INTRODUCTION
SOLID to liquid phase transformation involves a complex interplay of many physical effects. The solid/liquid interface is an active boundary from which latent heat is liberated during phase transformation. This heat is conducted away from the interface through the solid and liquid, establishing thermal boundary layers near the interface. If the liquid is not pure but contains solute, then preferential rejection or incorporation of solute occurs at the interface. For example, if there is a single solute present and its solubility is smaller in the (crystalline) solid than it is in the liquid, the solute will be rejected at the interface. This rejected material is diffused away from the interface through the solid and liquid setting up concentration boundary layers near the interface. The thermal and concentration boundary layer distributions determine, in part, whether there exist morphological instabilities of the interface. Fluid flow may also have a strong influence on interface morphology. In general, there are four different sources of flow within the melt. If the solidification process is occurring in a gravitational field, the thermal and solutal gradients induce buoyancy-driven convection that is known to greatly affect the interfacial patterns and hence the solidification microstructures that are present in the solidified material.m If the solidification process involves fluid-fluid interfaces, such as those found in containerless processing in a microgravity environment, then variations in surface tension along these interfaces may drive convection in the melt. Flow normal to the solidifying interface will be created by the expansion or contraction of material upon solidification. Finally, the presence of external forces may stir the melt. Brown[21 gives a broad survey of the processing configurations and the types of flows that occur.
S.H. DAVIS, Walter P. Murphy Professor of Applied Mathematics, is with the Department of Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, IL 60208. T.P. SCHULZE, Postdoctoral Fellow, is with the Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, England CB3 9EW. This article is based on a presentation made at the "Analysis and Modeling of Solidification" symposium as part of the 1994 Fall meeting of TMS in Rosemont, Illinois, October 2-6, 1994, under the auspices of the TMS Solidification Committee. METALLURGICALAND MATERIALSTRANSACTIONS A
There are several categories of problems that have received quantitative study. The first area, which we call prototype flows, involves simple flow geometries that correspond to well-studied hydrodynamic instabilities. If one (or more) of the rigid boundaries of such a geometry is replaced by a crystal interface, and if temperature and / or concentration gradients are posed to support this, then one can investigate the influence of heat and mass transfer on the morphology of the interface(s). Here, it is usually supposed that the liquids are "pu
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