Thermosolutal convection during directional solidification
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INTRODUCTION
THE effect of fluid flow on solidification and on the properties of the resulting solid has been an area of very active research in recent years. 1,2,3It is well known that horizontal temperature and concentration gradients give rise to fluid flow. In the absence of horizontal gradients, fluid flow also may occur due to thermosolutal instabilities, even if the fluid density decreases with height. The onset of convective and morphological instabilities during vertical directional solidification of a binary alloy has been determined by linear stability analysis. 4-9 For example, for the growth vertically upward of lead containing tin with a liquid temperature gradient of 200 K/cm, convective instability can occur for tin concentrations above 0.00032 wt pct. In this article we carry out a nonlinear numerical computation in order to determine the extent of solute segregation and the nature of the flow field caused by thermosolutal convection. Numerical solutions of the time-dependent differential equations for fluid flow, concentration, and temperature are obtained in two spatial dimensions for small Prandtl numbers and moderately large Schmidt numbers. We consider natural convection induced during unidirectional solidification of a binary alloy in the absence of imposed horizontal density gradients. The crystal is grown vertically upward at constant velocity V from a binary alloy. Figure 1(a) shows a schematic diagram of the configuration. The initial state is one in which the solid rejects or preferentially incorporates solute at the interface, giving rise to an exponential distribution of solute concentration ahead of the interface with a decay distance D/V as shown in Figure l(b). Also, an exponential temperature gradient exists in the liquid with a decay distance K/V and in the solid with characteristic distance Ks/V. Here, K and Ks
are the thermal diffusivities in the liquid and solid, respectively, and D is the diffusion coefficient for solute. A large solute concentration gradient will induce natural, or buoyant convection, often called double diffusive or thermosolutal convection,I~ if the solute rejected is lighter than the solvent or if a preferentially incorporated solute is heavier than the solvent. The temperature gradient, on the other hand, has a stabilizing influence since in the absence of solute the colder fluid will stably reside near the interface. It is worth emphasizing that thermosolutal convection may occur even though the net density decreases in the vertical direction. For a single component system, such a density distribution would indeed be stable, whereas for a doubly diffusive system the difference in diffusivities of the two components may allow small disturbances to amplify in time. To examine the natural convection during crystal growth, many simplifications are necessary. The simplifications will be enumerated and discussed briefly. During crystal growth morphological instability and convective instability will be coupled. Linear stability analysis 4-9 shows that th
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