Thermosolutal convection during dendritic solidification of alloys: Part II. Nonlinear convection
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INTRODUCTION
D U R I N G dendritic solidification of alloys, liquid flow is induced both by buoyancy forces and solidification shrinkage. Based on the experimental base of several investigators ( e . g . , Laxmanan et al., tq Sarazin and Hellawell, t2~ and Streat and Weinberg,[3]), there is strong evidence that the major reason for liquid flow is often the former, i.e., thermosolutal convection. This can be seen in Figure 1, which shows schematically the variation of the concentration of solute vs distance from the base of a directionally solidified ingot. Two curves are shown in Figure 1. One shows macrosegregation when thermosolutal convection is absent and flow of the interdendritic liquid is induced primarily by solidification shrinkage. This results in a positive segregation at the surface of the solidified ingot, which is often deemed "inverse segregation. "t4J On the other hand, when thermosolutal convection occurs, there is apparently an advection of solute from the enriched mushy zone to the all-liquid zone that results in negative segregation at the surface and a gradual increase of the solutal concentration in the completely solidified ingot. The major emphasis of this paper is to model the thermosolutal convection responsible for the latter type of macrosegregation described above. In an accompanying paper, tSl the thermosolutal convection was analyzed in terms of its linear stability. As a model system, Pb-20 wt pct Sn alloy solidifying at 0.002 c m - s -~ was selected. Marginal stability curves, in terms of the thermal gradient at the dendrite tips vs the horizontal wave number of the perturbed variables, were calculated for
gravitational constants of go, 0.5 go, 0.1 go, and 0.01 go. For 0.0001 go, the system was found by calculation to be stable for all thermal gradients (2.5 --< GL ---< 1 0 0 K . c m -~) and for all wave numbers (0-< to_< 130 cm-~). For the greater fractions of go, however, there were no minima in the marginal stability curves, so that the system was not found to be unconditionally stable for all wave numbers. The analyses did reveal that there are probable container widths, below which convection can be suppressed. In this paper, the same type of a system is formulated using dimensionless variables. Calculations are presented for various situations in which the linear stability analyses have predicted convection and no convection, t~] respectively. The nonlinear calculations presented herein agree with the predictions of the linear stability analyses. It is important to note, however, that the present nonlinear analysis does not permit the volume fraction of liquid in the mushy zone to deviate from its distribution determined for the nonconvecting and steady state. While this is a reasonable assumption to make for convection near the critical state, and it permits a prediction of the conditions in which thermosolutal convection is expected, such an assumption does not allow an investigation of the supercritical stage of convection that leads to the formation of localized segregat
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