Order of magnitude analysis of convective effects in dendritic solidification

The different types of convective phenomena which may occur during the dendritic solidification of metallic alloys are discussed from an order of magnitude analysis. Bulk thermal convection and/or interdendritic solutal convection have to be considered according to the values of the experimental data. Scaling laws for the solute boundary layer resulting from bulk thermal convection have already been derived. It is shown here that the interdendritic flow depends on a solutal Grashof number Gr based on the horizontal density gradient and a characteristic length L_s which is of the order of the liquid channels width. For Gr < 1, which is generally verified in practical cases, the interdendritic flow velocity U_r is proportional to the Grashof number. This a priori law compares favorably with the results of horizontal solidification experiments where the mean interdendritic flow velocity has been estimated from the resulting measured macrosegregation. In these experiments, as well as for most horizontal dendritic solidifications of metallic alloys at 1 g, the ratio $\text{U}{}_{\mathrm{r}}\text{R}$ (R is the growth rate) is of order one. In order to cancel the interdendritic flow effects, this ratio has to be lowered by one order of magnitude. According to our analysis, this can be obtained by performing the experiments either at a slightly reduced g level (～10^?1 g), or at 1 g in a vertical stable configuration with a sufficiently low residual horizontal thermal gradient.