Non-linear Interaction of Morphological and Convective Instabilities During Solidification of a Dilute Binary Alloy

We consider the stability of a horizontal, planar solid-melt interface during the solidification of a dilute binary alloy. Linear theory has shown that instability may occur due to either solutal convection in the melt or due to morphological effects. We investigate a system in which two modes of different horizontal wavenumber become unstable simultaneously. One mode represents the morphological instability and the other represents the convective instability. We carry out a weakly non-linear analysis, in order to determine the non-linear evolution equations for the amplitudes of these two modes. The coefficients of these equations determine the non-linear interaction which occurs near the onset of instability. We present an example of the application of the theory to the lead-tin alloy system, and show that the interaction is weak. 1