Pulsatile- and cellular-mode interaction in rapid directional solidification

Merchant and Davis performed a linear stability analysis on a model for the directional solidification of a dilute binary alloy valid for all speeds; in their work, the temperature field is decoupled from the analysis. The analysis revealed that, in addition to the Mullins-Sekerka cellular mode, there is an oscillatory instability whose preferred wave number is zero. In this paper, the nonlinear interaction of these two modes in the vicinity of their simultaneous onset is analyzed. A pair of coupled Landau equations is obtained that determines the amplitudes of the modes. Any combination of primary bifurcations may occur. Secondary bifurcations also occur; there exist stable mixed modes and regions of bistability. The results are discussed in terms of experimentally relevant variables.