WEAKLY NONLINEAR-ANALYSIS OF CONVECTION IN MUSHY LAYERS DURING THE SOLIDIFICATION OF BINARY-ALLOYS

We consider the solidification of a binary alloy in a mushy layer and analyse the system near the onset of buoyancy-driven convection in the layer. We employ a near-eutectic approximation and consider the limit of large far-field temperature. These asymptotic limits allow us to e xamine the rich dynamics of the mushy layer in the form of small devia tions from the classical case of convection in a horizontal porous lay er of uniform permeability. Of particular interest are the effects of the asymmetries in the basic state and the non-uniform permeability in the mushy layer, which lead to transcritically bifurcating convection with hexagonal planform. We obtain a set of three coupled amplitude e quations describing the evolution of small-amplitude convecting states in the mushy layer. These equations are analysed to determine the sta bility of and competition between tyro-dimensional roll and hexagonal convection patterns. We find that either rolls or hexagons can be stab le. Furthermore, hexagons with either upflow or downflow at the centre s can be stable, depending on the relative strengths of different phys ical mechanisms. We determine how to adjust the control parameters to minimize the degree of subcriticality of the bifurcation and hence ren der the system globally more stable. Finally, the amplitude equations reveal the presence of a new oscillatory instability.