Weakly-nonlinear Analysis of Convection in a Mushy Layer during the Solidiication of Binary Alloys

We consider the solidi cation 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 fareld temperature. These asymptotic limits allow us to examine the rich dynamics of the mushy layer in the form of small deviations from the classical case of convection in a horizontal porous layer of uniform permeability. Of particular interest are the e ects of the asymmetries in the basic state and the nonuniform permeability in the mushy layer, which lead to transcritically bifurcating convection with hexagonal planform. We obtain a set of three coupled amplitude equations describing the evolution of small-amplitude convecting states in the mushy layer. These equations are analysed to determine the stability of and competition between two-dimensional roll and hexagonal convection patterns. We nd that either rolls or hexagons can be stable. Furthermore, hexagons with either up ow or down ow at the centres can be stable, depending on the relative strengths of di erent physical mechanisms. We determine how to adjust the control parameters to minimize the degree of subcriticality of the bifurcation and hence render the system globally more stable. Finally, the amplitude equations reveal the presence of a new oscillatory instability.