NONLINEAR-ANALYSIS OF BUOYANT CONVECTION IN BINARY SOLIDIFICATION WITH APPLICATION TO CHANNEL FORMATION

We consider the problem of nonlinear thermal-solutal convection in the mushy zone accompanying unstable directional solidification of binary systems. Attention is focused on possible nonlinear mechanisms of chi mney formation leading to the occurrence of freckles in solid castings , and in particular the coupling between the convection and the result ing porosity of the mush. We make analytical progress by considering t he case of small growth Peclet number, delta, small departures from th e eutectic point, and infinite Lewis number. Our linear stability resu lts indicate a small O(delta) shift in the critical Darcy-Rayleigh num ber, in accord with previous analyses. We find that nonlinear two-dime nsional rolls may be either sub- or supercritical, depending upon a si ngle parameter combining the magnitude of the dependence of mush perme ability on solids fraction and the variations in solids fraction owing to melting or freezing. A critical value of this combined parameter i s given for the transition from supercritical to subcritical rolls. Th ree-dimensional hexagons are found to be transcritical, with branches corresponding to upflow and lower porosity in either the centres or bo undaries of the cells. These general results are discussed in relation to experimental observations and are found to be in general qualitati ve agreement with them.