Pattern selection in steady binary-fluid convection

Three-dimensional convection in a binary-fluid mixture is studied near the onset of the steady-state instability using symmetric bifurcation theory. Idealized boundary conditions are assumed in which the temperature and solute concentration are fixed at top and bottom, with stress-free boundary conditions on the velocity field. The effects of sidewalls are neglected. The problem is formulated as a bifurcation problem on a doubly periodic lattice, with two cases considered in detail: the square lattice and the hexagonal lattice. Symmetry considerations determine the form of the ordinary differential equations governing the dynamics of the neutrally stable modes. The relevant coefficients of these equations are calculated from the governing binary-fluid equations. The bifurcation diagrams are given for all physical values of the separation ratio, the Lewis number, and the Prandtl number. It is found that supercritical rolls are stable to all perturbations lying on the square and hexagonal lattices. Squares, hexagons, and triangles are never stable for the physically accessible regions of parameter space.