Abstract
Pattern selection in binary-fluid mixtures heated from below is studied for separation ratios S in the regime 0<S-S≪1, where S is the critical value for which the wavelength 2π/kc of the instability first becomes infinite. The basic equations are reduced to a scalar equation for the horizontal planform whose coefficients can be determined analytically for boundary conditions of experimental interest. Equivariant bifurcation theory is used to study pattern selection on both square and hexagonal lattices. The results depend strongly on the parameter β describing asymmetry in the boundary conditions at top and bottom. When β=0, squares are stable on the square lattice but are replaced by rolls with increasing β. The transition between stable squares and stable rolls takes place via a stable branch of new solutions called crossrolls. On the hexagonal lattice, hexagons are stable if β=0, but rolls are stable when β≠0.