Time-dependent pattern formation for convection in two layers of immiscible liquids

A linear and weakly nonlinear stability analysis is performed for speci c two-layer systems which have been examined experimentally in Degen et al (Phys Rev E 57, 6647 (1998)). For their water/silicone oil system, at the depth fraction where oscillations are predicted, many wavelengths are unstable. The neutral stability curve is rather at in this region, and oscillations begin slightly above the critical Rayleigh number. The neutral stability curves for their Fluorinert/silicone oil system show that oscillations are theoretically predicted in a very narrow parameter range. A 3D Hopf bifurcation on a hexagonal lattice is investigated for patterns that arise in extended domains. In both uid systems, travelling rolls, wavy rolls of type 1, and oscillating triangles are stable for most regimes. 47.20.Bp, 40.20.Ky, 47.54.+r. Typeset using REVTEX 1