Onset of oscillatory binary fluid convection in finite containers

The onset of oscillatory convection in binary fluid mixtures in a two-dimensional domain with realistic boundary conditions on all boundaries is determined as a function of the fluid parameters and the aspect ratio Gamma of the container. The first unstable mode has either odd or even parity under left-right reflection. Depending on Gamma and the separation ratio S, this mode has the form of a standing wave, or a "chevron,'' consisting of a Fair of waves propagating outwards from the cell center (or, in some cases, inwards towards it). Codimension-two points at which odd and even parity modes are simultaneously marginally stable are determined, as are various Takens-Bogdanov points. For fixed S<S-TB, all mode interactions among modes of like parity, arising as Gamma varies, are of the nonresonant Hopf-Hopf type; however, the details of the modal interchange are organized by resonant Hopf bifurcations with 1:1 resonance. Particular attention is paid to the asymptotic mode structure as Gamma-->x, and to the gap (in Rayleigh number and oscillation frequency) between successively unstable modes. The results quantify the parameter regime in which the weakly nonlinear dynamics of the system can be described in terms of the interaction of the first odd and even parity oscillatory modes.