Convection in binary fluids: Amplitude equations, codimension-2 bifurcation, and thermal fluctuations

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Abstract
The near-threshold behavior for thermal convection in binary fluid mixtures heated from below is determined for realistic (rigid and impervious) boundary conditions at top and bottom. We calculate up to third order the coefficients of the amplitude equations for the stationary, the traveling-wave, and the standing-wave convection. In all three cases, the bifurcation changes with decreasing separation ratio Ψ from forward to backward with the respective tricritical points occurring for negative Ψ. For small values of the Lewis number L, these tricritical points, together with the codimension-2 point, lie near Ψ=0 and for L=0 they all degenerate to Ψ=0, which describes the limit of normal fluids. Near the codimension-2 point, where the thresholds for the stationary and the Hopf bifurcation coincide, a generalized amplitude equation has to be considered, whose nonrescaleable coefficients are determined from the already-known ones of the conventional amplitude equations. Some special properties of the traveling waves, such as the Benjamin-Feir instability and the sign change of the group velocity, can be deduced from the codimension-2 amplitude equation. Finally, the influence of thermal fluctuations on the convection onset is considered. We determine the strength of the additive noise term appearing in the amplitude equation for free and previous boundary conditions as well as for rigid and impervious boundary conditions for both the stationary bifurcation and the traveling waves. The space-time-correlation function for the order parameter below threshold is derived. Our calculated noise strength and correlation function agree rather well with the data of a recent experiment [W. Schöpf and I. Rehberg, Europhys. Lett. 17, 321 (1992)].