Onset of time dependence at the secondary instability of a convecting binary mixture: Implications for wave-number selection

We present data for the onset of time dependence in convecting layers of a normal ${}_{}{}^3{}_{}{}^{}\mathrm{-}_{}^4{}_{}{}^{}$He mixture with positive separation ratio ψ. The boundaries of the fluid were cylindrical; data were obtained for aspect ratios Γ of 6≲Γ≲13. Time dependence typically first occurs as finite-amplitude periodic oscillations with a vanishing frequency at the onset Rayleigh number $R_1$ of the instability. The amplitude at onset A and $R_1$ show regular variations with Γ which reflect variations in the number of convection rolls. From A(Γ) and $R_1$(Γ) we deduce the typical wave vectors α of the convection rolls at $R_1$; α decreases only slowly with ψ. Yet, theory and estimates of ψ indicate that α at the onset of convection should change by a factor of 3. We argue that the values of α at $R_1$ are different from those predicted to occur at the onset of convection.