Exact solution of the linear-stability problem for the onset of convection in binary fluid mixtures

We solve the linearized problem for the onset of convection in binary fluid mixtures in a layer of infinite lateral extent for rigid-rigid, no-flux boundary conditions exactly. We present detailed data for the critical Rayleigh number and for the neutral frequency as a function of the separation ratio for the two experimentally most relevant systems: for ethanol-water and ${}_{}{}^3{}_{}{}^{}\mathrm{-}_{}^4{}_{}{}^{}$He mixtures. For both the oscillatory and the stationary onset we compare our results with those obtained for simple trial functions. The influence of the effect of rigid-rigid, no-flux boundary conditions on the location of the codimension-2 point is examined.