Exact calculation of convection thresholds in rotating binary liquid mixtures

The exact sine series method of Stein [Phys. Rev. A 43, 768 (1991)] has been extended to allow the calculation of critical Rayleigh numbers and wave numbers for both stationary and oscillatory convection states of rotating binary mixtures in Rayleigh–Bénard geometry. Checks on the accuracy of the method against previous calculations yield good agreement, provided these are multimode. Single-mode methods seriously underestimate Rc when the convection state is rotation-induced oscillatory. The method is used to obtain a hydrodynamic phase diagram showing a codimension-three point for a hypothetical mixture with Prandtl number 0.2 and Lewis number 0.0266. Threshold state calculations for a realistic 3He–4He mixture show that observation of a codimension-three point is impractical through the Prandtl number being too high.