Rayleigh–Bénard instability in nonreactive binary fluids. I. Theory

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Abstract
The hydrodynamic stability (to both stationary and oscillatory convection) of binary, Boussinesq fluids in Bénard geometry, with both the Soret and Dufour effects included is studied. A variational principle for the Rayleigh number Rcr at the onset of stationary convection is found; the complete exact solutions for Rcr are also given. Apart from trivial multiplicative factors, Rcr depends only on a single dimensionless parameter ψ which depends on the transport and thermodynamic coefficients. For ψ⩾0 (’’normal’’ Soret effect) the system becomes stable only if heated from below (R≳0), and for ψ<−1 instability may set in only if the fluid is heated from above (RRcr. No exact solutions are known for Rcr at the onset of oscillatory convection, but two integral expressions for Rcr are derived in this case. Expressions for the amplitude of convective motion and the Nusselt number as a function of ‖R‐Rcr‖ for both stationary and oscillatory convection are also derived. These expressions may be used to determine whether heat flux measurements are a useful tool for the study of the onset of instability in binary fluids.