Rayleigh--Benard instability in nonreactive binary fluids. I. Theory

The hydrodynamic stability (to both stationary and oscillatory convection) of binary, Boussinesq fluids in Benard geometry, with both the Soret and Dufour effects included is studied. A variational principle for the Rayleigh number R/sup cr/ at the onset of stationary convection is found; the complete exact solutions for R/sup cr/ are also given. Apart from trivial multiplicative factors, R/sup cr/ depends only on a single dimensionless parameter psi which depends on the transport and thermodynamic coefficients. For psi> or =0 (''normal'' Soret effect) the system becomes stable only if heated from below (R>0), and for psi<-1 instability may set in only if the fluid is heated from above (R<0). For intermediate values of psi there exist both negative and positive solutions for R/sup cr/. No exact solutions are known for R/sup cr/ at the onset of oscillatory convection, but two integral expressions for R/sup cr/ are derived in this case. Expressions for the amplitude of convective motion and the Nusselt number as a function of vertical-barR-R/sup cr/vertical-bar 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