Onset of convection in a variable-viscosity fluid

The Rayleigh number R, in a horizontal layer with temperature-dependent viscosity can be based on the viscosity at T0, the mean of the boundary temperatures. The critical Rayleigh number Roc for fluids with exponential and super-exponential viscosity variation is nearly constant at low values of the ratio of the viscosities at the top and bottom boundaries; increases at moderate values of the viscosity ratio, reaching a maximum at a ratio of about 3000, and then decreases. This behaviour is explained by a simple physical argument based on the idea that convection begins first in the sublayer with maximum Rayleigh number. The prediction of Palm (1960) that certain types of temperature-dependent viscosity always decrease Roc is confirmed by numerical results but is not relevant to the viscosity variations typical of real liquids. The infinitesimal-amplitude state assumed by linear theory in calculating Roc does not exist because the convection jumps immediately to a finite amplitude at R0c. We observe a heat-flux jump at R0c exceeding 10% when the viscosity ratio exceeds 150. However, experimental measurements of R0c for glycerol up to a viscosity ratio of 3400 are in good agreement with the numerical predictions when the effects of a temperature-dependent expansion coefficient and thermal diffusivity are included.