Onset of convection in variable viscosity fluids: Assessment of approximate viscosity–temperature relations

Abstract

The effect of the viscosity–temperature relation, μ(T), on the onset of convection in a horizontal fluid layer is discussed. Linear analysis shows that, for aqueous glycerol solutions, the critical Rayleigh numbers (Racrit) obtained using the Arrhenius approximation, μA(T), are in excellent agreement with those employing the actual μ(T) data. The results for the exponential approximation μe(T) differ to an extent that depends on the glycerol mass fraction and the temperature difference (ΔT) between the plates. The error associated with use of μe(T), while small, is of the same order as, or larger than, the uncertainty in careful experiments. For the broad class of liquids for which μA(T) is an excellent approximation to μ, we have assessed the errors in Racrit associated with the widely used μe(T). As ΔT and the viscosity contrast increase, the values of Racrit deviate increasingly from those predicted using μA(T). Also, the relative error in Racrit is much smaller than the maximum relative error in the coefficients [involving μ(T) and its derivatives with respect to T] of the linear disturbance equations.