Linear stability of experimental Soret convection

Abstract

Linear stability results for convection in binary fluid mixtures are given for experimental boundary conditions and parameter values. Normal ${}_{}{}^3{}_{}{}^{}\mathrm{-}_{}^4{}_{}{}^{}$He and ethanol-water mixtures are considered with no-slip boundary conditions on the velocity and no outward mass flux. Two cases are considered in detail: fixed temperature at top and bottom, and fixed temperature at the top and fixed thermal flux at the bottom. The role played by the Biot number is emphasized. The results are presented in a manner most useful to experimentalists. The errors incurred in determining the separation ratio S from experimental observations using scaled results with free boundaries are quantified. Neutral curves for both oscillatory and steady-state instabilities together with the corresponding critical wave numbers $k_c$ are determined as functions of S. For the latter, $k_c$ vanishes as S increases. The critical values of S for which this first occurs are determined analytically. Growth rates and frequencies for supercritical Rayleigh numbers are also given. The codimension-two point is found to be masked by steady-state instabilities that set in at smaller Rayleigh numbers, but the masking vanishes in the limit of small Lewis numbers and large Prandtl numbers. The lowest observable frequency along the Hopf curve is determined.