Coupled Marangoni-Benard/Rayleigh-Benard Instability With Temperature Dependent Viscosity

Abstract The onset of convection induced by coupled surface tension gradient and buoyancy forces is investigated with temperature dependent viscosity. Both surface tension and viscosity are assumed to vary linearly with temperature. The linear viscosity approximation was found to be an adequate representation of relevant published experimental data. One limiting case, Ma = 0, is the buoyancy driven convection problem typically referred to as the Rayleigh-Benard stability problem. The other limiting case, Ra = 0, is the surface tension (gradient) driven flow problem referred to as the Marangoni-Benard problem. Several investigations of the Rayleigh-Benard problem with temperature dependent viscosity have been reported. One stability investigation of the Marangoni-Benard problem with a linear viscosity variation has also been published. Results from that Marangoni-Benard study indicate that linear viscosity variation is destabilizing while similar studies for the variable-viscosity-Rayleigh-Benard problem suggest that the above viscosity profile is stabilizing. In this study the variable viscosity analysis is extended to the coupled problem, which bridges the above limiting cases. The equations and boundary conditions obtained from the linear analysis are solved numerically as a generalized eigenvalue problem. Neutral stability curves for different viscosity slopes are generated for the Marangoni-Benard and Rayleigh-Benard problems. It is shown that the curves can be collapsed to a single curve by appropriately scaling the results for each of the limiting cases. The critical Marangoni number is determined as a function of the slope of the viscosity temperature variation, ε, for different values of the Rayleigh number. Regression analyses of the numerical results are performed to provide a convenient means of computing the critical Marangoni number as a function of ε and the Rayleigh number. The linear temperature-dependent viscosity considered in this analysis gives results which are consistent with the Rayleigh-Benard studies employing an exponentially temperature-dependent viscosity. When the viscosity decreases linearly with temperature, the coupled buoyancy-surface tension problem, including the limiting special cases of Ra = 0 and Ma = 0, is found to be more stable than the constant viscosity case. The difference between this and the previous Marangoni-Benard study is explained.