Effect of pressure work and viscous dissipation in the analysis of the Rayleigh-Bénard problem

The classical Rayleigh-Bénard problem in an infinitely wide horizontal fluid layer with isothermal boundaries heated from below is revisited. The effects of pressure work and viscous dissipation are taken into account in the energy balance. A linear analysis is performed in order to obtain the conditions of marginal stability and the critical values of the wave number a and of the Rayleigh number Ra for the onset of convective rolls. Mechanical boundary conditions are considered such that the boundaries are both rigid, or both stress-free, or the upper stress-free and the lower rigid. It is shown that the critical value of Ra may be significantly affected by the contribution of pressure work, mainly through the functional dependence on the Gebhart number and on a thermodynamic Rayleigh number. While the pressure work term affects the critical conditions determined through the linear analysis, the viscous dissipation term plays no role in this analysis being a higher order effect. A nonlinear analysis is performed showing that the superadiabatic Rayleigh number replaces Ra in the functional dependence of the excess Nusselt number. Finally, a reasoning is proposed to show how the results obtained may be used as a test on the most appropriate formulation of the Oberbeck-Boussinesq model.