Rayleigh–Bénard stability and the validity of quasi-Boussinesq or quasi-anelastic liquid approximations

The linear stability threshold of the Rayleigh–Bénard configuration is analysed with compressible effects taken into account. It is assumed that the fluid under investigation obeys a Newtonian rheology and Fourier’s law of thermal transport with constant, uniform (dynamic) viscosity and thermal conductivity in a uniform gravity field. Top and bottom boundaries are maintained at different constant temperatures and we consider here mechanical boundary conditions of zero tangential stress and impermeable walls. Under these conditions, and with the Boussinesq approximation, Rayleigh (Phil. Mag., vol. 32 (192), 1916, pp. 529–546) first obtained analytically the critical value . For any arbitrary equation of state, the coefficients of the quadratic departure are determined analytically from the coefficients of the expansion of density up to degree three in terms of pressure and temperature.