A Spectral method determination of the first critical Rayleigh number in a cylindrical container

The onset of laminar axisymmetric Rayleigh–Bénard convection is investigated analytically for fluid in a cylindrical container. All surfaces are considered to be solid and no-slip for the flow, whereas for the thermal boundary conditions both a perfectly conducting and an insulated side wall are considered. The governing Boussinesq equations are perturbed and an approximate solenoidal flow field and a temperature field are determined, using the assumption of separation of variables. Subsequently, a Chebysev–Galerkin spectral method is employed to reduce the equations to a system of first-order nonlinear ordinary differential equations. The approximate representation of the flow and temperature fields make it possible to perform the calculations analytically. The first critical Rayleigh number (Ra_cr) is then calculated using local stability analysis. The resulting value of Ra_cr compares favorably with previous numerical and experimental studies. The analytical solution presented here allows for deeper insights into the physics of this extensively studied problem to be identified.