Analytical prediction of the number of Bénard cells manifesting in cylindrical(axi-symmetric) and two-dimensional rectangular enclosures for a given aspect ratio

The paper reports an analytical study of two problems of Rayleigh-Bénard convection(RBC): one in a Newtonian liquid occupying an axi-symmetric cylindrical enclosure(CE) and second in a two-dimensional rectangular enclosure(RE). Linear stability analysis is employed to obtain information on the onset of convection in terms of a critical Rayleigh number, Ra, as a function of the aspect ratio, Ro. The nature and the flow structures of the convective cells, at the onset of convection, are predicted theoretically by plotting the streamlines for different values of Ro. Not so shallow(Ro<1 or Ro≪1), and very shallow enclosures(Ro≫1) are considered. Concepts of cell-width and the wave number are respectively used in these two types of enclosures. An explicit analytical expression for the number of cells as a function of Ro is obtained theoretically. Exact prediction of the critical value of Ra, viz., Rac, as a function of Ro is also made. These are new findings in the case of both CE and RE. The fundamental differences between the problems of CE and RE are also highlighted.