Fast viscous bénard convection

Abstract

Abstract We have previously described a method for the precise study of steady two-dimensional convection problems in the asymptotic limit of large Rayleigh number R, assuming a very viscous fluid with the Prandtl number effectively infinite. The method here is applied to cellular convection between isothermal horizontal planes (Bénard convection). The result for the dimensionless heat flux, or Nusselt number, is where c is ⅓ for free-surface convection and ⅕ for fixed surface convection. The constant b is found numerically, as a function of the cell aspect ratio. Previous approximate analyses have obtained c values of ⅓ and ¼ for the two cases; b values were not determined.