A Perturbation Method for Studying Thermal Convection Problems

Abstract

When a layer of fluid is heated uniformly from below, and the Rayleigh number exceeds a critical value, convection takes place in it in a regular cellular pattern. A perturbation method is presented here for determining mathematically the pattern and amplitude of this steady convective motion. The essential point of the method is to expand functions which describe the velocity and temperature field in the fluid in a power series of a parameter e, while the Rayleigh number is put as a product of the critical value times (1+ee2). A set of inhomogeneous equations thus obtained can be solved by the perturbation method which is in use in non-linear oscillation problems. In the two-dimensional case, the slope of heat transport curve becomes steepened abruptly when the Rayleigh numher exceeds a critical value. Another problem which can be dealt with by this method is that of convection within a sphere. This forms an extension of CHANDRASEKHAR's linearized stability theory.Furthermore, a study is made of the steady thermal convection in a two-dimensional fluid layer when it is heated uniformly from below under a simultaneous constraint of nonuniform temperature given on its upper surface. Mathematically this is an application of the method mentioned above to a problem with inhomogeneous boundary conditions. it was found that the site of spontaneous convection cells is governed by the surface temperature disturbance having a critical wave length. Surface temperature disturbances having much longer or shorter wave lengths play very little part in this, while those having wave lengths close to the critical one have influential effect in determining the general feature of fluid motion.