Method of generalized similitude in free convection problems with arbitrary distribution of the temperature or heat flux on a vertical wall

Abstract

The self-similar problem of free convection near a heated vertical plate was solved for the first time in [1] for the simplest case of a constant wall temperature. In [2], Yang proved the existence of a self-similar solution to the problem of free convection for vertical plates and cylinders on the surfaces of which the temperature has a power-law distribution. In [3], Yang's solution was generalized to the case of free convection near a slender figure of revolution, but also only in the self-similar case of a power-law distribution of the temperature on the wall. In [4], this problem was solved in an extended nonsimilar formulation but by an artificial and not general method similar to Gertler's, the convergence of the approximations being slow. The present paper contains the solution to the problem of free convection near a vertical plate with arbitrary distribution of the temperature or heat flux on its surface. Rigorous application of the method of generalized similitude [5] leads in this case to universal equations that present insuperable computational difficulties, which forces one to use a simplified but fairly general method to solve this class of problems.