A Note on the Free Convection from Curved Surfaces

The self-similar free convection from the outer surface of a heated body is of great interest in several technical and environmental heat transfer processes that occur in practice, such as in meteorological devices, builing insulation systems, heat film sensors, energy storage in enclosures, etc. For a comprehensive review of this topic see e.g. Gebhart et al. [1], Gersten and Herwig [2], Bejan [3], and Schlichting and Gersten [4]. While the bulk of the classical research was concerned with simple geometries like flat plates, cylinders, and spheres, the general case of the free convective flow over a nonisothermal two-dimensional body of arbitrary geometric configuration has been attacked only a couple of years ago by Pop and Takhar [5]. The similar problem for nonisothermal curved surfaces embedded in fluid-saturated porous media has been discussed by Nakayama et al. [6– 8] and for the case of micropolar fluids by Char and Chang [9]. In both of these problems, the existence of a family of curved surfaces and of corresponding temperature distributions which permit similarity solutions of power-law type has been proven [5– 9]. The equation of the corresponding shape curves has been given in [5] and [6] in terms of a series expansion which only converges in the range 0 < n 1=2 of the shape exponent n. The aim of the present note is to show that (i) the two-dimensional curved surfaces which allow for self-similar free convection flows exist for any n > 0; (ii) their equation may be expressed in terms of Gauss’ hypergeometric function; and (iii) to discuss the main features of these surfaces as functions of n.