Influence of the prandtl number on second-order heat transfer due to surface curvature at a three-dimensional stagnation point

Abstract

The heat transfer at the stagnation point of a body in laminar, incompressible, three-dimensional flow is studied using second-order boundary-layer theory. The body is assumed to have two different principal curvatures at the stagnation point. The influence of the Prandtl number on the Nusselt number, in particular on the second-order terms resulting from longitudinal and transverse curvatures of the body surface, is shown. The conditions under which the contributions to the Nusselt number due to the transverse curvature effect and the longitudinal curvature effect cancel each other are demonstrated. Numerical results for a broad range of the Prandtl number (0.2 ≤ Pr ≤ 12) are given. The two limiting cases of very small and very large Prandtl numbers are considered using a perturbation analysis. Asymptotic expansions are obtained for both limiting cases; for very large Prandtl numbers, the method of matched asymptotic expansions is used to obtain the Nusselt number including both curvature effects. In the limiting case of very small Prandtl numbers, the first-order term for the Nusselt number, which is the result of Prandtl's boundary-layer theory, tends to zero as Prandtl numbers tend to zero. In contrast to this, the second-order contribution to the Nusselt number due to curvature approaches a finite, positive or negative value as Prandtl numbers tend to zero. Therefore, the second-order boundary-layer effect due to surface curvature gains significance in this case, and plays a key role for the heat transfer at stagnation points.