Solution of the Thermal Boundary-Layer Equation

The asymptotic formulation of the previous subsection has led not only to the important result given by (9-230) but also to a very considerable simplification in the structure of the governing equation in the thermal boundary-layer region. As a consequence, it is now possible to obtain an analytic approximation for 0. 

If a similarity solution exists, the coefficients of (9-232) must be either a constant or a function of f only. This means that 

The numerical value of the constant in (9-233) is arbitrary (provided it is nonzero) but is conveniently chosen as 2. However, the solution for 0 would be unchanged if any other nonzero value were chosen. The constant cannot be zero because the resulting equation for 0 does not have a solution that can satisfy both of the boundary conditions (9-226) and (9-227). Corresponding to (9-233), Eq. (9-232) for 0 now becomes 

Let us begin with (9 233). For this purpose, it is convenient to rewrite (9 233) in the form 

Then the left-hand side can be integrated directly to obtain the homogeneous solution g3 = c (1 — ri2 ri/2. It is left to the reader to verify that the general solution of (9-235) is 

---

Problem 11-1. Similarity Solutions. If ue=xm, find the most general functional form for the surface temperature, 9S = 6S (x), that allows a similarity solution of the thermal boundary-layer equation in two dimensions for large Re )S> 1 and Pr = 0(1). 

---