Channel with Constant Temperature of the Wall

Temperature field. We shall study the heat exchange in laminar flow of a fluid with parabolic velocity profile in a plane channel of width 2h. Let us introduce rectangular coordinates X, Y with the X-axis codirected with the flow and lying at equal distances from the channel walls. We assume that on the walls (at Y = h) the temperature is constant and is equal to Ti for X 0 and to T2 for X 0. Since the problem is symmetric with respect to the X-axis, it suffices to consider a half of the flow region, 0 Y h. 

The temperature distribution T, is described by the following equation and boundary conditions  

By analogy with the case of a circular tube, we seek the solution of problem (3.6.1)-(3.6.3) in the form of a series for separated variables  

The eigenvalues r k, Xm and the eigenfunctions gk, fm can be obtained by solving Eqs. (3.5.9) and (3.5.10) in which we omit the second terms (proportional to the first-order derivatives) and replace g by y. The boundary conditions remain the same. The coefficients Am and Bk can be obtained from the condition that the temperature and its derivatives are continuous at a = 0 [110]. 

In what follows, we present only the basic solutions of the problem for x 0. 

---