Mathematical Analysis of Two-Dimensional Heat Conduction

We first consider an analytical approach to a -two-dimensional problem and then indicate the numerical and graphical methods which may be used to advantage in many other problems. It is worthwhile to mention here that analytical solutions are not always possible to obtain indeed, in many instances they are very cumbersome and difficult to use. In these cases numerical techniques are frequently used to advantage. For a more extensive treatment of the analytical methods used in conduction problems, the reader may consult Refs. I, 2, 12, and 13. 

Consider the rectangular plate shown in Fig. 3-2. Three sides of the plate are maintained at the constant temperature 7 , and the upper side has some temperature distribution impressed upon it. This distribution could be simply a constant temperature or something more complex, such as a sine-wave distribution. We shall consider both cases. 

To solve Eq. (3-1), the separation-of-variables method is used. The essential point of this method is that the solution to the differential equation is assumed to take a product form 

The boundary conditions are then applied to determine the form of the functions X and T. The basic assumption as given by Eq. (3-4) can be justified only if it is possible to find a solution of this form which satisfies the boundary conditions. First consider the boundary conditions with a sine-wave temperature dis- 

Observe that each side of Eq. (3-6) is independent of the other because x and y are independent variables. This requires that each side be equal to some constant. We may thus obtain two ordinary differential equations in terms of this constant, 

---