Exercise 9.   Use a Schwarz-Christoffel transformation to find a conformal mapping    that will map the flow

in the upper half-plane onto the flow in a right-angled channel,

as indicated in Figure 11.107.                     Figure 11.107.

Solution 9.

See text and/or instructor's solution manual.

Answer.   ,   for convenience set  .  

Integrate and get   ,  

which will map the flow in the upper half-plane from a source at    onto the flow in a right-angled channel with  , as shown in Figure 11.107.

Solution.   The details are given in the solution of Example 11.28 in Section 11.9.  

Using the Schwarz-Christoffel transformation method, if we choose    and  ,  then the formula  

            

will determine a mapping   of the upper half-plane onto the domain indicated in Figure 11.74 (a).  

With  ,  we let ,  then    and  .  

    The limiting formula for the derivative becomes   

            

where  , which will determine a mapping   from the upper half plane onto the channel as indicated in Figure 11.74 (b) and Figure 11.107.

    Using integrals in Table 11.2 in Section 11.9, we obtain

           

    If we use the principal branch of the inverse sine function, then the boundary values    lead to the system  

               and   ,  

which we can solve to obtain  .  Hence the required solution is

            .    

We are done.   

Aside.  We can let Mathematica double check our work.

Recall.  In the calculus course we used trigonometric substitutions to integrate integrals like .

In particular, in calculus we introduced the trigonometric identities like  

                    .

Hence, we can use this substitution in    to obtain

                    .    

We are really done.   

        We can let Mathematica graph some of the streamlines.

                      The flow in a right-angled channel.

We are really really done.   

          We can use Mathematica to graph the conformal mapping   .

                      The conformal mapping   .   

This solution is complements of the authors.

(c) 2008 John H. Mathews, Russell W. Howell