Exercise 8.   Use a Schwarz-Christoffel transformation to find a conformal mapping    that will map the flow

in the upper half-plane onto the flow from a channel into a sector,

as indicated in Figure 11.106.                     Figure 11.106.

Solution 8.

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 from a channel with    into a sector, as shown in Figure 11.106.

Hint:  Set   and .  Use the change of variable    in the resulting integral.

Solution.   Along the x-axis use the points   .   The exterior angles are  ,  

and the formula for the derivative is  given by the Schwarz-Christoffel formula  

                       

For convenience set  .   

Now integrate and get    

                    .  

The first integral is easy to get

                    .  
            

The second integral can be found using the change of variable

                      
            
make these substitutions in the integral and get

                      

Next use the substitution      and get

                    

Now combine this with the first integral and get

                    
                    
Therefore,  

                    .

We are done.   

Aside.  We can let Mathematica double check our work.

Use the identity    and write the solution using logarithms

Therefore,  

                    .  

We are really done.   

Aside.  We can let Mathematica double check our work.

We are really really done.   

        We can let Mathematica graph some of the streamlines.

                      The flow from a channel    into a into a sector.

We are really really really done.   

Aside.  We can extend the flow into the third and fourth quadrants using symmetry.

                    The flow from a channel    into a into a sector.

We are really really really really done.   

          We can use Mathematica to graph the conformal mapping   .

                              The conformal mapping   .   

Aside.  We can extend the flow into the third and fourth quadrants using symmetry.

                                The conformal mapping   .   

We are really really really really really done.   

        Observe that the conditions     and  ,   are met.

The images of   ,   are   ,   respectively.

                    


                    
                    
                    
From calculus we have      so we will use      and write

                    .

Remark 1.1   If the computer algebra  Mathematica is used to perform the integration then the answer is

                      

which gives the correct result but uses the a specialized hypergeometric function.

Remark 1.2   Mathematica 7 will get the following formula for the integral

            

            

Remark 2.   If the computer algebra Maple is used to perform the integration then the answer is

                    .  

Summary of Formulas.   The following four mapping of the upper half-plane will produce the same results.

This solution is complements of the authors.

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