Exercise 10.   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 back into a quadrant,

as indicated in Figure 11.108,                     Figure 11.108.

where     is the endpoint of the ray  .

Hint.  Set   ,  ,  ,   and  ,  ,  ,   respectively, and let   .    

Solution 10.

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 onto the flow from a channel back into a quadrant, as shown in Figure 11.108,

where    ,  ,  ,   and  ,  ,  ,   respectively, and let   .      

The initial conditions can be used to obtain   and  ,  and the answer is

            .  

The logarithm term could also be written in the form  

            .  

And if the inverse hyperbolic functions are used then this can be written as

            .  

Solution.   The details are given in the solution of Exercise 15 in Section 11.9.  

Along the x-axis use the points   .   

The exterior angles are  ,  and the formula for the derivative is  

               

Integrate and get

            .  

The first integral is easy to get

            .

The second integral can be found using the suggested change of variable  

            
            
            

Make substitutions in the integral

              

Now use the substitution  ,  

            .

Now combine this with the first integral and get  

              

            
        
              

The initial conditions can be used to obtain   and  ,  and the answer is

            .  

The logarithm term could also be written in the form  

            .  

And if the inverse hyperbolic functions are used then this can be written as

            .  

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

            .  

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

            .  
            

Or if the second integral is treated separately, then Maple's answer will be  

            .  

We are done.   

Aside.  We can let Mathematica double check our work.

We are really done.   

        We can let Mathematica draw a graph of the streamlines.

                      The flow from a channel    into a quadrant.           

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