Exercise 7.   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 quadrant,

as indicated in Figure 11.105.                     Figure 11.105.

Solution 7.

See text and/or instructor's solution manual.

Answer.   Use a   ,   for convenience, set ,  and  .  

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 the first quadrant, as shown in Figure 11.105.

Solution.   The details are given in the solution of Exercise 8 in Section 11.9.   

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 ,  .  

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      and get  

                    .

Now combine this with the first integral and obtain  

                    
        
Therefore,   

                    .

We are done.   

Aside.  We can let Mathematica double check our work.

We are really done.   

We are really really done.   

        We can let Mathematica graph some of the streamlines.

                      The flow from a channel    into the first quadrant.

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 the right half-plane.

We are really really really really done.   

We can use Mathematica to check our work and 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.   

        There are other possible formulas for the solution.  The logarithm term could also be written in the form  

                     .

Aside.  We can let Mathematica double check our work.

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

                     .

        Observe that the conditions     and  ,   are met.

The images of   ,   are   ,   respectively.

                    

From calculus we have      so we will use      and write

                    

Remark 1.   If the computer algebra Mathematica is used to perform the integration then the answer is

                    .  

Remark 2.    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  

                    .  

Summary of Formulas.   The following five 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