Exercise 6.   Consider the complex potential   .   

6 (a).    Let  .    Show that      determines an ideal fluid flow around the domain  

                    ,   (as shown in Figure 11.56),  

which is a the flow around a quarter circle in the first quadrant.

Hint.  Use the conformal mapping   ,  and the results in Example 11.24.    

6 (b).    Show that the speed at the point    on the quarter-circle      is   

                    .  

6 (c).    Determine the stream function for the flow and sketch several streamlines.

Solution 6.

See text and/or instructor's solution manual.

Solution 6 (a).   The transformation      maps the angular region   

onto the upper half plane  ,  

where the complex potential for a flow around the unit circle in the -plane is   

                    .  

Substitute    and get the complex potential in the -plane  

                    .

Solution 6 (b).   In the -plane    and we have   .   

Use polar coordinates      and write the velocity vector in the w-plane  

                       

Using the result of Exercise 1, we have the fact that the velocity vector at the point  ,  on the unit circle,  (where  ),  is given by   

                    .   

Applying the chain rule to      we have      so that the velocity vector in the z-plane is  

                        

For points    on the unit circle where  ,   substitute the value    and obtain  

                    .   

A computation similar to that in Exercise 1 will reveal that

                      

and it is easy to see that  

                    .

Therefore, for points    on the quarter-circle   ,   where  ,  the speed is    

                    .  

We are done.   

Aside.  The details for the computation mentioned are  

                      

Solution 6 (c).

      The complex potential can be written as  

                    .  

Therefore, the stream function is

                    .

We are done.   

Aside. We can let Mathematica find the stream function.

Enter the complex potential and determine the stream function.  

We are really done.   

Aside.  We can make a plot of the stream function   .  For illustration purposes, we choose  .  

                    Flow around quarter circle in the first quadrant.

                    The stream function is   .  

                    The contour graph   .  

                    for the constants   .

We are really done.   

      The inverse of      is   

                    .  

                      A conformal branch of the mapping   .

This solution is complements of the authors.

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