Exercise 7.   Show that      is the complex potential for the ideal fluid flow inside the semi-infinite strip

                    ,    (as shown in Figure 11.57).    
  
Find the stream function.  

Solution 7.

See text and/or instructor's solution manual.

      The transformation      maps the semi-infinite strip      

onto the upper half plane  ,  where the complex potential in the -plane is   

                    .  

Substitute    and get the complex potential in the -plane:  

                    .

The stream function is

                    .   

Therefore, the stream function is  

                    .  

We are done.   

Aside.  We can let Mathematica double check our work.

We are really done.   

Aside.  For illustration we can make a plot of the stream function.

                    Flow Around the Inside of an Infinite Rectangle.

                    The stream function is   .  

                    The contour graph   .  

                    for the constants   .

We are really 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