Exercise 5.   Consider the ideal fluid flow, where the complex potential is  

                    ,     for     .  

5 (a).   Find the stream function   .

5 (b).   Sketch several streamlines of the flow in the angular region   ,   (as shown in Figure 11.55).

Solution 5.

See text and/or instructor's solution manual.

Solution 5 (a).   The stream function is   

                    ,   

or in polar coordinates use      and write  

                       

Solution 5 (b).

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

                     The streamlines   .     

We are done.   

                    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