Exercise 4.   Show that the stream function is given by      for an ideal fluid flow around the angular region  

                    ,   (as shown in Figure 11.54).   

Sketch several streamlines of the flow.  

Hint.  Use the conformal mapping   .  

Solution 4.

See text and/or instructor's solution manual.

      The transformation    maps the angular region    onto the upper half plane    

where the complex potential is  

                    .   

The complex potential in the plane is the composition  

                    .   

Use polar coordinates and write this as  

                    .  

Therefore the solution is  

                    .  

We are done.   

Aside. We can let Mathematica find the stream function.

Enter the complex potential and determine the stream function.  

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

                    The streamlines   ,       

                    for    .  

                    The contour graph   ,

                    for the constants    .  

We are really done.   

        The inverse of      is   

                    .    

                      The conformal mapping      for the flow.

This solution is complements of the authors.

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