Exercise 1.   Consider the ideal fluid flow for the complex potential   ,   where  .

1 (a).   Show that the velocity vector at the point  ,   on the unit circle,   (where  ),   is given by   

                    .  
                    
Solution 1 (a).

See text and/or instructor's solution manual.

The velocity vector    for the complex potential    is known to be  .  

                       

For points    on the unit circle where  ,   substitute the value  .  

Therefore,    

                    .   

We are done.   

Aside.  We can let Mathematica double check our work.

We are really done.   

Aside.  We can graph this flow.

                    Fluid flow around a circle, where the complex potential is  .

                    The streamlines are   .

                    The contour graph   ,   for  .

We are really really done.   

        The inverse of the mapping      is  

                    .  

                      A conformal branch of the mapping   .

This solution is complements of the authors.

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