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

1 (c).   Show that the speed at the point    on the unit circle is given by       

            and that the speed attains the maximum of      at the points      and is zero at the points   .   Where is the pressure the greatest?

Solution 1 (c).

See text and/or instructor's solution manual.

At the point    on the unit circle the speed    is

                          

Therefore,  

                    .  

        Clearly, the maximum speed occurs at the points      where we have   .  

At the point      the speed is      and

at the point      the speed is   .  

        Similarly, the speed is zero where we have      and   .   

At the point      the speed is   ,   and

at the point      the speed is   .  

        The pressure is greatest where the speed is least, which occurs at the points    and  ,  respectively.

We are done.   

Aside.  We can let Mathematica double check our work.

We are really done.   

Aside.  We can let Mathematica graph the speed at points on the unit circle.  

                                        Graph of the speed   .

This solution is complements of the authors.

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