Exercise 4.   Let the rays    and   and the segment    

form the walls of a containing vessel for a fluid flow in the semi-infinite strip    that is produced by

two sources of equal strength located at the points    and  .  

Find the complex potential for the flow shown in Figure 11.102.           Figure 11.102.

Hint.  Use the fact that   .

Solution 4.

See text and/or instructor's solution manual.

Answer.   Look carefully at the result in Example 11.32.  Recall that the mapping      is periodic with period    and that   .   

Furthermore,  .    Hence    

will produce streamlines where there are an infinite number of unit sources located at the points  .   Here you need to visualize that

the graph in Figure 11.96 is translated infinitely many times by  .  

        The rays    and   and the segment    are all portions of streamlines of   

                      

and form the walls of a containing vessel for a fluid flow in the semi-infinite strip  .

Therefore, the desired complex potential is   

                    .

Solution.   In Exercise 3 we saw that    is the complex potential for a source at  .  

The function    maps    onto  .  The composition    is a complex potential in the

infinite strip   ,  where a single source located at the point   and the lines    and    form the walls of the containing vessel.  

The trigonometric identity    can be used to see that a symmetric flow is produced in the infinite strip   .  

        Also, the rays    and   and the segment    

are all portions of streamlines of   

                      

and form the walls of a containing vessel for a fluid flow in the semi-infinite strip  .

        Therefore  the desired complex potential is

                    .  

We are done.   

Aside.  We can let Mathematica double check our work.

The velocity potential    is   

The stream function     is   

We are really done.   

        We can let Mathematica graph some of the streamlines and velocity potentials.

                    Some streamlines     for a fluid flow in the semi-infinite strip    

                    that is produced by two sources of equal strength located at the points    and  .  

                    Some velocity potentials     for a fluid flow in the semi-infinite strip    

                    that is produced by two sources of equal strength located at the points    and  .  

                    Streamlines and velocity potentials for a fluid flow in the semi-infinite strip    

                    that is produced by two sources of equal strength located at the points    and  .  

We are really really done.   

        Solving for the inverse of        we get  

                    .

We can use Mathematica to check our work and graph the conformal mapping   .

                      The conformal mapping   .  

We are really really done.   

        We can let Mathematica draw some other graphs of the streamlines and velocity potentials.

         A density plot of some velocity potentials   for a fluid flow in the semi-infinite strip   

         that is produced by two sources of equal strength located at the points    and  .  

          Some streamlines     for a fluid flow in the semi-infinite strip    

          that is produced by two sources of equal strength located at the points    and  .  

          Some velocity potentials    for a fluid flow in the semi-infinite strip   

         that is produced by two sources of equal strength located at the points    and  .  

This solution is complements of the authors.

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