Exercise 5.  Let the -axis be considered a wall of a containing vessel for a fluid flow in the right half plane  ,   

that is produced by a single source located at the point  .  

Find the complex potential for the flow shown in Figure 11.103.                Figure 11.103.

Solution 5.

See text and/or instructor's solution manual.

Answer.   .

Solution.   Consider Example 11.31 which involved two sources of equal strength located at the points    and  .  

In that example the complete -axis is a streamline.  

Hence, the -axis can be considered as a wall of a containing vessel for a fluid flow in the right half plane  .  

Therefore the complex potential is  .  

We are done.   

Aside.  We can let Mathematica double check our work.

The stream function     is   

The velocity potential    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 right half plane  ,   

                    that is produced by a single source located at the point  .

                    Some velocity potentials     for a fluid flow in the right half plane  ,   

                    that is produced by a single source located at the point  .  

                    Streamlines and velocity potentials for a fluid flow in the right half plane  ,   

                    that is produced by a single source located at the point  .  

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   .

                                A conformal branch of the mapping   .   

We are really 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 right half plane  ,   

          that is produced by a single source located at the point  .  

          Some streamlines     for a fluid flow in the right half plane  ,   

          that is produced by a single source located at the point  .

          Some velocity potentials     for a fluid flow in the right half plane  ,   

          that is produced by a single source located at the point  .  

This solution is complements of the authors.

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