Exercise 3.   Let the lines    and    form the walls of a containing vessel for a fluid flow in the

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

Find the complex potential for the flow in Figure 11.101.                     Figure 11.101.

Solution 3.

Answer.   Use the result in Example 11.32.  

Both positive and negative y-axes are streamlines for the complex potential   .

        Hence, the positive and negative y-axes could also be interpreted as walls for a fluid flow in the semi-infinite strip  .  

Therefore, the desired complex potential is    

                    .

Solution.   In Example 10.13 in Section 10.4,  we saw that the transformation    

is a one-to-one conformal mapping of the vertical strip    onto

the w-plane slit along the rays    and  .   

Also, the point    is mapped onto  ,  and the imaginary axis    in the z-plane

is mapped onto the imaginary axis    in the w-plane.

        In the complex w-plane      is the complex potential for a unit source at the origin   ,   and

the positive u-axis and both positive and negative v-axes are streamlines in for the complex potential   .

       Since     will map the the infinite strip    onto the right half-plane  ,  
       
the composition    is the desired 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.

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 infinite strip     

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

                    Some velocity potentials     for a fluid flow in the infinite strip     

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

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

                    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   .

                      The conformal 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 infinite strip     

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

                    Some streamlines     for a fluid flow in the infinite strip     

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

                   Some velocity potentials     for a fluid flow in the infinite strip     

                    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