Exercise 1.   Let the coordinate axes be walls of a containing vessel for a fluid flow in the first quadrant that is produced by  

a source of unit strength located at      and a sink of unit strength located at   .  

Show that      is the complex potential for the flow,

as shown in Figure 11.99.                     Figure 11.99.

Solution 1.

See text and/or instructor's solution manual.

Answer.   The velocity potential  ,  and the stream function is  .  

Remark.   Notice that the  streamlines    are identical to the level curves    that were constructed in Exercise 11 in Section 11.2   

where we investigated the function   .  

Also notice that the  streamlines      are identical to the isothermals      that were constructed in Exercise 3 in Section 11.5

where we investigated the function   .  

Solution.      is the complex potential for a source at      and sink at   .  

Except for the source    and sink  ,  the real axis    is a streamline for the complex potential  .   

Thus,    is the complex potential when the upper half-plane is the containing vessel, and the real axis    is a wall

for the fluid flow that is produced by a source of unit strength located at      and a sink of unit strength located at   .  

        The function      maps    and    onto    and ,   respectively.  

Therefore, the the desired complex potential is composition  

                    .

The velocity potential and stream function are  

                    ,    and  

                    ,   respectively.  

We are done.   

Aside.  The velocity potential and stream function can be written in the form  

                        

and  

                    

Therefore, the velocity potential and stream function are  

                    ,    and  

                    ,   respectively.  

We are really done.   

Aside.  We can let Mathematica double check our work.

The velocity potential    is   

The stream function     is   

We are really really done.   

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

                    Some streamlines     for a fluid flow in the first quadrant that is

                    produced by  a source of unit strength located at      and a sink of unit strength located at   .  

                    Some velocity potentials     for a fluid flow in the first quadrant that is

                    produced by  a source of unit strength located at      and a sink of unit strength located at   .  

                    Streamlines and velocity potentials for a fluid flow in the first quadrant that is

                    produced by  a source of unit strength located at      and a sink of unit strength located at   .  

We are really 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 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 first quadrant that is

                    produced by  a source of unit strength located at      and a sink of unit strength located at   .  

                    Some streamlines     for a fluid flow in the first quadrant that is

                    produced by  a source of unit strength located at      and a sink of unit strength located at   .  

                    Some velocity potentials     for a fluid flow in the first quadrant that is

                    produced by  a source of unit strength located at      and a sink of unit strength located at   .  

This solution is complements of the authors.

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