Exercise 2.   Let the coordinate axes be walls of a containing vessel for a fluid flow in the first quadrant that is produced by

two sources of equal strength located at the points      and   .  

Find the complex potential  for the flow in Figure 11.100.           Figure 11.100.

Solution 2.

See text and/or instructor's solution manual.

Answer.   

Solution.      is the complex potential for

two sources of equal strength located at      and sink at   .  

The function    maps   and   onto   and ,  respectively.  

Therefore, the the desired complex potential composition is  

                    .  

The velocity potential and stream function are

                    ,    and  

                    ,   respectively.  

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 first quadrant that is

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

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

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

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

                                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 first quadrant that is

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

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

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

                    Some velocity potentials     for a fluid flow in the first quadrant 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