Exercise 8.   Let      denote the branch of the inverse of      that is a

one-to-one mapping of the -plane slit along the segment      onto the domain   .  

Use the complex potential   

                    ,     in the -plane to show that the complex potential  

                    ,   

determines the ideal fluid flow around the segment  ,  

where the velocity at points distant from the origin is given by   ,   (as shown in Figure 11.58).  

Solution 8.

      Use the result of Exercise 2 where we see that the complex potential      determines the ideal fluid flow

around the unit circle  ,  where the velocity at points distant from the origin is given approximately by   ;  

that is, the direction of the flow for large values of is inclined at an angle with the axis.  The complex potential in the plane is  

                       

Now use the trigonometric identities        and    .  

This produces the desired complex potential in the z-plane  

                    .  

We are done.   

Aside.  For illustration we can make a plot of the stream function using   .  

                    Flow around a plate at a 45o angle.  

                    The complex potential is  .  

We are really done.   

      The inverse of      is   

                    .  

                      A conformal branch of the mapping   .

This solution is complements of the authors.

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