Exercise 2.   Show 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,  (as shown in Figure 11.53).

Solution 2.

See text and/or instructor's solution manual.

Answer.   

Solution.   The transformation      will rotate the plane through an angle in the clockwise direction.  

Use the complex potential     in the w-plane

                    ,   

for an ideal fluid flowing from left to right across the complex plane and around the unit circle  .  

Now use the substitution  ,  and obtain the complex potential in the -plane

                    .  

We are done.   

Aside.  We can graph this flow.

For illustration purposes we can  can explore the situation and choose    (which makes a angle).

                    Fluid flow around a circle at  45^o  angle.

                    Fluid flow around a circle, where the complex potential is  .

                    The streamlines are   .

                    The contour graph   ,   for  .

We are really done.   

        The inverse of the mapping      is  

                    .  

                      A conformal branch of the mapping   .

We are really really done.   

Aside.  For a second illustration we can explore the situation and choose    (which makes a angle).

                    Flow around a circle at  30^o  angle.

                    Fluid flow around a circle, where the complex potential is  .

                    The streamlines are   .

                    The contour graph   ,   for  .

We are really really really done.   

        The inverse of the mapping      is  

                    .  

                      A conformal branch of the mapping   .

This solution is complements of the authors.

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