Solution 8 (b).

See text and/or instructor's solution manual.

Solution.

        The transformation for the modified Joukowski airfoil can be written as the composition of three functions

,      and      and their composition is   

                      .    

        Let    be a circle that passes through the points    and has center      in the z-plane.  

As shown in Exercise 3, the image of the circle    under      is a line    that passes through the origin,

and the line is inclined at the angle      in the Z-plane  (as shown in Figure 11.69).

Furthermore, the image of the circle    is the circle    under  .  

        In Exercise 4 we investigated the situation where the function      is used.    

Now follow the steps using the mapping   .   For the given angle  ,  

the line    inclined at the angle    consists of two rays in the  Z-plane   

                    ,    and  
                    
                    .  

Using the polar coordinates    and  

for the mapping      we have      and     

                    .  

Thus the images of    and    are the two rays    and    respectively, where      

                    ,    and  
                    
                    .  

Observe carefully that in this case   .  In this case, the angle between    is  

                    .

Furthermore, the image of the circle    is the "cardioid like" curve    under  .  

        The transformation    maps the point      onto the point   .

For the function     we have   .   Since   ,   

by Theorem 10.1 in Section 10.1 , the mapping is conformal at    and "angles are preserved."

Hence, the angle of rotation at     is   .

        The rays     and    tangent to the "cardioid like" curve at    make angles    and  ,  respectively.

Then the images of    and    are the two arcs    and    respectively,  that pass through the points    and  .  

Their image arcs    and    will also make angles    and  ,  respectively, at  .  

Furthermore, the image of the "cardioid like" curve    is the modified Joukowski airfoil    under  .  

        Therefore, the angle at the trailing edge of this modified Joukowski airfoil is     radians.

We are really done.   

Aside.  We can explore some graphs.

                      The circles    and    in the z-plane.  

                      The line    and the circle    in the Z-plane.  

                      The rays    and    and the "cardioid like" curve    in the Z-plane.  

                      The arcs    and    and modified Joukowski airfoil    in the w-plane, where  .

Aside.  In the above construction we used the function    which makes the modified Joukowski airfoil    form an angle of    radians.

We can compare this by using the function      which makes the modified Joukowski airfoil    form an angle of    radians,

and by using the function      which makes the standard Joukowski airfoil    which form an angle of    radians.

                      The arcs    and    and modified Joukowski airfoil    in the w-plane, where  .  

                      The arcs    and    and Joukowski airfoil    in the w-plane, where  .  

This solution is complements of the authors.

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