Solution 7.

See text and/or instructor's solution manual.

Solution.

        The transformation  [Graphics:../Images/JoukowskiTransModHome_gr_382.gif]  maps the point   [Graphics:../Images/JoukowskiTransModHome_gr_383.gif]   onto the point   [Graphics:../Images/JoukowskiTransModHome_gr_384.gif].

For the function  [Graphics:../Images/JoukowskiTransModHome_gr_385.gif]   we have   [Graphics:../Images/JoukowskiTransModHome_gr_386.gif]   and   [Graphics:../Images/JoukowskiTransModHome_gr_387.gif].  

Since   [Graphics:../Images/JoukowskiTransModHome_gr_388.gif],   by Theorem 10.1 in Section 10.1 , the mapping is conformal at  [Graphics:../Images/JoukowskiTransModHome_gr_389.gif]  and "angles are preserved."

Since  [Graphics:../Images/JoukowskiTransModHome_gr_390.gif]  the rotation is  [Graphics:../Images/JoukowskiTransModHome_gr_391.gif].

        The rays tangent to the curve at  [Graphics:../Images/JoukowskiTransModHome_gr_392.gif]  make an angle  [Graphics:../Images/JoukowskiTransModHome_gr_393.gif]  and  [Graphics:../Images/JoukowskiTransModHome_gr_394.gif],  respectively.

The rays tangent to the image curve at  [Graphics:../Images/JoukowskiTransModHome_gr_395.gif]  will also make an angle  [Graphics:../Images/JoukowskiTransModHome_gr_396.gif]  and  [Graphics:../Images/JoukowskiTransModHome_gr_397.gif],  respectively.

Hence, the rays tangent to the image curve at  [Graphics:../Images/JoukowskiTransModHome_gr_398.gif]  will be pointing in the same direction.

        Therefore, that trailing edge of this Joukowski airfoil forms an angle of  0°.

 

We are done.   

 

Aside.  We can let Mathematica double check our work.

[Graphics:../Images/JoukowskiTransModHome_gr_399.gif]

[Graphics:../Images/JoukowskiTransModHome_gr_400.gif]

[Graphics:../Images/JoukowskiTransModHome_gr_401.gif]


[Graphics:../Images/JoukowskiTransModHome_gr_402.gif]

[Graphics:../Images/JoukowskiTransModHome_gr_403.gif]


[Graphics:../Images/JoukowskiTransModHome_gr_404.gif]

[Graphics:../Images/JoukowskiTransModHome_gr_405.gif]


We are really done.   

 

Aside.  We can explore some graphs.

The first exploration uses the circles   [Graphics:../Images/JoukowskiTransModHome_gr_406.gif]

and   [Graphics:../Images/JoukowskiTransModHome_gr_407.gif].

                    [Graphics:../Images/JoukowskiTransModHome_gr_408.gif]

                      The circles  [Graphics:../Images/JoukowskiTransModHome_gr_409.gif]  and  [Graphics:../Images/JoukowskiTransModHome_gr_410.gif]  in the z-plane.  

 

                    [Graphics:../Images/JoukowskiTransModHome_gr_411.gif]

                      The line  [Graphics:../Images/JoukowskiTransModHome_gr_412.gif]  and the circle  [Graphics:../Images/JoukowskiTransModHome_gr_413.gif]  in the Z-plane.  

 

                    [Graphics:../Images/JoukowskiTransModHome_gr_414.gif]

                      The ray  [Graphics:../Images/JoukowskiTransModHome_gr_415.gif]  and the cardioid  [Graphics:../Images/JoukowskiTransModHome_gr_416.gif]  in the W-plane.  

 

                      [Graphics:../Images/JoukowskiTransModHome_gr_417.gif]

                      The arc  [Graphics:../Images/JoukowskiTransModHome_gr_418.gif]  and Joukowski airfoil  [Graphics:../Images/JoukowskiTransModHome_gr_419.gif]  in the w-plane.  

 

Aside.   We can explore some more graphs.

The second exploration uses the circles   [Graphics:../Images/JoukowskiTransModHome_gr_420.gif],  

and   [Graphics:../Images/JoukowskiTransModHome_gr_421.gif].

                    [Graphics:../Images/JoukowskiTransModHome_gr_422.gif]

                      The circles  [Graphics:../Images/JoukowskiTransModHome_gr_423.gif]  and  [Graphics:../Images/JoukowskiTransModHome_gr_424.gif]  in the z-plane.  

 

                    [Graphics:../Images/JoukowskiTransModHome_gr_425.gif]

                      The line  [Graphics:../Images/JoukowskiTransModHome_gr_426.gif]  and the circle  [Graphics:../Images/JoukowskiTransModHome_gr_427.gif]  in the Z-plane.  

 

                    [Graphics:../Images/JoukowskiTransModHome_gr_428.gif]

                      The ray  [Graphics:../Images/JoukowskiTransModHome_gr_429.gif]  and the cardioid  [Graphics:../Images/JoukowskiTransModHome_gr_430.gif]  in the W-plane.  

 

                    [Graphics:../Images/JoukowskiTransModHome_gr_431.gif]

                      The arc  [Graphics:../Images/JoukowskiTransModHome_gr_432.gif]  and Joukowski airfoil  [Graphics:../Images/JoukowskiTransModHome_gr_433.gif]  in the w-plane.  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

This solution is complements of the authors.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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