Figure 135 — On an elliptic cylinder the 



calculated pressure is shown by a 



broken curve and observed 



pressures by small circles. 



-1.0- 



In steady motion the resultant force is a lift pFU per unit length, according to the 

 Kutta-Joukowski theorem proved in Section 73. Furthermore, comparison of Equation [83h] 

 with Equation [74h] shows that here y = a , 6, = U (a+ b) (b cos a + ia sin a)/2; hence 

 Equation [74k] gives for the torque per unit length on the cylinder about an axis through the 

 origin, in steady motion, 



N = np {a?- - b^) U^ sin 2a. 



[83t] 



Because of the sign, the torque tends to set the cylinder broadside to the stream. 



An elliptic cylinder in a converging stream Was considered by Oka^°^. 



(For notation and method; see Section 34; Reference 1, Section 71; Reference 2, 

 Sections 6.31, 6.32, 6.33, 6.42; Zahm and others, References 182 and 101.) 



84. ELLIPTIC CYLINDER IN TRANSLATION 



Let the cylinder described in the last section be itself in motion at velocity t/ in a 

 direction inclined at an angle a to the positive a;-axis or to the major axis of the ellipse, 

 and let the surrounding fluid be at rest at infinity. This case can be produced out of the pre- 

 ceding by imposing on everything a uniform velocity U in the required direction. Then, from 

 Equations [35a] and [82a], there is to be added in w the term 



-Uze- 



- cUe~'^ cosh C- 



After inserting exponentials in place of all hyperbolic cosines and eliminating ^ and c by 

 means of Equation [83d, e], from Equation [83a], 



Va + b _>■ iV 

 (6 cos a \ ia sina )e ^ + (^ - ^ ), 

 a - b 277 " 



[84a] 



201 



