Stagnation points occur where q^. = and qn = 0. The equation q^ = is solved either 

 by r = (2 or by cos 0=0. If the solution r = a is possible, stagnation points occur on the cyl- 

 inder at positions such that, to make qa - 0, 



r 



«in ^ = - 7—77 [69g] 



They are at A, B, in Figure 103. From [69g] and [69f] it is easily seen that the presence of 

 circulation shifts both of the stagnation points toward the side on which it reduces the fluid 

 velocity. 



Suppose that V and V have the same sign. Then, if F = ^naV , the two stagnation points 

 come together at = - 90°. If F/f/ > 4n-a, the equation for sin d cannot be solved; but now it 

 is possible to assume that cos 0=0, sin d = - 1, and to solve the equation qa = for /. Thus 

 as r/U is increased above 4n-a the stagnation point moves out along the radius 6 = - 90°, oc- 

 curring at 



= rr? I 1 + V^ [69h] 



The streamline that passes through such a stagnation point cuts itself perpendicularly and 

 encircles the cylinder, as in Figure 104; all fluid inside it remains inside, circling round the 

 cylinder along closed paths. 



Changing the sign of F reverses the flow pattern and alters all velocities as if by re- 

 flection in a mirror along the axis 6 - or n. Changing the signs of both F and U, however, 

 merely reverses all velocities without other change. 



If the motion is steady, so that the pressure is given by the Bernoulli equation, it is 

 easily seen from symmetry that the resultant force on the cylinder is a force transverse to the 

 direction of the stream, or a lift. Drag, in actual fluids, is an effect of viscosity, which is 

 here assumed to be absent. On an element of area of width a dd and of unit length in the di- 

 rection of the axis, the force is padO, directed toward the axis. Hence the total force in the 

 direction d = n^/2 on unit length of the cylinder is, substituting -pq^/2 for p and the value of 

 9g,2 for g2, 



^277 



L=- (__ p^2j asm Odd ^ pVU [69i] 



Here - sin d is introduced in taking a component of the force. 



If the velocity at infinity makes an angle y with the negative a;-axis, and if the axis of 

 the cylinder is displaced to the point 3 = s = a;. + iy , then 



155 



