and 



q^ = q^ + q^ = (l-— sin^^j [127e] 



On the sphere n^ = 11^(1- 3/4 sin^^), so that q has the minimum value 6'/3 at ^ = n/1, where 

 the fluid is moving directly backward. At front and rear q = U. 



The streamlines are illustrated for equally spaced values of (/» in Figure 212. 



The kinetic energy of the fluid, if its density is p, is 



r = - p / r^ dr I q^ ■ 9„ sin edO ^ — p a^ U^ [127f] 



" *i ^ 



Here, because of the symmetry, a ring-shaped element of volume, represented 



by 277^'^ sin ddddr, has been employed. (See Reference 1, Articles 92, 96; Keference 2, 



Section 15.32.) ■ i 



128. STREAMING FLOV/ PAST A SPHERE 



The flow around a stationary sphere when the fluid at infinity has a uniform velocity (J 

 is obtained from the results of the last section by imparting to everything an additional 

 uniform velocity -U. With appropriate terms added from [119a, b], in which r cos and r sin 6 

 replace x and Sf, the total potential and stream function are 



(P = U (r+ —\ cos e, ij/ = — U (r^-~\sin^d [128a, b] 



Here, for U > 0, the flow at infinity is toward 6 - n. 



Thus lA = when ^ = or = n-, and also if r = o. This shows that a streamline 

 approaches along the radius (? = 0, divides, passes around the sphere, reunites and continues 

 along the radius 6 - n. 



The velocity com,ponents of interest are 



q^ = - V (l- ~\cos d, qQ^u(l + ^\sine [128c,d] 



On the sphere, where ?• = a, '7 = | '7^1 = 3/2 |t^| sin d, so that q has a maximum value of 

 3f//2 on the equator at = n-/2. Stagnation points occur at = and 6 = n. 



If the motion is steady, the pressure on the sphere, by the Bernoulli equation, is 



320 



