Figure 109 — Symmetrical streamlines near 



a circular cylinder caused by a stream 



and two equal and opposite vortices. 



(Copied from Reference 28.) 



FORCES ON CYLINDERS 



72. THE DISTANT MOTION 



The fluid at infinity is usually assumed to be either at rest or in uniform motion. 

 Sometimes it is of interest to know how rapidly this condition is approached. 



Suppose, for generality, that at infinity the fluid is moving uniformly at speed U, and 

 that the motion is irrotational everywhere except perhaps inside a certain cylindrical surface 

 S, which may enclose within it one or more solid cylinders. The cross section of S need not 

 be circular. Then at large distances from S a closer approximation to the actual motion can 

 be secured by superposing the following three motions: 



(a) The uniform flow at speed V\ 



(b) A motion in which the velocity is everywhere directed along a radius from any chosen 

 axis inside S and is of magnitude 



?„ = 



2rrr 



[72a] 



(c) A circulatory motion in which the velocity is perpendicular to the radius drawn from 

 the chosen axis and of magnitude 



9r = 



2nT 



[72b] 



Here r denotes distance from the axis, 



V is the net volume per second that flows outward across S, per unit of its length, due 



to sources or sinks inside it, and 

 r is the circulation along any path encircling S once in the positive direction. 



^ > ?t,» Ivi ^nd r may be positive or negative. 



160 



