Yajnik and Lieber 



Hence as (x,y) approaches the origin, the ratio u:v approaches a:b. But if we 

 choose the unit vector n in the direction of the vector (a>b), we find that no 

 matter how small a neighborhood we choose, there is always a point where ve- 

 locity is normal to (a,b). 



If the continuity equation for an incompressible equation fluid is used, a 

 stronger result can be obtained. Let a vortex point be defined as a whirlpoint 

 where the value of i in the second requirement of the streamline segment is 

 unity. This ensures closed streamlines. 



Theorem II: In the plane flow of an incompressible fluid, any point of 

 zero velocity where the swirl is positive is a vortex point, and con- 

 versely, the swirl at a whirlpoint is positive if all the derivatives 



3u/3x, 3u/3y, Bv/3x, and 3v 3y do not vanish simultaneously. 



Consider any point of zero velocity where the swirl is positive: then it is a 

 whirlpoint according to Theorem I. Suppose the value of i in the second re- 

 quirement of the streamline is different from one. Then we can consider the 

 two-dimensional region bounded by the streamline segment and the radial seg- 

 ment joining the end points of the streamline. Since the flux across the radial 

 segment is zero on account of incompressibility, the tangential component of 

 velocity must be zero somewhere on the radial segment, and Eq. (27b) then im- 

 plies that angular velocity must vanish somewhere. But the neighborhood can 

 be chosen to be so small that swirl is positive everywhere and the angular ve- 

 locity of a line element cannot vanish. Hence follows the first part of the theo- 

 rem. The second part follows from Theorem I and the conclusion that (Bu/bx) 

 (3v/3y) - Ou/By)Ov/Bx) Cannot vanish at a whirlpoint when the velocity deriva- 

 tives do not vanish simultaneously. 



Consider the curvature of streamline at a point where velocity is different 

 from zero. We see from Eq. (20) that if the curvature is zero, a line element 

 tangential to the streamline has zero angular velocity. Hence the curvature of a 

 streamline cannot vanish in a region of positive swirl. 



The geometrical behavior of streamlines, then, is decisively influenced by 

 the angular velocities of line elements. The dynamic implications of these kine- 

 matic results will become clear in the following section. 



DEFINITION AND GENERAL PROPERTIES OF EDDIES 



The formulation of a clear idea of an eddy is prerequisite to a systematic 

 study of the properties of eddies. Since they are often identified in experimen- 

 tal investigations by closed or spiralling streamlines, it is but natural to seek a 

 formulation in terms of the angular velocities of line and surface elements 

 whose intimate relation with the streamline geometry has been pointed out in 

 the previous sections. This approach differs from the prevailing approach in 

 which eddies are regarded a priori as regions of concentrated vorticity and no 

 attempt is made to formulate an unambiguous notion of an eddy. 



448 



