studies on the Motion of Viscous Flows— I 



To extend the results of the linear case, let a whirl point be defined as a 

 point Xg of zero velocity which has a segment of a streamline x = x (h) , 

 hj 5 h 5 h2 in each of its neighborhoods, satisfying the following conditions: 



1. For any given unit vector t , there is a point x (h) on the segment so 

 that x(h) - xq = kt for some positive k . 



2. x^ = Cxj for some positive e , where x^ and xj are the ends of the 

 segment. = 



3. The segment does not pass through xq . i- ■ i j ''< . -• ' = .. > v. 



The definition essentially describes what is generally meant by streamlines 

 whirling around a point. The main result in this section, which is believed to be 

 new, is the following theorem. 



Theorem I: Any point of zero velocity where the swirl is positive is a 

 whirl point. Conversely, the swirl cannot be negative at a whirl point. 



Some preliminary results are required for its proof. Let the point of zero 

 velocity be chosen as the origin. The equations of streamlines in cylindrical 

 coordinates are 



dr/dh ^ u^, rd(9/dh = uq . (26) 



The conditions on the segment of the streamline for a whirl point are then that r 

 does not vanish anywhere on it, that for any given a there is a point on it with 

 the angular coordinate a + 2n-n , n being an integer, and that e^ and 6^ at the 

 ends differ by a multiple of imr . One can also estimate the velocity components 

 by using the mean value theorem 



dr 



Bu 



'"■'" dh Bi 

 and 



dh 3i 



< r < r 



< r_ < r 



(27a) 



(27b) 



Note that 3u /Br and Bugi/Br are the rate of deformation and the angular velocity 

 of a radial line element. . 



Now one can prove the first part of the theorem. If a neighborhood of the 

 point of zero velocity is specified, we can choose a circular neighborhood of 

 radius R within the given neighborhood such that the swirl is positive in the 

 circle. This is possible because the swirl given by Eq. (23) is a continuous 

 function. 



The streamline through any point in the circle, except the origin, can be 

 continued until it approaches a point of zero velocity or the circumference, 



445 



