Yajnik and Lieber 



since it is an integral curve of the differential equations in Eq. (22) (Hurewicz 

 [18]). It will be shown in the next section of this paper that velocity in the two- 

 dimensional region of positive swirl cannot vanish at two points. Hence the 

 streamline through any point other than the origin can be continued until it ap- 

 proaches the origin or the circumference. Now we want to show that a point can 

 be chosen within the circle so that the streamline through it turns around by l-n 

 before it approaches the origin or the circumference. Such a streamline will 

 clearly provide a segment satisfying the definition of a whirl point. 



Since the rate of deformation and the angular velocity of any radial line 

 element appearing in Eqs. (27a) and (27b) are continuous functions, they take 

 maximum and minimum values, say d^ , dj , wj , and wj , in the circle. Hence 

 we have 



di > (dr/dh)/r > d^ (28a) 



(28b) 



at any point in the circle of radius R . Since the angular velocity of a line ele- 

 ment cannot vanish in a region of positive swirl, w^ and wj are of the same 

 sign. One obtains from Eqs, (28a) and (28b) 



A > (dr/d(9)/r > B , (29) 



where A and B are equal to d^^wj and d^/v^^ if w^ is positive, or to dj/wj and 

 dj/wj if wj is negative. Hence for any point {r,0) on a streamline through 

 (tj, (9j) in the circle, we have 



A (0 - e^) > en(r/ri) > B (0 - 0^). (30) 



K A and B are zero, the streamlines are circles and the origin is a whirl- 

 point. If A is zero, the streamline through any point in the circle cannot cross 

 the circle, because r cannot increase and B is zero, so that A cannot approach 

 the origin. 



Suppose A is not zero. Consider a point (r^, 0^) with r^ = Re'-^"'^' inside 

 the circle. If the streamline through the point touches the circumference, Eq. 

 (30) would require that for the point on the circumference, A (0 - 0^) >2tt \A\, so 

 that and 0^ differ by more than 2n . Such a streamline can provide a seg- 

 ment meeting the requirements of the whirlpoint. If, on the other hand, the 

 streamline approaches the origin, B is different from zero and there is an in- 

 termediate point where r = r^ e'^ '^' . Then, from Eq. (30) we have 



- 277 |B| > B (0 - 0^) 



In this case, the streamline also would provide a segment required for the 

 whirlpoint. If A is zero, but B is not, the above argument can be applied to any 

 point rj <R . Thus, in any event, the point of zero velocity where the swirl is 

 positive is a whirlpoint. 



446 



