Yajnik and Lieber 



that of Bertrand in 1868 (15). (See also, Truesdell in 1954 (13), and Ericksen in 

 1955 (16).) As with Eqs. (5) and (12), the two equations here are conjugate. 



The number of possible directions of line elements of zero angular velocity 

 and of normals to surface elements of zero angular velocity at a given point is 

 influenced by the characteristic equation 



det |l - Vl = 0, 



or (18) 



- \^ + M\ - N = 0, 



where 



. ■ x:-!.?'^ '•', M = - V: V 2, N = det |v| , t^ V = 0. •'•■ -^• 



The quantity m^ + 27N2 is proportional to the discriminant of the cubic equa- 

 tion. It will be called "whirlicity," as it has an important role in deciding the 

 whirling character of streamlines. 



We will recall the following theorem which is mainly in accordance with 

 Thomson and Tait (1867), Bertrand (1868), and Truesdell (1954). At any point 

 and time, there is at least one line element and one surface element with zero 

 angular velocity. The number of distinct directions of such line elements or 

 normals of surface elements are: (a) three; or (b) one, two, or infinity; or (c) 

 a number determined according to whether the whirlicity is negative, zero, or 

 positive. A direction refers to a vector a or -a . The details of the proof can 

 be found elsewhere (Yajnik, 1964) (17). 



Next we examine the behavior near a point of zero velocity. Suppose that a 

 streamline approaches it in such a way that the unit tangent vector t = x' ap- 

 proaches to a limit, say a . Suppose further that the length parameter increases 

 as the point of zero velocity is approached. Then for any given e - o , there is 

 some Sj such that 



|t-al<e=s>si. (19) 



As t and a are unit vectors, we then obtain 

 and, by integration, we find 



< t • a < 1, 



1 "T"/ ^^ " ^0 5 (x - Xj) • a < (s - s^). 



where xi is the position vector at s = sj. Since the central term in the above 

 inequality approaches to a limit as the point of zero velocity is approached, s 



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