Studies on the Motion of Viscous Flows — I 

 ,^. ^*i: jk = Cejji u^ J + Cj^. Uj J )/2 , /. ,' ^/ '%'' 

 ... = (^ji "k6 + ^'«ki Vj,)/2, .,,, , ,,^..^,- (13) 



SO that ■ • - 



w*i = J^trjk "jf^k • (14) 



Thus each component of the angular velocity of the normal can also be written 

 as a quadratic form in the direction cosines of the normal and the coefficients 

 constitute a tensor of third rank. The tensor R*. j,^ will be called the adjoint 

 rotation tensor for reasons which will become apparent shortly. This tensor 

 and the angular velocity is also independent of expansion v- u . 



For brevity, we will call w the angular velocity of a surface element, al- 

 though it is the angular velocity of the normal. 



Vorticity w. can be readily expressed in terms of the rotation tensors 

 from Eqs. (7) and (13) as 



w. = R... = R*. .. . -, . A ': . ,;,.. (15) 



It is now easy to show that vorticity is an average measure of the angular veloc- 

 ities. In terms of the spherical polar angles and ^ (tj = cos 9 cos 0, t2 = 

 sin 6 sin 0, and t3 = cos (p), the average overall direction is 



J J Wj sin d6' d0 " ' ' • ^ ^ ■ ■■ 



= (Rj. jk^TT) j J tjt^ sin d0 d<^ 



I J ^i" 



(16) 



^i: jk ^k^^ = *i^3 



A similar computation can be made for w* . Vorticity is therefore three times 

 the average of the angular velocities of line or surface elements taken over all 

 possible directions. It is well-known that vorticity is proportional to different 

 types of averages, such as the average over three mutually normal directions 

 or all directions in a plane (Cauchy, 1841, and Truesdell, 1954) (12,13). 



Let us now consider a line element in the direction of t or a surface ele- 

 ment normal to n such that the angular velocity of the line or surface element 

 is zero. Then from Eqs. (5) and (12) 



t X (V • t) = 0, (n • V) X n = 0, 



or ' ' ' 



(\I - V) • t = 0, n ■ (\I - V) = (17) 



for some real k . The above conditions are necessary and sufficient for the an- 

 gular velocity of a line or surface element to be zero. The first condition was 

 essentially the contribution of Thomson and Tait in 1867 (14), and the second 



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