Studies on the Motion of Viscous Flows— I 



Bu^, 3r + u/r + Bu^/^z = • -^ ■ ' ' (39) 



introduces some differences. Since any element of a meridian plane has zero 

 angular velocity, the positive value of swirl » - Z?^ is a characteristic property 

 of an eddy where a and /3 as given by Eq. (32) can be expressed in the cylindri- 

 cal coordinates .^ ^, 



a = (Bu,/Br) (Bu/Bz) - (Bu^/Bz) (Bu/Br), (40) 



/S = - (Bu^'Br + Bu^/Bz)/2 = u^,'2r. . V: ,•..,-/ 



Hence, from Eqs. (33) and (38), 



V^p - pV • F = 2p (a - /32) - 6p,52. 



The characteristic dynamic property of such an eddy is then 



V-V - PV • F > - 3p (u^/r)2 /2 -•■- •' ' '-" (41) 



In particular, a particle at rest is in an eddy only if v2p _ pV • f is positive. 



Since Theorems I and II can easily be extended for the axisymmetric case, 

 all the kinematic properties, with minor modifications, are valid for axisym- 

 metric situations. -, , ^ 



CONCLUDING REMARKS - 



Mathematical representation for the angular velocity W of a fluid line ele- 

 ment and of the angular velocity of a fluid surface element are derived here. 

 These angular velocities are shown not to depend on expansion u; ^. Vorticity, 

 on the other hand, is shown to be three times the average of either velocity. 

 The relation of curvature of the fluid line element, its velocity, and its angular 

 velocity is derived, but to complete this fundamental relation it is necessary to 

 develop the equivalent of Frenet's formula in terms of the Eulerian derivative. 

 This has recently been obtained by Paul Lieber and his former student, Kirit 

 Yajnik, on the basis of the work presented in this paper. The results bring out 

 the differential-geometrical features of torsion of the fluid line element. The 

 behavior of the line element in the vicinity of a singular point is decisively in- 

 fluenced by the angular velocity W of the line elements at the singular point. 

 The new definition of an eddy given in the present paper and its relation to 

 whirlicity as it is defined here, has been deduced. In Theorem 3, which gives 

 the relation between whirlicity and eddy, we have considered whirlicity to be 

 positive. However, by considering it to be zero and/or negative, interesting 

 results have also been recently obtained by Lieber and his student, which evi- 

 dently concern the development of secondary and turbulent flows. These recent 

 results which rest incisively on the results of the present paper, will be pub- 

 lished in due course, and will point up the significance of the results presented 

 here. 



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