Studies on the Motion of Viscous Flows — I 



A fluid particle in a plane flow may be regarded as being in an eddy, de- 

 pending on whether or not the streamlines, relative to an inertial observer 

 which is at rest with the particle, possess the whirling property in the particle's 

 vicinity. Alternatively, we may decide whether or not a particle is in an eddy on 

 the basis of positive swirl or the absence of a line element having zero angular 

 velocity in the plane of motion. Note that any line element normal to the plane 

 and any surface element parallel to the plane have zero angular velocity. 



Generalization to arbitrary three-dimensional flow is facilitated by the 

 theorem of Thomson (14) and others (see the section, Analysis of Rotation, 

 which was presented earlier in this paper). The theorem asserts that there is at 

 least one line and one surface element having zero angular velocity. Note also 

 that in flow u= Ay ^ v= , and w= Az , the line elements parallel to the x or z 

 axis have zero angular velocity, and that there are planes in which line elements 

 with zero angular velocity are absent although the streamlines do not show any 

 whirling property. Consequently, it is necessary to examine the line elements 

 in a chosen plane. Hence the following definition: 



A fluid particle is said to be in an eddy at a given instant if, at 

 that time, all the line elements, which are parallel to a sur- 

 face element of zero angular velocity, have nonzero angular 

 velocity. 



Any particle in a region of positive swirl in a plane flow is certainly in an 

 eddy. In particular, any particle in a rigidly rotating fluid or the core of a Ran- 

 kine vortex is in an eddy. It will be shown later that no fluid particle in a plane 

 Couette flow is in an eddy. 



The advantages of the above definition are many. Since the properties 

 given in this section include connections with streamline geometry, vorticity, 

 and low-pressure spots, the definition pinpoints a characteristic property of the 

 phenomena. Furthermore, it provides a basis for deduction of properties and 

 interpretation of experimental data. Several qualitative observations about the 

 flow with eddies can be readily explained, as will be seen later. The analytical 

 content of the definition can also be readily extracted in the form of the follow- 

 ing theorem: 



Theorem III: A fluid particle is in an eddy at a given instant, if and 

 only if its whirlicity 4M-^ + 27n2 is positive. 



The theorem of Thomson (14) and others, as was described earlier, ensures 

 that there is at least one surface element of zero angular velocity. With the 

 z- axis normal to such a surface element, V^^^ and V^ vanish according to Eq. 

 (12), and the angular velocity of a line element in the x-y plane is parallel to 

 the z axis and is given by 



[(V - V ) + (V + V ) COS 261 + (V - V ) sin 26']/2, (31) 



L V yx xy/ \ yx xy/ \ yy xx' ■^^■' *• J *■ > > ' 



e being the angle made by the element with the x axis. The swirl in the xy 

 plane, being a product of maximum and minimum velocities, is a- 0^ , where 



449 



