Studies on the Motion of Viscous Flows — I 



connection between maxima of vorticity (or its magnitude) and closed stream- 

 lines. It was simple to construct two flows satisfying a continuity equation in 

 such a way that closed streamlines existed in the absence of maxima of vorticity 

 in one flow, while in the other flow, maxima of vorticity occurred in the absence 

 of closed streamlines. These flows are given by the stream functions 



and - '^: 



and associated vorticity fields are given by 



^j = [1 + (x2 + y2)] -: - 



and 



[1 - (x2 + y2)] 



In the first flow, streamlines are concentric circles and vorticity is minimum 

 at the origin. In the second flow (Fig. 1), there are no closed streamlines, al- 

 though vorticity is maximum at the origin. Thus if there exists at all a one-to- 

 one connection between concentration of vorticity and closed streamlines, it 

 cannot be on the basis of kinematics or conservation of mass alone. However, 

 hydrodynamics does not provide such a connection, as the computations of Hung 

 and Macagno (1966, 1967) (10,11) show that the regions of closed streamlines in 

 two-dimensional and axisymmetric flows of a Newtonian fluid in a channel or 

 pipe with a sudden expansion do not possess interior points of maximum vorticity. 



This paper is devoted to the formulation of a new concept of an eddy related 

 to closed streamlines and to the study of its fundamental properties. The new 

 results obtained are summarized in the following paragraphs. 



The angular velocities of elements of material curves and surfaces are de- 

 fined and are shown to be determined by two tensors of third rank. The tensors 

 bring out the connection between these angular velocities, the vorticity, and the 

 dissipation in a Newtonian fluid of constant properties. The tensors have more 

 information about rotation than the average measure vorticity. 



The curvature and torsion of streamlines are shown to be determined by 

 the angular velocities and the velocity. The whirling behavior of streamlines 

 near a point of zero velocity is also shown to be determined by the angular ve- 

 locities. 



General conclusions regarding closed or spiralling streamlines in plane 

 flows are established in terms of the product of maximum and minimum angular 

 velocities of an element of material curve in the plane of motion. It is shown 

 that if the product is positive at a point of zero velocity, the streamlines spiral 

 around the point, and conversely, if the streamlines spiral around a point, the 

 product cannot be negative. Also if the product is positive, the curvature of the 



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