Studies on the Motion of Viscous Flows— I 



eddy. In the case of incompressible fluids, if at least one streamline is closed 

 around a point of zero velocity in each of its neighborhoods, and if all the deriv- 

 atives of velocity do not vanish, the point is in an eddy. Furthermore, if a point 

 of zero velocity is in a plane eddy, there is at least one closed streamline 

 around it in each of its neighborhoods. Similar properties of axisymmetric 

 eddies are also discussed. 



These results thus describe the properties of eddies in terms of angular 

 velocities, vorticity, spiralling or closed streamlines, and the convexity of 

 streamlines and low-pressure spots on the basis of the new concept of eddy 

 which differs essentially from the prevailing notion of concentrated vorticity. 



ANALYSIS OF ROTATION 



Angular velocities of elements of material curves and surfaces are defined 

 and their properties are presented in this section. Let the location of a material 

 filament at time t be given by x = x(h, t), the parameter h remaining un- 

 changed for a given particle during the motion. The rate of change of a differ- 

 ential element dx = (3x/3h) dh is then given by '■ • 



dx = (32x/3hBt) dh = (3u/Bh) dh = dx • Vu , ' ' ' ' ' (1) 



where a superposed dot indicates a material derivative and u is the velocity 

 field. The rates of change of the length ds and the unit tangent vector t of the 

 line element are then governed by 



ds t + ds t = ds t" • Vu , (2) 



where 



dx = ds t , t • t = 1, t • t = . " ■' (3) 



The rotation of the element is most conveniently described by the angular ve- 

 locity w given by t x t, since it can be seen from Eq. (3) that 



t = W X t . (4) 



One can verify that the angular displacement in time interval t is equal to the 

 magnitude of wAt to the first order of t. 



The angular velocity w can be evaluated from Eqs. (2) and (3) as 



W = t X (t • Vu) =: tx (V • t) , (5) 



where the deviatoric part of the velocity gradient is given by ■•■ 



V = [Vu - (V -U) 1/3]''" (6) 



and I is the identity diadic. 



439 



