Yajnik and Lieber 



GEOMETRY OF STREAMLINES EST PLANE FLOWS 



We shall examine in this section closed or spiralling streamlines in the vi- 

 cinity of a point with zero velocity, in terms of angular velocities of line ele- 

 ments. The discussion is confined to plane flows which are, by definition, flows 

 in which the motion of the fluid takes place in parallel planes. The velocity is 

 assumed to be differentiable. Let an inertial observer be at rest relative to a 

 given fluid particle. With the z axis normal to the plane of motion, the equa- 

 tions of streamlines, which lie in the plane of motion, are 



dx/dh = u, dy/dh = v, 



(22) 



where the velocity components u and v relative to the observer depend on co- 

 ordinates X, y z and time t . The behavior of streamlines can be easily de- 

 scribed if the velocities are linear in x and y . From the results of ordinary 

 autonomous equations (Hurewicz, in 1958, for example) (18), one can describe 

 the streamline behavior in terms of 



S = flu ^ _ Im ^\ _ 1 fiiL + l^V 

 \Bx 3y 3y Bx/ 4 \3x ^y/. 



(23) 



If (3u/3x) (Bv/3y) - (3u/3y) (Bv/3x) is different from zero, the necessary and 

 sufficient condition for closed or spiralling streamlines is that S be positive at 

 the given particle. If, in addition, the flow satisfies the continuity equation of an 

 incompressible fluid, the streamlines are closed. It may be noted that nonzero 

 vorticity is necessary but not sufficient for positive S . 



The kinematic meaning of s , which we shall call swirl, can be readily un- 

 derstood. The angular velocity of a line element in the plane of motion can be 

 seen as parallel to the z axis from Eq. (8). It is given by 





- ^— I sin cos cp - ^— sm 



3v Bu 

 Bx By 



Bv , Bu \ r,,l^v Bu \ . „ 



B^ + ^) cos 20+ (^3^ - 3^) san 2 



(24) 



for an element inclined at an angle with the x axis. Its maximum and mini- 

 mum values are then 



\Bx By/ |\Bx Bxy \dy dx/ J J 



(25) 



Their product is the swirl, while their sum is the z -component of vorticity. 

 Note that positive swirl is associated with maximum and minimum angular ve- 

 locities of the same sign, and hence with the absence of any line element having 

 zero angular velocity in the plane of motion. 



444 



