520 XI. HYDRODYNAMICS. 



The same problem gives us the motion of two equal vortex- 

 couples approaching each other head on, or a single vortex- couple 

 approaching a plane boundary, showing how as they are stopped they 

 spread out. The beharior of vortex -couples will serve to illustrate 

 that of circular vortex rings, for the theory of which the reader is 

 referred to Helmholtz's original paper. The two opposite parts of a 

 circular vortex appear to be rotating in opposite directions if viewed 

 on their intersection by a diametral plane normal to the circle, thus 

 resembling a vortex -couple. It is found that the circular vortex 

 advances with a constant velocity in the direction of the fluid in the 

 center, maintaining its diameter, but that when approaching a wall 

 head on it spreads out like the vortex -couple. Two circular vortices 

 approaching each other do the same thing, but if moving in the 

 same direction the forward one spreads out, the following one 

 contracts and is sucked through the foremost vortex, when it in turn 

 spreads out and the one which is now behind passes through it, and 

 so on in turn, as may also be shown for two columnar vortex-couples 

 traveling in the same direction. 



Most of these properties of circular vortices may be realized 

 with smoke rings made by causing smoke to puff out through a 

 circular hole in a box, or mouth of a smoker, or smoke-stack of a 

 locomotive. The friction at the edge of the hole holds the outside 

 of the smoke back, while the inside goes forward, establishing thereby 

 the vortical rotation. As previously stated no vortex could be formed 

 if there were no friction. It is to be noticed that the direction of 

 the fluid on the inside of the vortex gives the direction of advance. 



196. Irrotational Motion. We shall now consider the non- 

 vortical motion of an incompressible fluid. We then have a velocity 

 potential qp and 



-t r\r>\ d<P dty d<P 



106) . u=z?-> v = ^-i w = ^-- 



ox oy dz 



The equation of continuity becomes 



107) 4y = 0, 



and the potential is harmonic at all points except where liquid is 

 being created (sources) or withdrawn (sinks). The volume of flow 

 per unit time outward from any closed surface S is 



108) - / / [u cos (nx) -f v cos (ny) -f- w cos (nzj] dS 



so that if this is not equal to zero, it is equal to the quantity created 

 in the space considered in unit time, 



