Formation of Coreless Vortices. 255 



ginary ; and the proper modification, from (7) forwards, gives 

 for these and such cases, instead of (14), the following: — 



tww+zf^m • • • {1V) - 



The result is easily written down for each of the two last 

 cases [Examples (5) and (6)]. 



XXVIII. On the Formation of Coreless Vortices by the Motion 

 of a Solid through an Inviscid Incompressible Fluid. By 

 Sir W. Thomson, LL.D., FM.S* 



FipAKE the simplest case : let the moving solid be a globe, 

 -«- and let the fluid be of infinite extent in all directions. 

 Let its pressure be of any given value, P, at infinite distances 

 from the globe, and let the globe be kept moving with a given 

 constant velocity, V. 



If the fluid keeps everywhere in contact with the globe, its 

 velocity relatively to the globe at the equator (which is the 

 place of greatest relative velocity) is |-V. Hence, unless 

 P>f Y 2 f, the fluid will not remain in contact with the globe. 



Suppose, in the first place, P to have been >f V 2 , and to 

 be suddenly reduced to some constant value < | V 2 . The fluid 

 will be thrown off the globe at a belt of a certain breadth, and a 

 violently disturbed motion will ensue. To describe it, it will 

 be convenient to speak of velocities and motions relative to the 

 globe. The fluid must, as indicated by the arrow-heads in fig. 1, 

 flow partly backwards and partly forwards, at the place, I, 

 where it impinges on the globe, after having shot off at a tan- 

 gent at A. The back-flow along the belt that had been bared 

 must bring to E some fluid in contact with the globe ; and 

 the free surface of this fluid must collide with the surface of 

 the fluid leaving the globe at A. It might be thought that 

 the result of this collision is a " vortex-sheet," which, in virtue 

 of its instability, gets drawn out and mixed up indefinitely, 

 and is carried away by the fluid further and further from the 

 globe. A definite amnout of kinetic energy would thus be 

 practically annulled in a manner which I hope to explain in 

 an early communication to the Royal Society of Edinburgh. 



But it is impossible, either in our ideal inviscid incom- 

 pressible fluid, or in a real fluid such as water or air, to 



* Communicated by the Author, having been read at the Meeting of 

 the Royal Society, 3rd February. 1887. 



f The density of the fluid is taken as unity. 



