Tulin and Shwartz 



causing annihilation. In the case of complete annihilation, A, = 0. 



Momentuni Conservation . The momentum, Mj, of the vortex 

 pair is conserved in motion through a homogeneous medium. It 

 may on account of similarity be expressed in the form, 



— z = K (4j) • W-R = const. (11) 



p m ' B 



where K(ijj) is a momentum coefficient. Its form may be deduced 

 through use of the identity. Lamb [ 1945] , pg. 229, 



P 



= ^^ytididi (12) 



The latter integral can be taken separately over the interior and 

 shear layer of the vortex pair, which yields terms upon making use 

 of similarity, 



H 



int. 



« 



Y; d^ d^ « WjR^ (13) 



YC di d^« (We - Wi)R^ (14) 



shear layer 



As a result, the form of KJ[ij() consistent with our assumptions, 

 is seen to be , 



KJ4;) = K, - Kgij; (15) 



where K. and K, are undetermined constants. 



Energy. The total kinetic energy in the vortex pair motion, 

 taking account of similarity, may be expressed as the sum of two 

 terms , 



^^' = K. wfR^ + K. wf R^ (16) 



p ' ' * • 



while the dissipation taJkes the form. 



332 



