Tulin and Shwartz 



in a medium of uniform density with Ap = 0. The dissipation coef- 

 ficient in nature, D = Cq/PK, may be determined by a comparison 

 between theoretical trajectories such as given by (31) and obser- 

 vations of vortex pair rise in homogeneous media. As shown in 

 Fig. 6, such a comparison leads to the conclusion that D is quite 

 small (D < 0.2). 



The trajectory given by (31) may be compared to the law which 

 would apply if momentum were conserved, 



z^*j «= t (3Z) 



which coincides with (31) only if D = 1. The variance of observed 

 trajectories from the momentum law (3 2) is clearly seen in Figs. 5 

 and 6. 



Volume conservation in a self-similar flow leads to a linear 

 relation between the nominal radius of the mass and the height of rise 

 from the virtual origin z = 0: 



R = (3z (33) 



Conservation of mass takes the form 



^[(|)\r^*'p,]=2'2.r'*'p.WP (34) 



where Pg , is the density of the surrounding fluid at any given height 

 z and Pj is the average density within the rising mass, defined by 



(ly ^R^'J[p.(z) - p^(z)] = y (P - Pe) de (dil)^ d; (35) 



where the integration is taken over the entire volume of the rising 

 mass. 



The formulation of conservation of energy is based upon (27) 

 and (28) 



^ [f p,wV*' + k(p, - p,)g.R^*' ] = - C^, -^'r^*^ (36) 



where W and R are the observed and measured gross properties 

 of the rising mass while K and k are the coefficients of the virtual 

 kinetic and potential energies, respectively, defined in two dimen- 

 sions , e.g., by 



344 



