20 



According to the available data, the relation of the drag coeffi- 

 cient to the Reynolds number for ellipsoidal bubbles is greatly affected by 

 the value of M. This is to be expected since in this range the bubble shape 

 is neither completely controlled by surface tension as is true for spherical 

 bubbles, nor almost completely free of it, as appears to be the case for spher- 

 ical caps. 



It is possible to surmise the role of the parameter M in determin- 

 ing the Reynolds number required for transition from ellipsoidal bubbles to 

 spherical caps. A little manipulation of M modifies it to the form 



if^fjVL)' 



Re 



r 7 r-y 



The terms in parentheses represent the ratios between pressure-gradient and 

 surface-tension forces and viscous and surface-tension forces, respectively. 

 For a given Reynolds number, M may be increased by Increasing either the pres- 

 sure gradient forces or viscous forces. If the pressure gradient is increased, 

 the buoyant force is increased. Since in the transition region, the drag co- 

 efficient has attained a constant value, the terminal velocity of the bubble 

 will increase. Therefore, in order to maintain the Reynolds number constant 

 the viscosity of the medium must be increased. Hence, increasing the pressure- 

 gradient forces results in an increase in the viscous forces. As a result, 

 surface-tension forces become relatively smaller as M is increased at a con- 

 stant Reynolds number. In addition, we know from experiment that increasing 

 the Reynolds number at constant M results in transition from ellipsoidal bub- 

 bles to spherical caps, Indicating relatively smaller surface-tension forces 

 with increasing Reynolds numbers. Since maintenance of the ellipsoidal shape 

 is due primarily to surface-tension forces, increasing M would result in tran- 

 sition to spherical caps occurring at a lower Reynolds number. The data sup- 

 port this conclusion fairly well. 



In most hydrodynamic problems we are concerned solely with the varia- 

 tion of drag coefficient with Reynolds number for a single liquid, water. The 

 form of M indicates that for a specific liquid at a given temperature, chang- 

 ing the pressure gradient is equivalent to varying M. Therefore in using dif- 

 ferent liquids we are obtaining the same sort of information as if we were to 

 vary the pressure gradient, provided our assumption concerning the relevant 

 physical variables is correct. 



