42 



surface of the bubble must vanish. The surface must then be able to hold molecules of the 

 pure liquid, just as in the case of tap water and surface-active substances the surface at- 

 tracts and holds a high concentration of particles or molecules of the surface-active sub- 

 stances. The molecules at the surface would travel with the bubble and hence would, in ef- 

 fect, give the same boundary conditions as a rigid surface. As the shear forces become 

 larger in comparison to the forces holding the molecules at the surface, "rigidity" at the 

 surface cannot be maintained; circulation inside the bubble ensues and the drag of the bubble 

 becomes smaller as compared to that of a rigid body. 



In addition, since it was not possible to correlate the results of the experiments on 

 the motion of bubbles in the gravity pressure field in terms of nondimensional parameters 

 formed from the usual liquid properties (viscosity, surface tension, density), further work on 

 freely rising bubbles is necessary before the results obtained from such tests can be utilized 

 and before the more complicated behavior of bubbles in variable pressure gradients can be 

 understood. Particularly, an understanding of the reason for the transition of bubbles from 

 fluid to "rigid" bodies as well as a criterion for this transition is most desirable, since such 

 transition might be influenced by the magnitude of the pressure gradient. In the region of 

 Reynolds numbers where the bubbles behave like rigid spheres, the pressure gradient probably 

 has no effect on the drag coefficient. 



SUMMARY 



As the size of the bubbles was increased in the tests, a change in bubble shape from 

 spherical to ellipsoidal to spherical cap shape was observed in all liquids. The volumes at 

 which these transitions occur, however, varied with the properties of the liquid. For spheri- 

 cal bubbles of given volume the results show that the viscosity of the liquid is the most im- 

 portant property determining the rate of rise. For ellipsoidal bubbles, the surface tension 

 assumes greater importance. Spherical cap bubbles rise independently of the properties of 

 the liquid. 



The results show that the motion of air bubbles rising at their terminal velocity in a 

 gravity field cannot be described completely by use of dimensionless parameters formed from 

 the usual liquid properties (viscosity, surface tension, density), the equivalent radius of the 

 bubble, the acceleration of gravity, and the terminal velocity. 



The drag coefficients of tiny spherical bubbles coincide with those of corresponding 

 rigid spheres. With increase in bubble size, a decrease in the drag as compared to that of 

 rigid spheres occurs in some liquids. This change in the drag is due to the development of 

 circulation inside the bubble. The drag curves of the spherical bubbles rising in various 

 liquids fall between two limiting curves, namely the drag curves of rigid and fluid spheres, 

 respectively. It was not possible to determine a criterion for the transition region of the 

 bubbles from "rigid" to fluid spheres. 



The region of ellipsoidal bubbles extends over different ranges of Reynolds numbers 



