41 



range of bubble sizes did not extend fully into the region of spherical cap bubbles, it is quite 

 certain that the presence of surface-active substances will not alter their rate of rise, which 

 was shown to be independent of all physical properties of the liquid. 



The difference in behavior of the air bubbles must be sought in the behavior at the 

 surface. A high concentration of molecules of the surface-active substances will be found 

 at the surface of the bubble. As for the case of the tap water, these molecules would travel 

 with the bubble and would impart, in effect, a "rigid" surface to the bubble, that is to say, 

 would impose the condition of zero velocity at the boundary. 



Thus, the results of the tests given in Figure 29 show that these surface-active sub- 

 stances increase the drag of the bubbles (in the region of bubbles having equivalent radii of 

 0.03 to 0.30 cm); beyond the critical concentrations, any increase in concentration has rela- 

 tively little influence on the drag of the bubbles. 



BUBBLES AS RIGID BODIES 



In previous sections it was shown that bubbles rising in pure liquids behave essen- 

 tially like fluid bodies over a large range of bubble size, but that below a certain critical 

 size (the size being different for various liquids), the bubbles behave like rigid bodies, that 

 is to say, the drag of the bubbles equals that of corresponding rigid bodies. 



A possible explanation of the anomaly of behavior of the gas bubbles is given by 

 Boussinesq's dynamic surface tension. 5 As pointed out under "Theoretical Solutions," 

 for small bubbles the effect of the dynamic increment of surface tension increases the drag 

 to the value of corresponding rigid bodies. With increase in bubble size, this effect becomes 

 negligible and the drag of the bubble equals that of a fluid body. The dynamic increment in- 

 cludes a constant of proportionality (surface viscosity) which is a function of the two fluids 

 composing the interface. Therefore, for air bubbles, the transition region from "rigid" to 

 fluid bodies would be different for the various liquids. There is, however, no experimental 

 evidence that dynamic surface tension, as postulated by Boussinesq, exists. 



From a "hydrodynamic" point of view, the reason for the transition of the bubbles to 

 "rigid" bodies is not clear. As mentioned earlier, the mere inclusion of surface tension as 

 a pressure drop in the boundary conditions does not alter the analytical solution for fluid 

 spheres. Hence, it appears that the presence of surface tension should have no effect on the 

 motion of the bubble, except in maintaining the spherical shape. Thus the anomalous behavior 

 of the bubbles cannot be explained in terms of "hydrodynamics," but must be sought in terms 

 of a surface phenomenon. If it could be shown analytically (if only in the region of slow flow) 

 that equality of drag of corresponding rigid bodies and bubbles also implies equality of bound- 

 ary conditions at the surface, then, as in the case of rigid bodies, the velocity of the entire 



♦Mixtures such as the 13 percent ethyl alcohol-water mixture are included in this category. 



