32 



density, are of primary importance in the motion of these bubbles. 



Thus, in this region the Reynolds number inside the bubble is of no importance in the 

 description of bubble rise, since correlation was obtained in terms of drag coefficient, Rey- 

 nolds number, and "M" number. 



In Figure 21 the drag coefficient of the bubbles was plotted as a function of Weber 

 number. It is seen that, for the liquids tested, transition to a constant value of drag coeffi- 

 cient (spherical cap region) is reached at a Weber number of about 20. 



SPHERICAL CAP BUBBLES 



When the viscous and surface tension forces become small relative to the hydrodynamic 

 forces, the shape assumed by the bubbles is that of the so-called spherical caps. Typical 

 shapes of these bubbles are shown in Figure 18. The upper surface is essentially spherical, 

 while the lower surface varies from a highly irregular one for liquids of low viscosity to a 

 smooth surface for very viscous liquids. The configuration of the upper surface results al- 

 most exclusively from the hydrodynamic forces. 



The geometric similarity of these bubbles was shown by Rosenberg 1 and Davies and 

 Taylor, 22 who determined a constant drag coefficient of 2.6 for them. The results of the 

 present tests in a number of liquids confirm this value; Figure 21. The velocity of spherical 

 cap bubbles of given size rising in any liquid can be determined from the constant value of 

 the drag coefficient or directly from the velocity curve (Figure 15). For C D = (8/3) gr /U 2 

 = 2.6, we obtain for the rate of rise of the spherical caps in all liquids 



v = i.02y^7 



Thus, the velocity of rise of these bubbles is a function of the bubble size only and 

 not of the properties of the liquid (see special case of "Dimensional Analysis"). 



PATH OF BUBBLES 



Figures 22-25 show representative paths and corresponding shapes of bubbles in the 

 various liquids. Three types of motion of the bubbles were observed in the experiments: 

 (1) rectilinear motion, (2) motion in a helical path, and (3) rectilinear motion with rocking. 

 The motion of spherical bubbles is either rectilinear or helical. For ellipsoidal and spherical 

 cap bubbles, all three types of motion can occur. It appears that the type of motion may be 

 predicted from the value of the Reynolds number at which the motion takes place. Below 

 Reynolds numbers of about 300 the motion is rectilinear. With increase in Reynolds number 

 spiraling begins and increases in amplitude and frequency until a maximum is reached. At 

 Reynolds numbers of about 3000, the spiraling disappears and only rectilinear motion with 



♦See, e.g., Reference 22. 



