27 



these transitions occur vary with the liquids. Photographs of typical shapes are shown in 

 Figure 18. It should be noted here that some of the shapes shown in these photographs are 

 instantaneous shapes, since the shape of large bubbles does not remain constant during the 

 ascent. An exception are bubbles rising in a highly viscous medium (e.g., mineral oil and 

 corn syrup). 



The results for the spherical bubbles only are plotted in terms of the drag coefficient 

 and the Reynolds number in Figure 19, with Reynolds numbers ranging up to about 400. The 

 drag curve for rigid spheres is also included. 48 From it, the following can be observed: The 

 drag curves of spherical bubbles in the various liquids fall between two limiting curves. As 

 upper limit, the drag curve of rigid spheres is obtained, while the lower limit is the drag curve 

 for fluid spheres. With decreasing Reynolds number, the rigid sphere curve connects with the 

 straight line of Stokes' Law, while the fluid sphere curve connects with the line of Hadamard- 

 Rybczynski's Law. The curve for the fluid spheres was obtained by drawing the lower en- 

 velope to the experimental curve; its accuracy can be confirmed by additional tests in other 

 liquids or by extension of the theoretical solution into regions beyond that of very slow flow. 



It will be noted from Figure 19 that the curve for mineral oil, for example, follows the 

 straight line of Hadamard-Rybczynski's Law over a certain region of Reynolds numbers. 

 This indicates that the boundary conditions assumed in the analytical solution for fluid 

 spheres are actually fulfilled and that circulation exists inside the bubble . Circulation in- 

 side bubbles has been observed experimentally. 49 



The experimental curves of Figure 19 also indicate an interesting aspect of the phe- 

 nomenon of bubble motion, namely that with decreasing Reynolds number, the drag coefficient 

 of the bubbles becomes equal to the drag of rigid spheres. This transition may occur at a 

 Reynolds number of about 40 (as for filtered and distilled water) or may not take place until 

 very low Reynolds numbers are reached, i.e., well within the region of slow flow (as for olive 

 oil 10 or very viscous syrup 12 ). Thus, from the experimental data available, it appears certain 

 that tiny air bubbles rising in any liquid follow Stokes' Law. 



For bubbles behaving like rigid bodies, thus indicating absence of motion inside the 

 bubble, the internal Reynolds number (although nonvanishing) is of no significance in de- 

 scribing the rising motion of the bubbles. Likewise, the internal Reynolds number cannot be 

 used to predict the transition point at which the drag of the bubbles becomes less than that 

 of corresponding rigid spheres. Beyond this transition point, the internal Reynolds number 

 might be of importance in describing the motion of the bubbles. 



Surface tension tends to make the- surface area of the bubble as small as possible. 

 For a given volume, the configuration of minimum surface area is a sphere. This effect of 

 surface tension would be most pronounced for bubbles of small radii. 



