6 



For bubbles rising at their terminal velocity the drag coefficient C D can then be written as 



(8/3) gr e 



c d = : — 



u 2 



instead of its usual form of D/(l/2)p V 2 A where A is the projected area. For the analytical 

 solutions of Stokes and Hadamard-Rybczynski the drag coefficient becomes, respectively: 



C D =— (Rigid sphere) 



C D =— (Fluid sphere) 



For the special case in which only four variables, namely the velocity, the accelera- 

 tion of gravity, the density of the fluid medium, and the equivalent radius, are taken as per- 

 tinent, only one dimensionless group, the drag coefficient, is obtained, i.e., 



C D = constant 



This solution will be shown to apply to the region of very large (spherical cap) bubbles. 



PREVIOUS EXPERIMENTAL WORK 



RATE OF RISE OF BUBBLES 



Interest in the motion of air bubbles has existed for many years. The work on bubbles 

 has, however, been mostly experimental in nature. Exceptions are an attempted theoretical 

 analysis by Theremin 7 in 1829, the analytical solution of Hadamard, 2 - 3 Rybczynski, 4 and 

 Boussinesq, 5,6 and the dimensional analysis of Schmidt 8 and Rosenberg. x 



The early experimental work on bubbles was largely concerned with very small bubbles 

 and was carried out for the purpose of determining the extent of Stokes region. Allen 9 deter- 

 mined the rate of rise of air bubbles in water and in aniline up to bubble radii of 0.04 and 

 0.06 cm, respectively. Arnold 10 measured velocities of small air bubbles in olive oil and in 

 aniline. Bond and Newton 11 investigated air bubbles in syrup and in water glass (sodium 

 silicate). 



The range of bubble sizes was extended by other investigators 12, 13 who were mainly 

 interested in the problem in connection with air-lift pumps, 14 " 17 gas absorption, 18 " 20 or 

 propagation of sound in liquids. 21 These experiments were carried out in water. In subse- 

 quent years, some investigations were also made in liquids other than water. Davies and 

 Taylor 22 used nitrobenzene as well as water and measured velocities of large bubbles. 

 Temperley and Chambers 23 extended the range of Taylor's experiments in water to bubbles 

 of equivalent radii up to approximately 6 cm. Bryn 24 made tests in various water-glycerine 

 and water-ethyl alcohol mixtures. Robinson 25 measured the rate of rise of small air bubbles 



