HISTORICAL BACKGROUND 



Considerable theoretical and experimental work has been done on the 

 motion of bubbles in liquids. However, no single report presents a compre- 

 hensive discussion of the subject . Most of the theoretical work has been con- 

 fined to a dimensional analysis with suggested solutions for the rate of rise 

 for bubbles in the Stokes region of flow. The experimental work has been de- 

 voted to measurements of the rate of rise of bubbles in various liquids with 

 some observations as to the bubble shape and path but there is considerable 

 question concerning the reliability of some of the techniques involved. 



DIMENSIONAL ANALYSIS 



In nearly all papers on the motion of bubbles in liquids, the au- 

 thors have recognized the following factors as pertinent: 



U the velocity of rise of the bubble, 



g the acceleration of gravity, 



p the density of the liquid, 



I the length parameter indicative of the bubble size, 



ft the coefficient of viscosity of the liquid, and 



y the surface tension for the bubble-liquid interface. 



Although this list does not complete the set of quantities needed to specify 

 the system, the effect of other factors such as absolute pressure and the vis- 

 cosity and density of the gas constituting the bubble are believed to be neg- 

 ligible. A convenient length parameter is the radius of the bubble. Since 

 large bubbles are not spherical, they are described in terms of the equivalent 

 spherical radius r which is the radius of a sphere having the same volume as 

 the bubble . 



Applying the methods of dimensional analysis, 4 it is possible to 

 group these factors into three dimensionless parameters in terms of which the 

 relation among the six physical quantities can be described. The usual choice 

 of combinations results in a statement of the following form: 



,gr 2Ur Pv U 2 

 1V U 2 u/p y ' 



For bubbles rising at their terminal velocity in liquids, the first 

 ratio will now be shown to be related to the drag coefficient 



s Drag force [2] 



U JpU 2 A 



