available.* However, MOller 44 showed, by means of a dye technique, that the flow about a 

 rigid sphere at a lower Reynolds number and larger boundary dimensions was identical to that 

 at a higher Reynolds number and smaller boundary dimensions and therefore that the effect 

 of the walls was to stabilize the flow about the sphere. These results for rigid spheres be- 

 yond the Stokes region of flow at least suggested the possibility of a similar effect for the 

 motion of gas bubbles. 



CYLINDRICAL BUBBLES 



At this point, it may be of interest to mention experiments on a special form of finite 

 boundary dimensions, i.e., the case of cylindrical bubbles. This term was first used by 

 Gibson 45 and applies to the type of bubble formed when a long cylindrical tube filled with 

 liquid is emptied from below or when a large amount of air is introduced through the bottom 

 of the tube. 



Gibson investigated the velocity and shape of these bubbles in water. Ward and Kess- 

 ler 16 conducted tests in pipes of various diameters. Hattori 46 was interested in the problem 

 in connection with the possibility of evaluating the surface tension of a liquid. Hence, he 

 was concerned with tubes of small diameter, since the so-called critical tube diameter (below 

 which the bubble no longer rises but remains stationary) is a function of the surface tension 

 of the liquid. Dumitrescu 47 obtained an analytical expression for the velocity of a cylindrical 

 bubble by neglecting viscous and surface tension forces, thus reducing the problem to one of 

 potential flow. The differential equation for the velocity potential together with the existing 

 boundary condition yields a solution for the velocity of rise as a function of the tube diameter 

 only. His experimental tests in water show that for a tube of sufficiently large diameter 

 (3 cm for water at room temperature) the measured velocities agree very closely with the 

 theoretical values. Therefore, for large bubbles, the physical properties of the liquid no 

 longer have any effect on the flow about the bubble and the bubbles are geometrically similar. 

 Davies and Taylor 22 investigated the shape and rate of rise of cylindrical bubbles in order 

 to obtain a better understanding of the pressure distribution of spherical cap bubbles in an 

 infinite medium. 



SCOPE OF THE PRESENT INVESTIGATION 



The present investigation was initiated in connection with a program of study of the 

 behavior of air bubbles in water at variable pressure gradients. Since extensive experimenta- 

 tion was required for direct experimental study of the motion of air bubbles in such pressure 



*A pap • by Coppock and Meiklejohn has recently come to the attention of the authors. From tests con- 

 ducted with air bubbles in water, they conclude that no wall effect exists for bubbles ranging in equivalent 

 radius from 0.01 to 0. 1 cm rising in a tube of 5 cm diameter. 



