71 



will be devoted to some qualitative comments on the physical 

 causes of this flattening and of finer-scale departures from 

 sphericity. 



An easy way to see why a rapidly moving bubble must be- 

 come flattened is to consider the non-uniformity in the distribu- 

 tion of pressure over the surface of a bubble which is constrained 

 to remain spherical. In Taylor's approximation, where the bubble, 

 though constrained to be spherical, is free to expand or con- 

 tract and to move up and down, it is clear that the average pres- 

 sure over the surface of the bubble will equal the gas pressure 

 and that the first moment of the pressure will be zero. It is not 

 hard to show further that the second moment of the pressure, i.e., 

 the coefficient of the second order spherical harmonic in the 

 expansion of the pressure at the surface of the bubble, must in 

 this approximation be the same as for a uniformly moving rigid 

 sphere whose size and translational velocity are the same as the 



instantaneous values for the bubble. This can be verified from 



25 

 Taylor's explicit expression for the pressure , in which the 



2 2 

 only terms in cos 9 or sin 6 are simply proportional to the 



square of the velocity of migration, independent of radial velocity 



and acceleration. Now it is well known that in the steady motion 



of a rigid sphere through an incompressible frictionless fluid the 



pressure is higher at the front and rear stagnation points than 



25 

 '^See the article by Taylor in this volume. 



