' ^ ' 439 



trajectory and a mean taken. The results are plotted in Figure 2, together with Taylor and tevies' 

 points. In one 'ilm the time-scale was uncertain, and the points obtained from it are shown 

 separately on an arbitrary velocity scale. They have Deen included Because one of these points was 

 from a doublo bubble of the kind described above. The results indicate that its upward velocity is 

 definitely less than that of a single one of the same mean radius (Figure 3). 



Leaving this doubtful film out of account it will be seen that the remaining 11 points agree 

 with Taylot and Cavies' relation within the experimental error. Taylor and Oavies' values for the 

 radii of the caps of the bubbles extended from 2 cm. to 6 cm., while ours extend from u cm. to 15 cm. 

 The question of the stability of Bubbles of various sizes was examined by measuring the velocities of 

 9 bubbles that ultiretely split up, but no correlation between velocity and stability could be traced, 

 the Bubbles appearing to split at all velocities in the range studied. This seems to indicate that 

 we are near the limit of stability, which is perhaps fixed by the relative importance of surface 

 tension and hydrodynamical forces. By tending to keep the surface small, surface tension would act 

 as a stabilising influence, but would become less important for larger bubbles. Taylor and Davies 

 (1) mention that conditions had to be adjusted carefully tu obtain their bubbles. This agrees with 

 what we have found with our rather larger ones. 



If this interpretation of the results is correct, it would seem that the final upward velocity 

 of the products of an explosion, after the oscillations have ceased, and the bubble has broken up into 

 small ones, is of the order of 2 - 3 feet per second, compared with 17.2 feet per second inferred by 

 Taylor and Oavies for a 30O lb. charge on the assumption that the explosion products remain'as one 

 bubble. 



It is perhaps worth mentioning that the rate of rise of bubbles of exhaust gas from a torpedo 

 is known to be of the order of 2 feet per second (again of the same order of magnitude). 



Theoretical Considerations . 



Although it is clear from Taylor and Oavies' and our photographs that these bubbles have a 

 wake, the motion of the remainder of the water is probagly irrotational and it is of interest to 

 examine whether there are any solutions of the hydrodynamical equations which enable the velocity 

 and pressure conditions to be satisfied along a cap of a sphere of limited angle. A start on this 

 problem is made by Taylor and Davies (1), who show that the ordinary solution for a sphere in 3 

 uniform stream satisfies the conditions for continuity of pressure as well as velocity as far as terms 

 involving & (where 9 is the polar angle referred to an axis through the cap of the bubble) provided 

 that velocity and radius are connected by the relation u = /vga which has been confirmed 

 experimentally. if we take a more general velocity potential of the form , , ^^^ n 



1^ = _2 + _! + U r cos d, 



r r^ 



representing a combined source and dipole in a uniform stream, it is possible to satisfy the continuity 

 conditions as far as terms involving t? , provided that we assume that the profile of the cap is of the 

 form R = a + bj (9^ + bj9 . (it is unnecessary to introduce odd powers of 9). Alternatively, if we 

 assume A, zero, so that the bubble behaves as a simple source rather than as a dipole source, we can 

 still satisfy the equations as far as terms of the order 0^. The various solutions are tabulated 

 in Table 1. 



