- 3 - 229 



Condition (u) is rrore difficult to satisfy. It could be satisfied if (a) the model and full-scale 

 Bubbles were both produced by c-xplosion of gases contained in thin spherical envelopes such as rubber 

 balloons, provided the initial pressures of the uni^xploded ijases in the two cases were the same as (or any 

 given multiple of) those of the surrounding fluids, and (b) the densities of the two fluids are the same, 

 i.e., p. ' Py Such experimental conditions would ensure that conditions (3) and (u) were both satisfied, 

 but it would be difficult to devise 2 model experiment se that (») was satisfied when the full-scale bubble 

 is due to the explosion of a s-.^lid explcsive and the small bubble t- a spark. Fortunately, h;wever, there 

 Is not much difference between the bubble produced by releasing all the energy at a very small radius and 

 that produced by releasing most of it at very small radius and the rest during the expansion. or. Comrie's 

 curves showing the radius a as a function of time are very insensitive to the constant "C" which determines 

 the value of G(a)/W. 



The value of G(a)/M for the bubble produced by d m.:;ss M of j given explosive depends only on the 

 absolute preesure in the bubble. |f similarity of bubbles on two scale"; is obtained by choosing the 

 atmospheric pressure, weight of explosive and depth so that conditions (1), (2) and (3) are satisfied, 

 G(a)/M win be less at a given stage of the expansion in the small-scale experiment than In the large- 

 scale explosion, owing to the fact that the pressure is less. It seems thst. In comparing large-scale 

 underwater explosions with small-scale sparks in a liquid under reduced pressure, one is comparing bubbles 

 in which the constant c in Comrie's calculation varies from 0.06 up to 0.1 with the case where C is so small 

 as to be negligible. The difference between the two c^ses is not great. 



4. The effect of vi sco sity . 



To satisfy condition (5) in a small-scale experiment designed to represent a bubble in water would 

 require a liquid whose viscosity is very much less than that of water. no such liquid is available. 

 Fortunately, however, it is not necessary to satisfy this condition, because the loss of energy due to the 

 viscous forces opposing the expansion or contraction of the bubble is very small compared with the whole 

 energy of the bubble, even when a liquid of much greater viscosity than water is used. 



The rate of dissipation of energy owing to visctsity is 



j' 



the integral being taken over the surface of the sphere, and q being the radial velocity of the fluid. 

 Since q = aa Vr^, equation (7) gives 

 F = 8 TT /xaa^ 

 and the total energy, 1^, dissipated during time t is 



* .2 r^ • 



Vfx = e TT fA J aa dt = e TT /J. ] .?,i da. 



° ^ 



Equation (2) of Report a is now modified to the form 



— TTpa^gz + 2n pa^ k^ * -r-rrpa^J^ + 577^/ aa da = w - G(a) • (s) 



' \ 



Reducing to the non-dimensional form of equation (5) of Report 4 equation (s) becomes 



-, , Bq'^ ds 

 iMjJhrir.a W 



yjf ^ 2 77 3'3 I W I 6 \df / 3 



(9) 



I 77^ f , — > 



(pVg) i o ( dt 



