131 



VERTICAL MOTION OF A SPHERICAL BUBBLE AND THE 

 PRESSURE SURROUNDING IT 



G. I. Taylor 



August 1942 



Summary, 



The upward motion of the spherical hollo* which Is forined in water by an explosion is 

 discussed on the assumption that the spherical 'orm is preserved. During the expanding stage 

 the buoyancy of the hollo* gives a large amount of vertical momentum to the surrounding water. 

 During the contracting stage this momentum is concentrated round a rapidly diminishing volume. 

 An intense concentration of vertical momentum is thus produced in the neighbourhood of the 

 vertical line through the charge. The pressure at points above the charge and near this line 

 rises to considerable values owing to two causes:- 



(a) Whun the hollow is near its greatest contraction at the end of its first oscillation 



its centre is nearer to points vertically abcwe the seat of the explosion than it was 

 at the time of the explosion (but further from points on the level of the explosion). 



(s) The pressure due to the combination of vertical motion and exoansion on the second 



expansion is in certain circumstances large. 



^or comparison with observation, the maximum displacements to be expected in a 

 rectangular plate 6 feet x « feet x 0.173 inches attacked by 4.65 lb. of T.N.T. placed 

 (a) 14 feet below the plate; (b) 14 feet away horizontally are calculated. They are found to 

 be 4,3 and 1.5 inches respectively. The observeu displacements were 4,35 and 1.15 inches. 



Though complete dynamical similarity is not possible when charges of varying, sizes are 

 exploded under water, yet in certain circumstances dynamical similarity should be very nearly 

 attained. It is shown, however, that there are large ranges of depth and charje weight which 

 could not be explored even approximately by model experiments unless the experimental apparatus 

 were so constructed that the atmospherical pressure above the water surface could be reduced. 



The radial motion of tne water near the seat of an explosion has been studied by many 



writers. Recently Conyers Herrings has analysed the small change in shape which the expanding 



spherical hollcw suffers owing to the varying hydrostatic pressure in the water due to gravity. 



He showed that during th" period whi;n this effect is sitbII it merely displaces the sphere vertically 



without changing its shape. This vertical motion may be regarded 3S being due to the vertical 



moiTientum given to the fluid surrounding the spherical hollow by tne floating power of the hollow. 



The momentum associated with a spherical noUow of radius a moving with velocity - St is - i 

 ,u J, a? ' dt 2 



(, TT-pa-^) g^ , z being the depth of its centre and p thp density of water, The rate at which 



upward momentum is communi edited by the floating power of thi hollow is -^tt p.i g, so that 



m 



j a^dt (1) 



Starting with a very small at t = 0, the time of the explosion, it will be seen that during 

 the early stages while a is increasing rapidly - dz/dt is less than 2gt, i,e. the hollow rmves 

 upwards more slowly then it would if a were constant. On the other hand, during the period while 

 the hollow is large J a dt acquires a considerable magnitude so that when the bubble contricts. 



