234 - 8 - 



Ttie hollow which appears in photograph No. 9 of Figure 3b subsequently disappears as the bubble 

 expands a second time but reappears on its second contraction. Photograph No. 19 of Figure 8c appears 

 to have a flat undersurface but the fact that it is really only a thin hollow shell will be seen in kg. 20, 

 where the bubole is at its second minimum (see Figure 9) and the shell has contracted to vanishing thick- 

 ness over a large part of its area. 



The very rapid rise near the minimum radius, which is so characteristic a feature of the simple 

 theory, depends on the fact that the virtual momentum of a spherical hollow is^7T/3 a^U • i U (mass of 



3 2 



liquid displaced), so that a decrease in a corresponds with a rapid increase in u. if the bubble becomes 



flattened, the virtual Inertia rapidly increases if it flattens without changing radius. The virtual 

 inertia of an infinitely thin disk is actually rather greater than that of a sphere of the same diameter. 

 Thus if the "radius" of the bubble is t-dken as i (the horizontal diameter) instead of i (horizontal 

 diameter + vertical diameter), the measured curve will not reach such a low minimum and the calculated 

 vertical velocity will be reduced. Figure 9 shows both methods of estimating a near the minimum; the 

 points calculated by the former method are surrounded with a triangle, and by the latter with a circle. 



Even making this assumption as to a, the theoretical calculation predicts a larger rise of the 

 bubble than is observed in this experiment. some, but not all, of this can be accounted for as being 

 due to the effect of tho free surface. It seems probable that a wake is formed behind the bubble shortly 

 after it begin* to contract and that this greatly decreases the vertical velocity, so that at the time the 

 contraction of radius becomes very rapid the bubble has far less virtual inertia in a form in which it can 

 be concentrated by the contraction than it has according to theory. It may be that the effect is 

 accentuated Oy the hijh viscosity of the oil used in these experiments. Further light is thrown on this 

 by experiments made with a smaller bubble. 



10, Experiments with a smaller bubble . 



Another set of measurements was made with the pressure at the surface equivalent to 6.5 cm. of oil 

 and the spark at 6.05 cm. below the surface, but with a spark of smaller energy, namely that given by the 

 discharge of a 0.2/-t F. condenser charged to a potential difference of UOOO volts. The stationary plate 

 camera was used, giving a set of single photographs of the bubbles produced by the different sparks. The 

 results are shown in Figure lO. It will be seen that a^^^ = 2.1 cm., so that a^ax^^o " 2.1/(6.5 + 6.05) » 

 0.167. This lies between the value o.m for z" = 3.0 and 0.25 for z' = 2.0. By interpolation the value 

 of z' corresponding with ci /z = 0.167 is z' = 2.8. For this value of z" interpolation from Comrie's 

 curves gives ^^J^ = O.iu so that l = 2.1/O.uu = u.76 cm. For this value of L the time scale is given 

 by t/t' = i/(u. 76/981) = 0.070. The du rat ion of the f i rst period of osci llation for z" = 2.8 is given by 

 interpolation as t' = 0.51, so that t = o.5l x 0.070 = 0.035 seconds. This is larger than what is observed, 

 namely about 0.O30 seconds. The effect of the surface, however, is to decrease the period in the ratio 



(*-i> 



1 and taking a = 1.9 cm. this ratio is 0.92, so that the calculated period would be 0.0325 seconds. 



NO better agreement with observation could have been expected in view of the fact that each point in this 

 series refers to a different bubble. 



Referring to Figure 10 it will be seen that, after the bubble has ceased to rise at the rate 

 calculated on the assumption that it remains spherical while pulsating, the rate of rise fluctuates with the 

 pulsations but attains roughly constant mean velocity of about 80 cm./sec. If the resistance of the 

 bubble is expressed in the form R = k tt a^ (i^U^) where k is a drag co-efficient, R is equal to the buoy- 

 ancy, so that 277 s?pq = k 77 a^ (* pu^) or k = 8ag/3U^, and since the mean radius during the period while 



3 , 



the bubble is rising approximately uniformly is 1 cm. , k = (a x 1 x 98") + (3 x 80 ) = O.m. This is of 



the order of magnitude that might be expected. The drag co-efficient of a solid sphere at the same vatue 

 of Reynold's number is 0.56. 



It is worth noticing that the drag co-efficient of a spherical bubble, for the small values of 

 Reynold's numbers in the region where stokes' Law holds, is two-thirds the drag co-efficient of a solid 

 sphere of the same radius. It this relationship holds at the larger value of Reynold's number appropriate 

 to our rising bubble, the drag co-efficient would be 0.374. 



