-3- 417 



For the larapst of trip snail DuBDies in the waKe, viscous strr-sses would produc; such a 

 oistortion that the fornu)a (1.2) woulo not be rxpcctta to apply. 



The jniforiiitj of th.- vc-locity of ris-: of thr suDDles, .'.nd tn( order of magnitude of thr 

 experimental error in tne mc.iSurement of th^ velocity tiv.y be jud-'/d from Figure 2, in which, t imo t, 

 and the vertical o ispl acwKht , X, of th;. two ijuSDles ire plottr<1 ,>s -bscissae and ordinates, respectively. 

 Tnr actual measured values of X and t for the bubbi? of Kigur' 1 urc indicated by .the circular dots 

 in Fiiture 2. - nd tf.ose for a second, l-.rycr bubblr t.y crosses; thQ straight lines of closest fit drawn 

 through the ucs.^rv.;d points cr- rt.-notcd by 'A' :nd 'B' nspectively. It will be seen that the scatter 

 of the experimental points is not t'xcessivo and that the velocity of rise jf the two bubbles is reasonably 

 constant over the interval mc.sured. 



The shape of the prvfilt of the bubbles was f und by mc:-suriny the films ' n a travelling 

 micr^sci.pe fittto with tw ' independent motions at right -■ncjles t- jne -nother. The results f-r the 

 Uwer ph^t. graph' ^f Figure 1 are shown graphically in Figure 3 where the circular dots represent points 

 on the central, regular portion of the profile of the bubble, deduced from the microscope readings. 

 In Figure 3, the vertical and horizontal axes are parallel to the corresponding axes in the tank and the 

 origin is taken at the uppermost point on the bubble; the dirvsnsions given in Figure 3 refer to the 

 actual size of the bubble. The crosses with vertical axes and with sxes at ns" to the vertical in 

 Figure 3 represent points on the profile of the same bubble, obtained from measurements of photographs 

 taken 105,7 and 132.5 milliseconds earlier than the lower photograph ef Figure 1. The agreement betwe(^n 

 the three sets cf points shows that the shape vf the cap if the bubble undergoes very little variatitn 

 ever the rjnge f time covered by the three phcit.-graptis. 



The curve in Figure 3 is an arc of circle of radius 3. 01 cm., drawn to pass through the origin 

 and since the scatter of the observed points around this curve is within the limits of the errors made in 

 measuring the film, the upper part of the bubble is e portion of a sphere within the experimental error. 

 It is worth noticing that the angle subtended at the centre of the circle by the arc in Figure 3 is about 

 75°, whilst the angular width cf the whole bubble in Figure 2 (referred to the centre) is ibout 90°. 



The vertical motion of a gas bubble with a spheri cat cap. 



Measurements of the pressure over the front part of the surface of a sphere exposed to a wind 

 shows that the distribution of pressure is very similar to that calculated assuming ideal hydrodynamic 

 flow. similar measurements over a solid spherical cap set with its vertex facing the wind show that 

 the removal of the rear part of the sphere does not greatly affect the pressure distribution over the 

 front except near tie rim of the cap. 



According to the hydrodynamic theory of ideal fluids, the pressure p at angle fp from the 

 stagnation point of vertex of a sphere (see inset in Figure 3) moving with uniform velocity u in a 

 fluid of density p is 



2 ' J 

 P = P^ + ipuMl-j sin^0) (2.1) 



where p is the pressure at the depth z, of the point concerned, at a great horizontal distance from 

 the sphere. 



Thus 



P^ = p^ + g pz (2.2) 



where p is the atmospheric pressure, and, at angle (p, for a sphere of radius R, 



z = d + R(i-cos(j;) (2.3) 



where d is the depth of the vertex. 



From equations (2.1), (2.2) and (2.3) 



, 9 , 



P = Pq + g /«) + 9 PR ( 1 - cos (^) ♦ ipu-'d-- sin^,^) ... (2.4) 



The 



