43 



for the various liquids. For liquids of low "M" number (say less than 10~ 3 ), a minimum in 

 the drag curve is reached at Reynolds numbers of the order of 250. These minima occur near 

 the transition from spherical to ellipsoidal shape. Such minima in the drag curve are not ob- 

 tained for liquids of high "<W" number. For the liquids used, transition to spherical caps is 

 completed at a Weber number of about 20. 



The drag coefficients of spherical cap bubbles are independent of bubble size and 

 have a constant value of 2.6- The rate of rise of these bubbles as a function of the equivalent 

 radius is given by the experimentally determined relation: 



t/ = 1.02J/£T 



For bubbles (ranging in equivalent radius from 0.03 to 0.25 cm) rising in tap water, an 

 increased drag as compared to bubbles in clean (filtered or distilled) water was observed. 

 The presence of certain surface-active substances in the water similarly increases the drag 

 of bubbles (ranging in equivalent radius from 0.03 to 0.30 cm) as compared to bubbles in 

 pure water. Beyond a certain critical concentration of these surface-active substances, an 

 increase in concentration has relatively little influence on the drag of the bubbles. 



Tests to determine the effect of the container walls on the velocity of rise indicate 

 the absence of such effect for the range of bubble volumes and container sizes tested. 



ACKNOWLEDGMENTS 



The authors wish to acknowledge the suggestions of Dr. Lawrence M. Kushner, National 

 Bureau of Standards, regarding the explanation of the behavior of bubbles in tap water and in 

 water containing surface-active materials. 



REFERENCES 



1. Rosenberg, Benjamin, "The Drag and Shape of Air Bubbles Moving in Liquids," 

 TMB Report 727, September 1950. 



2. Lamb, Horace, "Hydrodynamics," Dover Publications, New York, 1945, p. 599. 



3. Hadamard, J. "Mouvement permanent lent d'une sphere liquide et visqueuse dans un 

 liquide visqueux," Comptes Rendus, Acad. Sci., Paris, 1911, Vol. 152, pp. 1735-1738. 



4. Rybczynski, W., "Uber die fortschreitende Bewegung einer fltlssigen Kugel in einem 

 zahen Medium," Bulletin Academie de Sciences de Cracovie (Series A), 1911, pp. 40-46. 



5. Boussinesq, J., "Vitesse de la chute lente, devenue uniform, d'une goutte liquide 

 spherique, dans un fluide visqueux de poids specifique moindre," Ann. de Chimie et de Phys., 

 1913, Vol. 29, pp. 364-372, or Comptes Rendus, 1913, Vol. 156, pp. 1124-1129, also see 

 Comptes Rendus, 1913, Vol. 157, pp. 313-318. 



