Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-19T12:07:45.643Z Has data issue: false hasContentIssue false
  • English
  • Français

Distorted gas bubbles at large Reynolds number

Published online by Cambridge University Press:  29 March 2006

M. El Sawi
M. El Sawi
Affiliation:
Department of Mathematics, Imperial College, London
Present address: Department of Mathematics, University of Khartoum, Sudan.
Get access Rights & Permissions [Opens in a new window]

Abstract

The distortion of a gas bubble rising steadily in an inviscid incompressible liquid of infinite extent under the action of surface tension forces is investigated theoretically using an appropriate extension of the tensor virial theorem. A convenient parameter for distinguishing the bubble shape is the Weber number W. The virial method leads to an expression relating W and the axis ratio χ, of the transverse and longitudinal axes of the bubble. To first order in W, this relation agrees with the linear theory established by Moore (1959). Also, comparison of the results with his (1965) approximate theory reveals similar features and excellent agreement up to χ = 2. In particular, it confirms his prediction of the existence of a maximum Weber number. Although the present work does not consider the stability of these bubbles, it is interesting to note that the maximum value of 3.271 attained by W differs only by about 2.8% from the critical Weber number obtained by Hartunian & Sears (1957) for the onset of instability.

An approximate method for the study of slightly distorted spheroidal gas bubbles is also formulated and the resulting boundary-value problem solved numerically. The theory is then extended to include gravity. The joint effect of surface tension as well as gravitational forces has not been included in earlier theories. The shapes of the bubbles are traced and compared with the unperturbed spheroids. Comparisons for the velocity of bubble rise are made between the present predictions and some experimental results. In particular the results are compared with recent experimental data for the motion of gas bubbles in liquid metals.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 62 , Issue 1 , 8 January 1974 , pp. 163 - 183
Copyright
© 1974 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Chandrasekhar, S. 1969 Ellipsoidal Figures of Equilibrium. Yale University Press.
Chandrasekhar, S. & Lebovitz, N. R. 1963a Astrophys. J. 137, 1142–1161.
Chandrasekhar, S. & Lebovitz, N. R. 1963b Astrophys. J. 137, 1162–1171.
El Sawi, M. 1970 Ph.D. thesis, Imperial College, London.
Fox, L. 1957 Numerical Solution of Two-point Boundary Problems. Oxford University Press.
Haberman, W. L. & Morton, R. K. 1953 An experimental investigation of the drag and shape of air bubbles rising in various liquids. David Taylor Model Basin Rep. no. 802.Google Scholar
Harper, J. F., Moore, D. W. & Pearson, J. R. A. 1967 J. Fluid Mech. 27, 361–366.
Hartunian, R. A. & Sears, W. R. 1957 J. Fluid Mech. 3, 27–47.
Jones, D. R. M. 1965 Ph.D. dissertation, University of Cambridge.
Lamb, H. 1959 Hydrodynamics, 6th edn. Cambridge University Press.
Ledoux, P. & Pekeris, C. L. 1941 Astrophys. J. 94, 124–135.
Moore, D. W. 1959 J. Fluid Mech. 6, 113–130.
Moore, D. W. 1965 J. Fluid Mech. 23, 749–766.
Rosenkilde, C. E. 1967 J. Math. Phys. 8, 84–88.
Rosenkilde, C. E. 1969 Proc. Roy. Soc A 312, 473–494.
Saffman, P. G. 1956 J. Fluid Mech. 1, 249–275.
Shwerdtfeger, K. 1968 Chem. Engng Sci. 23, 937–938.
Siemes, W. 1954 Chem. Ing. Tech. 26, 614–630.
Weatherburn, C. E. 1930 Differential Geometry of Three Dimensions, vol. 2. Cambridge University Press.
Winnikow, S. & Chao, B. T. 1966 Phys. Fluids, 9, 50–61.