Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-19T03:04:09.341Z Has data issue: false hasContentIssue false
  • English
  • Français

On the equations of motion for mixtures of liquid and gas bubbles

Published online by Cambridge University Press:  28 March 2006

L. Van Wijngaarden
L. Van Wijngaarden
Affiliation:
Twente Institute of Technology, Enschede, The Netherlands
Get access Rights & Permissions [Opens in a new window]

Abstract

On the basis of previous work by the author, equations are derived describing one-dimensional unsteady flow in bubble-fluid mixtures. Attention is subsequently focused on pressure waves of small and moderate amplitude propagating through the mixture. Four characteristic lengths occur, namely, wavelength, amplitude, bubble diameter and inter-bubble distance. The significance of their relative magnitudes for the theory is discussed. It appears that for high gas content the dispersion is weak and then the conservation of mass and momentum lead to equations similar to the Boussinesq equations, describing long dispersive waves of finite amplitude on a fluid of finite depth. For waves propagating in one direction only, the corresponding equation is similar to the Korteweg–de Vries equation.

It is shown that for mixtures of low gas content the frequency dispersion is in most cases not small. Finally, solutions of the Korteweg–de Vries equation representing cnoidal and solitary waves in a bubble–liquid mixture are given explicitly.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 33 , Issue 3 , 2 September 1968 , pp. 465 - 474
Copyright
© 1968 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

Benjamin, T. BROOKE 1966 Discussion on Van Wijngaarden (1966), 6th Symposium on Naval Hydrodynamics Washington D.C.
Broer, L. J. F. 1964 On the interaction of nonlinearity and dispersion in wave propagation. Appl. Sci. Res. B 11, 273.Google Scholar
Campbell, I. J. & Pitcher, A. S. 1957 Shock waves in a liquid containing gas bubbles. Admiralty Research Laboratory Teddington, Middlesex, Rep. RI/G/HY/17/0.Google Scholar
Carstensen, E. L. & Foldly, L. L. 1947 Propagation of sound through a liquid containing bubbles J. Acous. Soc. Am. 19, 481.Google Scholar
Hsieh, D. Y. & Plesset, M. S. 1961 On the propagation of sound in a liquid containing gas bubbles. Physics Fluids, 4, 8, 970.Google Scholar
Korteweg, D. J. & DE VRIES, G. 1895 On the change of form of long waves advancing in a rectangular canal and on a new type of long stationary waves. Phil. Mag. 39, 240, p. 422.Google Scholar
Lamb, H. 1932 Hydrodynamics, 6th ed. Cambridge University Press.
Lighthill, M. J. 1966 Contributions to the theory of waves in non-linear dispersive systems J.I.M.A. 1, 269.Google Scholar
Meyer, E. & Skudzrijk, E. 1953 über die akustischen Eigenschaften von Gasblasenschleiern in Wasser Akust. Beih. 3, 434.Google Scholar
Morse, P. M. & Feshbach, H. 1953 Methods of Theoretical Physics, McGraw-Hill, p. 1498.
Plesset, M. S. 1962 Bubble dynamics. Report 85–23. Cal. Inst. Techn. Div. of Engin. and Appl. Sci., Pasadena, Cal.Google Scholar
Plesset, M. S. & Hsieh, DIN-YU 1960 Theory of gas bubble dynamics in oscillating pressure fields. Cal. Inst. Technology, Engng Div. Rept. 85–16.Google Scholar
Whitham, G. B. 1965a Non-linear dispersive waves. Proc. Roy. Soc. A 283, 238.Google Scholar
Whitman, G. B. 1965b A general approach to linear and non-linear dispersive waves using a Lagrangian. J. Fluid. Mech. 22, 273.Google Scholar
Whitham, G. B. 1967 Non-linear dispersion of water waves J. Fluid Mech. 27, 399.Google Scholar
Wijngaarden, L. VAN 1964 On the collective collapse of a large number of cavitation bubbles in water. Proc. 11th International Congress of Applied Mechanics, Munich 1964, p. 854. Editor H. Görtler, Springer Verlag.
Wijngaarden, L. VAN 1966 Linear and non-linear dispersion of pressure pulses in liquid bubble mixtures. 6th Symposium on Naval Hydrodynamics, Washington D.C.Google Scholar