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

L. van Wijngaarden

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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.
Original languageEnglish
Pages (from-to)465-474
JournalJournal of fluid mechanics
Volume33
Issue number3
DOIs
Publication statusPublished - 1968

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Bubbles (in fluids)
Equations of motion
equations of motion
bubbles
Liquids
liquids
Gases
gases
Fluids
Unsteady flow
Solitons
cnoidal waves
Conservation
Momentum
fluids
unsteady flow
planetary waves
elastic waves
Wavelength
conservation

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van Wijngaarden, L. / On the equations of motion for mixtures of liquid and gas bubbles. In: Journal of fluid mechanics. 1968 ; Vol. 33, No. 3. pp. 465-474.
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On the equations of motion for mixtures of liquid and gas bubbles. / van Wijngaarden, L.

In: Journal of fluid mechanics, Vol. 33, No. 3, 1968, p. 465-474.

Research output: Contribution to journalArticleAcademic

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PY - 1968

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AB - 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.

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