### Abstract

Original language | English |
---|---|

Pages (from-to) | 465-474 |

Journal | Journal of fluid mechanics |

Volume | 33 |

Issue number | 3 |

DOIs | |

Publication status | Published - 1968 |

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### Cite this

*Journal of fluid mechanics*,

*33*(3), 465-474. https://doi.org/10.1017/S002211206800145X

}

*Journal of fluid mechanics*, vol. 33, no. 3, pp. 465-474. https://doi.org/10.1017/S002211206800145X

**On the equations of motion for mixtures of liquid and gas bubbles.** / van Wijngaarden, L.

Research output: Contribution to journal › Article › Academic

TY - JOUR

T1 - On the equations of motion for mixtures of liquid and gas bubbles

AU - van Wijngaarden, L.

PY - 1968

Y1 - 1968

N2 - 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.

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.

U2 - 10.1017/S002211206800145X

DO - 10.1017/S002211206800145X

M3 - Article

VL - 33

SP - 465

EP - 474

JO - Journal of fluid mechanics

JF - Journal of fluid mechanics

SN - 0022-1120

IS - 3

ER -