Hostname: page-component-848d4c4894-mwx4w Total loading time: 0 Render date: 2024-06-27T04:40:20.943Z Has data issue: false hasContentIssue false

Travelling waves in bubbly liquid with continuous bubble-size distribution

Published online by Cambridge University Press:  26 September 2008

S. L. Gavrilyuk
Affiliation:
Theoretical Division, Lavrentyev Institute of Hydrodynamics, Novosibirsk, 630090, Russia

Abstract

An existence and uniqueness theorem for travelling waves in a bubbly liquid with a continuous bubble-size distribution is proved.

Type
Research Article
Copyright
Copyright © Cambridge University Press 1995

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

Benney, D. J. 1973 Some properties of long waves. Studies Appl. Math. 52, 4569.CrossRefGoogle Scholar
Birnir, B. & Smereka, P. 1990 Existence theory and invariant manifolds of bubble clouds. Commun. Pure Appl. Math. XLIII, 363413.CrossRefGoogle Scholar
Caflisch, R. E., Miksis, M. J., Papanicolaou, G. C. & Ting, L. 1985a Effective equations for wave propagation in a bubbly liquid. J. Fluid Mech. 153, 259273.CrossRefGoogle Scholar
Caflisch, R. E., Miksis, M. J., Papanicolaou, G. C. & Ting, L. 1985b Wave propagation in bubbly liquids at finite volume fraction. J. Fluid Mech. 160, 114.CrossRefGoogle Scholar
Caflisch, R. E., Miksis, M. J., Papanicolaou, G. C. & Ting, L. 1986 Wave propagation in bubbly liquid. Lectures in Applied Mathematics, American Mathematical Society, Providence, RI, 23, 327343.Google Scholar
Daletskii, Yu. L. & Krein, M. G. 1970 Stability of solutions of differential equations in Banach space. Moscow, Nauka (in Russian).Google Scholar
Gavrilyuk, S. L. 1989 Existence, uniqueness and Lax's stability of travelling waves in polydisperse bubbly liquid with dissipation. Partial Differential Equations, Siberian Div. of the USSR Academy of Sci., Inst. of Mathematics, Novosibirsk, 3353 (in Russian).Google Scholar
Gavrilyuk, S. L. & Fil'ko, S. A. 1991 Shock waves in polydisperse bubbly media with dissipation. Zh. Prikl. Mekh. Tekhn. Fiz. 5, 2634 (in Russian).Google Scholar
Gavrilyuk, S. L. 1991 Large-time asymptotics of solution of the Cauchy problem for equations of linear wave propagation in bubbly liquid with continuous bubble size distribution. Siberian Division of the USSR Academy of Sci., Lavrentyev Institute of Hydrodynamics, Novosibirsk, Preprint 1.Google Scholar
Gavrilyuk, S. L. 1992 Propagation of a signal in a liquid with continuous distribution of bubble sizes. Zh. Prikl. Mekh. Tekhn. Fiz. 4, 5460 (in Russian).Google Scholar
Gumerov, N. A. 1992 On quasimonochromatic weakly-nonlinear waves in bubbly liquid with a small dissipation. Prikl. Matem. Mekhan. 56(1), 5867 (in Russian).Google Scholar
Hartman, P. 1964 Ordinary Differential Equations. John Wiley.Google Scholar
Iordanskii, S. V. 1960 On the equations of motion for a liquid containing gas bubbles. Zh. Prikl. Mekh. Tekhn. Fiz. 3, 102110 (in Russian).Google Scholar
Kogarko, B. S. 1964 One-dimensional nonstationary motion of liquid with an initiation and progression of cavitation. Dokl. Akad. Nauk SSSR, 155 (4), 779782 (in Russian).Google Scholar
Miksis, M. J. & Ting, L. 1992 Effective equations for multiphase flows—waves in a bubbly liquid. Advances in Applied Mechanics, 28, 141260.CrossRefGoogle Scholar
Nigmatulin, R. I. 1987 Dynamics of multiphase flows. Moscow, Nauka (in Russian).Google Scholar
Palais, R. S. & Terng, C. 1988 Critical point theory and submanifold geometry. Lecture Notes in Mathematics 1353. Springer-Verlag.Google Scholar
Pugh, C. C. 1969 On a theorem of P. Hartman. Am. J. Math. 91 (2), 363367.CrossRefGoogle Scholar
Richtmayer, R. D. 1982 Principles of Advanced Mathematical Physics. Moscow, Mir (in Russian).Google Scholar
Skripnik, I. V. 1976 Solvability and properties of solutions of nonlinear elliptic equations. Current Mathematical Problems, VINITI, Moscow, 9.Google Scholar
Teshukov, V. M. 1991 Long wave approximation for vortex free boundary flows. International Series of Numerical Mathematics 99. Birkhäuser-Verlag, 413421.Google Scholar
Van Wijngaarden, L. 1968 On the equations of motion for mixtures of liquid and gas bubbles. J. Fluid Mech. 33, 465474.CrossRefGoogle Scholar