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Considerations on bubble fragmentation models

Published online by Cambridge University Press:  01 October 2010

C. MARTÍNEZ-BAZÁN,
J. RODRÍGUEZ-RODRÍGUEZ,
G. B. DEANE,
J. L. MONTAÑES and
J. C. LASHERAS
C. MARTÍNEZ-BAZÁN*
Affiliation:
Área de Mecánica de Fluidos, Departamento de Ingeniería Mecánica y Minera, Universidad Jaén, Campus de las Lagunillas, 23071 Jaén, Spain
J. RODRÍGUEZ-RODRÍGUEZ
Affiliation:
Área de Mecánica de Fluidos, Departamento de Ingeniería Térmica y de Fluidos, Universidad Carlos III de Madrid, 28911 Leganés, Spain
G. B. DEANE
Affiliation:
Marine Physical Laboratory, Scripps Institution of Oceanography, La Jolla, CA 92093-0238, USA
J. L. MONTAÑES
Affiliation:
Departamento de Motopropulsión y Termofluidodinámica, Universidad Politécnica de Madrid, Plaza del Cardenal Cisneros 3, 28040 Madrid, Spain
J. C. LASHERAS
Affiliation:
Department of Mechanical and Aerospace Engineering, University of California San Diego. La Jolla, CA 92093-0411, USA
*
†Email address for correspondence: cmbazan@ujaen.es
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Abstract

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In this paper we describe the restrictions that the probability density function (p.d.f.) of the size of particles resulting from the rupture of a drop or bubble must satisfy. Using conservation of volume, we show that when a particle of diameter, D0, breaks into exactly two fragments of sizes D and D2 = (D30−D3)1/3 respectively, the resulting p.d.f., f(D; D0), must satisfy a symmetry relation given by D22f(D; D0) = D2f(D2; D0), which does not depend on the nature of the underlying fragmentation process. In general, for an arbitrary number of resulting particles, m(D0), we determine that the daughter p.d.f. should satisfy the conservation of volume condition given by m(D0) ∫0D0 (D/D0)3f(D; D0) dD = 1. A detailed analysis of some contemporary fragmentation models shows that they may not exhibit the required conservation of volume condition if they are not adequately formulated. Furthermore, we also analyse several models proposed in the literature for the breakup frequency of drops or bubbles based on different principles, g(ϵ, D0). Although, most of the models are formulated in terms of the particle size D0 and the dissipation rate of turbulent kinetic energy, ϵ, and apparently provide different results, we show here that they are nearly identical when expressed in dimensionless form in terms of the Weber number, g*(Wet) = g(ϵ, D0) D2/30 ϵ−1/3, with Wet ~ ρ ϵ2/3D05/3/σ, where ρ is the density of the continuous phase and σ the surface tension.

JFM classification

Drops and Bubbles: Breakup/coalescence

Information

Type
Papers
Information
Journal of Fluid Mechanics , Volume 661 , 25 October 2010, pp. 159 - 177
Copyright
Copyright © Cambridge University Press 2010

References

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