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Bubbly flow and its relation to conduction in composites

Published online by Cambridge University Press:  26 April 2006

Peter Smereka  and
Graeme W. Milton
Peter Smereka
Affiliation:
Courant Institute, 251 Mercer St., New York, NY 10012, USA Department of Mathematics, University of California, Los Angeles, CA 90024-1555, USA.
Graeme W. Milton
Affiliation:
Courant Institute, 251 Mercer St., New York, NY 10012, USA
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Abstract

Following Wallis, the relation between non-viscous bubbly flow and conduction in composites is examined. The bubbles are treated as incompressible and correspond to non-conducting inclusions. A simple relation is found between the effective conductivity and the energy coefficient which is agreement with previous calculations. It is shown that the energy coefficient is frame dependent and, in the frame of zero volumetric flux, is equal to the virtual mass density. Zuber's virtual mass density corresponds to the conductivity of the Hashin–Shtrikman coated-sphere geometry. This connection is exploited to extend Zuber's result to ellipsoidal bubbles. The hyperbolicity of effective equations derived from a variational principle is analysed for various bubble configurations. Without bubble clustering the equations are ill-posed (unstable). However, when the bubbles group into ellipsoidal clusters the resulting effective equations are well-posed for a wide range of parameter values.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 233 , December 1991 , pp. 65 - 81
Copyright
© 1991 Cambridge University Press

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References

Batchelor, G. K. 1972 Sedimentation in a dilute dispersion of spheres. J. Fluid Mech. 52, 245.Google Scholar
Beek, Van P. 1982 O(α)-accurate modeling of the virtual mass effect in a liquid—bubble dispersion. Appl. Sci. Res. 38, 323.Google Scholar
Bergman, D. J. 1978 The dielectric constant of a composite material – A problem in classical physics. Phys. Rep. C43, 377.Google Scholar
Bergman, D. J. 1982 Rigorous bounds for the complex dielectric constant of a two-component composite. Ann. Phys. 138, 78.Google Scholar
Biesheuvel, A. & Gorissen, W. C. M. 1990 Void fraction disturbances in a uniform bubbly fluid. Intl J. Multiphase Flow 16, 211.Google Scholar
Biesheuvel, A. & Spolestra, S. 1989 The added mass coefficient of a dispersion of spherical gas bubbles in liquid. Intl J. Multiphase Flow 15, 911.Google Scholar
Biesheuvel, A. & Wijngaarden, L. Van 1984 Two-phase flow equations for a dilute dispersion of gas bubbles in liquid. J. Fluid. Mech. 148, 301.Google Scholar
Geurst, J. A. 1985 Virtual mass in two-phase flow. Physica 129 A, 233Google Scholar
Hashin, Z. & Shtrikman, S. 1962 A variation approach to the theory of the effective magnetic permeability of multiphase materials. J. Appl. Phys. 33, 3125.Google Scholar
Jeffrey, D. J. 1973 Conduction through a random suspension of spheres.. Proc. R. Soc. Lond. A 335, 355.Google Scholar
Kok, J. B. W. 1988 Kinetic energy and added mass of hydrodynamically interacting gas bubbles in liquid. Physica 148 A, 240Google Scholar
Lhuillier, D. 1986 Phenomenology of inertia effects in a dispersed solid-fluid mixture. Intl J. Multiphase Flow 11, 427.Google Scholar
Maxwell, J. C. 1981 A Treatise on Electricity and Magnetism, 2nd edn., vol. 1. p. 398. Clarendon.
Mcphedran, R. C. & McKenzie, D. R. 1978 The conductivity of lattices of spheres I. The simple cubic lattice. Proc. R. Soc. Lond. A 359, 45Google Scholar
Mercadier, Y. 1981 Contribution à l’étude des propagations de perturbations de taux de vide dans écoulements diphasiques eau—air à bulles. D.Ing. thesis. Université Sci. Medicale et l'Institu national Polytechnique de Grenoble.
Milne-Thomson, L. M. 1968 Theoretical Hydrodynamics. Macmillan.
Milton, G. W. 1981 Bounds on the complex permittivity of a two component composite material, J. Appl. Phys. 52, 5286.Google Scholar
Pauchon, C. & Smereka, P. 1991 Momentum interactions in dispersed flow: An averaging and a variational approach. Intl J. Multiphase Flow (in press).Google Scholar
Rayleigh, Lord 1991 On the influence of obstacles arranged in a rectangular order upon the properties of a medium. Phil. Mag. 34, 484.Google Scholar
Sangani, A. S. & Acrivos, A. 1983 The effective conductivity of a periodic array of spheres. Proc. R. Soc. Lond. A 386, 263Google Scholar
Smith, A. P. & Ashcroft, N. W. 1988 Rapid convergence of lattice sums and integrals in ordered and disordered systems. Phys. Rev. B 38, 942Google Scholar
Tartar, L. 1985 Estimations fines des coefficients homogenénéisés. In Ennio DeGiorgi's Colloquium (ed. P. Kree) Research Notes in Mathematics, vol. 125, pp. 168. Pitman.
Wallis, G. B. 1989a On Geurst's equation for inertial coupling in two-phase flow In Two Phase Waves in Fluidized Beds, Flowing Composites and Granular Media. Inst. for Math. and its Applications, University of Minnesota.
Wallis, G. B. 1989b Inertial coupling in two-phase flow: macroscopic properties of suspensions in an inviscid fluid. Multiphase Sci. Technol. 5, 239.Google Scholar
Wijngaarden, L. Van 1976 Hydrodynamic interaction between gas bubbles in liquid. J. Fluid Mech. 77, 27.Google Scholar
Zuber, N. 1964 On the dispersed two-phase flow in the laminar flow regime. Chem. Engng. Sci. 19, 897.Google Scholar