Transport in Porous Media

, Volume 20, Issue 1–2, pp 37–76 | Cite as

Surface tension models with different viscosities

  • I. Ginzbourg
  • P. M. Adler


Surface tension in ILB models for fluids with different viscosities and different numbers of rest populations is derived, starting from the so-called mechanical definition. It is shown that the standard perturbation, inserted into these models in order to create surface tension, should be slightly modified for models with different viscosities in order to avoid the dependence of surface tension upon the actual phase distribution. The analytical results are numerically confirmed by mechanical and bubble tests. It is demonstrated also that the perturbation of the lattice Boltzmann equation gives rise to the appearance of anisotropic terms in population solutions related to anomalous currents and density fluctuations. When particular values of the eigenvalues of the collision operators are used, these spurious currents are annihilated in the time-independent solutions of the mechanical tests in arbitrarily inclined channels when bounce-back conditions are imposed at the solid boundaries.

Key words

surface tension viscosity perturbation lattice gas model lattice-Boltzmann model immiscible lattice gas model 


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  1. Adler, C., d'Humières, D. and Rothman, D.: 1994, Surface tension and interface fluctuations in immiscible lattice gases,J. Phys. I, France 4, 29–46.Google Scholar
  2. Appert, C. and Zaleski, S.: 1990, A lattice gas with a liquid-gas transition,Phys. Rev. Lett. 64, 1–4.Google Scholar
  3. Appert, C., Rothman, D. H. and Zaleski, S.: 1991, A liquid-gas model on a lattice,Physica D,47, 85–96.Google Scholar
  4. Appert, C. and Zaleski, S.: 1993, Dynamical liquid-gas phase transition,J. Phys. II, France 3, 309–337.Google Scholar
  5. Appert, C.: 1993, Transition de phase dynamique de type liquide-gaz et création d'interfaces dans un gaz sur réseau, Thèse de Doctorat de l'Université Paris VI.Google Scholar
  6. Burgess, D., Hayot, F. and Saam, W. F.: 1988, Model for surface tension in lattice-gas hydrodynamics,Phys. Rev. A 38, 3589–3592.Google Scholar
  7. Chen, S., Wang, Z., Shan, X. and Doolen, G.: 1992, Lattice-Boltzmann computational fluid dynamics in three dimensions,J. Stat. Phys. 68, 379–400.Google Scholar
  8. Cornubert, R., d'Humières, D. and Livermore, D.: 1991, A Knudsen layer theory for lattice gases,Physica D,47, 241–259.Google Scholar
  9. Edwards, D. A., Brenner, H. and Wasan, D. T.: 1991, Interfacial transport processes and rheology,Butterworth-Heinemann Series in Chemical Engineering.Google Scholar
  10. Frisch, U., Hasslacher, B. and Pomeau, Y.: 1986, Lattice gas automata for the Navier-Stokes equation,Phys. Rev. Lett. 56, 1505–1508.Google Scholar
  11. Frisch, U., d'Humières, D., Hasslacher, B., Lallemand, P., Pomeau, Y. and Rivet, J. P.: 1987, Lattice gas hydrodynamics in two and three dimensions,Complex Systems,1, 649–707.Google Scholar
  12. Ginzbourg, I. and Adler, P. M.: 1994a, Boundary flow condition analysis for the three-dimensional lattice Boltzmann model,J. Phys. II, France,4, 191–214.Google Scholar
  13. Ginzbourg, I. and Adler, P. M: 1994b, Boundary conditions at a plane fluid-interface in the FCHC lattice Boltzmann model, in preparation.Google Scholar
  14. Grunau, D. W.: 1993, Lattice methods for modeling hydrodynamics, PhD Thesis, Colorado State University.Google Scholar
  15. Gunstensen, A. K., Rothman, D. H., Zaleski, S. and Zanetti, G.: 1991, Lattice-Boltzmann model of immiscible fluids,Phys. Rev. A 43, 107–114.Google Scholar
  16. Gunstensen, A. K. and Rothman, D. H.: 1991, A Galilean-invariant two-phase lattice gas,Physica D 47, 53–63.Google Scholar
  17. Gunstensen, A. K. and Rothman, D. H.: 1992, Microscopic modeling of immiscible fluids in three dimensions by a lattice-Boltzmann method,Europhys. Lett. 18, 157–161.Google Scholar
  18. Gunstensen, A.K.: 1992, Lattice-Boltzmann studies of multiphase flow through porous media, PhD Thesis, MIT.Google Scholar
  19. Hayot, F.: 1991, Fingering instability in a lattice gas,Physica D 47, 64–71.Google Scholar
  20. Higuera, F. J. and Jimenez, J.: 1989, Boltzmann approach to lattice gas simulations,Europhys. Lett. 9, 663–668.Google Scholar
  21. Higuera, F. J., Succi, S. and Benzi, R.: 1989, Lattice gas dynamics with enhanced collisions,Europhys. Lett. 9, 345–349.Google Scholar
  22. d'Humières, D., Lallemand, P. and Frisch, U.: 1986, Lattice gas models for 3D hydrodynamics,Europhys. Lett. 2, 291–297.Google Scholar
  23. d'Humières, D. and Lallemand, P.: 1987, Numerical simulations of hydrodynamics with lattice gas automata in two dimensions,Complex Systems 1, 599–632.Google Scholar
  24. McNamara, G. R. and Zanetti, G.: 1988, Use of the Boltzmann equation to simulate lattice-gas automata,Phys. Rev. Lett. 61, 2332–2335.Google Scholar
  25. Rem, P. C. and Somers, J. A.: 1989, Cellular automata on a transputer network, in R. Monaco (ed),Discrete Kinetic Theory, Lattice Gas Dynamics, and Foundation of Hydrodynamics, World Scientific, Singapore, pp. 268–275.Google Scholar
  26. Rothman, D. H. and Keller, J. M.: 1988, Immiscible cellular-automaton fluids,J. Statist. Phys. 5, 1119–1127.Google Scholar
  27. Rothman, D. H.: 1990, Macroscopic laws for immiscible two-phase flow in porous media: results from numerical experiments,J. Geophys. Res. B95, 8663–8674.Google Scholar
  28. Rowlinson J. and Widom, B.: 1982,Molecular Theory of Capillarity, Clarendon Press, Oxford.Google Scholar
  29. Somers, J. A. and Rem, P. C.: 1991, Analysis of surface tension in two phase lattice gases,Physica D47, 39–46.Google Scholar
  30. Succi, S., Foti, E. and Higuera, F.: 1989, Three-dimensional flows in complex geometries with the lattice Boltzmann method,Europhys. Lett. 10, 433–438.Google Scholar
  31. Zanetti, G.: 1991, The hydrodynamics of lattice gas automata,Physica D47, 30–35.Google Scholar

Copyright information

© Kluwer Academic Publishers 1995

Authors and Affiliations

  • I. Ginzbourg
    • 1
  • P. M. Adler
    • 1
  1. 1.Asterama 2LPTMChasseneuilFrance

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