Abstract
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.
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References
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.
Appert, C. and Zaleski, S.: 1990, A lattice gas with a liquid-gas transition,Phys. Rev. Lett. 64, 1–4.
Appert, C., Rothman, D. H. and Zaleski, S.: 1991, A liquid-gas model on a lattice,Physica D,47, 85–96.
Appert, C. and Zaleski, S.: 1993, Dynamical liquid-gas phase transition,J. Phys. II, France 3, 309–337.
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.
Burgess, D., Hayot, F. and Saam, W. F.: 1988, Model for surface tension in lattice-gas hydrodynamics,Phys. Rev. A 38, 3589–3592.
Chen, S., Wang, Z., Shan, X. and Doolen, G.: 1992, Lattice-Boltzmann computational fluid dynamics in three dimensions,J. Stat. Phys. 68, 379–400.
Cornubert, R., d'Humières, D. and Livermore, D.: 1991, A Knudsen layer theory for lattice gases,Physica D,47, 241–259.
Edwards, D. A., Brenner, H. and Wasan, D. T.: 1991, Interfacial transport processes and rheology,Butterworth-Heinemann Series in Chemical Engineering.
Frisch, U., Hasslacher, B. and Pomeau, Y.: 1986, Lattice gas automata for the Navier-Stokes equation,Phys. Rev. Lett. 56, 1505–1508.
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.
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.
Ginzbourg, I. and Adler, P. M: 1994b, Boundary conditions at a plane fluid-interface in the FCHC lattice Boltzmann model, in preparation.
Grunau, D. W.: 1993, Lattice methods for modeling hydrodynamics, PhD Thesis, Colorado State University.
Gunstensen, A. K., Rothman, D. H., Zaleski, S. and Zanetti, G.: 1991, Lattice-Boltzmann model of immiscible fluids,Phys. Rev. A 43, 107–114.
Gunstensen, A. K. and Rothman, D. H.: 1991, A Galilean-invariant two-phase lattice gas,Physica D 47, 53–63.
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.
Gunstensen, A.K.: 1992, Lattice-Boltzmann studies of multiphase flow through porous media, PhD Thesis, MIT.
Hayot, F.: 1991, Fingering instability in a lattice gas,Physica D 47, 64–71.
Higuera, F. J. and Jimenez, J.: 1989, Boltzmann approach to lattice gas simulations,Europhys. Lett. 9, 663–668.
Higuera, F. J., Succi, S. and Benzi, R.: 1989, Lattice gas dynamics with enhanced collisions,Europhys. Lett. 9, 345–349.
d'Humières, D., Lallemand, P. and Frisch, U.: 1986, Lattice gas models for 3D hydrodynamics,Europhys. Lett. 2, 291–297.
d'Humières, D. and Lallemand, P.: 1987, Numerical simulations of hydrodynamics with lattice gas automata in two dimensions,Complex Systems 1, 599–632.
McNamara, G. R. and Zanetti, G.: 1988, Use of the Boltzmann equation to simulate lattice-gas automata,Phys. Rev. Lett. 61, 2332–2335.
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.
Rothman, D. H. and Keller, J. M.: 1988, Immiscible cellular-automaton fluids,J. Statist. Phys. 5, 1119–1127.
Rothman, D. H.: 1990, Macroscopic laws for immiscible two-phase flow in porous media: results from numerical experiments,J. Geophys. Res. B95, 8663–8674.
Rowlinson J. and Widom, B.: 1982,Molecular Theory of Capillarity, Clarendon Press, Oxford.
Somers, J. A. and Rem, P. C.: 1991, Analysis of surface tension in two phase lattice gases,Physica D47, 39–46.
Succi, S., Foti, E. and Higuera, F.: 1989, Three-dimensional flows in complex geometries with the lattice Boltzmann method,Europhys. Lett. 10, 433–438.
Zanetti, G.: 1991, The hydrodynamics of lattice gas automata,Physica D47, 30–35.
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Ginzbourg, I., Adler, P.M. Surface tension models with different viscosities. Transp Porous Med 20, 37–76 (1995). https://doi.org/10.1007/BF00616925
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DOI: https://doi.org/10.1007/BF00616925