Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-25T14:47:23.844Z Has data issue: false hasContentIssue false
  • English
  • Français

Lattice-Boltzmann equations for describing segregation in non-ideal mixtures

Published online by Cambridge University Press:  26 October 2012

Paulo C. Philippi ,
Keijo K. Mattila ,
Diogo N. Siebert ,
Luís O. E. dos Santos ,
Luiz A. Hegele Júnior  and
Rodrigo Surmas
Paulo C. Philippi*
Affiliation:
Mechanical Engineering Department, Federal University of Santa Catarina, 88040-900 Florianópolis, SC, Brazil
Keijo K. Mattila
Affiliation:
Mechanical Engineering Department, Federal University of Santa Catarina, 88040-900 Florianópolis, SC, Brazil
Diogo N. Siebert
Affiliation:
State University of Santa Catarina, 88330-668 Camboriu, Brazil
Luís O. E. dos Santos
Affiliation:
Federal University of Santa Catarina, 89218-000 Joinville, Brazil
Luiz A. Hegele Júnior
Affiliation:
Mechanical Engineering Department, Federal University of Santa Catarina, 88040-900 Florianópolis, SC, Brazil
Rodrigo Surmas
Affiliation:
Petrobras, 21941-915 Rio de Janeiro, Brazil
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In fluid mechanics, multicomponent fluid systems are generally treated either as homogeneous solutions or as completely immiscible parts of a multiphasic system. In immiscible systems, the main task in numerical simulations is to find the location of the interface evolving over time, driven by normal and tangential surface forces. The lattice-Boltzmann method (LBM), on the other hand, is based on a mesoscopic description of the multicomponent fluid systems, and appears to be a promising framework that can lead to realistic predictions of segregation in non-ideal mixtures of partially miscible fluids. In fact, the driving forces in segregation are of a molecular nature: there is competition between the intermolecular forces and the random thermal motion of the molecules. Since these microscopic mechanisms are not accessible from a macroscopic standpoint, the LBM can provide a bridge linking the microscopic and macroscopic domains. To this end, the first purpose of this article is to present the kinetic equations in their continuum forms for the description of the mixing and segregation processes in mixtures. This paper is limited to isothermal segregation; non-isothermal segregation was discussed by Philippi et al. (Phil. Trans. R. Soc., vol. 369, 2011, pp. 2292–2300). Discretization of the kinetic equations leads to evolution equations, written in LBM variables, directly amenable for numerical simulations. Here the dynamics of the kinetic model equations is demonstrated with numerical simulations of a spinodal decomposition problem with dissolution. Finally, some simplified versions of the kinetic equations suitable for immiscible flows are discussed.

JFM classification

Complex Fluids: Complex Fluids Drops and Bubbles: Drops and Bubbles Interfacial Flows (free surface): Interfacial Flows (free surface)
Type
Papers
Information
Journal of Fluid Mechanics , Volume 713 , 25 December 2012 , pp. 564 - 587
Copyright
©2012 Cambridge University Press

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

Anderson, D. M., McFadden, G. B. & Wheeler, A. A. 1998 Diffuse-interface methods in fluid mechanics. Annu. Rev. Fluid Mech. 30, 139–165.CrossRefGoogle Scholar
Boltzmann, L. 1872 Weitere Studien über das Wärmegleichgewicht unter Gasmolekülen. Sitz. Akad. Wiss. 66, 275.Google Scholar
Cahn, J. W. & Hilliard, J. E. 1958 Free energy of a non-uniform system. Part 1. Interfacial free energy. J. Chem. Phys. 28, 258.CrossRefGoogle Scholar
Chapman, S. & Cowling, T. G. 1991 The Mathematical Theory of Non-uniform Gases, 3rd edn. Cambridge University Press.Google Scholar
Cristea, A., Gonnella, G., Lamura, A. & Sofonea, V. 2006 Finite-difference lattice Boltzmann model for liquid–vapour systems. Math. Comput. Simul. 72, 113–116.CrossRefGoogle Scholar
Facin, P. C., Philippi, P. C. & dos Santos, L. O. E. 2004 A nonlinear lattice-Boltzmann model for ideal miscible fluids. Future Gener. Comp. Syst. 20 (6), 945–949.CrossRefGoogle Scholar
Gunstensen, A. K. & Rothman, D. H. 1992 Microscopic modelling of immiscible fluids in three dimensions by a lattice-Boltzmann method. Europhys. Lett. 18, 157–161.CrossRefGoogle Scholar
Gunstensen, A. K., Rothman, D. H., Zaleski, S. & Zanetti, G. 1991 Lattice Boltzmann model of immiscible fluids. Phys. Rev. A 43 (8), 4320–4327.CrossRefGoogle ScholarPubMed
Guo, Z., Zheng, C. & Shi, B. 2002 Discrete lattice effects on the forcing term in the lattice Boltzmann method. Phys. Rev. E 65, 046308.CrossRefGoogle ScholarPubMed
Halliday, I., Hollis, A. P. & Care, C. M. 2007 Lattice Boltzmann algorithm for continuum multicomponent flow. Phys. Rev. E 76, 026708.CrossRefGoogle ScholarPubMed
He, X. & Doolen, G. D. 2002 Thermodynamic foundations of kinetic theory and lattice Boltzmann models for multiphase flows. J. Stat. Phys. 107 (1), 309–328.CrossRefGoogle Scholar
He, X. & Luo, L.-S. 1997 Theory of the lattice Boltzmann method: from the Boltzmann equation to the lattice Boltzmann equation. Phys. Rev. E 56 (6), 6811–6817.CrossRefGoogle Scholar
He, X., Chen, S. & Zhang, R. 1999 A lattice Boltzmann scheme for incompressible multiphase flow and its application in simulation of Rayleigh–Taylor instability. J. Comp. Phys. 152, 642–663.CrossRefGoogle Scholar
He, X., Shan, X. & Doolen, G. D. 1998 Discrete Boltzmann equation model for nonideal gases. Phys. Rev. E R 57 (1), 13–16.CrossRefGoogle Scholar
Holdych, D. J., Rovas, D., Georgiadis, J. G. & Buckius, R. O. 1998 An improved hydrodynamics formulation for multiphase flow lattice-Boltzmann models. Intl J. Mod. Phys. C 9 (8), 1393–1404.CrossRefGoogle Scholar
Inamuro, T., Konishi, N. & Ogino, F. 2000 A Galilean invariant model of the lattice Boltzmann method for multiphase fluid flows using free-energy approach. Comput. Phys. Comm. 129 (1–3), 32–45.CrossRefGoogle Scholar
Kikkinides, E. S., Yiotis, A. G., Kainourgiakis, M. E. & Stubos, A. K. 2008 Thermodynamic consistency of liquid–gas lattice Boltzmann methods: interfacial property issues. Phys. Rev. E 78, 036702.CrossRefGoogle ScholarPubMed
Kloubek, J. 1992 Development of methods for surface free energy determination using contact angles of liquids on solids. Adv. Colloid Interface 38, 99–142.CrossRefGoogle Scholar
Korteweg, D. J. 1901 Sur la forme que prennent les équations du mouvement des fluides si l’on tient compte des forces capillaires causées par des variations de densité. Arch. Neer. Sci. Exactes Ser. 6, 1–24.Google Scholar
Lee, T. & Fisher, P. F. 2006 Eliminating parasitic currents in the lattice Boltzmann equation method for nonideal gases. Phys. Rev. E 74, 046709.CrossRefGoogle ScholarPubMed
Lee, T. & Lin, C.-L. 2005 A stable discretization of the lattice Boltzmann equation for simulation of incompressible two-phase flows at high density ratio. J. Comp. Phys. 206, 16–47.CrossRefGoogle Scholar
Li, Q. & Wagner, A. J. 2007 Symmetric free-energy-based multicomponent lattice Boltzmann method. Phys. Rev. E 76 (3), 36701.CrossRefGoogle ScholarPubMed
Lishchuk, S. V., Care, C. M. & Halliday, I. 2003 Lattice Boltzmann algorithm for surface tension with greatly reduced microcurrents. Phys. Rev. E 67 (3), 036701.CrossRefGoogle ScholarPubMed
Luo, L.-S. 1998 Unified theory of lattice Boltzmann models for nonideal gases. Phys. Rev. Lett. 81 (8), 1618–1621.CrossRefGoogle Scholar
Maxwell, J. C. 1874 Van der Waals on the continuity of gaseous and liquid states. Nature 10, 477–480.Google Scholar
Philippi, P. C., Hegele, L. A. Jr, dos Santos, L. O. E. & Surmas, R. 2006 From the continuous to the lattice Boltzmann equation: the discretization problem and thermal models. Phys. Rev. E 73 (5), 56702.CrossRefGoogle Scholar
Philippi, P. C., dos Santos, L. O. E., Hegele, L. A. Jr, Pico, C. E., Siebert, D. N. & Surmas, R. 2011 Thermodynamic consistency in deriving lattice Boltzmann models for describing segregation in non-ideal mixtures. Phil. Trans. R. Soc. A 369, 2292–2300.CrossRefGoogle ScholarPubMed
Prausnitz, J. M., Lichtenthaler, R. N. & Azevedo, E. G. 1986 Molecular Thermodynamics of Fluid-Phase Equilibria. Prentice Hall.Google Scholar
dos Santos, L. O. E., Facin, P. C. & Philippi, P. C. 2003 Lattice-Boltzmann model based on field mediators for immiscible fluids. Phys. Rev. E 68 (5), 56302.CrossRefGoogle Scholar
Shan, X. 2006 Analysis and reduction of the spurious current in a class of multiphase lattice-Boltzmann models. Phys. Rev. E 73 (4), 47701.CrossRefGoogle Scholar
Shan, X. & Chen, H. 1993 Lattice Boltzmann model for simulating flows with multiple phases and components. Phys. Rev. E 47 (3), 1815–1819.CrossRefGoogle ScholarPubMed
Shan, X. & Chen, H. 1994 Simulation of nonideal gases and liquid–gas phase transitions by the lattice Boltzmann equation. Phys. Rev. E 49 (4), 2941–2948.CrossRefGoogle ScholarPubMed
Shan, X., Yuan, X.-F. & Chen, H. 2006 Kinetic theory representation of hydrodynamics: a way beyond the Navier–Stokes equation. J. Fluid Mech. 550, 413–442.CrossRefGoogle Scholar
Swift, M. R., Orlandini, E., Osborn, W. R. & Yeomans, J. M. 1996 Lattice Boltzmann simulations of liquid–gas and binary fluid systems. Phys. Rev. E 54 (5), 5041–5052.CrossRefGoogle ScholarPubMed
Swift, M. R., Osborn, W. R. & Yeomans, J. M. 1995 Lattice Boltzmann simulation of nonideal fluids. Phys. Rev. Lett. 75 (5), 830–833.CrossRefGoogle ScholarPubMed
Wagner, A. J. 2006 Thermodynamic consistency of liquid–gas lattice Boltzmann simulations. Phys. Rev. E 74 (5), 56703.CrossRefGoogle ScholarPubMed
Yuan, P. & Schaefer, L. 2006 Equations of state in a lattice Boltzmann model. Phys. Fluids 18, 042101.CrossRefGoogle Scholar