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Three-dimensional immiscible lattice gas: Application to sheared phase separation

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Abstract

A new lattice-gas cellular automaton model for simulating binary fluids in three dimensions is introduced. It is particularly suitable for modeling slow flows of mixtures with complicated interface geometries or within complicated boundaries, such as in the interior of a porous rock. Phase separation is triggered spontaneously in the model by statistical fluctuations and phase domains are approximately isotropic. The measured surface tension is large compared to that in analogous two-dimensional models. The model is applied to a study of the time-dependent effective viscosity of a phase-separating mixture in a simple shear flow. Results qualitatively match both experiment and theory: the viscosity increases rapidly, then decays gradually to a steady-state value which is larger than the viscosity of the pure fluids. The effective viscosity increases with increasing concentration and decreases with increasing strain rate.

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References

  1. 1.

    L. A. Utracki,Polymer Alloys and Blends (Hanser Publishers, 1990).

  2. 2.

    G. I. Taylor,Proc. R. Soc. Lond. A 138:41–48 (1932).

  3. 3.

    F. D. Rumscheidt and S. G. Mason,J. Colloid Sci. 16:238–261 (1961).

  4. 4.

    V. G. Levich,Physicochemical Hydrodynamics (Prentice-Hall Englewood Cliffs, New Jersey, 1962).

  5. 5.

    T. Gillespie, The effect of concentration on the viscosity of suspensions and emulsions, inRheology of Emulsions (Macmillan, (1963), pp. 115–124.

  6. 6.

    S. Torza, R. G. Cox, and S. G. Mason,J. Colloid Interface Sci. 2:395–411 (1972).

  7. 7.

    A. Onuki,Phys. Rev. A 35:5149 (1987).

  8. 8.

    M. Doi and T. Ohta,J. Chem. Phys. 95:1242–1248 (1991).

  9. 9.

    B. V. Boshenyatov and I. V. Chernyshev,Fluid Mech. Sov. Res. 20:124–129 (1991).

  10. 10.

    D. Beysens, F. Perrot, and T. Baumberger,Physica A 204:76–86 (1994).

  11. 11.

    D. A. Drew,Annu. Rev. Fluid Mech. 15:261 (1983).

  12. 12.

    B. Lafaurie, C. Nardone, R. Scardovelli, S. Zaleski, and G. Zanetti,J. Comput. Phys. 113:134–147 (1994).

  13. 13.

    T. Koga and K. Kawasaki,Physica A 196:389–415 (1993).

  14. 14.

    S. Puri and B. Dünweg,Phys. Rev. A 45:R6977-R6980 (1992).

  15. 15.

    D. H. Rothman and J. Keller,J. Stat. Phys. 52:1119 (1988).

  16. 16.

    C. Appert and S. Zaleski,Phys. Rev. Lett. 64:1 (1990).

  17. 17.

    C. Appert and S. Zaleski,J. Phys. II France 3:309–337 (1993).

  18. 18.

    D. H. Rothman and S. Zaleski,Rev. Mod. Phys. 66:1417–1479 (1994).

  19. 19.

    A. K. Gunstensen, D. H. Rothman, S. Zaleski, and G. Zanetti,Phys. Rev. A. 43:4320–4327 (1991).

  20. 20.

    A. K. Gunstensen and D. H. Rothman,Europhys. Lett. 18(2):157–161 (1992).

  21. 21.

    D. Grunau, S. Chen, and K. Eggert,Phys. Fluids A 5:2557–2562 (1993).

  22. 22.

    F. J. Alexander, S. Chen, and D. W. Grunau,Phys. Rev. B 48:634 (1993).

  23. 23.

    X. Shan and H. Chen,Phys. Rev. E 47:1815–1819 (1993).

  24. 24.

    T. Toffoli and N. Margolus,Physica D 47:263–272 (1991).

  25. 25.

    C. Adler, B. Boghosian, E. G. Flekkøy, N. Margolus, and D. H. Rothman,J. Stat. Phys., this issue.

  26. 26.

    K. Hamano, T. Ishii, M. Ozawa, J. V. Sengers, and A. H. Krall,Phys. Rev. E 51:1254–1262 (1995).

  27. 27.

    A. H. Krall, J. V. Sengers, and K. Hamano,Phys. Rev. Lett. 69:1963–1966 (1992).

  28. 28.

    B. Khuzhaerov,J. Eng. Phys. 1991:1564–1570.

  29. 29.

    P. C. Rem and J. A. Somers, Cellular automata on a transputer network, inDiscrete Kinetic Theory, Lattice-Gas Dynamics, and Foundations of Hydrodynamics, R. Monaco, ed. (World Scientific, singapore, 1989), p. 268.

  30. 30.

    C. Adler, D. d'Humières, and D. H. Rothman,Jour. Phys. I France 4:29–46 (1994).

  31. 31.

    A. K. Gunstensen and D. H. Rothman,Physica D 47:47–52 (1991).

  32. 32.

    D. H. Rothman and L. P. Kadanoff,Comput. Phys. 8:199–204 (1994).

  33. 33.

    U. Frisch, B. Hasslacher, and Y. Pomeau,Phys. Rev. Lett. 56:1505 (1986).

  34. 34.

    U. Frisch, D. d'Humières, B. Hasslacher, P. Lallemand, Y. Pomeau, and J.-P. Rivet,Complex Systems 1:648 (1987).

  35. 35.

    G. Zanetti,Phys. Rev. A 40:1539 (1989).

  36. 36.

    L. Kadanoff, G. McNamara, and G. Zanetti,Phys. Rev. A 40:4527 (1989).

  37. 37.

    S. Wolfram,J. Stat. Phys. 45:471 (1986).

  38. 38.

    A. K. Gunstensen and D. H. Rothman,Physica D 47:53–63 (1991).

  39. 39.

    D. d'Humières, P. Lallemand, and U. Frisch,Europhys. Lett. 2:291 (1986).

  40. 40.

    H. S. M. Coxeter,Regular Polytopes (Dover, New York, 1973).

  41. 41.

    M. Hénon,Complex Systems 1:475–494 (1987).

  42. 42.

    G. Westland, Optimizing a reduced collision table for the FCHC-lattice gas automaton, Master's thesis, State University of Utrecht (April 1991).

  43. 43.

    J. A. Somers and P. C. Rem, Obtaining numerical results from the 3D FCHC-lattice gas, inNumerical Methods for the Simulation of Multi-Phase and Complex Flow, T. M. M. Verheggen, ed. (Springer-Verlag, Berlin, 1992), pp. 59–78.

  44. 44.

    J. A. Somers and P. C. Rem,Physica D 47:39–46 (1991).

  45. 45.

    S. Chen, G. D. Doolen, K. Eggert, D. Grunau, and E. Y. Loh,Phys. Rev. A 43:7053–7056 (1991).

  46. 46.

    C. Appert, D. H. Rothman, and S. Zaleski,Physica D 47:85–96 (1991).

  47. 47.

    D. H. Rothman,Phys. Rev. Lett. 65:3305 (1990).

  48. 48.

    D. H. Rothman,Europhys. Lett. 14:337–342 (1991).

  49. 49.

    D. H. Rothman,J. Geophys. Res. 95:8663 (1990).

  50. 50.

    J. Rowlinson and B. Widom,Molecular Theory of Capillarity (Clarendon Press, Oxford, 1982).

  51. 51.

    C. Appert, J. F. Olson, D. H. Rothman, and S. Zaleski,J. Stat. Phys., this issue.

  52. 52.

    A. W. Lees and S. F. Edwards,J. Phys. C: Solid State Phys. 5:1921–1929 (1972).

  53. 53.

    J. M. Rallison,Annu. Rev. Fluid Mech. 16:45–66 (1984).

  54. 54.

    M. Hénon,Complex Systems 1:763–789 (1987).

  55. 55.

    A. H. Krall, J. V. Sengers, and K. Hamano,Int. J. Thermophys. 10:309 (1989).

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Olson, J.F., Rothman, D.H. Three-dimensional immiscible lattice gas: Application to sheared phase separation. J Stat Phys 81, 199–222 (1995). https://doi.org/10.1007/BF02179976

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Key Words

  • Multiphase flow
  • computational techniques
  • phase transitions