Advertisement

Fluid Dynamics of Mixtures of Incompressible Miscible Liquids

  • Daniel D. JosephEmail author
Chapter
Part of the Computational Methods in Applied Sciences book series (COMPUTMETHODS, volume 15)

Abstract

The velocity field of binary mixture of incompressible miscible liquids is non-solenoidal when the densities of the two liquids are different. If the mixture density is linear in the volume fraction, as is the case of liquids which satisfy the law of additive volumes, then the velocity can be decomposed into a solenoidal and expansion part. Here we propose a theory for liquids which do not satisfy the law of additive volumes. In this theory the mixture density is again given by a linear form but the densities of the liquids are scaled by the factor expressing the change of the volume of the mixture upon mixing. The dynamical theory of simple mixtures of incompressible liquids can be formed as the correct form of the Navier–Stokes equations in which the compressibility of the mixture is recognized. A rigorous form of the diffusion equation, different than the usual one, is also derived from first principles. The diffusion equation is based a non-linear form of Fick’s law, expressed in terms of gradients of the chemical potential. It is argued that the diffusion of species (of heat and in general) is impossible; signals must move with a finite speed though they may rapidly decay to diffusion. The underlying equation for the evolution of species and heat in the linear case is a damped wave equation rather than the conventional diffusion equation. The Navier–Stokes theory can be identified as a mass transport theory. The solenoidal part of the velocity satisfies an equation which can be shown to govern the transport of volume; it differs from the mass transport velocity by an irrotational expansion velocity associated with the dilitation of the mixture. The equations governing the transport of mass and volume differ from one another by well-defined mathematical transformations; the choice of one or the other is a matter of convenience. However, a genuine difference is associated with boundary conditions. The conventional assumption that the mass transport velocity vanishes is supported by calculations from molecular dynamics but these calculations employ entirely different assumptions and, hence, lack authority. The idea that gradients of composition ought to induce stresses and not just diffusion has been considered and is modeled by a second-order theory introduced by Korteweg 1901. There is not strong evidence that these stresses are important except in regions of strong gradients where a relaxation theory rather than a second-order theory ought to apply. A relaxation theory for stresses due to gradients of composition which relaxes into the second-order theory when the gradients are small is proposed and applied to explain observations of a transient interfacial tension which may be traced to a difference between the relaxation times for diffusion and stresses.

Keywords

Interfacial Tension Mixture Density Additive Volume Simple Mixture Miscible Displacement 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    F. T. Arecchi, P. K. Buah-Bassuah, F. Francini, C. Pérez-Garcia, and F. Quercioli. An experimental investigation of the break-up of a liquid drop falling in a miscible fluid. Europhys. Lett., 9(4):333–338, 1989.CrossRefADSGoogle Scholar
  2. 2.
    D. P. Barkey. Morphology selection and the concentration boundary layer in electrochemical deposition. J. Electrochem. Soc., 138(10):2912–2917, 1991.CrossRefGoogle Scholar
  3. 3.
    N. Baumann, D. D. Joseph, P. Mohr, and Y. Renardy. Vortex rings of one fluid in another in free fall. Phys. Fluids A, 4(3):567–580, 1991.CrossRefADSGoogle Scholar
  4. 4.
    H. Brenner. Navier–Stokes revisited. Phys. A, 349(1–2):60–132, 2005.MathSciNetGoogle Scholar
  5. 5.
    J. Cahn and J. Hilliard. Free energy of s nonuniform system I: Interfacial free energy. J. Chem. Phys., 28:258–267, 1958.CrossRefADSGoogle Scholar
  6. 6.
    J. Camacho and H. Brenner. On convection induced by molecular diffusion. Ind. Eng. Chem. Res., 34:3326–3335, 1995.CrossRefGoogle Scholar
  7. 7.
    H. S. Carslaw and J. C. Jaeger. Operational methods in applied mathematics. Dover, New York, 1963.zbMATHGoogle Scholar
  8. 8.
    C. Cattaneo. Sulla conduzione del calore. Atti Sem. Mat. Fis. Univ. Modena, 3:83–101, 1949.zbMATHMathSciNetGoogle Scholar
  9. 9.
    C. Chen and E. Meiburg. Miscible displacements in capillary tubes. Part 2: Numerical simulations. J. Fluid Mech., 326:57–90, 1996.Google Scholar
  10. 10.
    C. Y. Chen and E. Meiburg. Miscible displacements in capillary tubes: Influence of Korteweg stresses and divergence effects. Phys. Fluids, 14(7):2052–2058, 2002.CrossRefMathSciNetADSGoogle Scholar
  11. 11.
    H. T. Davis. A theory of tension at a miscible displacement front. In Numerical Simulation in Oil Recovery (Minneapolis, MN, 1986), pages 105–110, Berlin, 1988. Springer.Google Scholar
  12. 12.
    H. Freundlich. Colloid and capillary chemistry. Mathuen & Co. Ltd, London, 1926.Google Scholar
  13. 13.
    P. Galdi, D. D. Joseph, L. Preziosi, and S. Rionero. Mathematical problems for miscible incompressible fluids with Korteweg stresses. European J. Mech. B Fluids, 10(3):253–267, 1991.zbMATHADSGoogle Scholar
  14. 14.
    P. Garik, J. Hetrick, B. Orr, D. Barkey, and E. Ben-Jacob. Interfacial cellular mixing and a conjecture on global deposit morphology. Phys. Rev. Lett., 66(12):1606–1609, 1991.CrossRefADSGoogle Scholar
  15. 15.
    T. Graham. On the law of diffusion of gases. Philos. Mag. 2, 175, 194:269–276, 351–358, 1833. Reprinted in Chemical and Physical Researches, Edinburgh University Press, 1876, pp. 44–70.Google Scholar
  16. 16.
    H. H. Hu and D. D. Joseph. Miscible displacement in a Hele-Shaw cell. Z. Angew. Math. Phys., 43(4):626–644, 1992.zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    R. Jackson. Transport in porous catalysts. Elsevier, New York, 1977.Google Scholar
  18. 18.
    D. D. Joseph. Fluid dynamics of two miscible liquids with diffusion and gradient stresses. European J. Mech. B Fluids, 9(6):565–596, 1990.MathSciNetGoogle Scholar
  19. 19.
    D. D. Joseph, A. Huang, and H. Hu. Non-solenoidal velocity effects and Korteweg stresses in simple mixtures of incompressible liquids. Phys. D, 97 (1–3):104–125, 1996.Google Scholar
  20. 20.
    D. D. Joseph and L. Preziosi. Heat waves. Rev. Mod. Phys., 61(1):41–73, 1989.zbMATHCrossRefMathSciNetADSGoogle Scholar
  21. 21.
    D. D. Joseph and L. Preziosi. Addendum to the paper “Heat waves” [Rev. Mod. Phys. 61, 41 (1989)]. Rev. Mod. Phys., 62(2):375–391, 1990.Google Scholar
  22. 22.
    D. D. Joseph and Y. Renardy. Fundamentals of Two-Fluid Dynamics, Part II. Springer, New York, 1992.Google Scholar
  23. 23.
    M. Kojima, E. J. Hinch, and A. Acrivos. The formation and expansion of a toroidal drop moving in a viscous fluid. Phys. Fluids, 27(1):19–32, 1984.CrossRefADSGoogle Scholar
  24. 24.
    J. Koplik and J. R. Banavar. No-slip condition for a mixture of two liquids. Phys. Rev. Lett., 80(23):5125–5128, 1998.CrossRefADSGoogle Scholar
  25. 25.
    D. Korteweg. Sur la forme que prennant les équations du mouvement des fluides si l’on tient compte des forces capillaires causées par les variations de densité. Arch. Neerl. Sci. Ex. Nat., Ser. II, 6:1–24, 1901.Google Scholar
  26. 26.
    L. D. Landau and E. M. Lifshitz. Fluid Mechanics, volume 6 of Course of Theoretical Physics. Pergamon Press, Addison-Wesley Publishing Co., London, Reading, Mass., 1959.Google Scholar
  27. 27.
    L. D. Landau and E. M. Lifshitz. Course of theoretical physics. Vol. 6. Fluid mechanics. Pergamon Press, Oxford, 2nd edition, 1987.Google Scholar
  28. 28.
    T. Y. Liao and D. D. Joseph. Sidewall effects in the smoothing of an initial discontinuity of concentration. J. Fluid Mech., 342:37–51, 1997.zbMATHCrossRefMathSciNetADSGoogle Scholar
  29. 29.
    J. Lowengrub and L. Truskinovsky. Quasi-incompressible Cahn–Hilliard fluids and topological transitions. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 454(1978):2617–2654, 1998.zbMATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    J. Maddox. Heat conduction is a can of worms. Nature, 338:373, 1989.CrossRefADSGoogle Scholar
  31. 31.
    P. C. Malone and D. N. Wheatley. A bigger can of worms. Nature, 349:373, 1991.CrossRefADSGoogle Scholar
  32. 32.
    J. C. Maxwell. On the dynamical theory of gases. Philos. Trans Roy. Soc. London, 157:49, 1867.CrossRefGoogle Scholar
  33. 33.
    S. E. May and J. V. Maher. Capillary-wave relaxation for a meniscus between miscible liquids. Phys. Rev. Lett., 67(15):2013–2016, 1991.CrossRefADSGoogle Scholar
  34. 34.
    G. Mo and F. Rosenberger. Molecular-dynamics simulations of flow with binary diffusion in a two-dimensional channel with atomically rough walls. Phys. Rev. A, 44(8):4978–4985, 1991.CrossRefADSGoogle Scholar
  35. 35.
    J. E. Mungall. Interfacial tension in miscible two-fluid systems with linear viscoelastic rheology. Phys. Rev. Lett., 73(2):288–291, 1994.CrossRefADSGoogle Scholar
  36. 36.
    A. Narain and D. D. Joseph. Linearized dynamics for step jumps of velocity and displacement of shearing flows of a simple fluid. Rheol. Acta, 21(3):228–250, 1982.zbMATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    P. Petitjeans and T. Maxworthy. Miscible displacements in capillary tubes. Part 1. Experiments. J. Fluid Mech., 326:37–56, 1996.Google Scholar
  38. 38.
    J. A. Pojman, C. Whitmore, M. L. Turco Liveri, R. Lombardo, J. Marszalek, R. Parker, and B. Zoltowski. Evidence for the existence of an effective interfacial tension between miscible fluids: Isobutyric acid-water and 1-butanol-water in a spinning-drop tensiometer. Langmuir, 22:2569–2577, 2006.CrossRefGoogle Scholar
  39. 39.
    V. V. Pukhnachov. Mathematical model of natural convection under low gravity. Preprint 796, Institute for Mathematics and its Applications, Univ. of Minnesota, 1991.Google Scholar
  40. 40.
    G. Quinke. Die oberfachenspannung an der Grenze von Alkohol mit wasserigen Salzlosungen. Ann. Phys., 9(1), 1902.Google Scholar
  41. 41.
    S. Runge and G. H. Frischat. Stability of al2o3-containing droplets in glass melts. J. Non-Cryst. Solids, 102(1–3):157–164, 1988.CrossRefADSGoogle Scholar
  42. 42.
    J. B. Segur. Physical properties of glycerol and its solutions. In C. S. Miner and N. N. Dalton, editors, Glycerol, pages 238–334. Reinhold, 1953.Google Scholar
  43. 43.
    J. Serrin. The form of interfacial surfaces in Korteweg’s theory of phase equilibria. Quart. Appl. Math., 41(3):357–364, 1983/84.Google Scholar
  44. 44.
    P. G. Smith, M. Van Den Ven, and S. G. Mason. The transient interfacial tension between two miscible fluids. J. Colloid Interface Sci., 80(1):302–303, 1981.CrossRefGoogle Scholar
  45. 45.
    M. Van der Waals. Theorie thermodynamique de la capillarité dans l’hypothése d’une variation de densité. Arch. Neerl. Sci. Ex. Nat., 28:121–201, 1895.Google Scholar
  46. 46.
    B. Zoltowski, Y. Chekanov, J. Masere, J. A. Pojman, and V. Volpert. Evidence for the existence of an effective interfacial tension between miscible fluids. 2. Dodecyl acrylate-poly(dodecyl acrylate) in a spinning drop tensiometer. Langmuir, 23:5522–5531, 2007.Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.University of Minnesota, Minneapolis University of CaliforniaIrvineUSA

Personalised recommendations