Advertisement

A thermodynamic model of multiphase flows with moving interfaces and contact line

  • Yongqi WangEmail author
  • Martin Oberlack
Original Article

Abstract

In this paper, we develop a general continuum description for thermodynamic multiphase flows with intersecting dividing surfaces, and three-phase common contact line, taking the contribution of the excess surface and line thermodynamic quantities into account. Starting with the standard postulates of continuum mechanics and the general global balance statement for an arbitrary physical quantity in a physical domain of three bulk phases including singular material or non-material phase interfaces and a three-phase contact line, we derive, in addition to the classical local balance equations for each bulk phase, the local conservation equations on the phase interfaces and at the contact line. Then, these additional interface and line balance laws are specified for excess surface and line physical quantities, e.g., excess mass, momentum, angular momentum, energy, and entropy, respectively. Some simplified forms of these balance laws are also presented and discussed.

Keywords

Multiphase flows Three-phase contact line Surface balance equations Line balance equations 

References

  1. 1.
    Alts T., Hutter K.: Continuum description of the dynamics and thermodynamics of phase boundaries between ice and water, part I: surface balance laws and their interpretation in terms of three-dimensional balance laws averaged over the phase change boundary layer. J. Non-Equilib. Thermodyn. 13, 221–257 (1988)ADSzbMATHCrossRefGoogle Scholar
  2. 2.
    Amirfazli A., Kwok D.Y., Gaydos J., Neumann A.W.: Line tension measurements through drop size dependence of contact angle. J. Colloid Interface Sci. 205, 1–11 (1998)CrossRefGoogle Scholar
  3. 3.
    Amirfazli A., Keshavarz A., Zhang L., Neumann A.W.: Determination of line tension for systems near wetting. J. Colloid Interface Sci. 265, 152–160 (2003)CrossRefGoogle Scholar
  4. 4.
    Amirfazli A., Neumann A.W.: Status of the three-phase line tension: a review. Adv. Colloid Interface Sci. 110, 121–141 (2004)CrossRefGoogle Scholar
  5. 5.
    Babak V.G.: Generalised line tension theory revisited. Colloids Surf. A 156, 423–448 (1999)CrossRefGoogle Scholar
  6. 6.
    Bedeaux D.: Nonequilibrium thermodynamic description of the three-phase contact line. J. Chem. Phys. 120, 3744–3748 (2004)ADSCrossRefGoogle Scholar
  7. 7.
    Bier M., Chen W., Gowrishankar TR, Astumian RD, Lee RC: Resealing dynamics of a cell membrane after electroporation. Phys. Rev. E 66(6), 062905 (2002)ADSCrossRefGoogle Scholar
  8. 8.
    Cermelli P., Fried E., Gurtin M.: Transport relation for surface intergrals arising in the formulation of balance laws for evolving fluid interfaces. J. Fluid Mech. 544, 339–351 (2005)MathSciNetADSzbMATHCrossRefGoogle Scholar
  9. 9.
    Dussaud A., Vignes-Adler M.: Wetting transition of n-alkanes on concentrated aqueous salt solutions. Line tension effect. Langmuir 13, 581–589 (1997)CrossRefGoogle Scholar
  10. 10.
    Edwards D.A., Brenner H., Wasan D.T.: Interfacial Transport Processes and Rheology. Butterworth-Heinemann, Boston (1991)Google Scholar
  11. 11.
    Eggleton C.D., Stebe K.J.: An adsorption-desorption controlled surfactant on a deforming droplet. J. Colloid Interface Sci. 208, 68–80 (1998)CrossRefGoogle Scholar
  12. 12.
    Eggleton C.D., Tsai T.M., Stebe K.J.: Tip streaming from a drop in the presence of surfactants. Phys. Rev. Lett. 87, 048302 (2001)ADSCrossRefGoogle Scholar
  13. 13.
    Gatignol R., Prud’homme R.: Mechanical and Thermodynamical Modeling of Fluid Interfaces. World Scientific, Singapore (2001)CrossRefGoogle Scholar
  14. 14.
    Getta T., Dietrich S.: Line tension between fluid phases and a substrate. Phys. Rev. E 57, 655–671 (1998)ADSCrossRefGoogle Scholar
  15. 15.
    Gokhale S.J., Plawsky J.L., Wayner P.C. Jr: Effect of interfacial phenomena on dewetting in dropwise condensation. Adv. Colloid Interface Sci. 104, 175–190 (2003)CrossRefGoogle Scholar
  16. 16.
    Graham C., Griffith P.: Drop size distributions and heat transfer in dropwise condensation. Int. J. Heat Mass Transf. 16, 337–346 (1973)CrossRefGoogle Scholar
  17. 17.
    Gu Y.: Drop size dependence of contact angles of oil drops on a solid surface in water. J. Colloids Surf. A 181, 215–224 (2001)CrossRefGoogle Scholar
  18. 18.
    Gurtin M.E., Struthers A., Williams W.O.: A transport theorem for moving interfaces. Q. Appl. Math. 47, 773–777 (1989)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Gurtin M.E.: On thermomechanical laws for the motion of a phase interface. J. Appl. Math. Phys. 42, 370–388 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Indekeu J.O.: Line tension at wetting. J. Mod. Phys. B 8, 309–345 (1994)ADSCrossRefGoogle Scholar
  21. 21.
    Jin F., Gupta N.R., Stebe K.J.: The detachment of a viscous drop in a viscous solution in the presence of a soluble surfactant. Phys. Fluids 18, 022103 (2006)ADSCrossRefGoogle Scholar
  22. 22.
    Levich V.G.: Physicochemical Hydrodynamics. Prentice Hall, Englewood Cliffs (1962)Google Scholar
  23. 23.
    Marmur A.: Line tension and the intrinsic contact angle in solidliquidfluid systems. J. Colloid Interface Sci. 186, 462–466 (1997)CrossRefGoogle Scholar
  24. 24.
    Moeckel G.P.: Thermodynamics of an interface. Arch. Ration. Mech. Anal. 57, 255–280 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Morel C.: On the surface equations in two-phase flows and reacting single-phase flows. Int. J. Multiphase Flow 33, 1045–1073 (2007)CrossRefGoogle Scholar
  26. 26.
    Oliver J.F., Huh C., Mason S.G.: Resistance to spreading of liquids by sharp edges. J. Colloid Interface Sci. 59, 568–581 (1977)CrossRefGoogle Scholar
  27. 27.
    Osher S., Sethian J.A.: Fronts propagating with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys. 79, 12–49 (1988)MathSciNetADSzbMATHCrossRefGoogle Scholar
  28. 28.
    Otis D.R. Jr, Ingenito E.P., Kamm R.D., Johnson M.: Dynamic surface tension of surfactant TA: experiments and theory. J. Appl. Physiol. 77, 2681–2688 (1994)Google Scholar
  29. 29.
    Petryk H., Mroz Z.: Time derivatives of integrals and functionals defined on varying volume and surface domains. Arch. Mech. 38, 697–724 (1986)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Pompe, T.: Line tension behavior of a first-order wetting system. Phys. Rev. Lett. 89(7) Art.No. 076102 (2002)Google Scholar
  31. 31.
    Qu W., Li D.: Line tension of simple liquidliquidfluid systems. Colloids Surf. A 156, 123–135 (1999)CrossRefGoogle Scholar
  32. 32.
    Rodrigues J.F., Saramago B., Fortes M.A.: Apparent contact angle and triple-line tension of a soap bubble on a substrate. J. Colloid Interface Sci. 239, 577–580 (2001)CrossRefGoogle Scholar
  33. 33.
    Rusanov A.I.: Surface thermodynamics revisited. Surf. Sci. Rep. 58, 111–239 (2005)ADSCrossRefGoogle Scholar
  34. 34.
    Sagis L.M., Slattery J.C.: Incorporation of line quantities in the continuum description for multiphase, multicomponent bodies with intersecting dividing surfaces. I. Kinematics and conservation principles. J. Colloid Interface Sci. 176, 150–164 (1995)CrossRefGoogle Scholar
  35. 35.
    Shikhmurzaev Y.D.: The moving contact line on a smooth solid surface. Int. J. Multiphase Flow 19, 589–610 (1993)zbMATHCrossRefGoogle Scholar
  36. 36.
    Shikhmurzaev Y.D.: Capillary Flows with Forming Interfaces. Chapman & Hall/CRC, London, New York (2008)zbMATHGoogle Scholar
  37. 37.
    Slattery J.C., Sagis L., Oh E.-S.: Interfacial Transport Phenomena. Springer, New York (2007)zbMATHGoogle Scholar
  38. 38.
    Solomentsev Y., White L.R.: Microscopic drop profiles and the origins of line tension. J. Colloid Interface Sci. 218, 122–136 (1999)CrossRefGoogle Scholar
  39. 39.
    Sussman M., Fatemi E., Smereka P., Osher S.: An improved level set method for incompressible two-phase flows. Comput. Fluids 277, 663–680 (1998)CrossRefGoogle Scholar
  40. 40.
    Sussman M., Smereka P., Osher S.: A level set approach for computing solutions to incompressible two-phase flow. J. Comput. Phys. 114, 146–159 (1994)ADSzbMATHCrossRefGoogle Scholar
  41. 41.
    Szleifer I., Widom B.: Surface tension, line tension, and wetting. Mol. Phys. 5, 925–943 (1992)ADSCrossRefGoogle Scholar
  42. 42.
    Thomas T.Y.: Extended compatibility conditions for the study of surfaces of discontinuity in continuum mechanics. J. Math. Mech. 6, 311–322 (1957)MathSciNetzbMATHGoogle Scholar
  43. 43.
    Truesdell C.: A First Course in Rational Continuum Mechanics. Academic Press, New York (1977)zbMATHGoogle Scholar
  44. 44.
    Vera-Graziano R., Muhl S., Rivera-Torres F.: The effect of illumination on contact angles of pure water on crystalline silicon. J. Colloid Interface Sci. 170, 591–597 (1995)CrossRefGoogle Scholar
  45. 45.
    Vignes-Adler M., Brenner H.: A micromechanical derivation of the dierential equations of interfacial statics. III. Line tension. J. Colloid Interface Sci. 103, 11–44 (1985)CrossRefGoogle Scholar
  46. 46.
    Wallace J.A., Schürch S.: Line tension of sessile drops placed on a phospholipid monolayer at the water–fluorocarbon interface. Colloids Surf. 43, 207–221 (1990)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Chair of Fluid Dynamics, Department of Mechanical EngineeringTechnische Universität DarmstadtDarmstadtGermany
  2. 2.Center of Smart InterfacesTechnische Universität DarmstadtDarmstadtGermany
  3. 3.Graduate School of Computational EngineeringTechnische Universität DarmstadtDarmstadtGermany

Personalised recommendations