Journal of Materials Science

, Volume 51, Issue 4, pp 1820–1828 | Cite as

Underdamped capillary wave caused by solutal Marangoni convection in immiscible liquids

  • Fei Wang
  • Marouen Ben Said
  • Michael Selzer
  • Britta Nestler
HTC 2015


Using a Cahn–Hilliard–Navier–Stokes model with a capillary tensor to account for solutal Marangoni force, we observe an interfacial wave at the interface of two immiscible liquids. A Fourier analysis shows that the interfacial wave is induced by oscillatory modes of solutal Marangoni convection. A critical Marangoni number is defined above which the oscillatory modes of solutal Marangoni convection are able to occur. This material property is a function of the wave number for different Cahn numbers and is determined from numerical simulations.


  1. 1.
    Bush JWM (2004) Surface tension module. MIT OpenCourseWare, CambridgeGoogle Scholar
  2. 2.
    Wierschem A, Velarde MG, Linde H, Waldhelm W (1999) Interfacial wave motions due to Marangoni instability: II. Three-dimensional characteristics of surface waves in annular containers. J Colloid Interface Sci 212:365–383CrossRefGoogle Scholar
  3. 3.
    Linde H, Velarde MG, Waldhelm W, Wierschem A (2001) Interfacial wave motions due to Marangoni instability: III. Solitary waves and (periodic) wave trains and their collisions and reflections leading to dynamic network (cellular) patterns in large containers. J Colloid Interface Sci 236:214–224CrossRefGoogle Scholar
  4. 4.
    Wierschem A, Linde H, Velarde MG (2000) Internal waves excited by the Marangoni effect. Phys Rev E 62:6522CrossRefGoogle Scholar
  5. 5.
    Linde H, Velarde MG, Waldhelm W, Loeschcke K, Wierschem A (2005) On the various wave motions observed at a liquid interface due to Marangoni stresses and instability. Ind Eng Chem Res 44:1396–1412CrossRefGoogle Scholar
  6. 6.
    Michallet H, Barthelemy E (1998) Experimental study of interfacial solitary waves. J Fluid Mech 366:159–177CrossRefGoogle Scholar
  7. 7.
    Sternling CV, Scriven LE (1959) Interfacial turbulence: hydrodynamic instability and the Marangoni effect. AIChE J 5:514–523CrossRefGoogle Scholar
  8. 8.
    Reichenbach J, Linde H (1981) Linear perturbation analysis of surface-tension-driven convection at a plane interface (Marangoni instability). J Colloid Interface Sci 84:433–443CrossRefGoogle Scholar
  9. 9.
    Cahn JW, Hilliard JE (1958) Free energy of a nonuniform system. I. Interfacial free energy. J Chem Phys 28:258–267CrossRefGoogle Scholar
  10. 10.
    Wang F, Choudhury A, Selzer M, Mukherjee R, Nestler B (2012) Effect of solutal Marangoni convection on motion, coarsening, and coalescence of droplets in a monotectic system. Phys Rev E 86:066318CrossRefGoogle Scholar
  11. 11.
    Goldstein H, Poole C, Safko J (2001) Classical mechanics. Addison Wesley, New YorkGoogle Scholar
  12. 12.
    Wheeler AA, McFadden GB (1997) On the notion of a \(\xi \)-vector and a stress tensor for a general class of anisotropic diffuse interface models. Proc Roy Soc A 453:1611–1630CrossRefGoogle Scholar
  13. 13.
    Liu H, Zhang Y, Valocchi AJ (2012) Modeling and simulation of thermocapillary flows using lattice Boltzmann method. J Comput Phys 231:4433–4453CrossRefGoogle Scholar
  14. 14.
    Langer JS (1980) Instability and pattern formation in crystal growth. Rev Mod Phys 52:1–28CrossRefGoogle Scholar
  15. 15.
    Derby B, Favier JJ (1983) A criterion for the determination of monotectic structure. Acta Metall 31:1123–1130CrossRefGoogle Scholar
  16. 16.
    Egry I, Ratke L, Kolbe M, Chatain D, Curiotto S, Battezzati L, Johnson E, Pryds N (2010) Interfacial properties of immiscible Co–Cu alloys. J Mater Sci 45:1979–1985. doi:10.1007/s10853-009-3890-0 CrossRefGoogle Scholar
  17. 17.
    Kaptay G (2008) A Calphad-compatible method to calculate liquid/liquid interfacial energies in immiscible metallic systems. Calphad 32:338–352CrossRefGoogle Scholar
  18. 18.
    Mendelev MI, Mishin Y (2009) Molecular dynamics study of self-diffusion in bcc Fe. Phys Rev B 80:144111CrossRefGoogle Scholar
  19. 19.
    Assael MJ et al (2010) Reference data for the density and viscosity of liquid copper and liquid tin. J Phys Chem Ref Data 39:033105CrossRefGoogle Scholar
  20. 20.
    Griebel M, Dornseifer T, Neunhoeffer T (1997) Numerical simulation in fluid dynamics: a practical introduction. Society for Industrial and Applied Mathematics, PhiladelphiaGoogle Scholar
  21. 21.
    Moin P (2010) Fundamentals of engineering numerical analysis. Cambridge University Press, New YorkCrossRefGoogle Scholar
  22. 22.
    Gibbs JW (1957) The collected works of J. Willard Gibbs. Yale University Press, New YorkGoogle Scholar
  23. 23.
    Chattoraj DK, Birdi KS (1984) Adsorption and the Gibbs surface excess. Plenum Press, New YorkCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Fei Wang
    • 1
    • 2
  • Marouen Ben Said
    • 1
    • 2
  • Michael Selzer
    • 1
    • 2
  • Britta Nestler
    • 1
    • 2
  1. 1.Institute of Materials and ProcessesKarlsruhe University of Applied SciencesKarlsruheGermany
  2. 2.Institute of Applied Materials-Computational Materials ScienceKarlsruhe Institute of Technology (KIT)KarlsruheGermany

Personalised recommendations