Skip to main content
SpringerLink
Log in

Linear stability of a horizontal phase boundary subjected to shear motion

  • Regular Article
  • Published:
The European Physical Journal E Aims and scope Submit manuscript

Abstract

We investigate the stability of slowly smearing phase boundary that appears at the contact of two miscible liquids. A hydrodynamic flow is imposed along the boundary. The aim is to find out whether the slow diffusive smearing of a boundary can be overrun by faster mixing. The phase-field approach is used to model the evolution of the binary mixture. The linear stability in respect to 2D perturbations is studied. If the heavier liquid lies above the lighter liquid, the interface is unconditionally unstable due to the Rayleigh-Taylor and Kelvin-Helmholtz instabilities. The imposed flow accelerates the growth of the long-wave modes and suppresses the growth of the short-wave perturbations. Viscosity, diffusivity and capillarity reduce the growth of perturbations. If the heavier liquid underlies the lighter one, the interface can be stable. The stability boundaries are defined by the strength of gravity (density contrast) and the intensity of the imposed flow. Thinner interfaces are usually characterised by larger zones of instability. The thermodynamic instability, identified for the thicker interfaces with the thicknesses greater than the thickness of a thermodynamically equilibrium phase boundary, makes such interfaces unconditionally unstable. The zones of instability are enlarged by diffusive and capillary terms. Viscosity plays its stabilising role.

Graphical abstract

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. A. Kheniene, A. Vorobev, Phys. Rev. E 88, 022404 (2013).

    Article  ADS  Google Scholar 

  2. T. Babadagli, J. Petrol. Sci. Engin. 57, 221 (2007).

    Article  MATH  Google Scholar 

  3. M. Mukhopadhyay, Natural extracts using supercritical carbon dioxide (CRC Press LLC, 2000). .

  4. E.B. Nauman, A. Nigam, Trans. IChemE, Part A, Chem. Engin. Res. Design 85, 612 (2007).

    Article  Google Scholar 

  5. A. Vorobev, Curr. Opin. Colloid Interface Sci. 19, 300 (2014).

    Article  Google Scholar 

  6. S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability (Oxford University Press, 1961).

  7. P. Drazin, W. Reid, Hydrodynamic instability (Cambridge University Press, 1981).

  8. P.N. Guzdar, P. Satyanarayana, J.D. Huba et al., Geophys. Res. Lett. 9, 547 (1982).

    Article  ADS  Google Scholar 

  9. W. Zhang, Z. Wu, D. Li, Phys. Plasmas 12, 042106 (2005).

    Article  MathSciNet  ADS  Google Scholar 

  10. B.J. Olson, J. Larsson, S.K. Lee et al., Phys. Fluids 23, 114107 (2011).

    Article  ADS  Google Scholar 

  11. L.F. Wang, W.H. Ye, Y.J. Li, Phys. Plasmas 17, 042103 (2010).

    Article  ADS  Google Scholar 

  12. W.H. Ye, L.F. Wang, C. Xue et al., Phys. Plasmas 18, 022704 (2011).

    Article  ADS  Google Scholar 

  13. J. Holmboe, Geophys. Publ. 24, 67 (1962).

    Google Scholar 

  14. P.G. Drazin, J. Fluid Mech. 4, 214 (1958).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  15. S.A. Maslowe, R.E. Kelly, J. Fluid Mech. 48, 405 (1971).

    Article  ADS  MATH  Google Scholar 

  16. P. Hazel, J. Fluid Mech. 51, 39 (1972).

    Article  ADS  Google Scholar 

  17. L.N. Howard, S.A. Maslowe, Boundary-Layer Meteorol. 4, 511 (1973).

    Article  ADS  Google Scholar 

  18. P.G. Baines, H. Mitsudera, J. Fluid Mech. 276, 327 (1994).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  19. J.R. Carpenter, N.J. Balmforth, G.A. Lawrence, Phys. Fluids 22, 054104 (2010).

    Article  ADS  MATH  Google Scholar 

  20. G.A. Lawrence, S.P. Haigh, Z. Zhu, Coastal Estuarine Studies 54, 295 (1998).

    Article  Google Scholar 

  21. S.P. Haigh, G.A. Lawrence, Phys. Fluids 11, 1459 (1999).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  22. S.A. Thorpe, W.D. Smyth, L. Li, J. Fluid Mech. 731, 461 (2013).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  23. S. Alabduljalil, R.H. Rangel, J. Engin. Math. 54, 99 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  24. R. Barros, W. Choi, Phys. Fluids 23, 124103 (2011).

    Article  ADS  Google Scholar 

  25. S.A. Thorpe, J. Fluid Mech. 32, 693 (1968).

    Article  ADS  Google Scholar 

  26. S.A. Thorpe, Radio Sci. 4, 1327 (1969).

    Article  ADS  Google Scholar 

  27. A.M. Hogg, G.N. Ivey, J. Fluid Mech. 477, 339 (2003).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  28. G.M. Corcos, F.S. Sherman, J. Fluid Mech. 73, 241 (1976).

    Article  ADS  Google Scholar 

  29. W.D. Smyth, J. Fluid Mech. 497, 67 (2003).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  30. J. Lowengrub, L. Truskinovsky, Proc. R. Soc. London A 454, 2617 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  31. A. Vorobev, Phys. Rev. E 82, 056312 (2010).

    Article  ADS  Google Scholar 

  32. L.D. Landau, E.M. Lifshitz, Course of Theoretical Physics. Volume 5. Statistical Physics (Elsevier, Butterworth Heinemann, 1980).

  33. M.S.P. Stevar, A. Vorobev, J. Colloid Interface Sci. 383, 184 (2012).

    Article  Google Scholar 

  34. C. Pozrikidis, J. Fluid Mech. 351, 139 (1997).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  35. D. Bettinson, G. Rowlands, Phys. Rev. E 54, 6102 (1996).

    Article  ADS  Google Scholar 

  36. S.D. Conte, SIAM Rev. 8, 309 (1966).

    Article  MathSciNet  ADS  Google Scholar 

  37. L.M. Mack, J. Fluid Mech. 73, 497 (1976).

    Article  ADS  MATH  Google Scholar 

  38. A. Tumin, ASME J. Fluids Eng. 125, 428 (2003).

    Article  Google Scholar 

  39. R. Betchov, A. Szewczyk, Phys. Fluids 6, 1391 (1963).

    Article  ADS  MATH  Google Scholar 

  40. S.A. Maslowe, Hydrodynamic instabilities and the transition to turbulence, chapt. Shear flow instabilities and transition (Springer Berlin Heidelberg, 1981) pp. 181--228.

  41. A. Vorobev, O. Zikanov, J. Fluid Mech. 574, 131 (2007).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  42. C.-C.P. Caulfield, J. Fluid Mech. 258, 255 (1994).

    Article  MathSciNet  ADS  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Engineering and the Environment, University of Southampton, Southampton, UK

    A. Kheniene & A. Vorobev

Authors
  1. A. Kheniene
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. Vorobev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. Vorobev.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kheniene, A., Vorobev, A. Linear stability of a horizontal phase boundary subjected to shear motion. Eur. Phys. J. E 38, 77 (2015). https://doi.org/10.1140/epje/i2015-15077-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1140/epje/i2015-15077-4

Keywords

Search

Navigation