Three-dimensional simulations of instabilities in a Marangoni-driven, low Prandtl number liquid bridge with magnetic stabilization to verify linear stability theory

Regular Article


The Full-Zone model of a liquid bridge encountered in crystal growth is analyzed via linear stability analysis and three-dimensional spectral element simulations, neglecting gravitational forces, for Prandtl number 0.02. The base state is axisymmetric and steady state. Linear stability predicts the character of flow transitions and the value of ReFZ, the thermocapillary Reynolds number, at which instabilities occur. Previous linear stability findings show that application of a steady, axial magnetic field stabilizes the base state. Previous three-dimensional simulations with no magnetic field predict a first transition that agrees well with linear stability theory. However, these simulations also demonstrated that continued time integration at just slightly higher ReFZ leads to what appears to be periodic flow. Closer inspection and comparison with linear stability theory revealed that this apparent periodicity was actually competition between two steady modes with different axial symmetries. Here an axial magnetic field is applied in three-dimensional simulations and it is verified that the magnetic field does have the intended effect of stabilizing the flow and removing modal competition. The azimuthal flow shows excellent agreement with eigenvectors predicted by linear stability theory.


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  1. 1.
    A. Eyer, R. Nitsche, H. Zimmermann, J. Crystal Growth 47, 219 (1979)ADSCrossRefGoogle Scholar
  2. 2.
    M. Levenstam, G. Amberg, J. Fluid Mech. 297, 357 (1995)ADSCrossRefMathSciNetMATHGoogle Scholar
  3. 3.
    G. Chen, A. Lizee, B. Roux, J. Crystal Growth 180, 638 (1997)ADSCrossRefGoogle Scholar
  4. 4.
    M. Prange, M. Wanschura, H.C. Kuhlmann, H.J. Rath, J. Fluid Mech. 394, 281 (1999)ADSCrossRefMATHGoogle Scholar
  5. 5.
    M. Lappa, Computers Fluids 34, 743 (2005)CrossRefMATHGoogle Scholar
  6. 6.
    O. Bouizi, C. Delcarte, G. Kasperski, Phys. Fluids 19, 114102 (2007)ADSCrossRefGoogle Scholar
  7. 7.
    C.W. Lan, B.C. Yeh, J. Crystal Growth 262, 59 (2004)ADSCrossRefGoogle Scholar
  8. 8.
    C.W. Lan, B.C. Yeh, Fluid Dyn. Mater. Proc. 1, 33 (2005)Google Scholar
  9. 9.
    B.C. Houchens, J.S. Walker, J. Thermophysics Heat Trans. 137, 186 (2005)CrossRefGoogle Scholar
  10. 10.
    K.E. Davis, Y. Huang, B.C. Houchens, Phys. Fluids (to appear)Google Scholar
  11. 11.
    Y. Huang, B.C. Houchens, Eur. Phys. J. Special Topics 192, 47 (2011)ADSCrossRefGoogle Scholar
  12. 12.
    Y. Huang, B.C. Houchens, J. Mech. Materials Structures 6, 905 (2011)Google Scholar
  13. 13.
    A. Patera, J. Comp. Phys. 54, 468 (1984)ADSCrossRefMATHGoogle Scholar
  14. 14.
    Y. Maday, A. Patera, State-of-the-art Surveys Comput. Mech. 1, 71 (1989)Google Scholar
  15. 15.
    M.O. Deville, P.F. Fischer, E.H. Mund, High-Order Methods for Incompressible Fluid Flow Cambridge University Press, Cambridge, 2004Google Scholar

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© EDP Sciences and Springer 2013

Authors and Affiliations

  1. 1.Rice UniversityHoustonUSA

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