The European Physical Journal Special Topics

, Volume 227, Issue 5–6, pp 481–492 | Cite as

Prandtl number dependence of convective fluids in tall laterally heated slots

  • J. Sánchez UmbríaEmail author
  • M. Net
Regular Article
Part of the following topical collections:
  1. Nonlinear Phenomena in Physics: New Techniques and Applications


The influence of the Prandtl number on the stable stationary and periodic flows of the fluids contained in laterally heated slots under realistic conditions, namely non-slip boundaries, insulated top and bottom horizontal limits and perfectly conducting lateral sides, is analyzed by using continuation methods. The branches of solutions are computed by decreasing the Prandtl number, for four Rayleigh numbers. The dynamical behavior depends strongly on both parameters. For a Rayleigh number, Ra = 103, the steady flow remains stable in the wide range of Prandtl numbers computed. At Ra = 104 and O(105) the first bifurcations are of Hopf type giving rise to a type of oscillations that affects the bulk of the fluid, alternating from a general circulation to multi-vortex solutions, or the boundary layer, respectively. However, it is found that, in any case, the location of the shear determines the type of the oscillations. Moreover, at Ra = 104 the critical multipliers at the secondary bifurcations on the main branch of POs are real, giving rise to different kinds of very bounded stable periodic states of different symmetries and periods. At Ra of order 105 the instability of the periodic orbits gives rise directly to quasi-periodic flows.


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  1. 1.
    R. Feigelson (Ed.), 50 years Progress in Crystal Growth. A reprint collection (Elsevier, 2004) Google Scholar
  2. 2.
    M. Christon, P. Gresho, S. Sutton, Int. J. Numer. Methods Fluids 40, 953 (2002) ADSCrossRefGoogle Scholar
  3. 3.
    M. Lappa, Thermal Convection: Patterns Evolution and Stability (Wiley, Singapore, 2010) Google Scholar
  4. 4.
    C.K. Mamun, L.S. Tuckerman, Phys. Fluids 7, 80 (1995) ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    L.S. Tuckerman, D. Barkley, Bifurcation analysis for timesteppers, in Numerical Methods for Bifurcation Problems and Large-Scale Dynamical Systems, edited by E. Doedel, L.S. Tuckerman IMA Volumes in Mathematics and its Applications (Springer-Verlag, 2000), Vol. 119, pp. 453–466 Google Scholar
  6. 6.
    K.H. Winters, Int. J. Numer. Methods Eng. 25, 401 (1988) CrossRefGoogle Scholar
  7. 7.
    A.Y. Gelfgat, P.Z. Bar-Yoseph, A.L. Yarin, J. Fluid Mech. 388, 315 (1999) ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    S. Xin, P. Le Quéré, Numer. Heat Transfer. Part A 50, 437 (2006) CrossRefGoogle Scholar
  9. 9.
    D. Henry, H. BenHadid, Phys. Rev. E 76, 016314 (2007) ADSCrossRefGoogle Scholar
  10. 10.
    A. Bergeon, E. Knobloch, Physica D 237, 1139 (2008) ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    J. Sánchez, M. Net, B. García-Archilla, C. Simó, J. Comput. Phys. 201, 13 (2004) ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    J. Sánchez, M. Net, B. García-Archilla, C. Simó, Continuation of periodic orbits in large-scale dissipative systems, in Proceedings of the Equadiff-2003 Conference, edited by F. Dumortier, H. Broer, J. Mawhin, A. Vanderbauwhede, S.V. Lunel (World Scientific, Singapore, 2005), pp. 625–630 Google Scholar
  13. 13.
    M. Net, J. Sánchez, SIAM J. Appl. Dynam. Syst. 14, 674 (2015) CrossRefGoogle Scholar
  14. 14.
    S. Wakitani, Phys. Fluids 10, 1924 1998 Google Scholar
  15. 15.
    H. Yahata, J. Phys. Soc. Jpn 66, 3434 (1998) ADSCrossRefGoogle Scholar
  16. 16.
    S. Xin, P. Le Quéré, Fluid Dyn. Res. 44, 031419 (2012) ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    M. Net, J. Sánchez, Phys. Rev. E 95, 023102 (2017) ADSCrossRefGoogle Scholar
  18. 18.
    H. Ke, Y. He, Y. Liu, F. Cui, Int. J. Thermophys. 33, 1143 (2012) ADSCrossRefGoogle Scholar
  19. 19.
    J. Sánchez, M. Net, Eur. Phys. J. Special Topics 225, 2465 (2016) ADSCrossRefGoogle Scholar
  20. 20.
    R.B. Lehoucq, D.C. Sorensen, SIAM J. Matrix Anal. Appl. 17, 789 (1996) MathSciNetCrossRefGoogle Scholar

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© EDP Sciences and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Departament de Física, Universitat Politècnica de CatalunyaBarcelonaSpain

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