Advertisement

Journal of Hydrodynamics

, Volume 18, Issue 1, pp 260–264 | Cite as

Numerical research on the hydrodynamic stability of Blasius flow with spectral method

  • Ming-Liang Xie
  • Hong-Bing Xiong
  • Jian-Zhong Lin
Session B4

Abstract

Orr-Sommerfeld equation is solved numerically using expansions in Chebyshev polynomials. The numerical results show a good agreement with Howarth’s solution, with relatively low computational cost. This method is then applied to the stability of flat plate boundary layer flow compared with the finite difference method; our study shows that the expansions in Chebyshev polynomials are more suitable for the solution of hydrodynamic stability problems than the expansions in finite difference method.

Key words

linear stability Blasius equation Chebyshev polynomials spectral method 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    DAVID R., TIMOTHY J. J., LAURIE E. L. Imaging of electroosmotic flow in plastic microchannels[J]. Anal. Chem., 2001, 73: 2509–2515.CrossRefGoogle Scholar
  2. [2]
    MAO H. B., MATTHEW A. H., YOU M., PAUL S. C. Reusable platforms for high-throughput on-chip temperature gradient assays [J]. Anal. Chem., 2002, 74: 5071–5075.CrossRefGoogle Scholar
  3. [3]
    TANG G.Y., YANG C., CHAI J.C., GONG H.Q. Joule heating effect on electroosmotic flow and mass species transport in a microcapillary [J]. Int. J. Heat Mass Transfer, 2004, 47:215–227.CrossRefGoogle Scholar
  4. [4]
    XUAN X.C., SINTON D., LI D.Q. Thermal end effects on electroosmotic flow in a capillary [J]. Int. J. Heat Mass Transfer 2004, 47:3145–3157.CrossRefGoogle Scholar
  5. [5]
    ABRAHAM D.S., GEORGE M. W. Controlling flows in microchannels with patterned surface charge and topography [J]. Acc. Chem. Res., 2003, 36:597–604CrossRefGoogle Scholar
  6. [6]
    NICO V., JEROEN B., PIOTR G., GINO V. B., GERT D. Importance and reduction of the sidewall-induced band-broadening effect in pressure-driven microfabricated columns [J]. Anal. Chem., 2004, 76: 4501–4507.CrossRefGoogle Scholar
  7. [7]
    LI Z.H., LIN J.Z. NIE D.M. New approach to minimize dispersion induced by turn in the capillary electrophoresis channel flows [J]. Applied Mathematics and Mechanics, 2005, 26: 685–690.CrossRefGoogle Scholar
  8. [8]
    TAYLOR S. G. Dispersion of soluble matter in solvent flowing slowly through a tube [J]. Proc. Roy. Soc. A, 1953, 219:186–193.CrossRefGoogle Scholar
  9. [9]
    ARIS R. On the dispersion of a solute in a fluid flowing through a tube [J]. Proc. Roy. Soc. A, 1956, 235:67–74.CrossRefGoogle Scholar
  10. [10]
    EINSTEIN A. Investigation on the Theory of the Brownian Movement [M]. New York: Dover Publications, 1956.zbMATHGoogle Scholar
  11. [11]
    OROSAG S.T., Accurate solution of the Orr-Sommerfeld stability equation [J]. J. Fluid Mech., 1971, 50(4): 689–703.CrossRefGoogle Scholar
  12. [12]
    SCHLICHTING H., Boundary Layer Theory M]. McGraw Hill Book Company, Inc., New York, 1954: 43.Google Scholar
  13. [13]
    WANG L., A new algorithm for solving classical Blasius equation[J]. Applied Mathematics and Computation, 2004, 157: 1–9.MathSciNetCrossRefGoogle Scholar
  14. [14]
    YU L.T., CHEN C.K. The solution of the Blasius equation by the differential transformation method. Mathl. Comput. Modeling. 1998, 28(1): 101–111.MathSciNetCrossRefGoogle Scholar
  15. [15]
    GRIFFITHS D.F., WATSON G.A., Numerical analysis[M]. Longman Scientific & Technical, Essex, UK, 1987.Google Scholar
  16. [16]
    JORDINSON R., The flat plate boundary layer. Part 1. Numerical integration of the Orr-Sommerfeld equation[J]. J. Fluid Mech. 1970, 43(4): 801–811.CrossRefGoogle Scholar
  17. [17]
    MERCIER B., An introduction to the numerical analysis of spectral methods [M]. Springer, 1989.Google Scholar

Copyright information

© China Ship Scientific Research Center 2006

Authors and Affiliations

  • Ming-Liang Xie
    • 1
    • 2
  • Hong-Bing Xiong
    • 1
    • 2
  • Jian-Zhong Lin
    • 1
    • 2
  1. 1.Department of Mechanics, State Key Laboratory of Fluid Power Transmission and ControlZhejiang UniversityHangzhouChina
  2. 2.China Jiliang UniversityHangzhouChina

Personalised recommendations