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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
DAVID R., TIMOTHY J. J., LAURIE E. L. Imaging of electroosmotic flow in plastic microchannels[J]. Anal. Chem., 2001, 73: 2509–2515.
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.
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.
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.
ABRAHAM D.S., GEORGE M. W. Controlling flows in microchannels with patterned surface charge and topography [J]. Acc. Chem. Res., 2003, 36:597–604
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.
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.
TAYLOR S. G. Dispersion of soluble matter in solvent flowing slowly through a tube [J]. Proc. Roy. Soc. A, 1953, 219:186–193.
ARIS R. On the dispersion of a solute in a fluid flowing through a tube [J]. Proc. Roy. Soc. A, 1956, 235:67–74.
EINSTEIN A. Investigation on the Theory of the Brownian Movement [M]. New York: Dover Publications, 1956.
OROSAG S.T., Accurate solution of the Orr-Sommerfeld stability equation [J]. J. Fluid Mech., 1971, 50(4): 689–703.
SCHLICHTING H., Boundary Layer Theory M]. McGraw Hill Book Company, Inc., New York, 1954: 43.
WANG L., A new algorithm for solving classical Blasius equation[J]. Applied Mathematics and Computation, 2004, 157: 1–9.
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.
GRIFFITHS D.F., WATSON G.A., Numerical analysis[M]. Longman Scientific & Technical, Essex, UK, 1987.
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.
MERCIER B., An introduction to the numerical analysis of spectral methods [M]. Springer, 1989.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Xie, ML., Xiong, HB. & Lin, JZ. Numerical research on the hydrodynamic stability of Blasius flow with spectral method. J Hydrodyn 18 (Suppl 1), 260–264 (2006). https://doi.org/10.1007/BF03400456
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF03400456