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New algebraic approaches to classical boundary layer problems

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Abstract

Classical non-steady boundary layer equations are fundamental nonlinear partial differential equations in the boundary layer theory of fluid dynamics. In this paper, we introduce various schemes with multiple parameter functions to solve these equations and obtain many families of new explicit exact solutions with multiple parameter functions. Moreover, symmetry transformations are used to simplify our arguments. The technique of moving frame is applied in the three-dimensional case in order to capture the rotational properties of the fluid. In particular, we obtain a family of solutions singular on any moving surface, which may be used to study turbulence. Many other solutions are analytic related to trigonometric and hyperbolic functions, which reflect various wave characteristics of the fluid. Our solutions may also help engineers to develop more effective algorithms to find physical numeric solutions to practical models.

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References

  1. Ibragimov, N. H.: Lie Group Analysis of Differential Equations, Volume 2, CRC Handbook, CRC Press, 1995

  2. Ovsiannikov, L. V.: Group Analysis of Differential Equations, Academic Press, New York, 1982

    MATH  Google Scholar 

  3. Garaev, K. G.: Group properties of an equation for nonstationary space boundary layer of incompressible fluid. Trudy Kazan. Aviatsion. Inst., 119, 47–53 (1970)

    MathSciNet  Google Scholar 

  4. Vereshchagina, L. I.: Group stratification of space nonstationary boundary layer equations. Vestnik Leningr. Universiteta, 13(3), 82 (1973)

    Google Scholar 

  5. Daragan, M. A., Derbenev, S. A.: Group investigations of laminar boundary layer on rotating wing. Izvestiya Vuzov. Aviatsionaya Tekhnika, 4, 81 (1985)

    Google Scholar 

  6. Derbenev, S. A.: Group properties of boundary layer equations with magnetic field and chemical reactions. Trudy Kazan. Aviatsion. Inst., 119, 100–105 (1970)

    MathSciNet  Google Scholar 

  7. Garaev, K. G., Pavlov, Y. G.: Group properties of equations of optimally controlled boundary layer. Izvestiya Vuzov. Aviatsion. Tekhnika, 4, 5 (1970)

    Google Scholar 

  8. Lankerovich, M. Ya.: Group properties of three-dimensional boundary layer equations on arbitrary surfaces. Dinamika Splash. Sredi. Institute of Hydrodynamics, 7, 12 (1971)

    Google Scholar 

  9. Loitsyanskii, L. G.: Laminar Boundary Layer, Fizmatgiz, Moscow, 1962

    Google Scholar 

  10. Xu, X.: Stable-range approach to the equation of nonstationary transonic gas flows. Quart. Appl. Math., 65, 529–547 (2007)

    MathSciNet  MATH  Google Scholar 

  11. Xu, X.: Asymmetric and moving-frame approaches to Navier-Stokes equations. Quart. Appl. Math., 67, 163–193 (2009)

    MathSciNet  MATH  Google Scholar 

  12. Abel, M. S., Nandeppanavar, M. M.: Heat transfer in MHD viscoelastic boudary layer flow over a stretching sheet with non-uniform heat source/sink. Commun. Nonl. Sci. Num. Sim., 14, 2120–2131 (2009)

    Article  MathSciNet  Google Scholar 

  13. Abdel-Maleck, M. B., Helal, M. M.: Similarity solutions for magneto-forced-unsteady free convective laminar boundary-layer flow. J. Comp. Appl. Math., 218, 202–214 (2008)

    Article  Google Scholar 

  14. Almog, Y.: Thin boundary layer of chiral smectics. Calc. Var., 33, 299–328 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Bennis, A.-C., Chacón Rebollo, T., Gomez Marmol, M., et al.: Stability of some turbulent vertical models for the ocean mixing boundary layer. App. Math. Lett., 21, 128–133 (2008)

    Article  MATH  Google Scholar 

  16. Cheng, J., Liao, S., Mohapatra, R. N., et al.: Series solutions of nano boundary layer flows by means of the homotopy analysis method. J. Math. Anal. Appl., 343, 233–145 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  17. Cousteix, J., Mauss, J.: Interactive boundary layer models for channel flow. Euro. J. Mech. B/Fluids, 28, 72–87 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. Dennis, D. J., Nickels, T. B.: On the limitation of Taylor’s hypothesis in constructing long structures in a turbulent boundary layer. J. Fluid Mech., 614, 197–206 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. Dixit, S. A., Ramesh, O. N.: Pressure-gradient-dependent logarithmic laws in sink flow turbulent boundary layers. J. Fluid Mech., 615, 445–475 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  20. Egorov, I. V., Fedorov, A. V., Soudakov, V. G.: Receptivity of a hypersonic boundary layer over a flat plate with a porous coating. J. Fluid Mech., 601, 165–187 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  21. Ehrenstein, U., Gallaire, F.: Two-dimensional global low-frequency oscillations in a separating boundarylayer flow. J. Fluid Mech., 614, 315–327 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  22. Ilin, K.: Viscous boundary layers in flows through a domain with permeable boundary. Euro. J. Mech. B/Fluids, 27, 514–538 (2008)

    Article  MATH  Google Scholar 

  23. Ishak, A., Naar, R., Pop, I.: MHD boudary-layer flow of a micropolar fluid past a wedge with constant wall heat flux. Commun. Nonl. Sci. Num. Sim., 14, 108–118 (2009)

    Google Scholar 

  24. Ishak, A., Naar, R., Pop, I.: Dual solutions in mixed convection boudary layer flow of micropolar fluids, Commun. Nonl. Sci. Num. Sim., 14, 1324–1333 (2009)

    Article  Google Scholar 

  25. Lan, K. Q., Yang, G. C.: Positive solutions of the Falkner-Skan equation arising in the boundary layer theory. Canad. Math. Bull., 51(3), 386–398 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  26. Liao, S.-J.: A general approach to get series solution of non-similarity boundary-layer flows. Commun. Nonl. Sci. Num. Sim., 14, 2144–2159 (2009)

    Article  Google Scholar 

  27. Liu, Y., Zaki, T. A., Durbin, P. A.: Boundary-layer transition by interaction of discrete and continuous models. J. Fluid Mech., 604, 199–233 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  28. Merkin, J. H.: Free convective boundary-layer flow in a heat-generating porous medium: similarity solutions. Quart. Mech. Appl. Math., 61(2), 205–218 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  29. Naz, R., Mahomed, F. M., Mason, D. P.: Symmetry solutions of a third-order ordinary differential equation which arises from Prandtl boundary layer equations. J. Nonl. Math. Phys., 15,Supplement 1, 179–191 (2008)

    Article  MathSciNet  Google Scholar 

  30. Van D’ep, N.: On boundary layer equations for fluids with a moment stress. Prikladn. Matemanka i Mekhnika, 32(4), 748 (1989)

    Google Scholar 

  31. Oron, A., Gottlib, O., Novbari, E.: Weighted-residual integral boundary-layer model of temporally excited falling liquid films. Euro. J. Mech. B/Fluids, 28, 37–60 (2009)

    Article  MATH  Google Scholar 

  32. Ruban, A. I., Vonatsos, K. N.: Discontinuous solutions of the boundary-layer equations. J. Fluid Mech., 614, 407–424 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  33. Sakamoto, K., Akitomo, K.: The tidally induced bottom boundary layer in rotating frame: similarity of turbulence. J. Fluid Mech., 615, 1–15 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  34. Tsai, J.-C.: Similarity solutions for boundary layer flows with prescribed surface temperature. Appl. Math. Lett., 21, 67–73 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  35. Turkyilmazoglu, M.: Effects of suction/blowing on the lower branch modes of the incompressible boundary layer flow due to a rotation-disk. Z. Aangew. Math. Phys., 59, 676–701 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  36. Wells, A. J., Worster, M. G.: A geophysical model of vertical natural convection boundary layers. J. Fluid Mech., 609, 111–137 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  37. Xu, H., Pop, I.: Homotopy analysis of unsteady boundary-layer flow started impulsively from rest along a symmetric wedge. Z. Aangew. Math. Mech., 88(6), 507–514 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  38. Yang, G. C.: New results of Falkner-Skan equation arising in boubdary layer theory. Appl. Math. Comp., 202, 406–412 (2008)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Hua Loo-Keng Mathematical Laboratory, Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, P. R. China

    Xiao Ping Xu

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  1. Xiao Ping Xu
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Correspondence to Xiao Ping Xu.

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Supported by National Natural Science Foundation of China (Grant No. 10871193)

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Xu, X.P. New algebraic approaches to classical boundary layer problems. Acta. Math. Sin.-English Ser. 27, 1023–1070 (2011). https://doi.org/10.1007/s10114-011-9414-2

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  • DOI: https://doi.org/10.1007/s10114-011-9414-2

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