Skip to main content
SpringerLink
Log in

Self-similar solutions of the magnetohydrodynamic boundary layer system for a dilatable fluid

  • Published:
Acta Mechanica Aims and scope Submit manuscript

Summary

A rigorous mathematical analysis is given for a magnetohydrodynamic boundary layer problem, which arises in the two-dimensional steady laminar boundary layer flow for an incompressible electrically conducting dilatable fluid along an isolated surface in the presence of an exterior magnetic field orthogonal to the flow. In the self-similar case, the problem is transformed into a third-order nonlinear ordinary differential equation with certain boundary conditions, which is proved to be equivalent to a singular initial value problem for an integro-differential equation of first order. With the aid of the singular initial value problem, the uniqueness, existence, and nonexistence results for generalized normal solutions are established.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • H. I. Andersson (1992) ArticleTitleMHD flow of a viscous fluid past a stretching surface Acta Mech. 95 227–230 Occurrence Handle0753.76192 Occurrence Handle1179921 Occurrence Handle10.1007/BF01170814

    Article  MATH  MathSciNet  Google Scholar 

  • H. I. Andersson (1995) ArticleTitleAn exact solution of the Navier-Stokes equations for magnetohydrodynamic flow Acta Mech. 113 241–244 Occurrence Handle0863.76089 Occurrence Handle1361702 Occurrence Handle10.1007/BF01212646

    Article  MATH  MathSciNet  Google Scholar 

  • H. I. Andersson K. H. Bech B. S. Dandapat (1992) ArticleTitleMagnetohydrodynamic flow of a power-law fluid over a stretching sheet Int. J. Non-Linear Mech. 27 929–936 Occurrence Handle0775.76216 Occurrence Handle10.1016/0020-7462(92)90045-9

    Article  MATH  Google Scholar 

  • H. I. Andersson F. Irgens (1988) ArticleTitleGravity-driven film flow of power-law fluids along vertical walls Non-Newtonian Fluid Mech. 27 153–172 Occurrence Handle10.1016/0377-0257(88)85011-0

    Article  Google Scholar 

  • D. S. Djukic (1973) ArticleTitleOn the use of Croccos equation for the flow of power-law fluids in a transverse magnetic field AIChE J. 19 1159–1163 Occurrence Handle10.1002/aic.690190612

    Article  Google Scholar 

  • D. S. Djukic (1974) ArticleTitleHiemenz magnetic flow of power-law fluids Trans. ASME J. Appl. Mech. 41 822–823

    Google Scholar 

  • Falkner, V. M., Skan, S. W.: An analytic solution of a nonlinear singular boundary layer equations. Phil. Mag. 12, 865–896 (1931); Aeronautical Research Council Reports and Memoranda 1314 (1930), London, UK.

    Google Scholar 

  • P. Hartman (1982) Ordinary differential equations EditionNumber2 Birkhäuser Boston

    Google Scholar 

  • F. N. Ibrahim M. Terbeche (1994) ArticleTitleSolutions of the laminar boundary layer equations for a conducting power law non-Newtonian fluid in a transverse magnetic field J. Phys. D Appl. Phys. 27 740–747 Occurrence Handle10.1088/0022-3727/27/4/011

    Article  Google Scholar 

  • Oleinik, O. A., Samokhin, V. N.: Mathematical model in boundary layer theory, Differential Equations. Chapman & Hall/CRC: 1999.

  • K. B. Pavlov (1974) ArticleTitleMagnetohydynamic flow of an incompressible viscous fluid caused by deformation of a plane surface Magnitnaya Gidrodinamika 4 146–147

    Google Scholar 

  • K. B. Pavlov (1981) ArticleTitleOn the theory of non-stationary MHD-flows of non-Newtonian fluid dilatable fluids Magnitnaya Gidrodinamika 1 3–13 Occurrence Handle0506.68071

    MATH  Google Scholar 

  • Y. G. Sapunkov (1967) ArticleTitleSelf-similar solutions of the boundary layer problems for non-Newtonian fluid in magnetic field Izvest. Akad. Nauk SSSR, Mekh. Zhidk. Gaza 6 77–82

    Google Scholar 

  • Y. G. Sapunkov (1970) ArticleTitleRayleigh problem of non-Newtonian electroconductive fluids J. Appl. Tech. Phys. 2 50–55

    Google Scholar 

  • T. Sarpkaya (1961) ArticleTitleFlow of non-Newtonian fluids in a magnetic field AICHE. J. 7 324–328 Occurrence Handle10.1002/aic.690070231

    Article  Google Scholar 

  • H. Schlichting (1968) Boundary-layer theory EditionNumber6 McGraw-Hill New York

    Google Scholar 

  • R. A. Spinelli (1969) ArticleTitleOn the O.D.E of stagnation point flows of power law fluids ZAMP 20 479–486 Occurrence Handle0175.23801 Occurrence Handle10.1007/BF01595040

    Article  MATH  Google Scholar 

  • K. Vajravelu (1986) ArticleTitleHydromagnetic flow and heat transfer over a continuous, moving, porous, flat surface Acta Mech. 64 179–185 Occurrence Handle0614.76114 Occurrence Handle10.1007/BF01450393

    Article  MATH  Google Scholar 

  • K. A. Volosov K. B. Pavlov I. A. Fedotov (1980) ArticleTitleOn the theory of laminar boundary layer in a conducting non-Newtonian power law fluid in transversal magnetic field Magnitnaya Gidrodinamika 4 39–42

    Google Scholar 

  • B. Vujanovic A. M. Strauss D. S. Djukic (1972) ArticleTitleA variational solution of the Rayleigh problem for a power-law Non-Newtonian conducting fluid Ingenieur-Archiv 41 381–386 Occurrence Handle0243.76003 Occurrence Handle10.1007/BF00533141

    Article  MATH  Google Scholar 

  • J. Wang W. Gao (1999) ArticleTitleUniqueness, existence and nonexistence of normal solutions to a problem in boundary layer theory Tohoku Math. J. 51 539–553 Occurrence Handle0959.34016 Occurrence Handle1725625

    MATH  MathSciNet  Google Scholar 

  • Z. Zhang J. Wang W. Shi (2004) ArticleTitleA boundary layer problem arising in gravity-driven laminar film flow of power-law fluids along vertical walls ZAMP 55 769–780 Occurrence Handle1059.76007 Occurrence Handle2087764 Occurrence Handle10.1007/s00033-004-1122-7

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Xiamen University, Xiamen, 361005, P. R. China

    Z. Zhang & J. Wang

Authors
  1. Z. Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Z. Zhang.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zhang, Z., Wang, J. Self-similar solutions of the magnetohydrodynamic boundary layer system for a dilatable fluid. Acta Mechanica 188, 103–119 (2007). https://doi.org/10.1007/s00707-006-0396-6

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00707-006-0396-6

Keywords

Search

Navigation