Zeitschrift für angewandte Mathematik und Physik

, Volume 64, Issue 1, pp 123–143 | Cite as

Steady magnetohydrodynamic flow in a diverging channel with suction or blowing

  • G. C. Layek
  • S. G. Kryzhevich
  • A. S. Gupta
  • M. Reza
Article

Abstract

An analysis is made of steady two-dimensional divergent flow of an electrically conducting incompressible viscous fluid in a channel formed by two non-parallel walls, the flow being caused by a source of fluid volume at the intersection of the walls. The fluid is permeated by a magnetic field produced by an electric current along the line of intersection of the channel walls. The walls are porous and subjected to either suction (k > 0) or blowing (k < 0) of equal magnitude on both the walls. It is found that when the Reynolds number for the flow is large and the magnetic Reynolds number is very small, boundary layers are formed on the channel walls such that a sufficient condition for the existence of a unique boundary layer solution (without separation) in the case of suction is N > 2, N being the magnetic parameter. When k = 0, boundary layer exists without separation only when N > 2. Further, it is found that the necessary and sufficient condition for the existence of a unique solution for boundary layer flow (without separation) even in the presence of blowing (k < 0) is N > 2. For given value of k, velocity at a point increases with increase in N. It is also shown that when N > 2, blowing makes the boundary layer thinner. A similarity solution for steady temperature distribution in the divergent flow is also presented when the channel walls are held at variable temperature. It is found that for fixed value of wall suction, temperature at a point decreases with increase in N. It is further shown that when N > 2, steady distribution of temperature exists even in the case of blowing at the walls.

Mathematics Subject Classification

76D99 76W05 

Keywords

Diverging channel MHD boundary layer flow Suction/blowing Temperature distribution 

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References

  1. 1.
    Jeffery G.B.: The two-dimensional steady motion of a viscous fluid. Philos. Mag. Ser. 6 29, 455 (1915)MATHCrossRefGoogle Scholar
  2. 2.
    Hamel G.: Spiralförmige Bewegung zäher Flüssigkeiten. Jahresber. d. Dt. Mathematiker-Vereinigung 25, 34 (1916)MATHGoogle Scholar
  3. 3.
    Harrison W.J.: The pressure in a viscous liquid moving through channel with diverging boundaries. Proc. Camb. Philos. Soc. 19, 307 (1919)Google Scholar
  4. 4.
    Tollmien, W.: Gnenzschichtheoric, Handbuch der Experimental Physik, vol. 4, Teil 1. Akadeneisahe Verlagsgesellschaft, p. 241 (1921)Google Scholar
  5. 5.
    Noether, F.: Handbuch der Physikalischen und Technischen Mechanik, vol. 5. Leipzig, J. A. Barch, p. 733 (1931)Google Scholar
  6. 6.
    Dean W.R.: Note on the divergent flow of fluid. Philos. Mag. (7) 18, 759 (1934)MATHGoogle Scholar
  7. 7.
    Rosenhead L.: The steady two-dimensional radial flow of viscous fluid between two inclined plane walls. Proc. R. Soc. Lond. Sec. A 175, 436 (1940)CrossRefGoogle Scholar
  8. 8.
    Goldstein, S., (ed.): Modern Developments in Fluid Dynamics, vol. 1. Oxford University Press (1938)Google Scholar
  9. 9.
    Landau L., Lifshitz E.M.: Fluid Mechanics. Pergamon Press, NY (1959)Google Scholar
  10. 10.
    Dryden H., Murnaghan F., Bateman H.: Hydrodynamics. Dover, New York (1932)Google Scholar
  11. 11.
    Whitham G.B.: The Navier-Stokes equations of motion. In: Rosenhead, L. (ed.) Laminar Boundary Layers, pp. 114. Clarendon Press, Oxford (1963)Google Scholar
  12. 12.
    Batchelor G.K.: An Introduction to Fluid Dynamics. Cambridge University Press, Cambridge (1967)MATHGoogle Scholar
  13. 13.
    Millsaps K., Pohlhausen K.: Thermal distributions in Jeffery-Hamel flows between non-parallel plane walls. J. Aero. Sci. 20, 187 (1953)MathSciNetMATHGoogle Scholar
  14. 14.
    Jungclaus G.: Two-dimensional boundary layers and jets in magneto-fluid dynamics. Rev. Mod. Phys. 32, 823 (1960)MATHCrossRefGoogle Scholar
  15. 15.
    Chandrasekhar S.: Hydrodynamic and Hydromagnetic Stability. Oxford University Press, Oxford (1961)MATHGoogle Scholar
  16. 16.
    Cowling T.G.: Magnetohydrodynamics. Interscience Publishers, Inc., New York (1957)Google Scholar
  17. 17.
    Axford, W.L.: The magnetohydrodynamic Jeffery–Hamel problem for a weakly conducting fluid. Q. J. Mech. Appl. Math. 14(3), 335 (1961)Google Scholar
  18. 18.
    Shercliff J.A.: A Textbook of Magnetohydnamics. Pergamon Press, NY (1965)Google Scholar
  19. 19.
    Holstein, H.: Ähnliche laminare Reibungsschichten an durchlässigen Wänden. ZWB-VM, p. 3050 (1943)Google Scholar
  20. 20.
    Schlichting H., Gersten K.: Boundary Layer Theory, 8th revised edition. Springer, Berlin (2000)Google Scholar
  21. 21.
    Saaty T.L., Bram J.: Nonlinear Mathematics. McGraw Hill, New York (1964)MATHGoogle Scholar
  22. 22.
    Demazure M.: Bifurcations and Catastrophes, pp. 304. Springer, Berlin (2000)MATHCrossRefGoogle Scholar
  23. 23.
    Coppel W.A.: Dichotomies in Stability Theory, Lecture Notes in Mathematics No 629. Springer, Berlin (1978)Google Scholar
  24. 24.
    Bakker P.G.: Bifurcations in Flow Patterns. Kluwer, Dordrecht (1991)MATHCrossRefGoogle Scholar
  25. 25.
    Anderson D.A., Tannehill J.C., Pletcher R.H.: Computational Fluid Mechanics and Heat Transfer. Hemisphere Publishing Corporation, New York (1984)MATHGoogle Scholar
  26. 26.
    Gersten K., Körner H.: Wärmeübergang unter Berücksichtigung der Reibungswärme bei laminaren Keilströmungen mit veränderlicher Temperatur und Normalgeschwindigkeit entlang der Wand. Int. J. Heat Mass Transf. 11, 655 (1968)MATHCrossRefGoogle Scholar

Copyright information

© Springer Basel AG 2012

Authors and Affiliations

  • G. C. Layek
    • 1
  • S. G. Kryzhevich
    • 2
  • A. S. Gupta
    • 3
  • M. Reza
    • 4
  1. 1.Department of MathematicsThe University of BurdwanGolapbag, BurdwanIndia
  2. 2.Saint-Petersburg State University, PetrodvoretzSaint-PetersburgRussia
  3. 3.Mathematics DepartmentIndian Institute of TechnologyKharagpurIndia
  4. 4.Department of MathematicsNational Institute of Science and TechnologyBerhampurIndia

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