Advertisement

Zeitschrift für angewandte Mathematik und Physik

, Volume 61, Issue 1, pp 147–169 | Cite as

Transient hydromagnetic flow in a rotating channel permeated by an inclined magnetic field with magnetic induction and Maxwell displacement current effects

  • S. K. Ghosh
  • O. A. BégEmail author
  • J. Zueco
  • V. R. Prasad
Article

Abstract

Closed-form solutions are presented for the transient hydromagnetic flow in a rotating channel with inclined applied magnetic field under the influence of a forced oscillation. Magnetic Reynolds number is large enough to permit the inclusion of magnetic induction effects. The Maxwell displacement current effect is also included and simulated via a dielectric strength parameter. The governing momentum and magnetic induction conservation equations are normalized with appropriate transformations and the resulting quartet of partial differential equations are solved exactly. A parametric study is performed of the influence of oscillation frequency parameter (ω), time (T), inverse Ekman number, i.e. rotation parameter (K 2), square of the Hartmann magnetohydrodynamic (MHD) parameter (M 2), and magnetic field inclination (θ) on the primary and secondary induced magnetic field components (b x , b y ) and velocity components (u, v) across the channel. Network solutions are also obtained to validate the exact solutions and shown to be in excellent agreement. Applications of the study arise in planetary plasma physics and rotating MHD induction power generators and also astronautical flows.

Mathematics Subject Classification (2000)

76W05 76X05 65Z05 

Keywords

Unsteady flow Forced oscillation MHD Rotation Closed-form solutions Dielectric strength Maxwell displacement current Inclined magnetic field Network numerical simulation 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Takenouchi K.: Transient magnetohydrodynamic channel flow with axial symmetry at a supersonic speed. J. Phys. Soc. Japan. 54, 1329–1338 (1985)CrossRefGoogle Scholar
  2. 2.
    Triwahju Hardianto T., Sakamoto N., Harada N.: Computational study of diagonal channel magnetohydrodynamic power generation. Int. J. Energy Technol. Policy 6(1–2), 96–111 (2008)CrossRefGoogle Scholar
  3. 3.
    Narasimhan, M.N.: Transient magnetohydrodynamic flow in an annular channel, Technical Report, Wisconsin University-Madison, Mathematics Research Center, 24 p., February (1963)Google Scholar
  4. 4.
    Takhar H.S., Ram P.C.: Free convection in hydromagnetic flows of a viscous heat-generating fluid with wall temperature oscillation and Hall currents. Astrophys. Space Sci. 183, 193–198 (1991)zbMATHCrossRefGoogle Scholar
  5. 5.
    Ryabinin A.G., Khozhainov A.I.: Exact and approximate formulations of problems for unsteady flows of conducting fluids in MHD channels. Fluid Dyn. 2(4), 107–109 (1967)CrossRefGoogle Scholar
  6. 6.
    Barmin A.A., Uspenskii V.S.: Development of pulsation regimes in one-dimensional unsteady MHD flows with switching off of the electrical conductivity. Fluid Dyn. 21(4), 18–30 (1986)Google Scholar
  7. 7.
    Pop I., Soundalgekar V.M.: The Hall effect on an unsteady flow due to a rotating infinite disc. Nucl. Eng. Des. 44(3), 309–314 (1977)CrossRefGoogle Scholar
  8. 8.
    Sarojamma G., Krishna D.V.: Transient hydromagnetic convective flow in a rotating channel with porous boundaries. Acta Mech. 40(3–4), 277–288 (1981)zbMATHCrossRefGoogle Scholar
  9. 9.
    Kumari M., Takhar H.S., Nath G.: Non-axi-symmetric unsteady motion over a rotating disc in the presence of free convection and magnetic field. Int. J. Eng. Sci. 31, 1659–1668 (1993)zbMATHCrossRefGoogle Scholar
  10. 10.
    Ram P.C., Singh A.K., Takhar H.S.: Effects of Hall and ionslip currents on convective flow in a rotating fluid with a wall temperature oscillation. Magneto-Hydrodyn. Plasma Res. J. 5, 1–16 (1995)Google Scholar
  11. 11.
    Takhar H.S., Nath G.: Self-similar solution of the unsteady flow in the stagnation region on a rotating sphere with a magnetic field. Heat Mass Transf. 36, 89–96 (2000)CrossRefGoogle Scholar
  12. 12.
    Hayat T., Nadeem S., Asghar S., Siddiqui A.M.: MHD rotating flow of a third-grade fluid on an oscillating porous plate. Acta Mech. 152(1–4), 177–190 (2001)zbMATHCrossRefGoogle Scholar
  13. 13.
    Usha R., Götz T.: Spinning of a liquid film from a rotating disc in the presence of a magnetic field—a numerical solution. Acta Mech. 147(1–4), 137–151 (2001)zbMATHCrossRefGoogle Scholar
  14. 14.
    Roy S., Takhar H.S., Nath G.: Unsteady MHD flow on a rotating cone in a rotating fluid. Meccanica 39(3), 271–283 (2004)zbMATHCrossRefGoogle Scholar
  15. 15.
    Xu H., Liao S.-J.: Series solutions of unsteady MHD flows above a rotating disk. Meccanica 41(6), 20–40 (2006)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Bég O.A., Takhar H.S., Nath G., Chamkha A.J.: Mathematical Modeling of hydromagnetic convection from a rotating sphere with impulsive motion and buoyancy effects. Non-Linear Anal. Model. Control J. 11(3), 227–245 (2006)zbMATHGoogle Scholar
  17. 17.
    Naroua H., Takhar H.S., Ram P.C., Bég T.A., Bég O.A., Bhargava R.: Transient rotating hydromagnetic partially-ionized heat-generating gas dynamic flow with Hall/Ionslip current effects: finite element analysis. Int. J. Fluid Mech. Res. 34(6), 493–505 (2007)CrossRefGoogle Scholar
  18. 18.
    Pao H.P., Long R.R.: Magnetohydrodynamic jet-vortex in a viscous conducting fluid. Quart. J. Mech. Appl. Math. 19(1), 1–26 (1966)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Raptis A., Kafousias N., Tzivanidis G.: Hydromagnetic free convection effects on the oscillatory flow of an electrically conducting rarefied gas past an infinite vertical porous plate. Nucl. Eng. Des. 73(1–2), 53–68 (1982)CrossRefGoogle Scholar
  20. 20.
    Singh B., Lal J.: Finite element method in magnetohydrodynamic channel flow problems. Int. J. Numer. Methods Eng. 18(7), 1104–1111 (1982)zbMATHCrossRefGoogle Scholar
  21. 21.
    Sezgin M.: Magnetohydrodynamic flow in an infinite channel. Int. J. Numer. Methods Fluids 6(9), 593–609 (1986)zbMATHCrossRefGoogle Scholar
  22. 22.
    Polovko Y.A., Romanova E.P., Tropp É. A.: Onset of rotating stall in induction magnetohydrodynamic flows. J. Tech. Phys. 43(6), 673–677 (1998)CrossRefGoogle Scholar
  23. 23.
    Takhar H.S., Chamkha A.J., Nath G.: Unsteady flow and heat transfer on a semi-infinite flat plate with an aligned magnetic field. Int. J. Eng. Sci. 37, 1723–1736 (1999)CrossRefGoogle Scholar
  24. 24.
    Matsuo T., Tadamatsu A., Shimasaki M.: 3-D magnetohydrodynamic field computation of supersonic duct flow of weakly ionized plasma. IEEE Trans. Magn. 39(3), 1444–1447 (2003)CrossRefGoogle Scholar
  25. 25.
    Al-Khawaja M.J., Selmi M.: Highly accurate solutions of a laminar square duct flow in a transverse magnetic field with heat transfer using spectral method. ASME J. Heat Transf. 128(4), 413–417 (2006)CrossRefGoogle Scholar
  26. 26.
    Bég O.A., Bakier A.Y., Prasad V.R., Zueco J., Ghosh S.K.: Nonsimilar, laminar, steady, electrically-conducting forced convection liquid metal boundary layer flow with induced magnetic field effects. Int. J. Therm. Sci. 48(8), 1596–1606 (2009)CrossRefGoogle Scholar
  27. 27.
    Ghosh, S.K., Bég, O.A., Zueco, J.: Hydromagnetic free convection flow with induced magnetic field effects. Meccanica (2009, to appear)Google Scholar
  28. 28.
    Ghosh S.K.: A note on steady and unsteady hydromagnetic flow in a rotating channel in the presence of inclined magnetic field. Int. J. Eng. Sci. 29(8), 1013–1016 (1991)zbMATHCrossRefGoogle Scholar
  29. 29.
    Pop I., Ghosh S.K., Nandi D.K.: Effects of the Hall current on free and forced convection flows in a rotating channel in the presence of an inclined magnetic field. Magnetohydrodynamics 37(4), 348–359 (2001)Google Scholar
  30. 30.
    Ghosh S.K., Pop I.: An analytical approach to MHD plasma behaviour of a rotating environment in the presence of an inclined magnetic field as compared to excitation frequency. Int. J. Appl. Mech. Eng. 11(4), 845–856 (2006)Google Scholar
  31. 31.
    Prasad, V.R., Takhar, H.S., Zueco, J., Ghosh, S.K., Bég, O.A.: Numerical study of hydromagnetic viscous plasma flow with Hall current effects in rotating porous media. Invited paper, 53rd Congress ISTAM, University College of Engineering, Osmania University, Hyderabad, India, pp. 147–157 (2008)Google Scholar
  32. 32.
    Ghosh, S.K., Anwar Bég, O.A., Zueco, J.: Hydromagnetic convection flow in a rotating horizontal channel with inclined magnetic field and Hall current effects. Acta Astronautica (2009, under review)Google Scholar
  33. 33.
    Shercliff J.A.: A Textbook of Magnetohydrodynamics. Cambridge University Press, UK (1965)Google Scholar
  34. 34.
    Maxwell, J.C.: On physical lines of force. Part 1. The London, Edinburgh and Dublin Philosophical Magazine and Journal of Science, 4th series, pp. 161–175 (1861)Google Scholar
  35. 35.
    Ghosh, S.K., Bég, O.A.: Advanced Magneto-Fluid Mechanics: A Mathematical and Numerical Perspective. World Scientific Press, Singapore (2010, in preparation)Google Scholar
  36. 36.
    Kanayama, H., Tagami, H.D., Imoto, K., Sugimoto, S.: Finite element computation of magnetic field problems with the displacement current. In: Proceedings of the 6th Japan-China Joint Seminar on Numerical Mathematics, University of Tsukuba, Japan, 5–9 August, pp. 77–84 (2002)Google Scholar
  37. 37.
    Zhang D.M., Lie P.: Influence of displacement current on magnetic field distribution in ferrite core within kHz–MHz frequency range. J. Magn. Magn. Mater. 256(1–3), 183–188 (2003)CrossRefGoogle Scholar
  38. 38.
    Zhang D.: Permeability enhancement by induced displacement current in magnetic material with high permittivity. J. Magn. Magn. Mater. 313(1), 47–51 (2007)CrossRefGoogle Scholar
  39. 39.
    Ghosh S.K.: A note on unsteady hydromagnetic flow in a rotating channel permeated by an inclined magnetic field in the presence of an oscillator. Czech. J. Physics 51(8), 799–804 (2001)CrossRefGoogle Scholar
  40. 40.
    Landau L.D., Lifschitz E.M.: Electrodynamics of Continuous Media, International Course in Theoretical Physics. Pergamon, Oxford (1959)Google Scholar
  41. 41.
    Zueco J.: Numerical study of an unsteady free convective magnetohydrodynamic flow of a dissipative fluid along a vertical plate subject to constant heat flux. Int. J. Eng. Sci. 44, 1380–1393 (2006)CrossRefGoogle Scholar
  42. 42.
    Bég O.A., Takhar H.S., Zueco J., Sajid A., Bhargava R.: Transient Couette flow in a rotating non-Darcian porous medium parallel plate configuration: network simulation method solutions. Acta Mech. 200, 129–144 (2008)zbMATHCrossRefGoogle Scholar
  43. 43.
    Zueco J.: Network simulation method applied to radiation and viscous dissipation effects on MHD unsteady free convection over vertical porous plate. Appl. Math. Model. 31, 2019–2033 (2007)zbMATHCrossRefGoogle Scholar
  44. 44.
    Bég O.A., Zueco J., Takhar H.S.: Laminar free convection from a continuously-moving vertical surface in thermally-stratified non-Darcian high-porosity medium: network numerical study. Int. Comm. Heat Mass Transf. 35, 810–816 (2008)CrossRefGoogle Scholar
  45. 45.
    Bég O.A., Zueco J., Bhargava R., Takhar H.S.: Magnetohydrodynamic convection flow from a sphere to a non-Darcian porous medium with heat generation or absorption effects: network simulation. Int. J. Therm. Sci. 48(5), 913–921 (2009)CrossRefGoogle Scholar
  46. 46.
    Bég O.A., Zueco J., Takhar H.S.: Unsteady magnetohydrodynamic Hartmann–Couette flow and heat transfer in a Darcian channel with Hall current, ionslip, viscous and Joule heating effects: network numerical solutions. Commun. Nonlinear Sci. Numer. Simul. J. 14, 1082–1097 (2009)CrossRefGoogle Scholar
  47. 47.
    Pspice 6.0. Irvine, California 92718. Microsim Corporation, 20 Fairbanks (1994)Google Scholar
  48. 48.
    Zueco J.: Network method to study the transient heat transfer problem in a vertical channel with viscous dissipation. Int. Comm. Heat Mass Transf. 33, 1079–1087 (2006)CrossRefGoogle Scholar
  49. 49.
    Resler E.R. Jr, Sears W.R.: The prospects for magneto-aerodynamics. J. Aeronaut. Sci. 25, 235–246 (1958)MathSciNetGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2009

Authors and Affiliations

  • S. K. Ghosh
    • 1
  • O. A. Bég
    • 2
    Email author
  • J. Zueco
    • 3
  • V. R. Prasad
    • 4
  1. 1.Magnetohydrodynamics Program, Mathematics DepartmentNarajole Raj CollegeNarajole, Midnapore West DistrictIndia
  2. 2.Magneto-Aerodynamics and Fire Dynamics Research, Mechanical Engineering Program, Department of Engineering and MathematicsSheffield Hallam UniversitySheffield, South YorkshireUK
  3. 3.ETS Ingenieros Industriales Campus Muralla del Mar, Departamento de Ingeniería Térmicay FluidosUniversidad Politécnica de CartagenaCartagena (Murcia)Spain
  4. 4.Department of MathematicsMadanapalle Institute of Technology and ScienceMadanapalleIndia

Personalised recommendations