Skip to main content
SpringerLink
Log in

Network Electro-thermal Simulation of Non-isothermal Magnetohydrodynamic Heat Transfer from a Transpiring Cone with Buoyancy and Pressure Work

  • Original Paper
  • Published:
International Journal of Applied and Computational Mathematics Aims and scope Submit manuscript

Abstract

The steady, axisymmetric laminar natural convection boundary layer flow from a non-isothermal vertical circular porous cone under a transverse magnetic field, with the cone vertex located at the base, is considered. The pressure work effect is included in the analysis. The governing boundary layer equations are formulated in an (x, y) coordinate system (parallel and normal to the cone slant surface), and the magnetic field effects are simulated with a hydromagnetic body force term in the momentum equation. A dimensionless transformation is performed rendering the momentum and also heat conservation equations. The thermal convection flow is shown to be controlled by six thermophysical parameters—local Hartmann number, local Grashof number, pressure work parameter, temperature power law exponent, Prandtl number and the transpiration parameter. The transformed parabolic partial differential equations are solved numerically using the network simulation method based on the electrical-thermodynamic analogy. Excellent correlation of the zero Hartmann number case is achieved with earlier electrically non-conducting solutions. Local shear stress function (skin friction) is found to be strongly decreased with an increase in Prandtl number (Pr), with negative values (corresponding to flow reversal) identified for highest Pr with further distance along the streamwise direction. A rise in local Hartmann number, is observed to depress skin friction. Increasing temperature power law index, corresponding to steeper temperature gradient at the wall, strongly reduces skin friction at the cone surface. A positive rise in pressure work parameter decreases skin friction whereas a negative increase elevates the skin friction for some distance along the cone surface from the apex. Local heat transfer gradient is markedly boosted with a rise in Prandtl number but decreased principally at the cone surface with increasing local Hartmann number. Increasing temperature power law index conversely increases the local heat transfer gradient, at the cone surface. A positive rise in pressure work parameter increases local heat transfer gradient while negative causes it to decrease. A rise in local Grashof number boosts local skin friction and velocity into the boundary layer; local heat transfer gradient is also increased with a rise in local Grashof number whereas the temperature in the boundary layer is noticeably reduced. Applications of the work arise in spacecraft magnetogas dynamics, chemical cooling systems and industrial magnetic materials processing.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19

Similar content being viewed by others

References

  1. Witte, A.B., Harper, E.Y.: Experimental investigation and empirical correlation of local heat transfer rates in rocket-engine thrust chambers. Tech. Report 32-344, Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California (1962)

  2. Medford, J.E.: Transient radial heat transfer in uncooled rocket nozzles. Aerosp. Eng. 21(10), 15–21 (1962)

    Google Scholar 

  3. Chamkha, A.J., Quadri, M.A.: Combined heat and mass transfer by hydromagnetic natural convection over a cone embedded in a non-Darcian porous medium with heat generation/absorption effects. Heat Mass Transf. 38, 487–495 (2002)

    Article  Google Scholar 

  4. Bartz, D.R.: A simple equation for rapid estimation of rocket nozzle convective heat transfer coefficients. Jet Propuls. 37(1), 49–51 (1957)

    Google Scholar 

  5. Streif, M.L.: Heat transfer to a cone segment model. General Dynamics and Astronautics, Report, San Diego, California (1960)

  6. Braun, W.H., Ostrach, S., Heighway, J.E.: Free convection similarity flows about two-dimensional and axisymmetric bodies with closed lower ends. Int. J. Heat Mass Transf. 21(1–2), 121–135 (1961)

    Article  Google Scholar 

  7. Hering, R.G., Grosh, R.J.: Laminar free convection from a non-isothermal cone. Int. J. Heat Mass Transf. 5(11), 1059–1068 (1962)

    Article  Google Scholar 

  8. Hering, R.G.: Laminar free convection from a non-isothermal cone at low Prandtl numbers. Int. J. Heat Mass Transf. 8(10), 1333–1337 (1965)

    Article  Google Scholar 

  9. Roy, S.: Free convection over a slender vertical cone at high Prandtl numbers. ASME J. Heat Transf. 96(1–4), 174–176 (1974)

    Google Scholar 

  10. Na, T.Y., Chiou, J.P.: Laminar natural convection over a slender vertical frustrum of a cone. Heat Mass Transf. 12(2), 83–87 (1979)

    Google Scholar 

  11. Lin, F.N.: Laminar free convection from a vertical cone with uniform surface heat flux. Lett. Heat Mass Transf. 3(1), 49–58 (1976)

    Article  Google Scholar 

  12. Alamgir, M.: Overall heat transfer from vertical cones in laminar free convection: an approximate method. ASME J. Heat Transf. 101(1–4), 174–176 (1979)

    Article  Google Scholar 

  13. Pop, I., Takhar, H.S.: Compressibility effects in laminar free convection from a vertical cone. Appl. Sci. Res. 48, 71–82 (1991)

    Article  MATH  Google Scholar 

  14. Hossain, M.A., Munir, M.S., Takhar, H.S.: Natural convection flow of a viscous fluid about a truncated cone with temperature dependent viscosity. Acta Mech. 140, 171–181 (2000)

    Article  MATH  Google Scholar 

  15. Hossain, M.A., Paul, S.C.: Free convection from a vertical permeable cone with non-uniform surface temperature. Acta Mech. 151(1–2), 103–114 (2001)

    Article  MATH  Google Scholar 

  16. Takhar, H.S., Chamkha, A.J., Nath, G.: Effect of variable thermophysical quantities on the natural convection flow of gases over a vertical cone. Int. J. Eng. Sci. 42, 243–256 (2003)

    Article  MATH  Google Scholar 

  17. Chamkha, A.J., Takhar, H.S., Nath, G.: Unsteady compressible boundary layer flow over a circular cone near a plane of symmetry. Heat Mass Transf. 41, 632–641 (2005)

    Article  Google Scholar 

  18. Roy, S., Datta, P., Mahanti, N.C.: Non-similar solution of an unsteady mixed convection flow over a vertical cone with suction/injection. Int. J. Heat Mass Transf. 50(1–2), 181–187 (2007)

    Article  MATH  Google Scholar 

  19. Singh, P.J., Roy, S.: Unsteady mixed convection flow over a vertical cone due to impulsive motion. Int. J. Heat Mass Transf. 50(5/6), 949–959 (2007)

    Article  MATH  Google Scholar 

  20. Kumari, M., Nath, G.: Transient laminar compressible boundary layers over a permeable circular cone near a plane of symmetry. Int. J. Heat Mass Transf. 48(13), 2771–2778 (2005)

    Article  MATH  Google Scholar 

  21. Sluyter, M.M., Touryan, K.J.: The effect of a magnetic field on a rotating cone in compressible flow. Sandia National Labs Report, New Mexico, USA, AIAA-1969-721 (1969)

  22. Surma Devi, C.D., Takhar, H.S., Nath, G.: MHD flow past a cone. Acta Tech. Csav 31, 400–409 (1986)

    MATH  Google Scholar 

  23. Singh, S.N., Takhar, H.S., Ram, P.C.: Magnetohydrodynamic flow between coaxial rotating cone and a cylinder under the influence of radial a magnetic field. J. Magnetohydrodyn. Plasma Res. 6, 21–32 (1996)

    Google Scholar 

  24. Chamkha, A.J.: Non-Darcy hydromagnetic free convection from a cone and a wedge in porous media. Int. Commun. Heat Mass Transf. 23, 875–887 (1996)

    Article  Google Scholar 

  25. Chamkha, A.J., Khalid, A.-R.A., Al-Hawaj, O.: Simultaneous heat and mass transfer by natural convection from a cone and a wedge in porous media. J. Porous Media 3, 155–164 (2000)

    MATH  Google Scholar 

  26. Chamkha, A.J.: Coupled heat and mass transfer by natural convection about a truncated cone in the presence of magnetic field and radiation effects. Numer. Heat Transf. Part A 39, 511–530 (2001)

    Article  Google Scholar 

  27. Chamkha, A.J., Al-Mudhaf, A.: Unsteady heat and mass transfer from a rotating vertical cone with a magnetic field and heat generation or absorption effects. Int. J. Therm. Sci. 44, 267–276 (2005)

    Article  Google Scholar 

  28. Alam, M.M., Alim, M.A., Chowdhury, M.M.K.: Free convection from a vertical permeable circular cone with pressure work and non-uniform surface temperature. Nonlinear Anal. Model. Control 12(1), 21–32 (2007)

    MATH  Google Scholar 

  29. Gebhart, B.: Effects of viscous dissipation in natural convection. J. Fluid Mech. 14(2), 225–232 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  30. Takhar, H.S., Soundalgekar, V.M.: Effects of viscous dissipation and stress work on heat transfer in a boundary layer flow past a semi-infinite horizontal flat plate. L” Aerotechnica Missili E. Spazio 56, 210–212 (1978)

    MATH  Google Scholar 

  31. Soundalgekar, V.M., Takhar, H.S., Vignesam, N.V.: Combined free and forced convection flow past a semi-infinite vertical plate with variable surface temperature. Nucl. Eng. Des. 110, 95–98 (1988)

    Article  Google Scholar 

  32. Soundalgekar, V.M., Murty, T.V.R., Takhar, H.S.: Heat transfer in MHD unsteady stagnation point flow with a variable temperature. Indian J. Pure Appl. Math. 21, 384–389 (1990)

    MATH  Google Scholar 

  33. Ganesan, P., Ekambavanan, K., Soundalgekar, V.M., Takhar, H.S.: Transient free convection past a semi-infinite vertical plate with variable surface temperature. Int. J. Numer. Methods Heat Fluid Flow 7, 186–202 (1997)

    MATH  Google Scholar 

  34. Landau, L.D., Lifschitz, E.: Electrodynamics of Continuous Media, International Course in Theoretical Physics. Pergamon, Oxford (1959)

    Google Scholar 

  35. 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)

    Article  Google Scholar 

  36. Bég, O.A., Zueco, J., Norouzi, M., Davoodi, M., Joneidi, A.A., Elsayed, Assma F.: Network and Nakamura tridiagonal computational simulation of electrically-conducting biopolymer micro-morphic transport phenomena. Comput. Biol. Med. 44, 44–56 (2014)

    Article  Google Scholar 

  37. Pspice 6.0.: (1994) Irvine, California 92718. Microsim Corporation, 20 Fairbanks

  38. Cramer, K.C., Pai, S.I.: Magnetofluid Dynamics for Engineers and Applied Physicists. MacGraw-Hill, New York (1973)

    Google Scholar 

  39. Hill, P.G., Peterson, C.R.: In: Emmons, H.W., Penner, S.S. (eds.) Mechanics and Thermodynamics of Propulsion, Addison-Wesley Series in Aerospace Science, 2nd edition. Reading, Massachusetts (1967)

  40. Joshi, Y., Gebhart, B.: Effect of pressure stress work and viscous dissipation in some natural convection flows. Int. J. Heat Mass Transf. 24, 1577–1588 (1981)

    Article  MATH  Google Scholar 

  41. Ackroyd, J.A.D.: Stress work effects in laminar flat-plate natural convection. J. Fluid Mech. 62, 677–695 (1974)

    Article  MATH  Google Scholar 

  42. Rawat, S., Bhargava, R., Bhargava, R., Bég, O.A.: Transient magneto-micropolar free convection heat and mass transfer through a non-Darcy porous medium channel with variable thermal conductivity and heat source effects. Proc. IMechE Part C J. Mech. Eng. Sci. 223, 2341–2355 (2009)

    Article  Google Scholar 

  43. Bhargava, R., Sharma, S., Bég, O.A., Zueco, J.: Finite element study of nonlinear two-dimensional deoxygenated biomagnetic micropolar flow. Commun. Nonlinear Sci. Numer. Simul. 15, 1210–1233 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  44. Rana, P., Bhargava, R., Bég, Oa: Numerical solution for mixed convection boundary layer flow of a nanofluid along an inclined plate embedded in a porous medium. Comput. Math. Appl. 64, 2816–2832 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  45. Bég, O.A., Ghosh, S.K., Narahari, M.: Mathematical modelling of oscillatory MHD Couette flow in a rotating highly permeable medium permeated by an oblique magnetic field. Chem. Eng. Commun. 198, 235–254 (2010)

    Article  Google Scholar 

  46. Ghosh, S.K., Bég, O.A., Aziz, A.: A mathematical model for magnetohydrodynamic convection flow in a rotating horizontal channel with inclined magnetic field, magnetic induction and Hall current effects. World J. Mech. 1, 137–154 (2011)

    Article  Google Scholar 

  47. Bég, O.A., Tripathi, D.: Mathematica simulation of peristaltic pumping with double-diffusive convection in nanofluids: a bio-nano-engineering model. Proc. IMechE Part N J. Nanoeng. Nanosyst. 225, 99–114 (2012)

    Google Scholar 

  48. Prasannakumara, B.C., Krishnamurthy, M.R., Gireesha, B.J., Rama Gorla, S.R.: Effect of multiple slips and thermal radiation on MHD flow of Jeffery nanofluid with heat transfer. J. Nanofluids 5, 82–93 (2016)

    Article  Google Scholar 

  49. Prasannakumara, B., Gireesha, B., Gorla, R., Krishnamurthy, M.: Effects of chemical reaction and nonlinear thermal radiation on Williamson nanofluid slip flow over a stretching sheet embedded in a porous medium. ASCE J. Aerosp. Eng. (2016). doi:10.1061/(ASCE)AS.1943-5525.0000578,04016019

    Google Scholar 

Download references

Acknowledgments

Dr. O. Anwar Bég is grateful to the late Dr. Howard Brenner, formerly Willard Dow Professor of Chemical Engineering of MIT, USA for providing some excellent guidance on pressure work and dissipation fluid mechanics aspects. All the authors are also grateful to the reviewers for their constructive comments which have served to improve the present investigation.

Author information

Authors and Affiliations

  1. Fluid Mechanics, Spray Research Group, Petroleum/Gas Engineering Division, School of Computing, Science and Engineering, University of Salford, G77, Newton Building, Manchester, M54WT, England, UK

    O. Anwar Bég

  2. Departamento de Ingeniería Térmica y Fluidos, Universidad Politécnica de Cartagena, Cartagena, Murcia, Spain

    J. Zueco

  3. Spray Research Group, Petroleum/Gas Engineering Division, School of Computing, Science and Engineering, University of Salford, G79, Newton Building, Manchester, M54WT, England, UK

    Ali Kadir

  4. Renewable Energy and Geodynamics Research, Dickenson Road, Manchester, England, UK

    Tasveer A. Bég

  5. Magnetic Materials Research, Electrical/Electronic Engineering, Huddersfield University, Huddersfield, England, UK

    Umar F. Khan

Authors
  1. O. Anwar Bég
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. Zueco
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ali Kadir
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Tasveer A. Bég
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Umar F. Khan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to O. Anwar Bég.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bég, O.A., Zueco, J., Kadir, A. et al. Network Electro-thermal Simulation of Non-isothermal Magnetohydrodynamic Heat Transfer from a Transpiring Cone with Buoyancy and Pressure Work. Int. J. Appl. Comput. Math 3, 1525–1547 (2017). https://doi.org/10.1007/s40819-016-0192-5

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40819-016-0192-5

Keywords

Search

Navigation