Skip to main content
SpringerLink
Log in

Effect of induced magnetic field on natural convection in vertical concentric annuli

  • Research Paper
  • Published:
Acta Mechanica Sinica Aims and scope Submit manuscript

Abstract

In the present paper, we have considered the steady fully developed laminar natural convective flow in open ended vertical concentric annuli in the presence of a radial magnetic field. The induced magnetic field produced by the motion of an electrically conducting fluid is taken into account. The transport equations concerned with the considered model are first recast in the non-dimensional form and then unified analytical solutions for the velocity, induced magnetic field and temperature field are obtained for the cases of isothermal and constant heat flux on the inner cylinder of concentric annuli. The effects of the various physical parameters appearing into the model are demonstrated through graphs and tables. It is found that the magnitude of maximum value of the fluid velocity as well as induced magnetic field is greater in the case of isothermal condition compared with the constant heat flux case when the gap between the cylinders is less or equal to 1.70 times the radius of inner cylinder, while reverse trend occurs when the gap between the cylinders is greater than 1.71 times the radius of inner cylinder. These fields are almost the same when the gap between the cylinders is equal to 1.71 times the radius of inner cylinder for both the cases. It is also found that as the Hartmann number increases, there is a flattening tendency for both the velocity and the induced magnetic field. The influence of the induced magnetic field is to increase the velocity profiles.

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

Abbreviations

a :

Radius of inner cylinder (m)

b :

Radius of outer cylinder (m)

g :

Acceleration due to gravity (m/s2)

H′ z′ :

Induced magnetic field in z′-direction (A/m)

H :

Non-dimensional induced magnetic field in z-direction

J θ :

Induced current density (A/m2)

M :

Hartmann number

q′:

Heat flux (W/m2)

r′, θ′, z′:

Cylindrical coordinates (m), (°), (m)

r :

Non-dimensional radial distance

T′ f :

Ambient temperature (K)

T′ w :

Temperature at outer surface of inner cylinder (at r′ = a), (K)

T :

Temperature of the fluid in non-dimensional form (K)

u :

Fluid velocity in non-dimensional form along axial direction (m/s)

u′:

Velocity of fluid along axial direction (m/s)

U :

Characteristics velocity of fluid (m/s)

β :

Coefficient of thermal expansion

µ e :

Magnetic permeability (H/m)

η :

Magnetic diffusivity

ϑ :

Kinematic viscosity of the fluid (m2/s)

ρ :

Density of fluid (kg/m3)

λ :

Ratio of outer radius and inner radius, b/a

σ :

Conductivity of fluid (A2S−3/kgm3)

1:

Value at inner cylinder

λ :

Value at outer cylinder

θ :

Along θ-direction

References

  1. Ramamoorthy, P.: Flow between two concentric rotating cylinders with a radial magnetic field. The Physics of Fluids 4, 1444–1445 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  2. Arora, K. L., Gupta, P. R.: Magnetohydrodynamic flow between two rotating coaxial cylinders under radial magnetic field. The Physics of Fluid 15, 1146–1148 (1971)

    Article  Google Scholar 

  3. El-Shaarawi, M. A. I., Sarhan, A.: Developing laminar free convection in an open ended vertical annulus with a rotating inner cylinder. Journal of Heat Transfer 103, 552–558 (1981)

    Article  Google Scholar 

  4. Joshi, H. M.: Fully developed natural convection in an isothermal vertical annular duct. International Communications in Heat and Mass Transfer 14, 657–664 (1987)

    Article  Google Scholar 

  5. El-Shaarawi, M. A. I., Al-Nimb, M. A.: Fully developed laminar natural convection in open ended vertical concentric annuli. International Journal of Heat andMass Transfer 33, 1873–1884 (1990)

    Article  MATH  Google Scholar 

  6. Lee, Y. M., Kuo, Y. M.: Laminar flow in annuli ducts with constant wall temperature. International Communications in Heat and Mass Transfer 25, 227–236 (1998)

    Article  Google Scholar 

  7. Mahmud, S., Fraser, R. A.: Irreversibility analysis of concentrically rotating annuli. International Communications in Heat and Mass Transfer 29, 697–706 (2002)

    Article  Google Scholar 

  8. Leong, J. C., Lai, F. C.: Natural convection in concentric annulus with a porous sleeve. International Journal of Heat and Mass Transfer 49, 3016–3027 (2006)

    Article  MATH  Google Scholar 

  9. Shaija, A., Narasimham, G. S.V. L.: Effect of surface radiation on conjugate normal convection in a horizontal annulus driven by inner heat generating solid cylinder. International Journal of Heat and Mass Transfer 52, 5759–5769 (2009)

    Article  MATH  Google Scholar 

  10. Abu-Nada, E., Masoud, Z., Hijazi, A.: Natural convection heat transfer enhancement in horizontal concentric annuli using nanofluids. International Communications in Heat and Mass Transfer 35, 657–665 (2008)

    Article  Google Scholar 

  11. Jha, B. K.: MHD free-convection and mass transform through a porous media. Astrophysics and Space Science 175, 283–289 (1990)

    Article  Google Scholar 

  12. Singh, S. K., Jha, B. K., Singh, A. K.: Natural convection in vertical concentric annuli under a radial magnetic field. Heat and Mass Transfer 32, 399–401 (1997)

    Article  Google Scholar 

  13. Chandran, P., Sacheti, N. C., Singh, A. K.: Effects of rotation on unsteady hydrodynamic Couette flow. Astrophysics and Space Science 202, 1–10 (1993)

    Article  MATH  Google Scholar 

  14. Chandran, P., Sacheti, N. C., Singh, A. K.: Singh, Haydromagnetic flow and heat transfer past a continuously moving porous boundary. International Communications in Heat and Mass Transfer 23, 889–898 (1996)

    Article  Google Scholar 

  15. Chandran, P., Sacheti, N. C., Singh, A. K.: Unsteady hydromagnetic free convection flow with heat flux and accelerated boundary motion. Journal of Physical Society of Japan 67, 124–129 (1998)

    Article  MATH  Google Scholar 

  16. Chandran, P., Sacheti, N. C., Singh, A. K.: A unified approach to analytical solution of a hydromagnetic free convection flow. Scientiae Mathematicae Japonicae 53, 467–476 (2001)

    MathSciNet  MATH  Google Scholar 

  17. Singh A. K. Chandran, P. Sacheti, N. C.: Effects of transverse magnetic field on a flat plate thermometer problem. International Journal of Heat and Mass Transfer 43, 3253–3258 (2000)

    Article  MATH  Google Scholar 

  18. Barletta, A., Lazzari, A., Magyari, E., et al..: Mixed convection with heating effects in a vertical porous annulus with a radially varying magnetic field. International Journal of Heat and Mass Transfer 51, 5777–5784 (2008)

    Article  MATH  Google Scholar 

  19. Singh, R. K., Singh, A. K.: Hydromagnetic mixed convection between two vertical walls. Journal of Energy, Heat and Mass Transfer 31, 111–123 (2009)

    Google Scholar 

  20. Chandrasekhar, S.: Hydrodynamic and Hydromagnetic Stability, Oxford, England (1961)

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Central University of Bihar, Patna, India

    R. K. Singh

  2. Department of Mathematics, Banaras Hindu University, Varanasi, India

    A. K. Singh

Authors
  1. R. K. Singh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. K. Singh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. K. Singh.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Singh, R.K., Singh, A.K. Effect of induced magnetic field on natural convection in vertical concentric annuli. Acta Mech Sin 28, 315–323 (2012). https://doi.org/10.1007/s10409-012-0052-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10409-012-0052-4

Keywords

Search

Navigation