## Main Article Content

## Abstract

The steady free convective flow of a viscous incompressible and electrically conducting fluid in a two-dimensional cavity in the presence of a magnetic field applied normal to the plane of the cavity is investigated. The side vertical walls of the cavity are heated differentially while the horizontal walls are assumed to be insulated. The governing equations are re-formulated in terms of vorticity and stream function. The resulting boundary value problem is solved numerically using an alternating direction implicit (ADI) method. A number of plots illustrating the influence of Hartmann number and Rayleigh number on the streamlines and isotherms as well as the velocity and temperature profiles are shown. Furthermore, results for the average Nusselt number and the maximum absolute stream function have been obtained, and these are compared with the corresponding results in the literature when the magnetic field is applied along the cavity in the horizontal direction.

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## Keywords

## Article Details

* * References

- Batchelor, G.K. Heat transfer by free-convection across a closed cavity between vertical boundaries at different temperature. Quart. J. Appl. Math, 1954, 12, 209–233.
- Eckert, E.R.G. and Carlson, W.O. Natural convection in an air layer enclosed between two vertical plates with different temperatures. Int. J. Heat Mass Transf, 1961, 2, 106–120.
- Elder, J.W. Laminar free convection in a vertical slot. J. Fluid Mech,1965, 23, 77–98.
- Elder, J.W. Numerical experiments with free convection in a vertical slot. J. Fluid Mech, 1966, 24, 823–843.
- Gill, A.E. The boundary-layer for convection in a rectangular cavity. J. Fluid Mech.1966, 26, 515–536.
- Wilkes, J.O. and Churchill, S.W. The finite-difference computation of natural convection in a rectangular enclosure. AIChE J, 1966, 12, 161–166.
- de Vahl Davis, G. Laminar natural convection in an enclosed rectangular cavity. Int. J. Heat Mass Transf, 1968, 11, 1675–1693.
- Kimura, S. and Bejan, A. The boundary layer natural convection regime in a rectangular cavity with uniform heat flux from the side. J. Heat Transf, 1984, 106, 98–103.
- Hall, J.D., Bejan, A. and Chaddock, J.B. Transient natural convection in a rectangular enclosure with one heated side wall. Int. J. Heat Fluid Flow, 1988, 9, 396–404.
- Straughan, B. A nonlinear analysis of convection in a porous vertical slab. Geophys. Astrophys. Fluid Dyn, 1988, 42, 269–275.
- Hill, A.A. and Straughan, B. Linear and nonlinear stability thresholds for thermal convection in a box. Math. Meth. Appl. Sci, 2006, 29, 2123–2132.
- Barletta, A., Magyari, E., Pop, I. and Storesletten, L. Mixed convection with viscous dissipation in a vertical channel filled with a porous medium. Acta Mech, 2007, 194, 123–140.
- Oztop, H.F., Varol, Y. and Pop, I. Effects of wall conduction on natural convection in a porous triangular enclosure. Acta Mech, 2008, 200, 155–165.
- Tiwari, A.K., Singh, A.K., Chandran, P. and Sacheti, N.C. Natural convection in a cavity with a sloping upper surface filled with an anisotropic porous material. Acta Mech, 2012, 223, 95–108.
- Straughan, B. Energy Method, Stability and Nonlinear Convection. 2nd Ed. Appl. Math. Sci. Ser., vol. 91 Springer-Verlag, New York (2004)
- Oreper, G.M. and Szekely, J. The effect of an externally imposed magnetic field on buoyancy driven flow in a rectangular enclosure. J. Cryst. Growth, 1983, 64, 505–515.
- Garandet, J.P., Alboussiere, T. and Moreau, R. Buoyancy driven convection in a rectangular cavity with a transverse magnetic field. Int. J. Heat Mass Transf., 1992, 35, 741–748.
- Rudraiah, N., Barron, R.M., Venkatachalappa, M. and Subbaraya, C.K. Effect of a magnetic field on free convection in a rectangular cavity. Int. J. Engng. Sci., 1995, 33, 1075–1084.
- Alchaar, S., Vasseur, P. and Bilgen, E. Natural-convection heat transfer in a rectangular enclosure with a transverse magnetic field. J. Heat Transf., 1995, 117, 668–673.
- Kanafer, K. and Chamkha, A. J. Hydromagnetic natural convection from an inclined porous square enclosure with heat generation. Numer. Heat Transf. A., 1998, 33, 891–910.
- Pirmohammadi, M., Ghassemi, M. and Sheikhzadeh, G.A. Effect of a magnetic field on buoyancy-driven convection in differentially heated square cavity. IEEE Trans. Magnetics, 2009, 45, 407–411.
- Pirmohammadi, M., Ghassemi, M. and Sheikhzadeh, G.A. Effect of magnetic field on transient natural convection heat transfer. IEEE Trans. Magnetics, 2009, 45, 2788–2790.
- Al-Najem, N.M., Khanafer, K.M. and El-Refaee, M.M. Numerical study of laminar natural convection in tilted enclosure with transverse magnetic field. Int. J. Numer. Meth. Heat Fluid Flow, 1998, 8, 651–672.
- Ece, M.C. and Buyuk, E. Natural-convection flow under a magnetic field in an inclined rectangular enclosure heated and cooled on adjacent walls. Fluid Dyn. Res., 2006, 38, 564–590.
- Pirmohammadi, M. and Ghassemi, M. Effect of a magnetic field on convection heat transfer inside a tilted square enclosure. Int. Comm. Heat Mass Transf, 2009, 36, 776–780.
- Pirmohammadi, M., Ghassemi, M. and Hamedi, M. Effect of inclination angle on magneto-convection inside a tilted enclosure. IEEE Trans. Magnetics, 2010, 46, 3697–3700.
- Pirmohammadi, M. and Ghassemi, M. Numerical study of magneto-convection in a partitioned enclosure. IEEE Trans. Magnetics, 2009, 45, 2671–2674.
- Pirmohammadi, M., Ghassemi, M. and Keshtkar. A. Numerical study of hydromagnetic convection of an electrically conductive fluid with variable properties inside an enclosure. IEEE Trans. Plasma Sci., 2011, 39, 516–520.
- Mallinson, G.D. and de Vahl Davis, G. The method of the false transient for the solution of coupled differential equations. J. Comp. Phys., 1973, 12, 435–461.
- Samarskii, A.A. and Andreyev, V.B. On a high-accuracy difference scheme for an elliptic equation with several space variables. USSR J. Comp. Math. Math. Phys., 1963, 3, 1373–1382.

#### References

Batchelor, G.K. Heat transfer by free-convection across a closed cavity between vertical boundaries at different temperature. Quart. J. Appl. Math, 1954, 12, 209–233.

Eckert, E.R.G. and Carlson, W.O. Natural convection in an air layer enclosed between two vertical plates with different temperatures. Int. J. Heat Mass Transf, 1961, 2, 106–120.

Elder, J.W. Laminar free convection in a vertical slot. J. Fluid Mech,1965, 23, 77–98.

Elder, J.W. Numerical experiments with free convection in a vertical slot. J. Fluid Mech, 1966, 24, 823–843.

Gill, A.E. The boundary-layer for convection in a rectangular cavity. J. Fluid Mech.1966, 26, 515–536.

Wilkes, J.O. and Churchill, S.W. The finite-difference computation of natural convection in a rectangular enclosure. AIChE J, 1966, 12, 161–166.

de Vahl Davis, G. Laminar natural convection in an enclosed rectangular cavity. Int. J. Heat Mass Transf, 1968, 11, 1675–1693.

Kimura, S. and Bejan, A. The boundary layer natural convection regime in a rectangular cavity with uniform heat flux from the side. J. Heat Transf, 1984, 106, 98–103.

Hall, J.D., Bejan, A. and Chaddock, J.B. Transient natural convection in a rectangular enclosure with one heated side wall. Int. J. Heat Fluid Flow, 1988, 9, 396–404.

Straughan, B. A nonlinear analysis of convection in a porous vertical slab. Geophys. Astrophys. Fluid Dyn, 1988, 42, 269–275.

Hill, A.A. and Straughan, B. Linear and nonlinear stability thresholds for thermal convection in a box. Math. Meth. Appl. Sci, 2006, 29, 2123–2132.

Barletta, A., Magyari, E., Pop, I. and Storesletten, L. Mixed convection with viscous dissipation in a vertical channel filled with a porous medium. Acta Mech, 2007, 194, 123–140.

Oztop, H.F., Varol, Y. and Pop, I. Effects of wall conduction on natural convection in a porous triangular enclosure. Acta Mech, 2008, 200, 155–165.

Tiwari, A.K., Singh, A.K., Chandran, P. and Sacheti, N.C. Natural convection in a cavity with a sloping upper surface filled with an anisotropic porous material. Acta Mech, 2012, 223, 95–108.

Straughan, B. Energy Method, Stability and Nonlinear Convection. 2nd Ed. Appl. Math. Sci. Ser., vol. 91 Springer-Verlag, New York (2004)

Oreper, G.M. and Szekely, J. The effect of an externally imposed magnetic field on buoyancy driven flow in a rectangular enclosure. J. Cryst. Growth, 1983, 64, 505–515.

Garandet, J.P., Alboussiere, T. and Moreau, R. Buoyancy driven convection in a rectangular cavity with a transverse magnetic field. Int. J. Heat Mass Transf., 1992, 35, 741–748.

Rudraiah, N., Barron, R.M., Venkatachalappa, M. and Subbaraya, C.K. Effect of a magnetic field on free convection in a rectangular cavity. Int. J. Engng. Sci., 1995, 33, 1075–1084.

Alchaar, S., Vasseur, P. and Bilgen, E. Natural-convection heat transfer in a rectangular enclosure with a transverse magnetic field. J. Heat Transf., 1995, 117, 668–673.

Kanafer, K. and Chamkha, A. J. Hydromagnetic natural convection from an inclined porous square enclosure with heat generation. Numer. Heat Transf. A., 1998, 33, 891–910.

Pirmohammadi, M., Ghassemi, M. and Sheikhzadeh, G.A. Effect of a magnetic field on buoyancy-driven convection in differentially heated square cavity. IEEE Trans. Magnetics, 2009, 45, 407–411.

Pirmohammadi, M., Ghassemi, M. and Sheikhzadeh, G.A. Effect of magnetic field on transient natural convection heat transfer. IEEE Trans. Magnetics, 2009, 45, 2788–2790.

Al-Najem, N.M., Khanafer, K.M. and El-Refaee, M.M. Numerical study of laminar natural convection in tilted enclosure with transverse magnetic field. Int. J. Numer. Meth. Heat Fluid Flow, 1998, 8, 651–672.

Ece, M.C. and Buyuk, E. Natural-convection flow under a magnetic field in an inclined rectangular enclosure heated and cooled on adjacent walls. Fluid Dyn. Res., 2006, 38, 564–590.

Pirmohammadi, M. and Ghassemi, M. Effect of a magnetic field on convection heat transfer inside a tilted square enclosure. Int. Comm. Heat Mass Transf, 2009, 36, 776–780.

Pirmohammadi, M., Ghassemi, M. and Hamedi, M. Effect of inclination angle on magneto-convection inside a tilted enclosure. IEEE Trans. Magnetics, 2010, 46, 3697–3700.

Pirmohammadi, M. and Ghassemi, M. Numerical study of magneto-convection in a partitioned enclosure. IEEE Trans. Magnetics, 2009, 45, 2671–2674.

Pirmohammadi, M., Ghassemi, M. and Keshtkar. A. Numerical study of hydromagnetic convection of an electrically conductive fluid with variable properties inside an enclosure. IEEE Trans. Plasma Sci., 2011, 39, 516–520.

Mallinson, G.D. and de Vahl Davis, G. The method of the false transient for the solution of coupled differential equations. J. Comp. Phys., 1973, 12, 435–461.

Samarskii, A.A. and Andreyev, V.B. On a high-accuracy difference scheme for an elliptic equation with several space variables. USSR J. Comp. Math. Math. Phys., 1963, 3, 1373–1382.