Unsteady hydromagnetic convective flow in a vertical channel using Darcy–Brinkman–Forchheimer extended model with heat generation/absorption: Analysis with asymmetric heating/cooling of the channel walls Ajay Kumar Singh a,⇑ , Rajiv Kumar a , Usha Singh a , N.P. Singh a,b , Atul Kumar Singh c a Department of Mathematics, C.L. Jain College, Firozabad 283 203, India b Department of Applied Sciences and Humanities, Rama Institute of Engineering and Technology, Mandhana, Kanpur 209 217, India c Department of Mathematics, V.S.S.D. College, Kanpur 208 002, India article info Article history: Received 3 August 2010 Received in revised form 8 May 2011 Available online 19 August 2011 Keywords: Convective flow Vertical channel Porous medium Magnetic field abstract In the present paper transient as well as non-Darcian effects on laminar natural convection flow in a ver- tical channel embedded in a porous medium in the presence of uniform magnetic field applied normal to the flow region is studied. Forchheimer–Brinkman extended Darcy model is assumed to simulate momentum transfer within the porous medium. Approximate solutions are obtained using multi-param- eter perturbation technique. Variations in the velocity field with Darcy number, Grashof number, kine- matic viscosity ratio and variations in the temperature distribution are obtained and depicted graphically. Expression for the skin-friction and the rate of heat transfer at the channel walls are also derived and the numerical values for various physical parameters are presented in tabular form. Ó 2011 Elsevier Ltd. All rights reserved. 1. Introduction Considerable interest in the study of the phenomena of convec- tive flow through a fluid-saturated porous media stems both from fundamental considerations as well as from physical point of view. The research activities in this area have found its attention due to a broad range of applications in science and technology such as geo- thermal energy utilization, thermal insulation engineering, insula- tion of high temperature gas-cooled reactor vessels, thermal energy storage system, heat exchangers, nuclear based repositories and chemical catalytic reactor, petroleum reservoirs, pollutants dispersion in aquifers, fiber and granular insulation including structures for high power density machines, combustion in situ in underground reservoirs for enhancement of oil recovery, reduc- tion of hazardous combustion products using catalytic porous beds, ceramic radiant porous burners used by industrial firms as efficient heat transfer devices, storage of grain, food and vegeta- bles, industrial and agricultural water distribution, solar collectors with porous absorbers and porous bearings, blood flow in lungs or in arteries, porous heat pipes, solidification of casting, buried electrical cables, cores of chemical reactors, chemical catalytic connectors etc. These applications have attracted the attention of engineers and scientists from diversified disciplines. A great deal of experimental and theoretical research work carried out in this topic in the last three/four decades is available in the books of Kaviany [1], Pop and Ingham [2], Ingham and Pop [3], Nield and Bejan [4], Vadasz [5] and Vafai [6]. The earliest reference to fluid flowing through a porous media dates back to Darcy [7]. However its importance and ramifica- tions in process design and operation have been recognized only during the last three/four decades. Irmay [8] presented an analy- sis on the theoretical derivation of Darcy and Forchheimer mod- els. Consequently, Neale and Nader [9] presented an important analysis for the flow in a channel having fluid and porous regions and shown that the Darcy model with BJ-condition gives the same results as obtained by using the Brinkman model consider- ing continuity of the velocity and shear stress at the interface. Va- fai and Kim [10] obtained an exact solution for forced convection in a channel filled with porous medium using boundary layer approximation for fully developed flow subject to a constant- heat-flux boundary condition. Nakayama et al. [11] have studied Forchheimer free convection over a non-isothermal body of arbi- trary shape in a saturated porous medium. Consequently, the mechanism of convective heat transfer in a porous medium in a variety of processes and unit operations involving a range of geo- metric configurations has occurred in the literature. Indeed, sev- eral momentum transfer models have been proposed and used during the last few years, such as the Forchheimer-extended Darcy model and the general extended Darcy model which is Brinkman–Forchheimer extended Darcy model with an additional convectional inertial term. 0017-9310/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2011.06.009 ⇑ Corresponding author. Address: H. No. 236, Durga Nagar, Firozabad 283 203, U.P., India. E-mail addresses: aksinghnpsingh@rediffmail.com, aksinghnpsingh@yahoo.co.in (A.K. Singh). International Journal of Heat and Mass Transfer 54 (2011) 5633–5642 Contents lists available at ScienceDirect International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt