99 Nomenclature density.bo=Br(T) n coordinates 4BB nanent magnets(T) dimensionless fluid temperature.= pefficient (Wm-2K-) solid temperature. netic nsit wm aticviscos average Nusselt number mass magnenss m e(K) reference value velocity vector solid Wakayama and cow orkers [26-28]studi d the ransfe uation for nagnetic conve ing a m milar to ic d app cubi 13-34 ed Finite Eleme t Method (C M). The results ind ases ile an opposite trend d that the magn e can be us un erical availabe from Neody mium-Iro out hat t f air or they found tha natural magne and depends on the relati rate of pa dients theoretically and experim ntally and found that th on of par s.How wa onal fo ces on the natura cated in n in n. e stability of p uid layer by the action im u diamagnetic field.Gray et al.125]obtained the nu erica number and magnet the esult ifcant effect on the flow field and heat transfer in a paramagnet influenced by the inclination angle. Especially, when the aspect ratio is less or more than 1, it is found that the inclination angle plays a great role on flow and heat transfer. The natural convection heat transfer of Cu–water nanofluid in a cold outer circular enclosure containing a hot inner sinusoidal cir￾cular cylinder in the presence of horizontal magnetic field is inves￾tigated numerically by Sheikholeslami et al. [17] using the Control Volume based Finite Element Method (CVFEM). The results indi￾cate that in the absence of magnetic field, enhancement ratio decreases as Rayleigh number increases while an opposite trend is observed in the presence of magnetic field. In recent years, with the development of a superconducting magnet providing strong magnetic induction of 10 T or more, the natural convection of paramagnetic fluid like oxygen gas and air under magnetic field has been an interesting research topic inves￾tigated by many researchers [18,19]. The effect of the magnetic buoyancy force on the convection of paramagnetic fluids was first reported by Braithwaite et al. [20], they found that the effect of magnetic buoyancy force on convection depends on the relative orientation of the magnetic force and the temperature gradient. Carruthers and Wolfe [21] studied the thermal convection of oxy￾gen gas in a rectangular container with thermal and magnetic field gradients theoretically and experimentally, and found that the magnetic buoyancy force cancel out the influence of gravity buoy￾ancy force when rectangle enclosure heated from one vertical wall and cooled from opposing wall was located in horizontal magnetic field with vertical magnetic field gradient. Huang et al. [22–24] studied the stability of paramagnetic fluid layer by the action of a non-uniform magnetic field and the thermomagnetic convection of the diamagnetic field. Gray et al. [25] obtained the numerical similarity solutions for the two-dimensional plums driven by the interaction of a line heat source and a non-uniform magnetic field. Wakayama and coworkers [26–28] studied the magnetic control of thermal convection. Tagawa’s group [29–31] derived a model equation for magnetic convection using a method similar to the Boussinesq approximation and studied natural convection of para￾magnetic, diamagnetic and electrically conducting fluids in a cubic enclosure with thermal and magnetic field gradients at different thermal boundary. Ozoe and co-workers [18,32–34] studied natu￾ral convection of paramagnetic and diamagnetic fluids in a cylinder under gradient magnetic field at different thermal boundary and found that the magnetic force can be used to control heat transfer rate of paramagnetic and diamagnetic fluids. Yang et al. [35,36] numerically and experimentally investigated magnetothermal convection of air or oxygen gas in a enclosure by using the gradient magnetic field available from Neodymium–Iron–Boron permanent magnet systems, and pointed out that the enhancement or sup￾pression of magnetothermal convection of air or oxygen gas can be achieved by gradient magnetic field. Tomasz and co-workers [37–40] studied natural convection of paramagnetic fluids in a cubic enclosure under magnetic field by an electric coil and ana￾lyzed that the effect of inclined angle of electric coil, location of electric coil, Ra number on heat transfer rate of paramagnetic fluid. Above studies are concerned with the effect of magnetic force on natural convection of paramagnetic fluids. However, only a few studies are paid on the combined effects of both magnetic and gravitational forces on the natural convection of paramagnetic fluids in porous medium. Natural convection in an enclosure filled with a paramagnetic or diamagnetic fluid-saturated porous med￾ium under strong magnetic field was numerically investigated by Wang et al. [41–43]. Considering the effect of Darcy number, Rayleigh number and magnetic force number, the results of numerical investigation showed that the magnetic force had a sig￾nificant effect on the flow field and heat transfer in a paramagnetic Nomenclature b magnetic flux density (T) b0 reference magnetic flux density, b0 ¼ Br (T) B dimensionless magnetic flux Br magnetic flux density of permanent magnets (T) C C ¼ 1 þ 1 T0b cp fluid specific heat at constant pressure (J kg1 K1 ) Da Darcy number, j L2 fm magnetic force g gravitational acceleration (m s2 ) h solid-to-fluid heat transfer coefficient (W m2 K1 ) H dimensionless solid-to-fluid heat transfer coefficient, hL2 ekf H magnetic field intensity kf fluid thermal conductivity (W m1 K1 ) ks solid thermal conductivity (W m1 K1 ) L length of the enclosure, (m) Num average Nusselt number p pressure, Pa P dimensionless pressure p0 pressure at reference temperature, Pa p0 pressure difference due to the perturbed state, Pa Pr Prandtl number, Pr ¼ mf af Ra Rayleigh number, Ra ¼ gbðThTc ÞL3 af mf T0 T0 ¼ ThþTc 2 (K) Tc cold wall temperature (K) Tf fluid temperature (K) Th hot wall temperature (K) Ts solid temperature (K) u,v velocity components (m s1 ) U,V dimensionless velocity components U velocity vector x,y Cartesian coordinates X,Y dimensionless Cartesian coordinates Greek symbols af fluid thermal diffusivity (m s1 ) b thermal expansion coefficient (K1 ) c dimensionless magnetic strength parameter, c ¼ v0b2 0 lmgL hf dimensionless fluid temperature, hf ¼ Tf T0 ThTc hs dimensionless solid temperature, hs ¼ TsT0 ThTc e porosity l0 magnetic permeability of vacuum (H m1 ) lm magnetic permeability (H m1 ) lf fluid kinematic viscosity (kg m1 s 1 ) mf fluid dynamic viscosity (m2 s1 ) qf fluid density (kg m3 ) qs solid density (kg m3 ) v mass magnetic susceptibility (m3 kg1 ) v0 reference mass magnetic susceptibility (m3 kg1 ) vm volume magnetic susceptibility j Permeability (m2 ) K dimensionless thermal conductivity, K ¼ ekf ð1eÞks um scalar magnetic potential Subscripts 0 reference value c cold f fluid h hot s solid C. Jiang et al. / International Journal of Heat and Mass Transfer 91 (2015) 98–109 99