Pramana – J. Phys. (2020) 94:156 © Indian Academy of Sciences https://doi.org/10.1007/s12043-020-02015-7

Effect of magnetic field on the mixed Fe3O4/water ferrofluid flow in a horizontal porous channel

AMIRA JARRAY, ZOUHAIER MEHREZ∗ and AFIF EL CAFSI

Laboratoire d’Energétique et des Transferts Thermique et Massique (LETTM), Département de Physique, Faculté des Sciences de Tunis, Université d’el Manar, Tunis, Tunisia ∗Corresponding author. E-mail: [email protected]; [email protected]

MS received 7 May 2020; revised 11 July 2020; accepted 29 July 2020

Abstract. The effect of an external magnetic field on the mixed convection Fe3O4/water ferrofluid flow in a horizontal porous channel was studied numerically. The governing equations using the Darcy–Brinkman– Forchheimer formulation were solved by employing the finite volume method. The computations were carried out for a range of volume fractions of nanoparticles 0 ≤ ϕ ≤ 0.05, magnetic numbers 0 ≤ Mn ≤ 100, Reynolds numbers 100 ≤ Re ≤ 500, Darcy numbers 10−3 ≤ Da ≤ 10−1 and porosity parameters 0.7 ≤ ε ≤ 0.9 while fixing the at 104. Results show the formation of recirculation zone in the vicinity of the magnetic source under the influence of Kelvin force. It grows as the magnetic number increases. The friction factor increases by increasing the magnetic number and diminishes with the increase in Darcy number. The flow accelerates as the magnetic field intensifies. The rate increases by increasing the volume fraction of the nanoparticles and the magnetic number. The effect of magnetic field on the hydrodynamic and thermal behaviours of the ferrofluid flow considerably intensifies by increasing and Darcy number. The combined effect of ferromagnetic nanoparticles and magnetic field on the enhancement rate of heat transfer becomes more pronounced at high values of Reynolds number, permeability and/or porosity parameter.

Keywords. Mixed convection; porous medium; ferrofluid; magnetic field.

PACS Nos 44.15.+a; 44.25.+f; 44.27.+g; 47.65.Cb

1. Introduction modification of the flow field and the temperature distribution. It is created by swirling the zones in the Several techniques are there to increase heat transfer vicinity of the magnetic sources, modifying thus the in engineering applications using cooling and heating characteristics of the thermal boundary layer. This fact systems such as electronic and microelectronic devices, largely affects the heat transfer rate in the ferrofluid flow. heat exchangers, engines and automobiles, solar energy, In recent years, the problem of ferrofluid flow in the refrigerator-freezers, nuclear reactors and transformers. presence of magnetic field, called ferrohydrodynamic The most recent technique is the one where thermal (FHD), is largely studied in various configurations by conductivity of the carrier fluid is increased by adding many researchers. Mokhtari et al [1] showed that the nanosized metallic or metallic oxide particles. The heat transfer rate can reach 30% by applying a mag- obtained suspension, called nanofluid, has shown its netic field in the ferrofluid flow inside the tube with effectiveness in terms of heat transfer enhancement in twisted tapes. Hassan et al [2] numerically investi- many engineering configurations. When the suspended gated FHD in stretchable rotating disk. They showed nanoparticles are magnetisable, a particular nanofluid that the maximum heat transfer rate is obtained with called ferrofluid is obtained. Apart from its improved prolate iron nanoparticles and by applying oscillat- thermophysical properties, the heat transfer and the ing magnetic field. Ashouri and Shafii [3] studied ferrofluid flow behaviour can be modified by apply- the magnetoconvection ferrofluid flow in a permanent ing external non-uniform magnetic field. In fact, under magnet-inserted cavity. They proved that heat rate will the effect of magnetic field, the magnetic moments be maximum for an optimum size of the permanent of the particles follow the field lines, leading to the magnet. Kamıs and Atalık [4] studied the influence

0123456789().: V,-vol 156 Page 2 of 12 Pramana – J. Phys. (2020) 94:156 of thermomagnetic effect on the stability of Taylor– enhancement, especially at high frequency. Izadi et al Couette ferrofluid flow under an azimuthal magnetic [20] analysed the combined FHD–MHD natural con- field. They observed the stabilising effect of the strong vection of a hybrid carbon nanotube–Fe3O4 nanofluid magnetic field which can be amplified by increasing flow in an inversed T-shaped porous enclosure. Results the volume fraction and the size of the ferroparticles. show that increasing the Hartmann number restrains the The effect of FHD in the heat transfer enhancement convective motion and energy transport, and reduces the of a fin-and-tube compact heat exchanger was investi- heat transfer rate. However, the increment of the mag- gated by Bezaatpour and Rostamzadeh [5]. They found netic field strength intensifies the circulation and thermal that the local and average heat transfer coefficients transmission. Fadaei et al [21] investigated the forced- increase around all tubes by increasing the magnetic convection heat transfer problem in a pipe partially filled field strength. Gibanov et al [6] studied the MHD with porous medium under the influence of a magnetic and FHD effects on the mixed convection heat trans- field. They showed that the convective heat transfer rate fer ferrofluid flow in a lid-driven cavity containing increases up to 30% in the presence of magnetic field a heat-conducting solid backward facing step. They induced by a solenoid traversed by a current of inten- found that the heat transfer rate increases with the sity 10 A. Bezaatpour and Goharkhah [22] investigated increment of magnetic number and volume fraction of the FHD effect on the Fe3O4/water ferrofluid flow in a nanoparticles, whereas it decreases by increasing Hart- porous fin heat sink. They indicated that the heat transfer mann number. Sheikholeslami et al [7] investigated the rate enhances by increasing the volume fraction of fer- impact of variable magnetic forces on the magnetis- roparticles, the magnetic field intensity and fin porosities able hybrid nanofluid heat transfer through a circular whereas it is weakened with the Reynolds number. Izadi cavity. They demonstrated that adding hybrid nanopar- et al [23] showed that for certain pertinent parameters, ticles of MWCNT–Fe3O4 enhances the heat transfer. the heat transfer can increase by applying two mag- This depends on the pertinent parameters such as mag- netic fields on the magnetisable hybrid nanofluid inside a netic strengths ratio parameter, magnetic number and porous enclosure. Ghalambaz et al [24] emphasised the Hartmann number. Selimefendigil et al [8] examined the combined effect of FHD and MHD on the heat and mass effect of variable magnetic field on the forced convection transfers of magnetic nanofluid flow inside a hexago- Fe3O4–water nanofluid in a bifurcating channel. They nal cavity. They found that there is an increase in rates revealed that the heat transfer enhancements were in the of heat and mass transfers by increasing the magnetic range of 12–15% and 9–12% in the absence and pres- number but a decrease with the increase in Hartmann ence of magnetic field, respectively. Many investiga- number. tions studying the FHD of clear medium in channels and The aforementioned literature shows that only the microchannels were published (see refs [9–18]). In these effects of magnetic field on the ferrofluid flow in a clear works, various methods were used and different empir- medium in a horizontal channel have been studied in ical and theoretical correlations modelling the magnetic some recent works. However, this problem is not stud- moment and the Kelvin force were employed. The com- ied yet in a porous medium. The present study deals with mon point between the results is the strong effect of the FHD effect on the hydrodynamic and thermal char- magnetic field on the heat transfer enhancement. acteristics of mixed convection Fe3O4/water ferrofluid Porous medium is a rigid solid matrix having commu- flow in a horizontal porous saturated channel. nicant voids (pores) and containing one or more fluid phases (gas or liquid) which can flow and exchange the energy and the matter between them and with the 2. Physical configuration solid phase. Porous media play important roles in many industrial domains and in various field of science such as Figure 1 presents the studied configuration with bound- petroleum engineering, chemical engineering and elec- ary conditions. It is a horizontal channel with an aspect trochemistry, hydrogeology, geothermal energy, thermal ratio of L/H = 10 where L and H are the length and engineering, civil engineering, medicine, biochemistry the height of the channel respectively. A fully developed and nuclear engineering. Thus, the FHD in porous parabolic velocity profile and low temperature (TC ) are medium is considerably important because it can influ- deployed at the inlet. No-slip conditions are prescribed ence the transport and exchange of energy. In this regard, at the channel walls. At the outlet, convective boundary Amani et al [19] conducted an experimental investi- conditions velocity and temperature are imposed. The gation on the influence of variable magnetic field on lower wall is maintained isothermal at high temperature / the hydrothermal behaviour of Fe3O4 water ferrofluid (TH ) whereas the upper wall is thermally insulated. The flow through the metal foam. They showed the strong channel is filled by glass balls as the porous medium. effect of variable magnetic field on the heat transfer The ferrofluid is the pure water as the carrier fluid where Pramana – J. Phys. (2020) 94:156 Page 3 of 12 156

Figure 1. Physical configuration. Table 1. Thermophysical properties of the base fluid, solid coordinates are given as follows [25,27]: nanoparticles and glass balls. ∂ ∂v u + = Physical properties Water Fe3O4 Glass balls ∂ ∂ 0(4) x  y   ρ(kg/m3) 997.1 5200 2700 1 ∂u 1 ∂u ∂u / ρ + u + v cp (J kg K) 4179 670 840 nf ε ∂t ε2 ∂x ∂y k (W/m K) 0.613 6 1.05   −5 −1 ∂ μ ∂2 ∂2v β × 10 (K ) 21 1.3 0.9 =− p + nf u + ∂x ε ∂x2 ∂y2  ferronanoparticles of Fe3O4 are suspended. Table 1 dis- μnf 1.75ρnf − u − √ u2 + v2 u + F (5) plays the thermophysical properties of glass balls, pure 3 kx K 150 K ε 2 water and nanopowder of Fe O . The channel is sub-    3 4 1 ∂v 1 ∂v ∂v jected to the effect of a magnetic source placed in the ρ + u + v = nf ε ∂t ε2 ∂x ∂y vicinity of the lower heated wall at the coordinates x a   and y = b. In the current study, a and b are fixed at L/2 ∂ μ ∂2 ∂2v =− p + nf u + and 0, respectively. The components of magnetic field ∂y ε ∂x2 ∂y2 (Hx , Hy) and the magnitude of the magnetic field H can  μ 1.75 ρ be expressed as [25,26] − nf v − √ nf u2 + v2 v K ε 3 I y − b 150 K 2 Hx (x, y) = , (1) +(ρβ) g (T − Tc) + Fky (6) 2π ( − )2 + ( − )2 nf   x a y b ∂ ∂ ∂ − ρ T + ρ T + v T I x a cp , cp u Hy(x, y) =− , (2) nf m ∂t nf ∂x ∂y 2π (x − a)2 + (y − b)2    ∂2 ∂2 I 1 T T ( , ) = 2 + 2 =  , = knf,m + H x y Hx Hy ∂ 2 ∂ 2 2π (x − a)2 + (y − b)2 x y  ∂ M ∂ H ∂ H (3) −μ T u + v . (7) 0 ∂T ∂x ∂y where a and b are the coordinates of magnetic source in meters. The magnetic source is created by passing Here, u and v are the components of velocity in x and 3 constant electrical current I through the conductor wire. y directions (m/s), ρnf is the density (kg/m ) of the nanofluid, t is time (s), p is pressure (Pa), ε is the poros- ity parameter, K is the permeability of the porous media, 3. Mathematical formulation g is the gravitational acceleration (m/s2), T is the tem- perature (K), β is the coefficient of thermal expansion In the current investigation, it is considered that the flow (1/K), cp is the heat capacity (J/kg K), k is the ther- is two-dimensional, laminar, steady and incompressible mal conductivity (W/mK). The terms Fkx and Fky in and the fluid is assumed to be Newtonian. The porous (5)and(6) respectively, denote the components of mag- medium is assumed to be homogeneous and isotopic. netic force per unit volume, called also, in the FHD The Brinkman–Forchheimer–Darcy model is used to domain as Kelvin force. This force depends on spatial study mixed convection nanofluid flow in a porous variation of the magnetic field, i.e. magnetic gradients. medium. Taking into account Boussinesq approxima- It is expressed as tion and these assumptions, the equations of continuity   (4), momentum (5), (6),andenergy(7) in Cartesian Fk = Fkxux + Fkyuy = μ0 M∇ H, (8) 156 Page 4 of 12 Pramana – J. Phys. (2020) 94:156 where M represents the magnetisation vector. The fol- • Effective thermal diffusivity: lowing empirical expression is used to calculate the magnetisation [25]: knf,m   α = . (17) M = K H T − T . (9) e (ρ ) C c p nf   In (9) K is a constant and TC represents the Curie tem- Here ε represents the porosity parameter of the porous perature. It is the temperature from which the matter medium, the indices s and m indicate the solid phase, loses its permanent magnetisation, and it then becomes i.e. porous matrix and effective properties related to the paramagnetic. It is equal to 858 K for Fe3O4. nanofluid saturated porous medium, respectively. The term   The following non-dimensional variables are used to ∂ M ∂ H ∂ H obtain the final dimensionless form of the FHD equa- −μ T u + v 0 ∂T ∂x ∂y tions: x y tu u v in eq. (7) denotes the volumetric thermal power due to X = , Y = ,τ= 0 , U = ; V = , the magnetocaloric effect [25]. H H H u0 u 0 The effective density, the heat capacitance and the − − ρcp θ = T Tc = T Tc , = p ,σ= nf,m , thermal expansion coefficient of the magnetic nanofluid, P 2 TH − Tc T ρ u ρc estimated by using the mixture law, are defined as f 0 p nf  H ρ = ϕ ρ + ( − ϕ)ρ H = nf p p 1 bf (10) H0 (ρcp)nf = ϕp(ρcp)p + (1 − ϕ)(ρcp)bf (11) where I (ρβ) = ϕ (ρβ) + ( − ϕ)(ρβ) , H = H (a, 0) = . nf p p 1 bf (12) 0 2π |b| where ϕ is the volume fraction of the nanoparticles. Here X and Y are non-dimensional coordinates, τ is The Brinkman’s model [28] is used to calculate the the dimensionless time, u0 is the velocity at the entrance effective dynamic viscosity: (m/s) and θ is the non-dimensional temperature. The dimensionless governing equations are expressed μ 1 nf = . as follows: 2.5 (13) μbf (1 − ϕ) ∂U ∂V The effective thermal conductivity of the nanofluid is + = 0 (18) ∂ X ∂Y estimated by using the Maxwell’s model [29]:   ∂ ∂ ∂ 1 U + 1 U + U knf k p + 2kbf − 2ϕ(kbf − k p) U V = . (14) ε ∂τ ε2 ∂ X ∂Y kbf k p + 2kbf + ϕ(kbf − k p) ρ ∂ =− bf P The Brinkman and Maxwell models are used in this ρ ∂ X nf   work because of the absence, in the literature, of formu- 1 ρ μ ∂2U ∂2U lations modelling the viscosity and thermal conductivity + bf nf + ε ρ μ ∂ 2 ∂ 2 of Fe O /water. Even though these models are very old, Re nf bf X Y 3 4  they are still used because of their effectiveness, espe- 1 ρbf μnf 1.75 − U − √ U 2 +V 2 U ρ μ 3 cially for low volume fractions of the nanoparticles. Re Da nf bf 150 Da ε 2 The thermophysical properties of the nanofluid in the  ρbf  ∂ H saturated porous medium are defined as follows [27]: +Mn (δ − δ − θ) H (19) ρ C H ∂ X nf  • Thermal conductivity: 1 ∂V 1 ∂V ∂V + U + V ε ∂τ ε2 ∂ X ∂Y   2 2 ρbf ∂ P 1 ρbf μnf ∂ V ∂ V knf,m = εknf + (1 − ε) ks . (15) =− + + 2 2 ρnf ∂Y εRe ρnf μbf ∂ X ∂Y •  Heat capacitance: 1 ρ μ 1.75 − bf nf V − √ U 2 +V 2 V Re Da ρ μ ε 3 nf bf 150 Da 2 ρ = ε ρ + ( − ε) ρ . ρ (ρβ) Gr cp nf,m cp nf 1 cp s (16) + bf nf θ ρ (ρβ) 2 nf bf Re Pramana – J. Phys. (2020) 94:156 Page 5 of 12 156

 ρbf  ∂ H The heat transfer between the lower heated wall and +Mn (δC − δH − θ) H (20) ρnf ∂Y ferrofluid is quantified by the average :

∂θ ∂θ ∂θ L/H σ + U + V = 1 , ∂τ ∂ X ∂Y Num Nu dX (28)   L/H 0 α 1 ∂2 θ ∂2θ = e + where Nu is the local Nusselt number. It is expressed as α Pr Re ∂ X 2 ∂Y 2   bf   ∂θ (ρ )   hH knf CP bf ∂ H ∂ H Nu = =− . (29) +Mn Ec U + V ∂ ρ ∂ ∂ k f k f Y Y =0 ( CP )nf X Y  Here h is the coefficient of heat transfer (W/m2 K).The ×H (δH + θ) , (21) combined effect of the magnetic field and nanoparticles where on the heat transfer enhancement is characterised by K calculating the normalised Nusselt numbers Nu∗ .Itis Da = (Darcy number), m H 2 expressed as follows: μbf (ϕ = . ; = ) Pr = (), ∗ = Num 0 05 Mn 100 . ρ α Nu m (30) bf bf Num(ϕ = 0; Mn = 0) ρ u H = bf 0 The local friction factor along the lower wall is given by Re μ (Reynolds number), the following relation: bf   β 3ρ2 2 ∂U g bf TH bf C f = . (31) Gr = (Grashof number), Re ∂Y μ2 Y =0 bf The average friction factor is obtained by integrating eq. μ 2  0 H0 K T (31) as follows: Mn = (magnetic number), ρ 2 L/H bf u0 1 C = C dX. (32) 2 fm / f ρbf u L H 0 Ec = 0 (), (ρCP ) T bf 4. Numerical method, mesh sensitivity and  T T validation test δ = C and δ = H . C T H T Please note that subscripts nf, bf, e, H and C stand for The dimensionless governing equations with the associ- nanofluid, base fluid, effective, hot and cold respectively. ated boundary conditions are discretised on the standard The dimensionless boundary conditions are defined uniform staggered grid, by using the finite volume as follows: method. The time integration is performed using the time-splitting algorithm [30,32]. A centred scheme is • At the inlet: used for the diffusive fluxes, and for the convective terms a second-order upwind finite difference method • ( ) = ( − ) = U Y 4Y 1 Y and V 0(22)by means of quadratic upstream interpolation convec- • θ = 1. (23) tive kinematics (QUICK) scheme is adapted [31]. More details concerning the numerical procedure are already • At the outlet: given in [27,33–35]. Table 2 shows the mesh sensitivity test. It is performed ∂U ∂V ∂θ for the FHD ferrofluid mixed convection in a horizontal = 0, = 0and = 0. (24) = −2 ε = . ϕ = . ∂ X ∂ X ∂ X porous channel for Da 10 , 0 7, 0 05,

Table 2. Mesh sensitivity test. • At the solid walls: Grid Nu %Error C Re % Error • U = 0andV = 0. (25) m fm × • 102 58 7.56 – 25.14 – At the lower wall: θ = 1. (26) 166 × 70 8.17 8.07 26.43 5.13 × • ∂θ 228 82 8.40 2.81 26.99 2.11 At the upper wall: = 0. (27) 282 × 98 8.42 0.24 27.05 0.22 ∂Y 156 Page 6 of 12 Pramana – J. Phys. (2020) 94:156

Table 3. Average Nusselt number for Re = 10, (Ra) = 104 and Pecklet number (Pe) = 20/3. Reference Present Evans and Paolucci [36]Cominiet al [37] Nourollahi et al [38] Sahraoui et al [39]

Num 2.544 2.558 2.574 2.487 2.550

(a)

(b)

(c)

Figure 2. Streamlines (up) and isotherms (down) for ϕ = 0.01, ε = 0.7, Re = 500 and Da = 10−2.(a)Mn= 0, (b)Mn= 50 and (c)Mn= 100.

Re = 500 and Mn = 100. The difference between factor, average and normalised Nusselt numbers, for the third and the fourth grids are less than 0.25% in various Reynolds numbers (100 ≤ Re ≤ 500),mag- terms of average Num and modified averaged skin fric- netic numbers (0 ≤ Mn ≤ 100), porosity parameters −3 −1 tion C fmRe. Therefore, all the subsequent simulations (0.7 ≤ ε ≤ 0.9), Darcy numbers (10 ≤ Da ≤ 10 ) were performed by using the third grid of size 228 × 82 and volume fractions of nanoparticles (0 ≤ ϕ ≤ 0.05). which is optimum in terms of results occurrence and The Grashof number (Gr), the Eckert number (Ec) and computational time. the Prandtl number (Pr) are set to 104, 10−3 and 6.2, To validate the present house code, a comparison is respectively. done by comparing the present results with the previ- Figure 2 illustrates the effect of magnetic field inten- ous works [36–39] investigating the flow in a horizontal sity, presented by the magnetic number Mn, on the channel (see table 3). A good agreement with these streamlines (on the top) and isotherms (on the bottom) works is observed. This code is also successfully val- for ϕ = 0.01, Re = 500 and Da = 10−2. The stream- idated for other physical problems [40–42]. lines show the presence of a recirculation zone in the vicinity of the magnetic source, containing one eddy which grows as Mn increases. The ferromagnetic par- ticles are magnetised and their magnetic dipoles are 5. Results and discussion oriented along the lines of the magnetic field. This causes the deviation of ferrofluid particles which rotate In the present study, the results of the mixed convec- near the magnetic source under the influence of intense tion ferrofluid flow under the influence of magnetic Kelvin force. This rotating effect can be explained by field are presented using streamline function, temper- the magnetisation of nanoparticles which diminishes ature field, velocity profiles, local and average friction near the bottom wall at high temperature. The warm Pramana – J. Phys. (2020) 94:156 Page 7 of 12 156

Da = 10-3 (a)

(b)

(c)

Da = 10-1

(a)

(b )

(c)

Figure 3. Effect of Da on the streamlines (up) and isotherms (down) for ϕ = 0.01, ε = 0.7, Mn = 100. (a)Re= 100, (b) Re = 300 and (c)Re= 500. 156 Page 8 of 12 Pramana – J. Phys. (2020) 94:156 ferrofluid near the bottom wall moves under the influ- (a) ence of thermomigration phenomenon towards the 1.0 cooler regions where their magnetisation increases. Thus, the non-uniformity of the magnetisation of nanopar- 0.5 ticles is responsible for the formation of the recirculation zone. By increasing the magnetic field strength, this 0.0 phenomenon becomes more appreciable and the recircu-

lation zone intensifies. The corresponding temperature U fields reveal the existence of thermal boundary layer, in -0.5 the vicinity of the heated wall, where the isotherms are parallel indicating the domination of conduction heat -1.0 transfer mode. This characteristic of the thermal layer Mn = 0 Mn = 50 is disturbed by applying and intensifying the magnetic -1.5 Mn = 100 field. Indeed, in the recirculation zone, the isotherms become distorted showing the domination of convective 0.0 0.2 0.4 0.6 0.8 1.0 heat transfer mode due to the movement of ferrofluid Y under the effect of thermodiffusion phenomenon. In the downstream of recirculation, the isotherms become nar- (b) 1.5 rower by increasing the magnetic number. This is due to 1.0 the downward main flow distortion after recirculation, thus impacting the heated wall. This fact leads the tem- 0.5 perature gradient and so the local heat transfer rate to 0.0 increase. Figure 3 examines the effect of magnetic field on the -0.5 streamlines and isotherms for different values of Da and U -1.0 Re at ϕ = 0.01 and Mn = 100. It is clear that the effect of magnetic field on the flow field and temperature -1.5 distribution, discussed in the last paragraph, strongly depends on the permeability of the porous medium and -2.0 Mn = 0 Mn = 50 inertia effect. Indeed, the effect of magnetic field on -2.5 Mn = 100 the recirculation zone becomes more apparent by aug- menting Da and/or the Re. Increasing the permeability 0.0 0.2 0.4 0.6 0.8 1.0 facilitates the circulation of the ferrofluid and acceler- Y ates it considerably. This fact enhances the transport (c) and diffusion phenomena intensifying thus the effect 1.5 of magnetic field on the flow behaviour. Also, intensi- 1.0 fying the inertia force diminishes the thermal boundary 0.5 layer thickness; this increases the Kelvin force effect, previously described. Thus, in the presence of magnetic 0.0 field, the convective heat transfer mode enhances in the -0.5 recirculation zone by increasing Da and Re.

U -1.0 Figure 4 represents the profiles of horizontal compo- nent of velocity, plotted at X = 5H where the magnetic -1.5 source is placed, for various values of Da and Mn when -2.0 Re = 500 and ε = 0.7. For all Da, the U-velocity pro- Mn = 0 -2.5 file is quasiparabolic in the absence of magnetic field Mn = 50 (Mn = 0). This parabolic profile is distorted under the -3.0 Mn = 100 influence of magnetic field, presenting so a peak located 0.0 0.2 0.4 0.6 0.8 1.0 in the vicinity of the heated wall. The peak location Y moves up by increasing Da and Mn. Negative values of velocity appeared in the presence of magnetic field Figure 4. U-velocity profiles at the middle section of the revealing the formation of reverse flow near the bot- channel for different magnetic numbers at ϕ = 0.01, ε = 0.7, tom wall, confirming the presence of recirculation zone Re = 500. (a)Da = 10−3,(b)Da = 10−2 and (c) − rotating in the clockwise sense. By increasing Mn, the Da = 10 1. Pramana – J. Phys. (2020) 94:156 Page 9 of 12 156

35 (a) 250 ϕ = 0.01 Mn = 0 ϕ = 0.03 200 Mn = 50 ϕ = 0.05 Da = 10-3 Mn = 100 30 150 25 Da = 10-2 100 Re fm C 50 20 Re f C 0 15

-50 Da = 10-1 10 -100 020406080100 Mn -150 0246810Figure 6. Averaged friction factor vs. magnetic number for X various values of ϕ when Re = 500, ε = 0.7. (a)Da= 10−3, (b)Da= 10−2 and (c)Da= 10−1. (b) 200 Mn = 0 150 Mn = 50 Mn = 100 U-velocity intensity and gradients increase everywhere 100 in the canal section because of the rise of the Kelvin force. This fact becomes more important by increasing 50 Da due to the acceleration of the ferrofluid. Figure 5 depicts the friction factor profiles (eq. (31)) 0 along the heated wall for different values of Da and Mn Re f

C = ε = . -50 at Re 500 and 0 7. For all the cases, the fric- tion factor is positive and constant in the upstream of -100 the recirculation zone. It abruptly increases in this zone until a positive peak is reached, and then dramatically -150 decreases towards a negative peak. It regains its constant value in the downstream of the recirculation region. This -200 0246810behaviour is due to the inversion of the velocity gradi- X ent in the vicinity of the magnetic source. The friction factor profiles are similar in the fully-developed regime (c) 100 Mn = 0 regardless of Mn. However, their peaks become more Mn = 50 apparent by intensifying the magnetic field due to the Mn = 100 growth of the velocity gradients. It is noted also that the local friction factor diminishes everywhere and their 50 peaks weaken by increasing Da due to the augmen- tation of permeability which facilitates the movement of ferrofluid, thereby decreasing the local velocity 0 Re f gradients. C Figure 6 portrays the average friction factor (eq. (32)) as a function of Mn for various volume fractions of -50 nanoparticles and different Da at Re = 500 and ε = 0.7. The average friction factor increases by increasing Mn because of the increase of the pressure drop caused by

-100 the growth of the recirculation zone. It decreases by 0246810increasing the volume fraction of nanoparticles due to X the decrease of the velocity gradients. It is observed also that the average friction factor clearly decreases by aug- Figure 5. Local friction factor profiles for different magnetic menting Da. Augmenting the permeability of the porous − numbers at ϕ = 0.01, ε = 0.7, Re = 500. (a)Da= 10 3, medium decreases the flow resistivity, weakening thus = −2 = −1 (b)Da 10 and (c)Da 10 . the average friction factor. 156 Page 10 of 12 Pramana – J. Phys. (2020) 94:156

1.40 (a) ϕ = 0.01 (a) ε = 0.7 ϕ = 0.03 ε = 0.8 Da = 10-1 8.5 ϕ = 0.05 1.36 ε = 0.9

1.32 *

8.0 m 1.28 Nu -2 m Da = 10

Nu 1.24

7.5 1.20 Da = 10-3

1.16 100 200 300 400 500 Re 7.0 0 20406080100 ∗ ε Mn Figure 8. Num against Re for various values of Da and .

8.5 (b) ϕ = 0.01 Figure 7 shows the variations of Num (eq. (28)) vs. ϕ = 0.03 Mn for different values of Da and various nanopar- ϕ = 0.05 ticle volume fractions for Re = 500 and ε = 0.7. For all the cases, the intensification of the convective 8.0 heat transfer rate can be seen by increasing the vol- ume fraction of nanoparticles due to the enhancement of thermal conductivity of the ferrofluid. By compar- m ing the three cases it can be noted that, for high values Nu of Mn, Num increases by increasing Da due to the 7.5 intensification of the local temperature gradients in the vicinity of the heated wall. For Da = 10−3 the effect of Mn on the heat transfer enhancement is not con- siderable. It becomes much more visible by increasing (b) 7.0 Da because of the rise in permeability which favours 020406080100both natural and forced convection phenomena intensi- Mn fied by the Kelvin force. So, as seen in figure 7,Num is substantially increased by increasing Mn. This is essen- (c) ϕ = 0.01 tially due to the recirculation created by the magnetic 8.5 ϕ = 0.03 field which considerably modified the thermal behaviour ϕ = 0.05 of the ferrofluid flow. This fact has a positive effect on the mixed convection and so on the heat transfer 8.0 rate. Figure 8 exhibits the combined effect of both nanopar-

m ticles and magnetic field on the rate of heat transfer 7.5

Nu enhancement, presented by the normalized Nusselt number Nu∗ (eq. (30)), for various values of Re, Da and m ∗ porosity parameters. It can be observed clearly that Num 7.0 increases by increasing Da due to the positive effect of magnetic field on the heat transfer which intensifies by (c) increasing the permeability. The effect of Re on the heat 6.5 ∗ transfer enhancement rate depends on Da. Indeed, Num 0 20 40 60 80 100 = −3 Mn is insensible to the variation of Re for Da 10 .How- ever, it increases with Re for Da = 10−2 and 10−1.So, Figure 7. Averaged Nusselt number vs. magnetic number the combined effect of magnetic field and nanoparticles, for various values ϕ when Re = 500. (a)Da= 10−3,(b) on the heat transfer enhancement rate grows by increas- Da = 10−2 and (c)Da= 10−1. ing Re. This fact is more intensified by increasing Da Pramana – J. Phys. (2020) 94:156 Page 11 of 12 156 because the inertia effect increases with the permeability [9] H S Seo, J C Lee, I J Hwang and Y J Kim, Mater. Res. of the porous medium, thus favouring the forced convec- Bull. 58, 10 (2014) ∗ [10] V M Job and S R Gunakala, Int. J. Mech. Sci. 144, 357 tion heat transfer. It is also noted that Num increases with the porosity parameter and this becomes more visible (2018) by augmenting Da. Augmenting the porosity parameter [11] B Ghorbani, S Ebrahimi and K Vijayaraghavan, Int. J. leads to the increase of thermal diffusivity of the sat- Heat Mass Transf. 127, 544 (2018) [12] M M Bhatti, M A Yousif, S R Mishra and A Shahid, urated porous medium. This intensifies the combined Pramana – J. Phys. 93(6), 88 (2019) effect of the nanoparticles and magnetic field on the heat [13] W Nessab, H Kahalerras, B Fersadou and D Hammoudi, transfer enhancement. This can be accentuated further Appl. Therm. Eng. 150, 271 (2019) with the increase of permeability of the porous medium [14] A Salehpour and M Ashjaee, J. Magn. Magn. Mater. due to the enhancement of both natural and forced 480, 112 (2019) convection. [15] M Bahiraei, M Hangi and A Rahbari, Appl. Therm. Eng. 147, 991 (2019) [16] Y Cheng and D Li, Appl. Therm. Eng. 163, 114306 6. Conclusion (2019) [17] S Nadeem, S Ahmad and N Muhammad, Pramana – J. The convective FHD problem in a porous horizontal Phys. 94(1): 1 (2020) channel was studied numerically by using the finite vol- [18] R Djeghiour, B Meziani and O Ourrad, Pramana – J. Phys. 94(1): 50 (2020) ume method. The Kelvin force, induced by the magnetic [19] M Amani, M Ameri and A Kasaeian, Int. J. Therm. Sci. field, affects the dynamic structure of the ferrofluid flow 127, 242 (2018) by creating a recirculation zone. In this zone, which is [20] M Izadi, H F Oztop, M A Sheremet, S A M Mehryan formed by a single swirl rotating in the clockwise sense, and N A Hamdeh, Heat Transf. 76(6), 479 (2019) the convective heat transfer mode is favoured. The fer- [21] F Fadaei, M Shahrokhi, A M Dehkordi and Z Abbasi, J. rofluid flow accelerates under the influence of magnetic Magn. Magn. Mater. 475, 304 (2019) field. These facts are accentuated by increasing Mn, Re [22] M Bezaatpour and M Goharkhah, J. Magn. Magn. and Da. The friction factor increases by increasing Mn, Mater. 476, 506 (2019) Re, volume fraction of nanoparticles and decreases by [23] M Izadi, R Mohebbi, A A Delouei and H Sajjadi, Int. J. Mech. Sci. 151, 154 (2019) increasing Da. Num considerably increases by increas- ing nanoparticles volume fraction and magnetic field [24] M Ghalambaz, M Sabour, S Sazgara, I Pop and R Trâmbi¸ta¸s, J. Magn. Magn. Mater. 497, 166024 (2020) intensity. 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