Arab J Sci Eng (2014) 39:8331–8340 DOI 10.1007/s13369-014-1405-z
RESEARCH ARTICLE - MECHANICAL ENGINEERING
Free Convection Boundary Layer Flow from a Vertical Flat Plate Embedded in a Darcy Porous Medium Filled with a Nanofluid: Effects of Magnetic Field and Thermal Radiation Hamza Ali Agha · Mohamed Najib Bouaziz · Salah Hanini
Received: 5 October 2013 / Accepted: 30 June 2014 / Published online: 14 October 2014 © King Fahd University of Petroleum and Minerals 2014
Abstract In this work, a boundary layer analysis is presented for a vertical flat plate embedded in a porous medium saturated with a nanofluid. A radiation heat flux is included in the energy equation, and a uniform magnetic field is applied. The model used for the nanofluid includes the effects of Brownian motion and thermophoresis, while for the porous medium the Darcy model is used. Formulation of the problem is achieved by appropriate similarity transformations. To validate the numerical results, comparison is made with those available and a fairly good agreement is obtained. A parametric study of physical parameters is conducted to display their influence on the various profiles. Other quantities of interest are computed. It can be concluded that the radiation effect due to the heat flux combined with the magnetic parameter becomes prominent through the presence of the nanoparticles and through the rise of the Lorentz force. Keywords Darcy porous medium · Nanofluid · Natural convection · MHD · Thermal radiation
H. Ali Agha (B) Department of Mechanical Engineering, Faculty of Technology, University of A. MIRA, 06000 Bejaïa, Algeria e-mail:
[email protected] H. Ali Agha · M. N. Bouaziz · S. Hanini Laboratory of Biomaterials and Transport Phenomena, University of Medea, BP 164, 26000 Medea, Algeria
Nomenclature C Solutal concentration CT Temperature difference Brownian diffusion coefficient DB Thermophoretic diffusion coefficient DT DSm Solutal diffusivity of the porous medium f Dimensionless rescaled nanoparticle volume fraction g Gravitational acceleration K Permeability of the porous medium Effective thermal conductivity of the porous medium km Le Regular Lewis number Le p Nanofluid Lewis number M Magnetic parameter Nr Buoyancy ratio parameter Nb Brownian motion parameter Nc Regular buoyancy ratio Nt Thermophoresis parameter p Pressure
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qr R Rax s T Tw T∞ u, v x, y
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Radiative heat flux Radiation parameter Local Rayleigh number Dimensionless stream function Temperature Temperature at the vertical plate Ambient temperature attained as y tends to infinity Velocity components in the x and y directions Cartesian coordinates
Greek symbols αm Thermal diffusivity of the porous medium Volumetric thermal expansion coefficient of the fluid βT Volumetric solutal expansion coefficient of the fluid βC γ Dimensionless solutal concentration ε Porosity η Similarity variable θ Dimensionless temperature μ Viscosity of the fluid Fluid density ρf Nanoparticle mass density ρp (ρc)f Heat capacity of the fluid (ρc) p Effective heat capacity of the nanoparticle material σ Electrical conductivity of the fluid Stefan–Boltzman constant σ∗ χ The mean absorption coefficient φ Nanoparticle volume fraction Nanoparticle volume fraction at the vertical plate φw Ambient nanoparticle volume fraction attained as y φ∞ tends to infinity ψ Stream function
1 Introduction The study of natural convective heat and mass transfer in porous media has been the interest of several researches owing to its wide applicability in engineering problems such as in oil recovery technology, in the use of fibrous materials for thermal insulations, in the design of aquifers as an energy storage system, and also in the resin transfer molding process, in which fiber-reinforced polymeric parts are produced in final shape. Excellent reviews of the natural convection flows in porous media have been presented by many authors [1–4]. Nanofluids refer to a liquid containing a dispersion of nanoparticles of nanometer size. The nanoparticles are different from conventional particles in that they keep suspended in the base fluid without sedimentation [5]. Because the thermal conductivity of the ordinary heat transfer fluids is not adequate to meet today’s cooling rate requirements, the use of additives defines the nanofluid. These performed fluids have
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been shown to increase the thermal conductivity and convective heat transfer performance of the base liquids [6]. Numerous applications are noticed as in convective heat transfer fluids, ferromagnetic fluids, biomedical fluids, polymer nanocomposites, gain media in random lasers and as building blocks for electronic and optoelectronic devices [7]. The investigations of boundary layer flow, heat and mass transfer over a flat plate embedded in porous media containing nanofluids are important due to its applications in industries and many manufacturing processes. Nield and Kuznetsov [8] have presented similar solutions of the Cheng– Minkowycz problem for the double-diffusive natural convective boundary layer flow in a porous medium. Their work is based on the effects of Brownian motion and thermophoresis, investigated as dominant mechanisms by Buogiorno [9]. The same authors studied thermal instability and concluded that the primary contribution of the nanoparticles is via a buoyancy effect coupled with the conservation of nanoparticles [10]. Natural convective flow of a nanofluid over a vertical plate with a constant surface heat flux is investigated by Khan and Aziz [11]. Mixed convection boundary layer flow from a vertical flat plate embedded in a porous medium filled with nanofluids is investigated by Hamad and Pop using different types of nanoparticles [12]. In the other hand, the magnetic field effects on the fluid flow are now confirmed and largely used to affect the mechanical and thus the thermal boundary layers. Hamad et al. [13] discuss similarity reductions for problems of magnetic field effects on the free convection flow of a nanofluid past a semiinfinite vertical flat plate. They demonstrate that increasing the values of the magnetic parameter leads to a decrease in the velocity profiles and to an increase in the thermal profiles for fixed values of the solid volume fraction parameter. Nanofluids with Cu and Ag nanoparticles have the highest cooling performance. Hamad [14] derive an analytical solution of natural convection flow of a nanofluid over a linearly stretching sheet in the presence of a magnetic field. The result deduced is that the increase in the magnetic parameter, the momentum boundary layer thickness decreases, while the thermal boundary layer thickness increases. As the nanoparticle volume fraction parameter increases, the heat transfer rates decrease. Similarly, for a selected value of volume fraction of nanoparticle parameter, the heat transfer rates decrease as magnetic parameter increases. Zhenga et al. [15] have considered the MHD flow and heat transfer over a porous shrinking surface with velocity slip and temperature jump. Kandasamy et al. [16] examined scaling group transformation for MHD boundary layer flow of a nanofluid past a vertical stretching surface in the presence of suction/injection. The temperature profile of the fluid increases, whereas the nanoparticle volume fraction profile decreases with the increase in the Brownian motion and thermophoretic parameter. Yazdi et al. [17] studied the MHD mixed convec-
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tion stagnation-point flow over a stretching vertical plate in porous medium filled with a nanofluid in the presence of thermal radiation. Another interesting effect is the effect of the thermal radiation. Thermal radiation effects become important when the difference between the surface and the ambient temperature is large. In this case, radiation heat transfer beside the convective heat transfer plays a very important role which cannot be neglected. Abdul-kahar et al. investigated the steady boundary layer flow of a nanofluid past a porous vertical stretching surface with variable stream conditions in the presence of chemical reaction and radiative heat flux. They pointed out the impact of chemical reaction and thermal radiation on the heat transfer [6]. Prasad et al. [18] presented the thermal radiation effects on magnetohydrodynamic-free convection heat and mass transfer from a sphere in a variable porosity regime. Increasing porosity serves to accelerate the flow but reduce temperatures and concentration values. However, increasing the radiation parameter decreases the velocity and the temperature profiles but increases the concentration profile. Radiation effect on viscous flow of a nanofluid and heat transfer over a nonlinearly stretching sheet was investigated by Hady et al. [19] using the numerical solution scheme. Hamad et al. [20] examined the radiation effects on the heat and mass transfer in MHD stagnation-point flow over a permeable flat plate with thermal convective surface boundary condition, temperature-dependent viscosity and thermal conductivity. The unsteady Hiemenz flow of Cu-nanofluid over a porous wedge in the presence of thermal stratification due to solar energy radiation using lie group transformation has been investigated by Kandasamy et al. [21]. They found that the temperature of a nanofluid is decelerated significantly as compared to that of the base fluid with an increase in convection, radiation and thermal conductivity of the nanofluid. This offers the potential of improving the radiative properties of the liquids, leading to an increase in the efficiency of direct absorption solar collectors [21]. Motivated by the above-mentioned investigations, the present paper looks at the influence of the coupled parameters and mechanisms on the heat and mass transfers. Hence, the aim of the present work is to study the combined effect of MHD, and thermal radiation on the steady-free convection heat and mass transfer, over a vertical flat plate embedded in a Darcy porous medium containing a nanofluid which includes the effects of Brownian motion and thermophoresis. Based on the model developed by Nield and Kuznetsov [22], and on the Rosseland approximation for the radiative heat flux, nonlinear ordinary differential similarity equations are obtained and solved numerically. The effects of relevant parameters, mainly the magnetic and radiation parameters, on the profiles have been shown graphically and discussed. Interesting features of the solutions as local Nusselt and Sherwood numbers are tabulated.
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2 Problem Statement Let us consider the steady MHD natural convection–radiation boundary layer flow past a vertical semi-infinite flat plate embedded in the Darcy porous medium filled with a viscous nanofluid. The physical model, direction of gravitational acceleration and coordinate system are shown in Fig. 1. It is assumed that a uniform and constant magnetic field of strength B0 is applied normally to the plate. The plate receives an incident radiation flux of intensity qr , which is absorbed by the porous medium and transferred to the nanofluid containing nanoparticles. The temperature T , the solute concentration C and the nanoparticle fraction φ at the plate surface are Tw , Cw and φw , respectively. The corresponding ambient values, attained as y tends to infinity, are denoted by T∞ , C∞ and φ∞ . One realistic assumption is that the magnetic Reynolds number is very small, and hence, the induced magnetic field is negligible, as compared to the applied magnetic field. Thus, the Lorentz force can be simplified and vary linearly with the Darcy velocity of the nanofluid in the x direction. The Oberbeck–Boussinesq approximation and boundary layer for the nanofluid [22] are assumed valid. The local thermal equilibrium in the homogeneous porous medium is also supposed, i.e., no slip occurs between the base fluid and the nanoparticles. As mentioned before, only the Brownian motion and thermophoresis effects are retained in the present configuration. Under the above of these assumptions, the boundary layer equations governing the flow, thermal and concentration field can be written in dimensional form as ∂v ∂u + =0 ∂x ∂y
(1)
x qr
Tw Cw Φ
T∞ C∞ Φ
B0
g u
v Nanofluid in porous medium y
Fig. 1 Physical model and coordinate system
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μ ∂p = − u + [(1 − φ∞ ) ρf∞ g {βT (T − T∞ ) ∂x K + βC (C − C∞ )} − ρ p − ρf∞ g (φ − φ∞ ) − σ B02 u (2) ∂p =0 ∂y ∂T ∂T +v u ∂x ∂y ε (ρc) p ∂2T = αm 2 + ∂y (ρc) f −
(3)
∂φ ∂ T + DB ∂y ∂y
DT T∞
∂T ∂y
2
1 ∂qr (ρc) f ∂ y
∂C ∂C ∂ 2C 1 u +v = Dsm 2 ε ∂x ∂y ∂y 2 ∂φ ∂φ ∂ φ 1 DT ∂ 2 T u +v = DB 2 + ε ∂x ∂y ∂y T∞ ∂ y 2
(4) (5) (6)
(7)
where, σ ∗ is the Stefan–Boltzmann constant and χ is the mean absorption coefficient. Assuming that the temperature differences within the flow are sufficiently small such T 4 may expressed as a linear func3 T −3T 4 , where the highertion of temperature, so T 4 = 4T∞ ∞ order terms of the expansion are neglected. The boundary conditions for these above equations are At y = 0 : v = 0, T = Tw , C = Cw , φ = φw
(8)
As y → ∞ : u = 0, T → T∞ , C → C∞ , φ → φ∞
(9)
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ψ y 1/2 (T − T∞ ) Rax ; s = ; ; θ= 1/2 x (Tw − T∞ ) αm Rax (C − C∞ ) (φ − φ∞ ) γ = ;f = (Cw − C∞ ) (φw − φ∞ ) η=
Here, p is the pressure, u and v, the Darcy velocity components, T the temperature, C the concentration and φ the nanoparticle volume fraction. The physical parameters are noted as K the permeability of the medium with porosity ε, ρf the density, μ the viscosity, βT the volumetric thermal expansion coefficient of the fluid, βC the analogous solutal coefficient, σ is the electrical conductivity of the fluid and g the gravitational acceleration. The density of the particles is ρ P , (ρC) P the effective heat capacity of the nanoparticle material, while (ρC)f the heat capacity of the base fluid. αm is the thermal diffusivity of the porous medium. The Brownian diffusion coefficient is denoted DB , the thermophoretic diffusion coefficient DT , and the solutal diffusivity DSm for the porous medium. It should be noted that the flow is assumed to be slow so that a Forchheimer quadratic drag term does not appear in the momentum equation. It is assumed also that the solute does not affect the transport of the nanoparticles. Under the justified state that the nanofluid is optically thick fluid, the Rosseland diffusion approximation is employed, so the radiative heat flux qr is given by 4σ ∗ ∂ T 4 qr = − 3χ ∂ y
The study of the invariance of the above system with respect to one-parameter Lie group of point transformations is useful. The infinitesimal transformations can lead to a scaling all the system, and similarity solutions corresponding to symmetries can be obtained. The case where heat transfer dominates over mass transfer and the case where the Raleigh number is large are studied by Nield and Bejan [23]. The developed scale analysis indicates that the following transformation of variables is appropriate.
(10)
where Rax =
(1 − φ∞ ) ρf∞ K gβT (Tw − T∞ ) x αm μ
(11)
ψ is the stream function defined as u = ∂ψ/∂y and v = −∂ψ/∂x,
(12)
The Darcy velocity components become αm αm 1/2 Rax s ; v = − Rax s − ηs u= x 2x Equations (1–6) with the above similarity variables can be further reduced to a set of ordinary differential equations. (13) s + Ms = θ + N cγ − Nr f
4 1 θ + N b f θ + N tθ 2 + R (θ + C T )3 θ + sθ = 0 3 2 (14) 1 (15) γ + Leγ s = 0 2 N t 1 θ + Le p s f = 0 (16) f + Nb 2 Subject to the boundary conditions (8–9) which become: s(0) = 0, θ (0) = γ (0) = f (0) = 1s (∞) = θ (∞) = γ (∞) = f (∞) = 0
(17)
where primes denote the differentiation with respect to η. In the above equations, M =
σ B02 K μ
is the magnetic para-
w −C ∞ ) meter, N c = ββcT(C (Tw −T∞ ) is the regular buoyancy ratio, Nr (ρ p −ρf∞ )(φw −φ∞ ) (1−φ∞ )ρf∞ βT (Tw −T∞ ) is the nanofluid buoyancy ratio, N b
= =
ε(ρc) p DB (φw −φ∞ ) is the Brownian motion parameter, N t = (ρc) f αm ε(ρc) p DT (Tw −T∞ ) is the thermophoresis parameter, R = (ρc) f αm T∞ 4σ ∗ (Tw −T∞ )3 ∞ is the radiation parameter and C T = (TwT−T χ km ∞) is the difference temperature, while Le = ε αDmsm is the usual Lewis number, Le p = εαDmB is the nanofluid Lewis number.
The physical quantities of interest are the local Nusselt number N u x , the regular local Sherwood number Sh x and
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Table 1 Comparison of the maximum relative error ε Le p
Cr
Cb
Ct
ε [21]
ε [23]
ε present work
1
−0.309
−0.060
−0.166
0.154
0.154014
0.154
2
−0.230
−0.129
−0.162
0.147
0.147800
0.146
5
−0.148
−0.209
−0.152
0.126
0.126384
0.126
10
−0.111
−0.245
−0.150
0.119
0.119509
0.120
20
−0.086
−0.268
−0.149
0.114
0.114009
0.114
50
−0.064
−0.288
−0.149
0.110
0.110179
0.110
100
−0.053
−0.298
−0.148
0.108
0.108142
0.107
200
−0.045
−0.304
−0.148
0.107
0.106641
0.104
500
−0.039
−0.310
−0.148
0.106
0.105992
0.105
1,000
−0.036
−0.313
−0.148
0.107
0.107415
0.105
the nanofluid Sherwood number Sh x which are defined as xqw xqw , Sh x = , k (Tw − T∞ ) Dsm (Cw − C∞ ) xqw Sh x = DB (φw − φ∞ )
N ux =
(18)
where the surface heat flux qw , the surface solutal mass flux qw and the surface particle mass flux qw are given by ∂T 4σ ∗ ∂ T 4 qw = −k − , ∂ y y=0 3χ ∂ y y=0 ∂C ∂C , and qw = −DB qw = −Dsm ∂ y y=0 ∂ y y=0 (19) using the dimensionless variables, we obtain the reduced corresponding numbers N ux 4 3 = −θ (0) 1 + R (θ (0) + C T ) , 1/2 3 Rax Sh x Sh x = −γ (0) and = − f (0) (20) 1/2 1/2 Rax Rax
3 Results and Discussion A comprehensive numerical parametric study is conducted, and the results are reported graphically and in the tables. The set of the coupled ordinary differential Eqs. (13)–(16) is highly nonlinear and cannot be solved analytically. Together with the boundary conditions (17), they form a two point boundary value problem which can be solved for some values of the governing parameters. The finite difference method that implements the 3-stage Lobatto collocation formula, and the collocation polynomial provides a continuous solution that is fourth-order accurate uniformly in the interval of integration. Mesh selection and error control are based on the residual of the continuous solution. The collocation tech-
nique uses a mesh of points to divide the interval of integration into subintervals. The flow regions are controlled by physical, thermal and solutal parameters, namely M, Nc, Nr, Nb, Nt, R, Le, Le p and Ct. Numerical computations are carried out for different values of the parameters shown in all figures. Preliminary calculations are conducted to check the numerical results. The Darcy velocity, temperature, mass and nanoparticle volume fraction profiles are presented for a reasonable range of magnetic parameter. In order to check the accuracy of the used method, we compare in Table 1 the present results to those obtained by Nield and Kuznetsov [21] and by Mahdy and Ahmed [24] through the Nusselt number. The error defined as the relative difference between the different results. The coefficients reported in this table are the estimated values of the linear regression presented in [21]. It is clear from this table that the present results are in good agreement with those reported by the cited works in a large range of Le p . Figures 2, 3 and 4 display the effects of the magnetic parameter M on the dimensionless velocity distribution. Figure 2 shows that the increase in the regular buoyancy ratio Nc leads to fluid flow accelerates near the wall in the region η < 1.5. This is due to the fact that when the buoyancy effects increase, the concentration between the plate and the external boundary layer concentration increases. This trend is nil from η becomes larger than 1.5. Then, the velocity profiles are superimposed regardless the value of Nc. It is observed that as the magnetic parameter M increases, the Darcy velocity decreases. The Lorentz force opposes the flow increases also and leads to the strong deceleration of the flow. Figure 3 exhibits an inverse behavior of the velocity of the nanofluid as the nanofluid buoyancy ratio Nr increases. The fluid becomes sufficiently dilute in terms of particles near the plate. The simultaneous increase in M and Nr imposed a more decrease in the Darcy velocity. Figure 4 presents the results relative to the Brownian motion effect on the velocity profile. As expected, the increase
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Nr = 0.2, Nb = 0.2, Nt = 0.2, Le = 10.0, R = 0.5, Ct = 0.1, Lep = 10.0
Nc = 1.0, Nr = 1.0, Nt = 1.0, Le = 10.0, R = 0.5, Ct = 0.5, Lep = 10.0
1.8
1
1.6
0.9 0.8
1.4 Nc = 0.1
Nc = 1.0
1 0.8
Nb = 0.5
0.6
Nb = 2.0
0.5 0.4
0.6
M = 0.0, 1.0, 3.0
0.3
M = 0.0, 2.0, 5.0
0.2
0.4
0.1
0.2
0
0
Nb = 0.1
0.7
Nc = 0.5
s'(η)
s'(η)
1.2
0
2
4
6
η
8
0
2
4
10
Fig. 2 Variation of velocity profiles with η, for varying N c and M
6
η
8
10
Fig. 4 Variation of velocity profiles with η, for varying N b and M Nc = 1.0, Nr = 1.0, Nb = 1.0, Le = 10.0, R = 0.5, Ct = 0.5, Lep = 10.0
Nc = 1.0, Nb = 1.0, Nt = 1.0, Le = 10.0, R = 0.5, Ct = 0.5, Lep = 10.0
1
1.8
0.9
1.6
0.8
1.4
0.7
Nr = 0.5
1
Nr = 1.0
0.5 0.4
0.8 M = 0.0, 2.0, 5.0 0.6
0.3
0.4
0.2
0.2
0.1
0
Nt = 2.0
0.6
θ (η)
s'(η )
Nr = 0.1 1.2
Nt = 0.1 Nt = 1.0
0 0
2
4
η
6
8
10
Fig. 3 Variation of velocity profiles with η, for varying Nr and M
M = 0.0, 1.0, 3.0 0
2
4
6
η
8
10
12
Fig. 5 Variation of temperature profiles with η, for varying N t and M Nc = 0.5, Nr = 0.5, Nb = 0.5, Le = 10.0, Nt = 0.5, Ct = 0.2 Lep = 10.0
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1 0.9
R = 0.0
0.8
R = 2.0
0.7
R = 4.0 0.6
θ (η)
in the Brownian motion is accompanied by an increase in the velocity, but at the medium of the boundary layer. This is temporized as the magnetic parameter is low. It is noteworthy here that the maximum values of the velocity decrease about 50 % at M = 1.0 relative to the results without a magnetic field, and the same at M = 3 relatively when M = 1.0. As the value of M increases, the velocity profile becomes independent with the porosity of the medium. We can conclude that the presence of the magnetic field and a porous medium can be seen as strong tools to regulate the flow. Figures 5, 6 and 7 illustrate the magnetic effect on the temperature profiles, respectively, for various values of Nt, R and Ct. Figure 5 shows the effect of thermophoresis parameter Nt and M on the some profiles. Here, again, it is seen that the magnetic effect obviously affects the temperature distributions as it has shown in the velocity profiles. However, the thermal boundary layer thickness increases, while the velocity of the fluid at the plane decreases dramatically and the
0.5 0.4
M = 0.0, 2.0, 5.0
0.3 0.2 0.1 0
0
2
4
6
η
8
10
12
Fig. 6 Variation of temperature profiles with η, for varying R and M
mechanical boundary layer remains the same, as M increases. This is also observed when the thermophoresis parameter increases, attributed to a relative great convective transfer.
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Nc = 1.0, Nr = 1.0, Nb = 1.0, Le = 10.0, Nt = 1.0, R = 1.0, Lep = 10.0
1
0.9
0.9
0.8 0.7
Ct = 0.1
0.8
Ct = 0.5 Ct = 0.7
0.7
Nc = 0.1
0.6
Nc = 0.5 Nc = 1.0
γ (η)
0.6
θ (η)
Nr = 0.5, Nb = 0.5, Le = 10.0, Nt = 0.5, R = 0.5, Ct = 0.5, Lep = 10.0
1
0.5 0.4
0.5 0.4
M = 0.0, 1.0, 3.0 0.3
0.3
0.2
0.2
0.1
0.1
0
0
2
4
6
η
8
10
0
12
Fig. 7 Variation of temperature profiles with η, for varying Ct and M
M = 3.0, 10.0, 20.0
0
1
2
3
4
5
η
6
7
8
Fig. 8 Variation of concentration profiles η, for varying N c and M Nr = 0.5, Nb = 0.5, Nc = 0.5, Nt = 0.5, R = 1.0, Ct = 0.5, Lep = 10.0 1 0.9 0.8
Le = 5.0 0.7
Le = 7.0
γ (η)
0.6
Le = 9.0 0.5 0.4 0.3
M = 0.0, 5.0, 15.0
0.2 0.1 0
0
1
2
3
4
η
5
6
7
8
9
Fig. 9 Variation of concentration profiles with η, for varying Le and M Le = 10.0, Nb = 0.5, Nc = 0.5, Nt = 0.5, R = 1.0, Ct = 0.5, Lep = 10.0 1 0.9
Nr = 0.1
0.8
Nr = 0.7
0.7
Nr = 1.0
0.6
f(η)
Then, the simultaneous effects of magnetic field and the thermophoresis parameter play an important role in the surface heat transfer. The greatest difference in temperature profiles occurs at some distance from the wall. Figure 6 reveals the effect of the incident heat flux on the thermal boundary layer. The plane received more heat which is transferred to porous medium which in turn to the nanofluid, causing a rise in the temperature at each point away from the surface. The conduction effects play an important role here due to the presence of the particles in addition to the base fluid. It is also noticed that the momentum boundary layer thickness decreases while the thermal boundary layer thickness increases with the increasing value of R. It can be seen that the involved effect is relatively strong compared to the preceding thermophoresis effect. For the present parameter, this is easily explained by the relation between R and the difference (Tw − T∞ ) presented an exponent of three. The same remark is extended here about the thermal boundary layer thickness which increases, as M increases. Obviously, the magnetic field exerts a retarding force on the free-convective flow which promotes the conduction and affects the reached temperature. Consequently, the simultaneous effects of magnetic field and the radiation parameter are found to relatively change greatly the thermal profile, and thus, it is anticipated that this result will be the key to improve the heat transfer. It can be concluded that the radiation effect due to the heat flux becomes prominent through the presence of the nanoparticles. More is the thermal conductivity of the particle or of the porous medium, the more is the temperature of the nanofluid. Figure 7 represents the effect of combined the magnetic parameter and temperature difference on the temperature profiles for varying the difference temperature CT . As a clear result, it can be seen that the increase in the temperature
0.5 0.4 0.3
M = 0.0, 2.0, 6.0 0.2 0.1 0
0
1
2
3
4
η
5
6
7
8
9
Fig. 10 Variation of volume fraction profiles with η, for varying Nr and M
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Nr = 0.2, Nb = 0.2, Nc = 0.2, Nt = 0.2, Le = 10.0, Ct = 0.2, Lep = 10.0 1 0.9
f(η)
0.8 0.7
R = 0.0
0.6
R = 1.0
0.5
R = 3.0
0.4 0.3
M = 0.0, 5.0, 10
0.2 0.1 0
0
2
4
η
6
8
10
Fig. 11 Variation of volume fraction profiles η, for varying R and M
difference enhances the temperature fields. Due to the large difference, the convective flux is more important, and this lead to the rise in the temperature profile. As reported in the previous analysis, the combined of M and CT acts to more heat in the nanofluid. The following Figs. 8 and 9 demonstrate the concentration profiles boundary layer for the different and combined value of the magnetic parameter M, the regular buoyancy ratio Nc and the regular Lewis number Le. From Fig. 8, it is noted that if the regular buoyancy ratio Nc increase, the dimensionless mass profiles decrease near the wall while it has a small effect far away from it. Otherwise, it is shown that the increase in the concentration is related to the increase in the magnetic parameter M. In Fig. 9, since the larger values of regular Lewis number make the molecular diffusivity smaller or the thermal diffusivity greater, it decreases the concentration field. On the other hand, the magnetic fields enhance lower and thicken the concentration boundary layer near the wall. Figures 10 and 11 display the nanoparticle volume fraction profiles for different and combined value of the mag-
1/2
1/2
Table 2 Values of N u x /Rax , Sh x /Rax 0.5, Ct = 0.1, Le p = 10.0) M
Le
1/2
for selected values of M, Le and Nb with (N c = 0.5, Nr = 0.5, N t = 0.5, R = 1/2
N u x /Rax
In this paper, we have studied the influence of magnetic fields and thermal radiation on natural convection boundary layer flow for heat and mass transfer in a porous medium saturated
Sh x /Rax
Sh x /Rax
1/2
(regular)
(nanofluid)
Nb = 0.5
Nb = 1.0
Nb = 0.1
Nb = 0.3
Nb = 0.5
Nb = 0.1
Nb = 0.5
Nb = 1.0
0.8472
1.2292
1.7111
1.6394
1.6667
1.6679
0.6567
1.3031
1.3595
1.4661
2.1283
2.9631
2.8396
2.8868
2.8889
1.1384
2.2572
2.3547
1.8926
2.7478
3.8252
3.6659
3.7269
3.7295
1.4697
2.9140
3.0400
5
1.7522
2.5343
3.5259
2.3257
2.3726
2.3763
1.3385
2.6721
2.7890
10
1.6929
2.4577
3.4214
3.2789
3.3334
3.3358
1.3145
2.6064
2.7190
20
1.6522
2.3975
3.3304
4.6061
4.6688
4.6696
1.2559
2.5213
2.6319
10
4.0 3.0
1/2
4 Conclusions
Nb = 0.1 0.0 2.0
and Sh x /Rax
netic parameter M, the buoyancy ratio parameter Nr and the radiation parameter R. Figure 10 shows the variation of the nanoparticle volume fraction distribution with Nr and M. When the magnetic and Buoyancy ratio parameter increase, the volume fraction of nanoparticle in the fluid increases and the corresponding boundary layer becomes larger. Figure 11 illustrates the substantial rise of the volume fraction of the nanoparticle distribution in the porous medium when the radiation decreases. This can be explained by the strong coupling between the mechanisms in the dynamic, thermal and mass transfer linked to the free nanoparticles. As physically comprehensive, the increase in M tends to decelerate the velocity which governed the profile of the nanoparticles. Now, the variations of local Nusselt, regular Sherwood and nanofluid Sherwood numbers are concerned related to the different values of the magnetic parameter M, Lewis number Le, Brownian motion parameter Nb, the parameter of thermal radiation R, nanofluid Lewis number Le p and thermophoresis parameter Nt. The main results are illustrated in Tables 2 and 3. It can be seen that the magnetic field, radiation flux, the Brownian motion and nanofluid Lewis number enhance the heat transfer rate in terms of Nusselt number. It can be noticed that the regular Lewis number and the thermophoresis parameter reduce the heat transfer rate. Further, there is an enhancement of the both of the local Sherwood number and nanofluid Sherwood number as the magnetic parameter, Brownian motion parameter and nanofluid Lewis number increase. However, the both of local Sherwood number and nanofluid Sherwood numbers decrease when the thermal radiation parameter and thermophoresis increase.
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Table 3 Values of N u x /Rax , Sh x /Rax 0.5, Ct = 0.1, Le = 10.0) R
Le p
0.1 0.3
10
0.5 0.5
and Sh x /Rax
1/2
1/2
for selected values of R, Le p , and Nt with (M = 3.0, N c = 0.5, Nr = 0.5, N b = 1/2
N u x /Rax
Sh x /Rax Nt = 0.1
Nt = 0.5
Sh x /Rax
(nanofluid)
Nt = 0.1
Nt = 0.5
1/2
(regular)
Nt = 0.1
Nt = 0.5
Nt = 1.0
Nt = 1.0
Nt = 1.0
0.6117
0.6103
0.6083
3.4331
3.4253
3.4151
3.4250
3.3811
3.3170
1.1219
1.1058
1.0857
3.4124
3.4010
3.3867
3.3671
3.1777
2.9491
2.5441
2.4577
2.3546
3.3583
3.3358
3.3583
3.2052
2.6064
1.9346
5
2.2338
2.1217
1.9892
3.2751
3.2483
3.2166
2.1123
1.5693
0.9844
10
2.5441
2.4577
2.3546
3.3583
3.3358
3.3085
3.2052
2.6064
1.9346
15
2.7200
2.6472
2.5596
3.4123
3.3926
3.3685
4.0545
3.4243
2.7036
by a nanofluid past a semi-infinite vertical plate, via a model in which Brownian motion and thermophoresis are taking into account. We have used the Darcy model in the momentum equation, and we have assumed the simplest possible boundary conditions. The differential partial equations are transformed into the ordinary differential equations using the local similarity solution, validated by some results showed in the literature. The quantities of interest and the various fields depends on nine dimensionless parameters, namely a regular Lewis number Le, a nanofluid Lewis number Le p , a buoyancy ratio parameter Nr, a Brownian motion parameter Nb, a regular buoyancy ratio Nc, a thermophoresis parameterNt, a magnetic parameter M, a radiation parameter R and a temperature difference CT . One can conclude with the following note: – The regular buoyancy ratio parameter Nc increases the velocity close to the wall while it has no effect far from it and reduces the volume fraction of nanoparticles. – The buoyancy ratio Nr reduces the velocity and expands the volume fraction in nanofluid. Similarly, Lewis number Le enhances the concentration distributions. – Both the thermophoresis parameter Nt and temperature difference Ct enhance the temperature fields and thicken the thermal boundary layer. – The magnetic field and radiative heat flux play an important role in modifying and regulate the flow, heat and mass transfer. On one side, the magnetic parameter decreases the velocity profiles, and conversely increases the each of the temperature, concentration and the nanoparticle profiles. On the other side, the radiation parameter augments the temperature distributions and decreases the volume fraction of nanoparticle. These parameters have a profound effect on the Nusselt number and less effect on the Sherwood numbers. – The radiation effect due to the heat flux combined with the magnetic parameter becomes prominent through the presence of the nanoparticles and through the rise of the Lorentz force.
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