Journal of Mechanical Science and Technology 28 (9) (2014) 3885~3893 www.springerlink.com/content/1738-494x
DOI 10.1007/s12206-014-0852-5
Solar radiation assisted mixed convection MHD flow of nanofluids over an inclined transparent plate embedded in a porous medium† Meisam Habibi Matin1,* and Reza Hosseini2 1
Department of Mechanical Engineering, Kermanshah University of Technology, Kermanshah, Iran 2 School of Mechanical Engineering, Amirkabir University of Technology, Tehran, Iran (Manuscript Received August 15, 2013; Revised April 7, 2014; Accepted June 14, 2014)
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Abstract The present paper investigates analytically and numerically the magneto-hydrodynamic (MHD) mixed convection flow of nanofluid over a nonlinear stretching inclined transparent plate embedded in a porous medium under the solar radiation. The two-dimensional governing equations are obtained considering the dominant effect of boundary layer and also in presence of the effects of viscous dissipation and variable magnetic field. These equations are transformed by the similarity transformation to two coupled nonlinear transformed equations and then solved using a numerical implicit method called Keller-Box. The effect of various parameters such as nanofluid volume fraction, magnetic parameter, porosity, effective extinction coefficient of porous medium, solar radiation flux, plate inclination angle, diameter of porous medium solid particles and dimensionless Eckert, Richardson and Prandtl numbers have been studied on the dimensionless temperature and velocity profiles. Also the results are presented based on Nusselt number and Skin friction coefficient. Keywords: MHD; Mixed convection; Nanofluid; Porous medium; Solar radiation; Transparent plate ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
1. Introduction In the last three decades, fluid convection in a porous medium has been one of the interesting subjects in the heat transfer field. Researches show that the presence of porous medium makes the thermal conditions much better. Furthermore, another subject in the heat transfer field which has been considerably taken into account by scientists and engineers is the use of nanofluids for the enhancement of conductive heat transfer coefficient and finally increasing the convective heat transfer rate. The convective heat transfer of fluid over an inclined plate which is embedded in a porous medium due to solar radiation has many applications such as petroleum material production, separation processes in chemical engineering, solar collectors, thermal insulation systems, buildings and nuclear reactors. Cheng and Minkowycz [1] studied the natural convection over a plate embedded in porous medium with surface temperature variation. Bejan and Polikakos [2] investigated the free convective boundary layer in porous medium for nonDarcian regime. The mixed convective flow boundary layer over a vertical plate in porous medium was analysed by *
Corresponding author. Tel.: +98 831 7264338, Fax.: +98 831 7267530 E-mail address:
[email protected] † Recommended by Associate Editor Dae Hee Lee © KSME & Springer 2014
Merkin [3]. Kim and Vafai [4] studied the natural convective flow over a vertical plate embedded in porous medium. Chamkha [5] investigated the free convective flow in porous medium with uniform porosity ratio due to solar radiation. Shamara [6] investigated the effects of Ohmic heating and viscous dissipation on steady MHD flow near a stagnation point on an isothermal stretching sheet. The MHD mixed convective flow over a vertical porous plate in porous saturated medium and assuming non-Darcian model was studied by Takhar and Beg [7]. Sunder et al. [8] studied MHD free convection-radiation interaction along a vertical surface embedded in Darcian porous medium in presence of Soret and Dufour’s effects. Ranganathan and Viskanta [9] investigated the fluid mixed convective boundary layer over a vertical plate embedded in porous medium. They claimed that the viscous effects are significant and cannot be neglected. Ghasemi and Aminossadati [10] also presented a non-similarity solution for natural convective flow over an inclined plate in porous medium due to solar radiation. Forced convection over a vertical plate in a porous medium was studied by Murthy et al. [11] with a non-Darcian model. They showed that the increase of solar radiation flux and also suction causes the increase of Nusselt number and heat transfer rate. In recent years, the nanofluid heat transfer area has been given much attention by scientists. The fluids such as water, oil and ethylene glycol mixture have a low convective heat transfer coefficient which
3886
M. H. Matin and R. Hosseini / Journal of Mechanical Science and Technology 28 (9) (2014) 3885~3893
is due to their low thermal conductivity. Therefore, the recent researches have tried to enhance the convective heat transfer coefficient by adding nano or micro particles. Chio [12] was the first who worked on this area. Chio et al. [13] showed that the addition of a few nanoparticles to the flowing fluid causes the thermal conductivity enhancement such that the addition of about one percent nanoparticles doubles the conductivity. Recently several works have investigated the nanofluid free convection such as Ghasemi et al. [14] who studied the waterCu nanofluid free convection over an inclined surface. Maiga et al. [15] investigated the nanofluid effect on forced convective heat transfer. In a paper by Wang and Mujumdar [16], the nanofluid has been reconsidered and the performed numerical analyses have been reported. In the last decade, the nanofluid convective heat transfer problem in the porous medium has been given more attention by researchers. Ahmad and Pop [17] studied the mixed convective flow boundary layer over a vertical plate embedded in a porous medium filled with a nanofluid. They presented a similarity solution for their problem. The effect of viscous dissipation and variable magnetic field on mixed convection of nanofluids along the stretching sheet was investigated by Habibi Matin et al. [18]. Nield and Kuznetsov [19] solved the Cheng-Minkowycz problem for double-diffusive free convective boundary layer in a porous medium filled with nanofluid with similarity method assuming Darcian model and uniform porosity. In the present paper, the magneto hydrodynamic (MHD) mixed convection flow of nanofluid has been studied over a non-linearly stretching inclined transparent plate embedded in a porous medium with uniform porosity due to solar radiation. The boundary layer equations have been transformed by similarity transformation to two coupled non-linear ordinary differential equations. These equations have been reduced to five first order nonlinear ordinary differential equations and then have been solved with an implicit method called Keller-Box.
2. Formulation of the problem Non-Darcian two-dimensional steady-state boundary layer mixed convection MHD flow of nanofluid has been considered over a smooth non-linearly stretching inclined transparent plate embedded in a porous medium with constant porosity due to solar radiation and assuming viscous dissipation and variable magnetic field. An incompressible nanofluid with electrical conductivity in presence of magnetic field B(x) that is perpendicular to the plate has been considered. Fig. 1 shows the schematics of the physical model and system coordinates. It is assumed that the x and y coordinates are the flow directions on the plate and perpendicular to the plate respectively. The plate temperature (Tw) is assumed constant and it is considered higher than the ambient temperature (T∞). The basic fluid composing the nanofluid has Prandtl number of one. The nanoparticles added to the basic fluid are copper particles. Assuming incompressible viscous fluid and Boussinesq ap-
proximation, the governing equations are as follows: ¶u ¶v + =0 ¶x ¶y u +
¶u ¶u m nf ¶ 2u m nf s B( x) 2 +v = u - C (e )u 2 u 2 ¶x ¶y r nf ¶y r nf K (e ) r nf g ( rb ) nf
r nf
(1)
(2)
(T - T¥ )cos( b )
u
kef ¶ 2T 1 ¶T ¶T ¶qrad +v = + ¶x ¶y ( r Cp ) nf ¶y 2 ( r Cp ) nf ¶y
+
¶u m nf ( )2 . ( r Cp ) nf ¶y
(3)
The boundary conditions are: u = uw = bx m , v = 0, T = Tw , aty = 0 u ® 0, T ® T¥ , aty ® ¥
(4)
where b is a constant. u and v are the velocity components in x and y directions respectively. σ is the electric conductivity, β is the plate inclination angle, μnf, ρnf and βnf are the effective dynamic viscosity, effective density and effective thermal expansion coefficient of nanofluid respectively, which have been considered in the following relations [20]: r nf = (1 - j ) r f + jrs
mf m nf = (1 - j ) 2.5 ( rb ) nf = (1 - j )( rb )f + j ( rb )s .
(5) (6) (7)
Here φ is the solid volume fraction, μf is the dynamic viscosity of basic fluid, βf and βs are the thermal expansion coefficient of the basic fluid and nanoparticles respectively. ε is the porosity and also ρf, ρs, (Cp)f and (Cp)s are the basic fluid density, nanoparticles density, specific heat of the basic fluid and specific heat of the nanoparticles. Also K(ε) and C(ε) are the porous medium permeability and inertia coefficient which have the following relations for uniform porosity [21]:
K (e ) = C (e ) =
d p2e 3 175(1 - e ) 2 1.75(1 - e ) . d pe 2
(8) (9)
kef is the effective thermal conductivity of porous medium and the Pr number is obtained using this effective conductivity and qrad is the solar radiation flux. ε is the porosity of porous medium which is constant assuming uniform distribution of solid components and dp is the diameter of porous medium solid particles. Assuming that some of the solar radiation energy reaching the plate surface is absorbed by the fluid, the
3887
M. H. Matin and R. Hosseini / Journal of Mechanical Science and Technology 28 (9) (2014) 3885~3893
Beer law can be used in radiation absorption and written: q¢¢( y ) = q¢¢(0).(1 - exp(- ay ))
(10)
where q¢¢( y ) is the radiation flux reached to the distance y from the transparent plate, q¢¢(0) is the incident flux to the plate and a is the extinction coefficient of nanofluid. Also here the magnetic field function has been considered as follows [22, 23]: B ( x) = B0 x m -1 .
(11)
The following similarity variables have been used to transform the governing equations to ordinary differential equations:
Fig. 1. Schematic of the physical model and coordinate system.
q ηη + ((1 - j ) + j ( h=
1 y m +1 (Re x ) 2 x 2
(12)
where:
+
)) Pr. f q η
Ec.Pr . f ηη2 (1 - j ) 2.5
R -a h .exp( e ) = 0 . Re Re
(18)
And the transformed boundary conditions become:
Re x =
r f uw ( x) x. mf
(13)
The dimensionless stream function and dimensionless temperature are: 1
f (h ) =
q (h ) =
y ( x, y )(Re x ) 2
(14)
uw ( x) T - T¥ . Tw - T¥
(15)
Such that the stream function satisfies the continuity equation:
f η (0) = 1, f (0) = 0,q (0) = 1
¶y ¶y ,v = . ¶y ¶x
The dimensionless parameters of R, ae, Mn, Dp, Rex, Pr, Ec, Gr/Rex2 are radiation parameter, extinction (absorption) parameter, magnetic parameter, porous medium geometric parameter and dimensionless Reynolds, Prandtl, Eckert and Richardson numbers respectively, which are introduced here: Re x =
-(1 - j ) 2.5 ((1 - j ) + j (
rs m + 1 2 ))( ) . ff ηη 2 rf
1.75(1 - e ) 2 175(1 - e ) 2 rs )).(m + ). f η - ( 2 2 d pe 2 d p e .Re rf
+(1 - j ) 2.5 Mn). f η + (1 - j ) 2.5 ((1 - j ) + j (
( r Cp ) s ( r Cp ) f
))(
Gr )cos( b ).q = 0 Re 2
(17)
uf
x,
uw ( x) 2 , Cp (Tw - T ¥ )
Cf = -
f ηηη + (1 - j ) 2.5 ((1 - j ) + j (
uw ( x)
R = Nur ae ,
(16)
By the use of similarity parameters and their substitution in momentum and energy equations, the governing equations become:
(19)
f η (¥) = 0,q (¥) = 0 .
Ec = u=
( r Cp ) s ( r Cp ) f
2(m + 1) f ηη (0) Re x
ae = ax ,
Mn =
Dp = 2 0
sB , rfb
Pr =
dp x ( r Cp ) f kef
uf
Gr g (Tw - T¥ ) b f = Re x 2 uw ( x) 2 x 2 Nu = -
m +1 Re x . 2
(20)
3. Numerical method Two dimensional equations of momentum and energy for a non-linear stretching inclined transparent plate have been considered. These equations include the viscous dissipation and variable (non-linear) MHD. Then, they are transformed into similarity form. From similarity transformation, two nonlinear coupled ordinary differential equations are derived. These two equations are converted into five first order ordinary differential equations. Then the system of first-order equations are solved numerically using an efficient implicit
3888
M. H. Matin and R. Hosseini / Journal of Mechanical Science and Technology 28 (9) (2014) 3885~3893
Table 1. Skin friction and Nusselt number for different values of the physical parameters. m = 1.0, Pr = 1.0, x = 0.10, Re = 500. Ec 0
0.1
1
ae 0.5
1
0
Dp 20
20
φ
Mn
0.2
0.1
0.1
20
0
0
0.2
β 0
30
90
Nur 10
10
10
Gr/Re2
ε = 0.1
ε = 0.2
ε = 0.3
Cf
Nu
Cf
Nu
Cf
Nu
0.1
0.2517
9.3400
0.1213
11.9048
0.0798
13.1458
1.0
0.2400
10.0824
0.1027
12.7836
0.0571
14.0872
10
0.1423
13.0653
0.0436
16.4664
0.1174
18.1121
0.1
0.2428
6.0508
0.1172
10.2613
0.0775
11.9652
1
0.2283
7.3768
0.0944
11.7393
0.0497
13.4611
10
0.1105
12.5287
0.0810
16.2607
0.1610
16.9225
0.1
0.2114
9.5346
0.1079
2.9784
0.0769
6.7887
1
0.2113
9.5279
0.1078
2.9851
0.0769
6.7976
10
0.2111
9.4541
0.1075
3.0590
0.0765
6.8692
Table 2. Skin friction and Nusselt number for different values of the physical parameters. m = 1.0, Pr = 1.0, x = 0.10, Re = 500. Ec
ae
φ
Mn
β
Nur
Gr/Re2
Dp Cf
Nu
Cf
Nu
Cf
Nu
0
0.5
0
0.1
0
10
0.1
10
0.2936
8.4568
0.1394
11.7260
0.0893
13.1704
15
0.2388
9.7582
0.1143
12.4124
0.0747
13.6646
0.1
0.2
1
0
0
0
0
0
30°
90°
10
10
1
10
ε = 0.1
ε = 0.2
ε = 0.3
20
0.2066
10.3060
0.0997
12.8484
0.0664
13.9575
10
0.2758
7.1151
0.1100
11.7438
0.0518
13.8055
15
0.2182
8.5552
0.0814
12.7366
0.0338
14.4785
20
0.1838
9.4965
0.0643
13.3515
0.0232
14.8743
10
0.2944
3.2065
0.1399
9.6732
0.0892
12.1485
15
0.2395
5.6751
0.1146
10.8807
0.0741
12.9423
20
0.2073
6.8893
0.0998
11.6163
0.0655
13.4030
finite-difference scheme known as the Keller-Box method. The non-linear discretized system of equations is linearized using the Newton’s method [24-26]. The system of obtained equations is solved using the block-tri-diagonal-elimination technique. A step size of Δη = 0.005 is selected to satisfy the convergence criterion of 10-4 in all cases. In this solution, h¥ = 5 is sufficient to apply the perfect effect of boundary layer.
4. Results and discussion In this study, the two-dimensional steady state boundary layer magneto hydrodynamic (MHD) mixed convection flow of nanofluid has been considered over a smooth non-linearly stretching inclined transparent plate embedded in a porous medium due to solar radiation and with viscous dissipation and variable magnetic field. The dimensionless temperature and velocity diagrams are plotted in terms of similarity variable for different values of governing parameters and in x = 0.1 and have been discussed in details. Some tables have also been presented for Nusselt number Nu and skin friction coefficient Cf . Tables 1 and 2 present the values of Nu number and skin friction coefficient for some physical parameters of the problem. The fluid and nanoparticles properties are also given in
Table 3. Thermo-physical properties of water and nanoparticles [27]. Physical properties
Fluid phase (water)
Cu
ρ (kg.m-3)
997.1
8933
Cp(J.kg-1.K-1)
4179
385
β Ï 10 (K )
21.0
1.67
k (W.m-1.K-1)
0.613
401
5
-1
Table 3. Fig. 2 shows the dimensionless velocity profile for various values of porosity and nanoparticles volume fraction. As it can be seen, the increase of porosity causes the increase of velocity in boundary layer. That is reasonable because when the porosity increases, the fluid can move more freely along the plate and fewer barriers resist the momentum transfer along the plate. In controversy, when the nanoparticles volume fraction increases, the velocity in the boundary layer decreases and this is because of the more collisions between solid particles and consequently the reduction of nanofluid effective velocity. According to the diagram, it can be claimed that the porosity of porous medium and nanoparticles volume fraction have no effect on the hydrodynamic boundary layer thickness. Fig. 3 shows the effect of radiation Nusselt number parameter on dimensionless velocity profile for various values of
M. H. Matin and R. Hosseini / Journal of Mechanical Science and Technology 28 (9) (2014) 3885~3893
3889
Fig. 2. Dimensionless velocity profiles for different values of porosity and volume fraction.
Fig. 4. Dimensionless velocity profiles for different values of volume fraction and Eckert number.
Fig. 3. Dimensionless velocity profiles for different values of porosity and Radiation Nusselt number.
Fig. 5. Dimensionless velocity profiles for different values of porosity and Richardson number.
porosity. It can be seen that the radiation flux magnitude does not have much effect on velocity profile. Fig. 4 shows the effect of Eckert number on velocity profile for some values of nanoparticles volume fraction. It can be seen that the increase of Eckert number causes the increase of velocity magnitude inside the porous medium. Furthermore, with the decrease of nanoparticles volume fraction the maximum velocity does not occur on the plate but it occurs near the plate and inside the nanofluid. The effect of Richardson number Gr/Rex2 on the velocity profile has been shown in Fig. 5 for various values of porosity. As it is expected when the Richardson number increases, the hydrodynamic boundary layer thickness increases because when the Richardson number increases, the natural convection dominates the forced convection and the buoyancy forces play more effective role than the friction forces. The interesting point which should be mentioned is that in higher values of Richardson number the medium porosity has greater effect on
velocity profile, i.e. using the porous medium is more effective in natural convection conditions than the forced convection conditions. Also it can be seen that in higher porosity the friction effect continues to further distances of the plate while in small porosity the friction effect does not go far beyond the near points of the plate. Fig. 6 shows the effect of transparent plate inclination angle on dimensionless velocity profile. It is seen that the decrease of the plate inclination angle with respect to verticality shifts the velocity profiles upper and this is because the more the plate goes to verticality, the easier the fluid can move over the plate and this is due to the increase in cos(β) term in momentum equation which causes the fluid momentum to increase. Another point is that in the angle range of 0~30 degrees with respect to horizontal state, there is the main increase in velocity profile and in the angles near to horizontal state, the variation of the angle of inclination has approximately no effect on velocity.
3890
M. H. Matin and R. Hosseini / Journal of Mechanical Science and Technology 28 (9) (2014) 3885~3893
Fig. 6. Dimensionless velocity profiles for different values of porosity and plate inclination angle.
Fig. 8. Dimensionless temperature profiles for different values of volume fraction and Eckert number.
Fig. 7. Dimensionless temperature profiles for different values of porosity and nanoparticles volume fraction.
Fig. 9. Dimensionless temperature profiles for different values of volume fraction and effective extinction coefficient.
Fig. 7 shows the effect of nanoparticles volume fraction along with porosity on dimensionless temperature profile. As it is expected the reduction of porosity causes the increase of thermal boundary layer thickness and this is because the increase of the ratio of solid particles in porous medium (decrease of porosity) resists the movement of nanofluid and this result in warmer nanofluid and thicker thermal boundary layer. By the way, nanofluid volume fraction has an enhancement effect on temperature and this is in fact the reason of using nanofluid because the existence of nanoparticles increases the effective conductivity and finally thickens the boundary layer. The dimensionless temperature profile has been plotted in Fig. 8 for various values of Eckert number and nanoparticles volume fraction. It is obvious that the increase of Eckert number causes the increase of nanofluid temperature in thermal boundary layer and physically this can be totally verified, because when the friction on the plate increases due to fluid viscosity, more heat is generated and as a result the nanofluid
temperature increases. The influence of effective absorption coefficient of porous medium containing nanofluid on temperature profile has been shown in Fig. 9. It can be observed that when the effective absorption coefficient increases, the boundary layer thickness increases and this is because when the effective absorption coefficient increases, the solar radiation flux by surroundings increases and as a result the nanofluid temperature increases. It should be noted that the presence of nanofluid and also the dark color of solid particles of porous medium can increase the effective absorption coefficient and finally the temperature of the thermal boundary layer. Fig. 10 shows the effect of Richardson number on temperature for some values of porosity. It is seen that the temperature reduces in high Richardson numbers for which the natural convective flow is dominant. Fig. 11 shows the effect of radiation Nusselt number on temperature profile. It is seen that when this dimensionless number increases, the temperature in the boundary layer in-
M. H. Matin and R. Hosseini / Journal of Mechanical Science and Technology 28 (9) (2014) 3885~3893
3891
tion. (2) In the angle range of 0~30 degrees with respect to horizontal state, there is the main increase in velocity profile and in the angles near to horizontal situation, the variation of the angle of inclination has approximately no effect on velocity. (3) The nanofluid volume fraction has an enhancement effect on temperature and this is in fact the reason of using nanofluid. (4) When the effective absorption coefficient increases, the thermal conditions improve. The presence of nanofluid and also the dark color of solid particles of porous medium can increase the effective absorption coefficient and finally the temperature of thermal boundary layer. Fig. 10. Dimensionless temperature profiles for different values of porosity and Richardson number.
Nomenclature-----------------------------------------------------------------------a ae b B(x) B0 C Cf (Cp)f dp Dp Ec f
Fig. 11. Dimensionless temperature profiles for different values of porosity and plate Nusselt number.
creases while the boundary layer thickness does not vary considerably.
5. Conclusions In the present paper, the magneto hydrodynamic (MHD) mixed convection flow of nanofluid has been studied over a non-linearly stretching inclined transparent plate embedded in a porous medium with uniform porosity under solar radiation flux. The governing equations are transformed into two nonlinear coupled ordinary differential equations using similarity transformations. These two equations are then converted into five first order ordinary differential equations. The system of first-order equations are solved using the numerical implicit Keller-Box method. The results obtained are as follows: (1) The higher values of Richardson number have greater effect on velocity profile, i.e. using the porous medium in natural convection is more effective than the forced convec-
g Gr K k m Mn Nu Nur Pr qrad R Rex T Tw T∞ ΔT u uw v X Y
: Absorption or extinction coefficient of fluid, (m-1) : Effective extinction coefficient of porous medium, (m-1) : Stretching rate, positive constant, (-) : Magnetic field, (Tesla) : Magnetic rate, positive constant, (-) : Porous medium inertia coefficient, (m-1) : Skin friction coefficient (= -2 f ηη (0) / Re x ), (-) : Specific heat at constant pressure of the basic fluid, (J kg-1 K-1) : Particle diameter, (m) : Geometric parameter of porous medium, positive, (-) : Eckert number (= uw ( x) 2 / Cp (Tw - T ¥ ) ), (-) : Dimensionless velocity variable (=y ( x, y )(Re x )0.5 / uw ( x) ), (-) : Gravitational acceleration, (m s-2 ) : Grashof number (= g (Tw - T¥ ) b / n 2 ), (-) : Porous medium permeability, (m2) : Thermal conductivity, (W m-1 K-1) : Index of power law velocity, positive consrant, (-) : Magnetic parameter (= s B02 / r ¥b ), (-) : Nusselt number (= - Re x q η (0) ), (-) : Nusselt number based on radiation heat flux, (-) : Prandtl number (= mCp / k ), (-) : Radiation flux distribution, (W m-2) : Radiation parameter, (=Nur. ae), (-) : Local Reynolds number (= r uw ( x) x / m ), (-) : Temperature variable, (K) : Given temperature at the sheet, (K) : Temperature of the fluid far away from the sheet, (K) : Sheet temperature, (K) : Velocity in x-direction, (m s-1) : Velocity of the sheet, (m s-1) : Velocity in y-direction, (m s-1) : Horizontal distance, (m) : Vertical distance, (m)
Greek symbols ψ
: Stream function, (m2 s-1)
3892
η υ β ε μ ρ σ θ
j α β ρCp
M. H. Matin and R. Hosseini / Journal of Mechanical Science and Technology 28 (9) (2014) 3885~3893
: Similarity variable (= y (Re x )0.5 / x ), (-) : Kinematic viscosity, (m2 s-1) : Thermal expansion coefficient, (K-1) : Porosity of porous medium, positive, (-) : Dynamic viscosity, (N s m-2) : Density, (kg m-3) : Electric conductivity, (mho s-1) : Dimensionless temperature variable (= T - T¥ / Tw - T¥ ), (-) : Solid volume fraction, positive, (-) : Thermal diffusivity, (m2 s-1) : Plate inclination angle, degrees, (-) : Heat capacitance, (J m-3 K-1)
Subscripts ∞ f nf s w
: Infinity : Fluid : Nanofluid : Nanoparticle : Plate surface
References [1] P. Cheng and W. J. Minkowycz, Free convection about a vertical plate embedded in a porous medium with application to heat transfer from a dike, Journal of Geophysical Research, 82 (1977) 2040-2044. [2] A. Bejan and D. Poulikakos, The non-Darcy regime for vertical boundary layer natural convection in a porous medium, International Journal of Heat and Mass Transfer, 27 (1984) 717-722. [3] J. H. Merkin, Mixed convection boundary layer flow on a vertical surface in a saturated porous medium, Journal of Engineering Mathematics, 14 (1980) 301-313. [4] S. J. Kim and K. Vafai, Analysis of natural convection about a vertical plate embedded in a porous medium, International Journal of Heat and Mass Transfer, 32 (1989) 665-677. [5] A. J. Chamkha, Solar radiation assisted natural convection in uniform porous medium supported by a vertical flat plate, ASME Journal of Heat Transfer, 119 (1997) 35-43. [6] P. R. Shamara, Effects of Ohmic heating and viscous dissipation on steady MHD flow near a stagnation point on an isothermal stretching sheet, Thermal Science, 13 (2010) 5-12. [7] H. S. Takhar and O. A. Bèg, Effects of transverse magnetic field, Prandtl number and Reynolds number on non-Darcy mixed convective flow of an incompressible viscous fluid past a porous vertical flat plate in a saturated porous medium, International Journal of Energy Research, 21 (1997) 87-100. [8] S. Sunder Tak, R. Mathur, R. Kumar Gehlot and A. Khan, MHD free convection-radiation interaction along a vertical surface embedded in Darcian porous medium in presence
of Soret and Dufour’s effects, Thermal Science, 14 (2010) 137-145. [9] P. Ranganathan and R. Viskanta, Mixed convection boundary-layer flow along a vertical surface in a porous medium, Numerical Heat Transfer, 7 (1984) 305-317. [10] A. J. Chamkha, C. Issa and K. Khanafer, Natural convection from an inclined plate embedded in a variable porosity porous medium due to solar radiation, International Journal of Thermal Sciences, 41 (2002) 73-81. [11] P. V. S. N. Murthy, S. K. Mukherjee, D. Srinivasacharya and P. V. S. S. S. R. K. Warangal, Combined radiation and mixed convection from a vertical wall with suction/injection in a non-Darcy porous medium, Acta Mechanica, 168 (2004) 145-156. [12] S. U. S. Choi, Enhancing thermal conductivity of fluids with nanoparticles, Proc. of ASME International Mechanical Engineering Congress and Exposition, San Francisco, California, USA (1995) 99-105. [13] S. U. S. Choi, Z. G. Zhang, W. Yu, F. E. Lockwood and E. A. Grulke, Anomalously thermal conductivity enhancement in nanotube suspensions, Applied Physics Letters, 79 (2001) 2252-2254 . [14] B. Ghasemi and S. M. Aminossadati, Natural convection heat transfer in an inclined enclosure filled with a waterCuO nanofluid, Numerical Heat Transfer; Part A: Applications, 55 (2009) 807-823. [15] S. E. B. Maiga, S. J. Palm, C. T. Nguyen, G. Roy and N. Galanis, Heat transfer enhancement by using nanofluids in forced convection flow, International Journal of Heat and Fluid Flow, 26 (2005) 530-546. [16] X.-Q. Wang and A. S. Mujumdar, A review on nanofluids - Part I: theoretical and numerical investigations, Brazilian Journal of Chemical Engineering, 25 (2008) 613-630. [17] S. Ahmad and I. Pop, Mixed convection boundary layer flow from a vertical flat plate embedded in a porous medium filled with nanofluids, International Communications in Heat and Mass Transfer, 37 (2010) 987-991. [18] M. Habibi Matin, M. Dehsara and A. Abbasi, Mixed convection MHD flow of nanofluid over a non-linear stretching sheet with effects of viscous dissipation and variable magnetic field, Mechanika, 18 (2012) 415-423. [19] D. A. Nield and A. V. Kuznetsov, The ChengMinkowycz problem for the double-diffusive natural convective boundary layer flow in a porous medium saturated by a nanofluid, International Journal of Heat and Mass Transfer, 54 (2011) 374-378. [20] S. M. Aminossadati and B. Ghasemi, Natural convection cooling of a localised heat source at the bottom of a nanofluid-filled enclosure, European Journal of Mechanics B/Fluids, 28 (2009) 630-640. [21] D. Poulikakos and K. Renken, Forced convection in a channel filled with porous medium, including the effects of flow inertia, variable porosity, and Brinkman friction, ASME Journal of Heat Transfer, 109 (1987) 880-888. [22] T. C. Chiam, Hydromagnetic flow over a surface with a
M. H. Matin and R. Hosseini / Journal of Mechanical Science and Technology 28 (9) (2014) 3885~3893
power law velocity, International Journal of Engineering Science, 33 (1995) 429-435. [23] S. P. Anjali Devi and M. Thiyagarajan, Steady non-linear hydromagnetic flow and heat transfer over a stretching surface with variable temperature, Heat and Mass Transfer, 42 (2006) 671-677. [24] T. Cebeci and P. Bradshaw, Momentum transfer in boundary layers, Hemisphere Publishing Corporation, New York, USA (1977). [25] T. Cebeci and P. Bradshaw, Physical and computational aspects of convective heat transfer, Springer-Verlag, New York, USA (1988). [26] M. Z. Salleh, R. Nazar and S. Ahmad, Numerical solutions of the forced boundary layer flow at a forward stagnation point, European Journal of Scientific Research, 19 (2008) 644-653. [27] F. S. Ibrahim and M. A. A. Hamad, Group method analysis of mixed convection boundary layer flow of a micropolar fluid near a stagnation point on a horizontal cylinder, Acta Mechanica, 181 (2006) 65-81.
3893
Meisam Habibi Matin is a lecturer of Mechanical Engineering at Kermanshah University of Technology. He received his M.Sc. degree in 2011 from Amirkabir University of Technology, Tehran, Iran and B.Sc. degree in 2009 from Razi University, Kermanshah, Iran. His fields of interests include MicroNanofluidics, heat transfer, and CFD. Reza Hosseini is an associate professor of Mechanical Engineering at Amirkabir University of Technology (Tehran Polytechnic). He received his Ph.D. in 1981 and M.Sc. degree in 1977 from Brunel University, London, U.K. and B.Sc. degree in 1971 from Ferdowsi University, Mashhad, Iran. His interests are heat transfer, radiation heat transfer, direct energy conversion and renewable energies.
