Transp Porous Med (2012) 91:423–435 DOI 10.1007/s11242-011-9852-4
Laminar Free Convection Over a Vertical Wavy Surface Embedded in a Porous Medium Saturated with a Nanofluid A. Mahdy · Sameh E. Ahmed
Received: 26 September 2010 / Accepted: 25 August 2011 / Published online: 16 September 2011 © Springer Science+Business Media B.V. 2011
Abstract Numerical analysis is performed to examine laminar free convective of a nanofluid along a vertical wavy surface saturated porous medium. In this pioneering study, we have considered the simplest possible boundary conditions, namely those in which both the temperature and the nanoparticle fraction are constant along the wall. Non-similar transformations are presented for the governing equations and the obtained PDE are then solved numerically employing a fourth order Runge–Kutta method with shooting technique. A detailed parametric study (nanofluid parameters) is performed to access the influence of the various physical parameters on the local Nusselt number and the local Sherwood number. The results of the problem are presented in graphical forms and discussed. Keywords
Wavy surface · Nanofluid · Porous medium · Natural convection
List of symbols Variables a Amplitude of the wavy surface D Brownian diffusion coefficient D¯ Thermophoretic diffusion coefficient F Dimensionless stream function g Acceleration due to gravity k Thermal conductivity K Modified permeability of the porous medium Le Lewis number
A. Mahdy · S. E. Ahmed (B) Mathematics Department, Faculty of Science, South Valley University, Qena, Egypt e-mail:
[email protected] A. Mahdy e-mail:
[email protected]
123
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Nb Nr Nt Nu P qw qm Ra S Sh T u, υ x, y
A. Mahdy, S. E. Ahmed
The Brownian motion parameter The buoyancy ratio The thermophoresis parameter Local Nusselt number Pressure Heat transfer rate Mass transfer rate Rayleigh number Rescaled nanoparticle volume fraction Local Sherwood number Temperature of the fluid Components of velocity of the fluid Coordinate axes
Greek symbols α Thermal diffusivity of porous medium β Volumetric coefficient of thermal expansion of fluid μ Viscosity of the fluid γ The ratio between the effective heat capacity of the nanoparticle material and heat capacity of the fluid The characteristic length of the wavy surface ρf Fluid density ρp Nanoparticle mass density (ρc)f Heat capacity of the fluid (ρc)p Effective heat capacity of nanoparticle material ψ Stream function θ Dimensionless temperature φ Nanoparticle volume fraction
Subscripts w Conditions at the wall ∞, • Conditions in the free stream
1 Introduction In recent years, some interest has been given to the study of convective transport of nanofluids. The term “nanofluid” was coined by Choi (1995) for referring to suspension of nanoparticles in base fluid. Nanofluids have attracted enormous interest from researchers due to their potential for high rate of heat exchange incurring either little or no penalty in pressure drop. The convective heat transfer characteristic of nanofluids depends on the thermo-physical properties of the base fluid and the ultra fine particles, the flow pattern and flow structure, the volume fraction of the suspended particles, the dimensions and the shape of these particles. Xuan et al. (2005) have examined the transport properties of nanofluid and have expressed that thermal dispersion, which takes place due to the random movement of particles, takes a
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major role in increasing the heat transfer rate between the fluid and the wall. This requires a thermal dispersion coefficient, which is still unknown. Brownian motion of the particles, ballistic phonon transport through the particles and nanoparticles clustering can also be the possible reason for this enhancement (Keblinski et al. 2002). Das et al. (2003) has observed that the thermal conductivity for nanofluid increases with increasing temperature. They have also observed the stability of Al2 O3 –water and Cu–water nanofluid. Experiments on heat transfer due to natural convection with nanofluid have been studied by Putra et al. (2003) and Wen and Ding (2006). They have observed that heat transfer decreases with increase in concentration of nanoparticles. The viscosity of this nanofluid increases rapidly with inclusion of nanoparticles as shear rate decreases. In addition, a comprehensive survey of convective transport in nanofluids was made by Buongiorno (2006). Moreover, nanofluids in porous medium are of significant interest to researchers because of its applications in different fields. Recently, Nield and Kuznetsov (2009) presented an analytical study for natural convection past a vertical plate in a porous medium saturated by a nanofluid. A boundary layer analysis for the natural convection past an isothermal sphere in a Darcy porous medium saturated with a nanofluid was investigated by Chamkha et al. (2010). Kuznetsov and Nield (2010) developed a theory of double-diffusive nanofluid convection in porous media to investigating the onset of nanofluid convection in a horizontal layer of a porous medium saturated by a nanofluid for the case when the base fluid of the nanofluid is it self a binary fluid such as salty water. On the other hand, the study of heat transfer near irregular surfaces is of fundamental importance because it is often found in many industrial applications. The presence of irregular surface not only alters the flow field but also alters the heat transfer characteristics. Hady et al. (2006) discussed the problem of MHD free convection flow along a vertical wavy surface with heat generation or absorption effect. Kumar and Shalini (2004) studied the non-Darcy free convection induced by a vertical wavy surface in a thermally stratified porous medium. Molla et al. (2004) examined the natural convection flow along a vertical wavy surface with uniform surface temperature in the presence of heat generation or absorption. Hossain and Rees (1999) studied the heat and mass transfer in natural convection flow along a vertical wavy surface with constant wall temperature and concentration in Newtonian fluids. Cheng (2000) presented the solutions of the heat and mass transfer in natural convection flow along a vertical wavy surface in porous medium saturated with Newtonian fluids. Mahdy (2009) investigated the effect of Soret and Dufour numbers on MHD non-Darcian free convection from a vertical wavy surface. Motivated by the investigations mentioned above, the purpose of the present work is to consider the problem of boundary-layer free convection along a vertical wavy surface in a porous medium saturated by a nanofluid. The effects of Brownian motion and the thermophoresis are included for the nanofluid.
2 Formulation and Analysis A two-dimensional steady free convection boundary layer flow over a vertical wavy plate placed in a nanofluid saturated porous medium problem is considered. The coordinate system is chosen such that measures the distance normal along the wavy surface and measures the distance normal outward as shown in Fig. 1. The wavy surface profile is given by: y = δ(x) = a sin(π x/)
(1)
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Fig. 1 Schematic diagram and coordinate system
where a is the amplitude of the wavy surface and 2 is the characteristic length of the wavy surface. At this boundary (i.e., y = δ), the fluid temperature T and the nanoparticle fraction φ take constant values Tw and φw , respectively. The ambient values, attained as y tends to infinity, of T and φ are denoted by T∞ and φ∞ , respectively. Homogeneity and local thermal equilibrium in the porous medium are assumed. With introducing Oberbeck–Boussinesq and boundary layer approximations, the equations governing the steady state conservation of mass, momentum, thermal energy and nanoparticles for nanofluids using Darcy’s flow through a homogeneous porous medium near the vertical wavy surface can be written as follows
u=−
u
K μ
∂T ∂x
u
∂u ∂υ + =0 ∂x ∂y (ρp − ρf∞ )K g ∂P (1 − φ∞ )ρf∞ β K g + (T − T∞ ) − (φ − φ∞ ) ∂x μ μ K ∂P υ=− μ ∂y 2 2 ∂φ ∂ T ∂T ∂ T ∂φ ∂ T ∂ T +γ D +υ =α + + ∂y ∂x2 ∂ y2 ∂x ∂x ∂y ∂y 2 ¯ ∂T 2 + ∂∂Ty + TD∞ ∂x ∂φ ∂φ +υ =D ∂x ∂y
∂ 2φ ∂ 2φ + 2 2 ∂x ∂y
+
2 D¯ ∂ T ∂2T + T∞ ∂ x 2 ∂ y2
(2) (3) (4)
(5) (6)
The pressure can be eliminated from Eqs. 3 and 4 and one can obtain (ρp − ρf∞ )K g ∂φ ∂v (1 − φ∞ )ρf∞ β K g ∂ T ∂u − = − ∂y ∂x μ ∂y μ ∂y
(7)
where ρf is the density of the base fluid, while ρp is the density of the particles. α, β, and μ are the effective thermal diffusivity and volumetric volume expansion coefficient of the
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nanofluid and the dynamic viscosity of the fluid. The gravitational acceleration is denoted by g. In addition, u and υ are the volume-averaged velocity components in the x- and ydirections, respectively. K is the Darcy permeability of the porous medium. The coefficients that appear in Eqs. 3 and 4 are the Brownian diffusion coefficient D and the thermophoretic ¯ γ is the ratio between the effective heat capacity of the nanoparticle diffusion coefficient D, material and heat capacity of the fluid (i.e., γ = (ρc)p /(ρc)f ). The boundary conditions are taken to be υ = 0, T = Tw , φ = φw at y = δ(x) u = v = 0, T → T∞ , φ → φ∞ at y → ∞
(8)
The previous equations may be converted to non-dimensional form by considering the following new variables T − T∞ Tw − T∞ φ − φ∞ μψ , S= ψˆ = (1 − φ∞ )ρf∞ K gβ(Tw − T∞ ) φw − φ∞ ˆ = −1 (x, y, a, δ), θ = (x, ˆ yˆ , a, ˆ δ)
(9) (10)
The stream function ψ is defined as usual (u = ∂ψ/∂ y, υ = −∂ψ/∂ x). Using above transformation then Eqs. 5–7 turn into the following form ∂ 2 ψˆ ∂S ∂ 2 ψˆ ∂θ + = − Nr 2 2 ∂ xˆ ∂ yˆ ∂ yˆ ∂ yˆ ∂ S ∂θ ∂ 2θ ∂ ψˆ ∂θ ∂ ψˆ ∂θ 1 ∂ 2θ ∂ S ∂θ + + N − = + b ∂ yˆ ∂ xˆ ∂ xˆ ∂ yˆ Ra ∂ xˆ 2 ∂ yˆ 2 ∂ xˆ ∂ xˆ ∂ yˆ ∂ yˆ ∂θ 2 ∂θ 2 +Nt ∂ xˆ + ∂ yˆ
∂ ψˆ ∂ S ∂ ψˆ ∂ S ∂2 S Nt ∂ 2 θ ∂ 2θ 1 ∂2 S Le − + + + = ∂ yˆ ∂ xˆ ∂ xˆ ∂ yˆ Ra ∂ xˆ 2 ∂ yˆ 2 Nb ∂ xˆ 2 ∂ yˆ 2
(11)
(12) (13)
where the parameters are defined by Ra = Nr = Nb = Nt = Le =
(1 − φ∞ )ρf∞ K gβ(Tw − T∞ ) is the Rayleigh number μα (ρp − ρf∞ )(φw − φ∞ ) is the buoyancy ratio ρf∞ β(Tw − T∞ )(1 − φ∞ ) (ρc)p D(φw − φ∞ ) γ D(φw − φ∞ ) = is the Brownian motion parameter α (ρc)f α ¯ w − T∞ ) ¯ w − T∞ ) (ρc)p D(T γ D(T = is the thermophoresis parameter αT∞ (ρc)f αT∞ α is the Lewis number D
The effect of the wavy surface can be transferred from the boundary conditions into the governing equations by means of the coordinate transformation given by x˜ = x, ˆ
1/2 ˆ , ψ˜ = Ra 1/2 ψˆ y˜ = ( yˆ − δ)Ra
(14)
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Applying the previous transformation into Eqs. 11–13 with Ra → ∞, we obtain the governing equations: ∂S ∂ 2 ψ˜ ∂θ − Nr = 2 ∂ y˜ ∂ y˜ ∂ y˜
2 2 ˜ ˜ ∂ θ ∂ S ∂θ ∂θ ∂ ψ ∂θ ∂ ψ ∂θ 2 + Nb − = 1 + δ˙ + Nt ∂ y˜ ∂ x˜ ∂ x˜ ∂ y˜ ∂ y˜ 2 ∂ y˜ ∂ y˜ ∂ y˜
∂2 S ∂ ψ˜ ∂ S Nt ∂ 2 θ ∂ ψ˜ ∂ S − = 1 + δ˙2 + Le ∂ y˜ ∂ x˜ ∂ x˜ ∂ y˜ ∂ y˜ 2 Nb ∂ y˜ 2 (1 + δ˙2 )
(15) (16)
(17)
Now, Eqs. 15–17 may be reduced to a form more convenient for numerical solution by considering the following transformation ξ = x, ˜ η = ξ −1/2 (1 + δ˙2 )−1 y˜ , ψ˜ = x 1/2 F(ξ, η)
(18)
Thus, we get the boundary layer equations as following F = θ − Nr S
1 2 ∂θ ∂F θ + Fθ + Nb θ S + Nt θ = ξ F −θ 2 ∂ξ ∂ξ ∂S ∂F 1 Nt − S θ = ξ Le F S + LeF S + 2 Nb ∂ξ ∂ξ
(19) (20) (21)
Subject to the boundary conditions F = 0, θ = 1, S = 1, at η = 0 F = 0, θ → 0, S → 0, at η → ∞
(22)
The results of practical interest in many applications are the heat and mass transfer coefficients. The heat and mass transfer coefficients are expressed in terms of the Nusselt and Sherwood numbers respectively, which are given by: Nu =
xqw xqm , Sh = k(Tw − T∞ ) D(Cw − C∞ )
(23)
where qw and qm are the wall heat and mass fluxes, respectively, and defined by: qw = −kn · ∇T, qm = −Dn · ∇C (24) ˙ 1 + δ˙2 , 1/ 1 + δ˙2 is the unit vector normal to the wavy plate. k is the and n = −δ/ effective thermal conductivity. Employing previous transformation, we get the local Nusselt and Sherwood numbers from the following expressions
N u Ra −1/2 = − Sh Ra
123
−1/2
=−
x 1 + δ˙2 x 1 + δ˙2
∂θ ∂η ∂S ∂η
(25) η=0
(26) η=0
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Table 1 Comparison of the maximum relative error ε Le
Cr
Cb
Ct
ε Nield and Kuznetsov (2009)
ε present
1
−0.309
−0.060
−0.166
0.154
0.154014
2
−0.230
−0.129
−0.162
0.147
0.147800
5
−0.148
−0.209
−0.152
0.126
0.126384
10
−0.111
−0.245
−0.150
0.119
0.119509
20
−0.086
−0.268
−0.149
0.114
0.114009
50
−0.064
−0.288
−0.149
0.110
0.110179
100
−0.053
−0.298
−0.148
0.108
0.108142
200
−0.045
−0.304
−0.148
0.107
0.106641
500
−0.039
−0.310
−0.148
0.106
0.105992
1000
−0.036
−0.313
−0.148
0.107
0.107415
3 Results and Discussion Equations 19–21 are nonlinear equations and it is difficult to get a closed form solution for these systems of equations. Therefore, these equations subject to the boundary conditions (22) are solved numerically by fourth order Runge–Kutta method with shooting technique. Now, in order to check the accuracy of the solution, we compare the maximum relative error defined by ε = (N u est − N u)/N u applicable for Nr , Nb , Nt each in [0.0,0.5] where 1/2 N u est /Rax = 0.444 + Cr Nr + Cb Nb + Ct Nt , here Cr , Cb , Ct are the coefficients in the linear regression estimate with those obtained by Nield and Kuznetsov (2009). Table 1 shows the maximum relative error when x = 0 for the case of flat plate for various values Lewis number Le. It is clear from Table 1 that the present results are in excellent agreement with those reported by Nield and Kuznetsov (2009). In this section, numerical results for the heat and mass transfer rates represented by local Nusselt number N u Ra −1/2 and local Sherwood number Sh Ra −1/2 for various values of amplitude wave-length ratio (0 ≤ a ≤ 0.4), Brownian motion parameter (0.1 ≤ Nb ≤ 0.5), buoyancy ratio number (0.1 ≤ Nr ≤ 0.5), thermophoresis parameter (0.1 ≤ Nt ≤ 0.5), and Lewis number(1(5.0 ≤ Le ≤ 50.0)) are presented. The results of the current problem are presented in Figs. 2, 3, 4, 5, 6, 7, 8, 9. In all the results to be reported, the range for length of the stream-wise coordinate ξ is chosen to be 0 ≤ ξ ≤ 4.0. Figures 2 and 3 show the effect of amplitude wave-length ratio a, on the local Nusselt number N u Ra −1/2 and local Sherwood number Sh Ra −1/2 , respectively. It is found that, in general, both of local Nusselt number, and local Sherwood number follow the geometry of the wavy wall by taking a wave-like behavior inside the boundary layer region. In addition, greater fluctuation of Nusselt and Sherwood numbers is obtained as the wavy amplitude increases. Typical variations in the distributions of the local Nusselt number N u Ra −1/2 and distributions of local Sherwood number Sh Ra −1/2 for different values of Brownian motion parameter Nb for flat surface a = 0 and wavy surface (a = 0.2) are plotted in Figs. 4 and 5, respectively. It can be observed that, there are two opposite behaviors are taken for local Nusselt number and local Sherwood number. These behaviors represented by decrease local Nusselt number and increase local Sherwood number by increasing the Brownian motion parameter Nb .
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A. Mahdy, S. E. Ahmed 0.6 Nb = 0.5 Nr = 0.3 Nt = 0.1 Le = 5.0
Nu/Ra
1/2
0.4
0.2 a = 0.0, 0.1, 0.2, 0.3, 0.4
0.0 0
1
2
3
4
ξ Fig. 2 Effect of the amplitude of the wavy surface on dimensionless heat transfer rates 2.1 Nb = 0.5 Nr = 0.3 Nt = 0.1 Le = 5.0
1.8
Sh/Ra
1/2
1.5 1.2 0.9 a = 0.0, 0.1, 0.2, 0.3, 0.4
0.6 0.3 0.0 0
1
2
3
4
ξ Fig. 3 Effect of the amplitude of the wavy surface on dimensionless mass transfer rates
The effect of thermophoresis parameter Nt on the local Nusselt and Sherwood numbers for flat surface a = 0 and wavy surface (a = 0.2) are depicted in Figures 6 and 7. The results show that, increasing the thermophoresis parameter Nt causes retardation of the heat and mass transfer rates represented by general decrease in the local Nusselt number and local Sherwood number. With the help of Figs. 8 and 9, the effect of buoyancy ratio number Nr on the heat and mass transfer rates can be observed. The reference case for these figures is Nb = 0.3, Nt = 0.1, and Le = 5.0. It is clear that, increasing the buoyancy ratio parameter Nr leads to decrease not only local Nusselt number but also local Sherwood number. This can be explained as:
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0.8 Nr = 0.3 Nt = 0.1 Le = 5.0
Nu/Ra
1/2
0.6
0.4 Nb = 0.1, 0.2, 0.3, 0.4, 0.5
0.2 a = 0.0 a = 0.2
0.0 0
1
2
3
4
ξ Fig. 4 Effect of Nb number on dimensionless heat transfer rates
2.1 Nr = 0.3 Nt = 0.1 Le = 5.0
1.8
Sh/Ra
1/2
1.5 1.2 0.9
Nb = 0.1, 0.2, 0.3, 0.4, 0.5
0.6 a = 0.0 a = 0.2
0.3 0.0 0
1
2
3
4
ξ Fig. 5 Effect of Nb number on dimensionless mass transfer rates
the presence of buoyancy force in the flow region gives large temperature and concentration distributions which decays the heat and mass transfer rates. Figures 10 and 11 display the effect of Lewis number Le on the concentration profiles and local Sherwood number. The Lewis number is an important parameter in heat and mass transfer processes as it characterizes the ratio of thicknesses of the thermal and concentration boundary layers. Its effect on the species concentration has similarities to the Prandtl number effect on the temperature. Therefore, as expected, it is observed that as the Lewis number increases, the concentration decreases. In addition, increasing the Lewis number tends to
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0.6
Nr = 0.3 Nb = 0.3 Le = 5.0
Nu/Ra
1/2
0.4
Nt = 0.1, 0.2, 0.3, 0.4, 0.5
0.2
a = 0.0 a = 0.2
0.0 0
1
2
3
4
ξ Fig. 6 Effect of Nt number on dimensionless heat transfer rates 2.1
Nr = 0.3 Nb = 0.3 Le = 5.0
1.8
1.5
Sh/Ra
1/2
1.2 Nt = 0.1, 0.2, 0.3, 0.4, 0.5
0.9
0.6
a = 0.0 a = 0.2
0.3
0.0 0
1
2
3
4
ξ Fig. 7 Effect of Nt number on dimensionless mass transfer rates
decrease the concentration boundary layer thickness, thus increasing the mass transfer rate between the porous medium and the surface.
4 Conclusion The problem of natural convection of a nanofluid along a vertical wavy surface saturated porous media was studied. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. The governing equations were obtained and transformed into a non-similar form. The non-similar equations were solved numerically by a
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Nt = 0.1 Nb = 0.3 Le = 5.0
0.4
Nu/Ra
1/2
0.6
Nr = 0.1, 0.2, 0.3, 0.4, 0.5
0.2 a = 0.0 a = 0.2
0.0 0
1
2
3
4
ξ Fig. 8 Effect of Nr number on dimensionless heat transfer rates 2.1
Nt = 0.1 Nb = 0.3 Le = 5.0
1.8
Sh/Ra
1/2
1.5 1.2 Nr = 0.1, 0.2, 0.3, 0.4, 0.5
0.9 0.6 a = 0.0 a = 0.2
0.3 0.0 0
1
2
3
4
ξ Fig. 9 Effect of Nr number on dimensionless mass transfer rates
fourth order Runge–Kutta method with shooting technique. From the results of the problem, the following observations were found ∗ ∗ ∗ ∗
As the amplitude wave-length ratio increases the amplitude of local Nusselt number and local Sherwood number. The heat and mass transfer rates decreases by increasing either buoyancy ratio number or thermophoresis parameter. Increasing the values of Brownian motion parameter leads to increase the local Nusselt whereas the local Sherwood number takes the opposite behavior. As the Lewis number increases, the concentration boundary layer thickness decreases, whereas the local Sherwood number increases.
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Nr Nt Nb a
0.8
= = = =
0.3 0.1 0.3 0.2
S
0.6
0.4 Le = 5.0, 10.0, 20.0, 50.0
0.2
0.0 0
2
4
η Fig. 10 Effect of Le number on dimensionless concentration profiles 7
Nr = 0.3 Nt = 0.1 Nb = 0.3
6
Sh/Ra
1/2
5
4
3
2
1
a = 0.0 a = 0.2
Le = 5.0, 10.0, 20.0, 50.0
0 0
1
2
3
4
ξ Fig. 11 Effect of Le number on dimensionless mass transfer rates
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