Mixed-Convection Flow of Nanofluids and Regular Fluids in Vertical

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Mixed-Convection Flow of Nanofluids and Regular Fluids in Vertical Porous Media with Viscous Heating Mohammad Memari, Ayub Golmakani, and Asghar Molaei Dehkordi* Department of Chemical and Petroleum Engineering, Sharif University of Technology, P.O. Box 11155-9465, Tehran, Iran ABSTRACT: In this article, the problem of combined forced and natural convection in a vertical porous channel for both regular fluids and nanofluids has been solved by perturbation and numerical methods, taking into account the influences of viscous heating and inertial force. In this regard, various types of viscous dissipation models, including the Darcy model, the power of drag force model, and the clear fluid compatible model, were considered to account for viscous heating. In addition, the mass flux of nanoparticles was also considered in terms of Brownian and thermophoresis mechanisms. The velocity and temperature distributions of both the regular fluid and nanofluid and the Nusselt number values were determined by considering isothermal boundary conditions in terms of Grashof, Reynolds, Forchheimer, Brinkman, and Darcy numbers. Moreover, the results of the numerical method were validated against those predicted by the perturbation method for small values of the Forchheimer drag term and the Brinkman number. In addition, the results obtained for both the nanofluid and the regular fluid were compared in terms of dimensionless parameters such as Brinkman number, Forchheimer drag term, pressure drop, Grashof-to-Reynolds-number ratio, Darcy number, and nanoparticle mass flux. The predicted results clearly indicate that the presence of nanoparticles enhances the Nusselt number values significantly.

1. INTRODUCTION The term ‘‘nanofluid”, coined by Choi,1 refers to a liquid containing a dispersion of nanoparticles. The important feature of nanofluids is the enhancement of thermal conductivity, a phenomenon observed by Masuda et al.2 This phenomenon suggests the possibility of using nanofluids in various fields such as advanced nuclear systems3 and nanodrug delivery.4 Nanofluids have recently found applications in the gas and oil industries. Nanoparticles in nanofluids can flow through permeable media, and these flows can improve oil recovery; thus, nanoparticles offer a means of controlling the oil recovery processes. Nanoparticles are also used for the determination of changes in fluid saturation and reservoir properties during oil and gas production. Nanofluids can be used for improving oil recovery particularly for viscous oils. To improve the recovery of viscous oils, a fluid such as water is injected into the porous medium to displace the oil, as the viscosity of water is often less than that of oil; however, increasing the viscosity of the injected fluid (e.g., by using nanofluids) would significantly increase the recovery efficiency.5 Buongiorno6 published a comprehensive article on convective transport in nanofluids. According to his survey, there are seven slip mechanisms in nanofluids, namely, inertia, Brownian diffusion, thermophoresis, diffusiophoresis, Magnus effect, fluid drainage, and gravitational settling. After examining all of these effects, Buongiorno came to the conclusion that, in the absence of turbulence, Brownian diffusion and thermophoresis are the most important effects.6 A number of investigations have been conducted based on the transport equations derived by Buongiorno. For example, Tzou7,8 investigated the Benard instability of a quiescent nanofluid between two parallel walls. Numerous studies have been conducted on the convective transport of regular fluids in porous media. For example, Vafai r 2011 American Chemical Society

and Tien9 studied boundary and inertial effects on the fluid flow and heat transfer in porous media. Umavathi et al.10,11 conducted numerical studies on mixed convection in a vertical porous channel by considering the effect of viscous dissipation. In contrast, few studies have been conducted on the convective transport of nanofluids in porous media. Nield and Kuznetsov12 analytically studied the onset of convection in a horizontal layer of a porous medium saturated with a nanofluid. They considered the effects of Brownian motion and thermophoresis in the governing equations and found that, in the presence of nanofluids, the critical thermal Rayleigh number could be reduced or increased significantly. Ahmad and Pop13 analytically studied steady-state mixed-convection boundary-layer flow past a vertical flat wall embedded in a porous medium filled with nanofluids using various types of nanofluids such as Cu, Al2O3, and TiO2. Moreover, Nield and Kuznetsov14 analytically studied the Cheng Minkowycz problem of natural convection flow past a vertical wall in a porous medium saturated with a nanofluid. They applied the Darcy model to predict the onset of nanofluid convection and presented a similarity solution. Furthermore, Kuznetsov and Nield15 developed the theory of double-diffusive nanofluid convection in porous media and then applied this theory to study the onset of nanofluid convection in the horizontal layer of a porous medium saturated with a nanofluid in which the base fluid was a binary fluid such as salty water. In the present work, a comprehensive investigation has been conducted on the combined forced and natural convection flow of regular fluids and nanofluids in a vertical porous medium by Received: February 26, 2011 Accepted: June 17, 2011 Revised: June 13, 2011 Published: June 17, 2011 9403

dx.doi.org/10.1021/ie2003895 | Ind. Eng. Chem. Res. 2011, 50, 9403–9414

Industrial & Engineering Chemistry Research

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where the first term accounts for the pressure drop in the porous medium, the second term is the viscous term, the third term is the buoyancy term, the fourth term is the Darcy term, and finally the last term accounts for the inertia force. Moreover, the momentum equation for a regular fluid in the y direction becomes ∂p ¼0 ∂y

ð3Þ

This equation shows that the pressure varies only in the x direction; hence, eq 2 can be rewritten as

dp d2 u μ F m CF 2 ffiffiffiffi u ¼ 0 + μeff 2 + Fm gkm ðT T0 Þ m u p dx dy K K

If one assumes that dp/dx is constant (i.e., dp/dx = a),10,11 then taking the derivative of eq 4 with respect to x gives

Figure 1. Schematic diagram of the porous medium.

considering the inertial force and viscous heating, and the results obtained for regular fluids and nanofluids were then compared. Toward this end, the temperature and velocity profiles of nanofluids and regular fluids in a vertical porous medium were obtained by both perturbation and numerical methods.

2. MATHEMATICAL MODEL We concentrate our attention on combined forced and natural convection in a porous medium taking into account the effects of inertia and viscous dissipation for two cases: a regular fluid and a nanofluid. The temperature and velocity profiles and Nusselt number values were obtained for both cases and compared. A schematic diagram of the problem is shown in Figure 1. As shown in this figure, there are two vertical parallel walls at y = H and y = +H. The x axis was chosen as the direction of fluid flow against the gravitational field in the middle part of the porous medium, whereas the y axis was transverse to the walls. The temperatures of the left and right walls were set to T1 and T2, respectively, with T2 > T1. The physical properties (e.g., thermal conductivity, dynamic viscosity, and thermal expansion coefficient) of the fluid were assumed to be constant for both the regular fluid and nanofluid, whereas the OberbeckBoussinesq approximation6 was used for the fluid density. Furthermore, the physical properties of nanofluids reported by Buongiorno were used throughout the present work.6 Finally, for both cases, the fluid flow was considered to be fully developed, Newtonian, and at the steady state.

dT ¼0 dx

in which the only nonzero component of the velocity field is its longitudinal component, u (i.e., x component of velocity field). Under these conditions, the continuity equation becomes

keff

d2 T + Φv ¼ 0 dy2

ð6Þ

with

2 μm 2 d2 u du Fm CF 3 ffiffiffiffi u u C1 μeff u 2 + C2 μm + p Φv ¼ dy dy K K

ð7Þ

where Φv is the viscous heating of a regular fluid in the porous medium. Three models proposed for viscous heating in porous media are as follows:17 (1) For the “clear fluid compatible” model, C1 = 0 and C2 = 1. (2) For the “power of drag force” model, C1 = 1 and C2 = 0. (3) For the “Darcy” model, C1 = 0 and C2 = 0. By taking the second derivative of T with respect to y in eq 4 and substituting the result into the thermal energy equation (i.e., eq 6), we obtain " 2 # d4 u μ m d2 u 2Fm CF d2 u du ¼ + pffiffiffiffi u 2+ 4 2 dy dy μeff K dy K μeff dy μm Fm gkm 2 F gkm d2 u u C1 m u 2 dy keff μeff K keff 2 μ F gkm du F 2 CF gkm 3 + m m C2 + pmffiffiffiffi u ð8Þ dy keff μeff K keff μeff +

subject to the following boundary conditions:

ð1Þ

This equation shows that u is a function of y only. Note that, if one considers the Darcy velocity, uD, then the fluid velocity in the porous medium (u) is given by u = uD/ε. The momentum equation for the regular fluid in the x direction can be expressed as16

ð5Þ

Equation 5 clearly shows that T is a function of y alone as well. Considering eq 5, the thermal energy equation for the regular fluid in the porous medium is given by

2.1. Mixed Convection of a Regular Fluid in a Porous Medium. Consider a steady-state fluid flow in a porous medium

∂u ¼0 ∂x

ð4Þ

at y ¼ H,

u ¼ 0, T ¼ T1

ð9Þ

at y ¼ + H,

u ¼ 0, T ¼ T2

ð10Þ

Using eq 4 as well, we obtain

∂p ∂2 u μ F m CF 2 ffiffiffiffi u ¼ 0 + μeff 2 + Fm gkm ðT T0 Þ m u p ∂x ∂y K K ð2Þ 9404

at y ¼ H,

d2 u a Fm gkm ðT1 T0 Þ ¼ dy2 μeff

ð11Þ

at y ¼ + H,

d2 u a Fm gkm ðT2 T0 Þ ¼ dy2 μeff

ð12Þ

dx.doi.org/10.1021/ie2003895 |Ind. Eng. Chem. Res. 2011, 50, 9403–9414


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To transform these equations into dimensionless forms, we introduced the following dimensionless parameters: μeff y u T T0 , M ¼ , Y ¼ , U ¼ , θ¼ H um T2 T1 μm ðFCp Þp H Da υ Da ¼ pffiffiffiffi, σ ¼ pffiffiffiffiffi, γ ¼ , Pr ¼ , R ðFCp Þf K M R εγ F m CF u m H Le ¼ , β¼ , F ¼ , CBT H Le μm gkm ðT2 T1 ÞH 3 Fm u m H μm um 2 Gr ¼ , , Re ¼ , Br ¼ μm υ2 keff ðT2 T1 Þ τF ¼

2F Da, M

τBr ¼ GR Br,

GR ¼

Gr , Re

A¼

H2 a μeff um

nanofluids and regular fluids in the porous medium are the same. The continuity equation for nanoparticles in the absence of a chemical reaction can be expressed as12 ∂φ ∇T + Un 3 ∇φ ¼ ∇ 3 DB ∇φ + DT ð21Þ ∂t T where ϕ is the nanoparticle volume fraction; DB and DT are the Brownian and thermophoresis coefficients, respectively; and Un is the nanofluid velocity. According to Buongiorno,6 the nanoparticle volume fraction (ϕ) is strongly dependent on T, and because the fluid temperature in the present work varies only in the y direction, ϕ varies only in the y direction as well. Thus, eq 21 reduces to ∂ ∂φ DT ∂T DB + ¼0 ð22Þ ∂y ∂y T ∂y

ð13Þ Note that most of these dimensionless parameters, such as Re, Pr, Gr, Br, Da, and Le, are well-known, but the new dimensionless parameters introduced in this article, such as F, β, τF, and τBr, are defined in the Nomenclature section. Using these dimensionless parameters, eq 8 becomes " 2 # 2 2 d4 U d U d U dU σ 2 2 ¼ τF U 2 + dY 4 dY dY dY " # d2 U C2 dU 2 τF 3 2 2 + U + τBr σ U C1 U 2 + ð14Þ dY M dY 2 The velocity distribution can be obtained by integrating eq 14 subject to the appropriate boundary conditions (i.e., eqs 912). Moreover, the x component of the momentum equation (i.e., eq 4) in dimensionless form becomes d2 U GR τF θ σ2 U U 2 ¼ 0 A+ dY 2 M 2

ð15Þ

The temperature distribution of a regular fluid flow in the porous medium can be determined using eq 15 whenever the velocity distribution can be obtained as ! M d2 U τF 2 2 2 +A+σ U + U θ¼ ð16Þ GR dY 2

DB

1 2

ð17Þ

1 2

ð18Þ

at Y ¼ 1,

d2 U GR ¼ A+ dY 2 2M

ð19Þ

at Y ¼ + 1,

d2 U GR ¼ A dY 2 2M

ð20Þ

U ¼ 0, θ ¼

at Y ¼ + 1,

U ¼ 0, θ ¼ +

2.2. Mixed Convection of Nanofluids in a Porous Medium. Note that the continuity and momentum equations for both

¼ CBT

ð23Þ

where CBT is a constant. Moreover, the equations of motion for the nanofluid in the porous medium are the same as those for regular fluids (eqs 3 and 4). However, the physical properties of the nanofluid, such as density, dynamic viscosity, and thermal conductivity, can be estimated on the basis of nanoparticle volume fraction (ϕ).6 The thermal energy equation for a nanofluid in the porous medium taking into account the inertial force and viscous term can be expressed as ∂T + Un 3 ∇T ¼ ∇ 3 ðkeff ∇TÞ FCp ∂t ∇T 3 ∇T ð24Þ + εðFCp Þp DB ∇φ 3 ∇T + DT + Φv T Considering the assumptions made in the present work, eq 24 reduces to d2 T dφ DT dT dT + + Φv ¼ 0 ð25Þ keff 2 + εðFCp Þp DB dy dy T dy dy Equation 25 is the same as the thermal energy equation for a regular fluid. However, the second term accounts for the effect of the nanoparticles, whereas Φv is the viscous heating due to viscous dissipation given by eq 7. Substituting eq 23 into eq 25 yields

In addition, taking the mean temperature of the right and left-hand walls as the reference temperature, T0 = (T1 + T2)/2, the dimensionless boundary conditions (i.e., eqs 912) can be expressed as at Y ¼ 1,

∂φ DT ∂T + ∂y T ∂y

keff

d2 T dT + Φv ¼ 0 + εðFCp Þp CBT dy2 dy

ð26Þ

Next, substituting d2T/dy2 and dT/dy from the momentum equation into eq 26 gives " 2 # d4 u μm d2 u 2Fm CF d2 u du pffiffiffiffi u 2+ dy4 μeff K dy2 dy dy K μeff εðFCp Þp CBT μm du εðFCp Þp CBT 2Fm CF du pffiffiffiffi u dy keff μeff K keff μeff dy K εðFCp Þp CBT d3 u μm Fm gkm 2 Fm gkm d2 u u + + C u 2 1 dy3 dy keff keff μeff K keff 2 μ F gkm du F 2 CF gkm 3 m m C2 pmffiffiffiffi u ¼0 ð27Þ dy keff μeff K keff μeff 9405



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Equation 27 can be transformed into the following dimensionless form by using the dimensionless parameters introduced earlier

UðY Þ ¼ U0 ðY Þ + τF U1 ðY Þ + τF 2 U1 ðY Þ + :::

2 d 4 Un d 3 Un dUn 2 d Un + β σ βσ2 dY 4 dY 3 dY 2 dY " # d 2 Un dUn 2 dUn + + βU ¼ τ F Un dY 2 dY dY

" + τBr σ 2 Un 2 C1 Un

d2 Un C2 dUn 2 τF 3 + + Un dY 2 M dY 2

solution to these equations for small values of τF can be expressed as

¼

# ð28Þ

2 d4 Ur0 2 d Ur0 σ ¼0 dY 4 dY 2

" # 2 d4 Ur1 d2 Ur0 dUr0 2 2 d Ur1 σ ¼ Ur0 + dY 4 dY 2 dY 2 dY

3. PERTURBATION SOLUTION 3.1. Case I. Negligible Forchheimer and Brinkman Terms (i.e., τF = 0 and τBr = 0). In the case where both the inertial force

Ur ¼ a1r + a2r Y + a3r coshðσY Þ + a4r sinhðσY Þ

ð29Þ

Then, the temperature distribution of the regular fluid can be determined using eq 16 and substituting the obtained velocity distribution (i.e., eq 29). Moreover, the velocity distribution of the nanofluid can be determined by integrating eq 28 with both τF and τBr set equal to 0. Note that the dimensionless boundary conditions for both the nanofluid and regular fluid flow are the same (i.e., eqs 1720). Thus, the velocity distribution for a nanofluid is Un ¼ a1n + a2n eσY + a3n e+σY + a4n eβY

2 d4 Un0 d3 Un0 dUn0 2 d Un0 ¼0 + β σ βσ2 4 3 2 dY dY dY dY

d4 Un1 d3 Un1 d2 Un1 dUn1 +β σ2 βσ 2 4 3 dY dY dY 2 dY " # 2 2 d Un0 dUn0 dUn0 + + βUn0 ¼ Un0 dY 2 dY dY

at Y ¼ 1,

ð35Þ

ð36Þ

ð37Þ

Ur0 ¼ Ur1 ¼ 0, Un0 ¼ Un1 ¼ 0, θr ¼ θn ¼

1 2

ð38Þ at Y ¼ 1,

Ur0 ¼ Ur1 ¼ 0, Un0 ¼ Un1 ¼ 0, θr ¼ θn ¼ +

ð30Þ

1 2

ð39Þ d2 Ur0 GR d2 Ur1 , ¼ A+ ¼ 0, 2 2M dY 2 dY d2 Un0 GR d2 Un1 , ¼ A + ¼0 2M dY 2 dY 2

at Y ¼ 1,

d2 Ur0 GR d2 Ur1 , ¼ A ¼ 0, 2M dY 2 dY 2 d2 Un0 GR d2 Un1 , ¼ A ¼0 2M dY 2 dY 2

ð40Þ

at Y ¼ + 1,

ð31Þ

ð41Þ

Solving eqs 34 and 35 for n = 0 and 1 yields Ur ðY Þ ¼ Ur0 ðY Þ + τF Ur1 ðY Þ

d4 U n d3 U n d 2 Un dUn +β σ2 βσ 2 4 3 dY dY dY 2 dY " # 2 2 d Un dUn dUn ¼ τ F Un + + βUn dY 2 dY dY

ð34Þ

subject to the boundary conditions

where air (i = 14) and ain (i = 14) are constants of integration that can be adjusted using the appropriate boundary conditions. The temperature distribution of a nanofluid in the porous medium can be obtained from eq 16 by substituting the velocity distribution (i.e., eq 30). 3.2. Case II. Negligible Brinkman Term Only (i.e., τF 6¼ 0 and τBr = 0). In this case, only the effect of inertia is taken into account, and the viscous heating is neglected. The governing equations for both the regular fluid and nanofluid are " 2 # 2 d4 U r d2 U r dUr 2 d Ur σ ¼ τ F Ur + dY 4 dY 2 dY 2 dY

ð33Þ

Upon substitution of eq 33 for n = 0 and 1 (i.e., including only the first two terms of the expansion on the right-hand side) into eqs 14 and 28, the momentum equations for the regular fluid and nanofluid become

Note that eq 28 reduces to eq 14 (i.e., the equation for a regular fluid) for β = 0.

and viscous heating are negligible, the velocity distribution of a regular fluid in the porous medium (Ur) can be determined analytically by integrating eq 14 subject to the dimensionless boundary conditions given by eqs 1720 with both τF and τBr set equal to 0. The velocity distribution for this case becomes

∞

τF n Un ðY Þ ∑ n¼0

Ur0 ¼ d1 + d2 Y + d3 coshðσY Þ + d4 sinhðσY Þ ð32Þ

ð42Þ ð43Þ

Ur1 ¼ e1 + e2 Y + e3 Y 2 + e4 coshðσY Þ + e5 sinhðσY Þ + e6 Y coshðσY Þ + e7 Y sinhðσY Þ + e8 Y 2 coshðσY Þ + e9 Y 2 sinhðσY Þ + e10 sinhð2σY Þ + e11 coshð2σY Þ

For small values of τF, these equations can be approximately solved by a regular perturbation method. The approximate 9406

ð44Þ



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2 d4 Vr1 d2 Ur0 C2 dUr0 2 τF 3 2 2 d Vr1 2 σ ¼ σ U C U + + Ur0 r0 1 r0 dY 4 dY 2 dY 2 M dY 2

Ur ¼ d1 + d2 Y + d3 coshðσY Þ + d4 sinhðσY Þ + τF ½e1 + e2 Y + e3 Y 2 + e4 coshðσY Þ + e5 sinhðσY Þ + e6 Y coshðσY Þ + e7 Y sinhðσY Þ + e8 Y 2 coshðσY Þ + e9 Y 2 sinhðσY Þ + e10 sinhð2σY Þ + e11 coshð2σY Þ

ð54Þ ð45Þ

Furthermore, solving eqs 36 and 37 yields Un0 ¼ f1 + f2 eσY + f3 e+σY + f4 eβY

ð46Þ

Un1 ¼ g1 + g2 eσY + g3 e+σY + g4 eβY + g5 Y e+σY + g6 Y eσY + g7 e+2σY + g8 e2σY + g9 e2βY + g10 eð β + σÞY + g11 eðβ + σÞY ð47Þ Un ¼ f1 + f2 eσY + f3 e+σY + f4 eβY + τF ½g1 + g2 eσY + g3 e+σY + g4 eβY + g5 Y e+σY + g6 Y eσY + g7 e+2σY + g8 e2σY + g9 e2βY + g10 eð β + σÞY + g11 eðβ + σÞY

ð48Þ

Once again, the temperature distributions for both the regular fluid and nanofluid can be obtained by using eq 16 and substituting the obtained velocity distributions. 3.3. Case III. Non-Negligible Forchheimer and Brinkman Terms (i.e., τF 6¼ 0 and τBr 6¼ 0). In this case, the inertial force and viscous heating are both taken into account. The governing equations for the regular fluid and nanofluid, respectively, are " 2 # 2 2 d 4 Ur d U d U dUr r r σ2 ¼ τ F Ur + dY 4 dY 2 dY 2 dY " # d2 Ur C2 dUr 2 τF 3 2 2 + τBr σ Ur C1 Ur + + U ð49Þ dY 2 M dY 2 2 d4 U n d3 U n dUn 2 d Un + β σ βσ 2 4 3 2 dY dY dY dY " # 2 2 d Un dUn dUn ¼ τ F Un + + βUn dY 2 dY dY " # d2 Un C2 dUn 2 τF 3 2 2 + τBr σ Un C1 Un + + U dY 2 M dY 2

∞

# d Ur1 d Ur0 dUr0 2 2 d Ur1 σ ¼ Ur0 + dY 4 dY 2 dY 2 dY 4

2

"

2

d4 Vn1 d3 Vn1 d2 Vn1 dVn1 +β σ2 βσ2 4 3 dY dY dY 2 dY " # 2 2 d U C dU τ n0 2 n0 F + + Un0 3 ¼ σ 2 Un0 2 C1 Un0 dY 2 M dY 2

ð56Þ

ð57Þ

For both the regular fluid and nanofluid, the boundary conditions are at Y ¼ 1,

Ur0 ¼ Ur1 ¼ Vr1 ¼ 0, Un0 ¼ Un1 ¼ Vn1 ¼ 0, 1 θr ¼ θn ¼ ð58Þ 2

at Y ¼ + 1,

Ur0 ¼ Ur1 ¼ Vr1 ¼ 0, Un0 ¼ Un1 ¼ Vn1 ¼ 0, 1 θr ¼ θn ¼ + ð59Þ 2

at Y ¼ + 1,

ð50Þ

ð51Þ

By substituting eq 51 into eq 14 for n = 0 and 1, the momentum equation for the regular fluid becomes d4 Ur0 d2 Ur0 σ2 ¼0 4 dY dY 2

2 d4 Un1 d3 Un1 dUn1 2 d Un1 + β σ βσ2 4 3 2 dY dY dY dY " # d2 Un0 dUn0 2 dUn0 + + βUn0 ¼ Un0 dY 2 dY dY

ð55Þ

d2 Ur0 GR d2 Ur1 d2 Vr1 , ¼ A + ¼ 0, ¼ 0, 2M dY 2 dY 2 dY 2 d2 Un0 GR d2 Un1 d2 Vn1 , ¼ A + ¼ 0, ¼0 ð60Þ 2M dY 2 dY 2 dY 2

UðYÞ ¼ U0 ðY Þ + τF U1 ðY Þ + τBr V1 ðY Þ + τF 2 U2 ðY Þ + τBr 2 V1 2 ðY Þ + :::

∑ ½τF n UnðYÞ + τBrnVnðY Þ n¼1

2 d4 Un0 d3 Un0 dUn0 2 d Un0 ¼0 + β σ βσ 2 dY 4 dY 3 dY 2 dY

at Y ¼ 1,

For small values of τF and τBr, eqs 49 and 50 can be approximately solved by a regular perturbation method. The approximate solution to these equations for small values of τF and τBr is given by

¼ U0 ðYÞ +

Moreover, by substituting eq 51 into eq 28 for n = 0 and 1, the velocity distribution for the nanofluid can be determined using the equations

ð52Þ

d2 Un0 dY 2

d2 Ur0 GR d2 Ur1 d2 Vr1 , ¼ A ¼ 0, ¼ 0, 2M dY 2 dY 2 dY 2 GR d2 Un1 d2 Vn1 , ¼ A ¼ 0, ¼0 ð61Þ 2 2M dY dY 2

The velocity distribution of the regular fluid for n = 0 and 1 becomes Ur ðY Þ ¼ Ur0 ðY Þ + τF Ur1 ðY Þ + τBr Vr1 ðY Þ

ð62Þ

Ur0 ¼ m1 + m2 Y + m3 coshðσY Þ + m4 sinhðσY Þ

ð63Þ

with

Ur1 ¼ n1 + n2 Y + ðn3 + n4 C1 + n5 C2 ÞY 2 + n6 Y 3 + n7 Y 4 + ðn8 + n9 C1 + n10 C2 Þ coshðσY Þ + ðn11 + n12 C1 + n13 C2 Þ sinhðσY Þ + ðn14 + n15 C1 + n16 C2 ÞY coshðσY Þ + ðn17 + n18 C1 + n19 C2 ÞY sinhðσY Þ + ðn20 + n21 C1 ÞY 2 coshðσY Þ + ðn22 + n23 C1 ÞY 2 sinhðσY Þ

ð53Þ

+ ðn24 + n25 C1 + n26 C2 Þ sinhðσ2 YÞ + ðn27 + n28 C1 + n29 C2 Þ coshðσ2 Y Þ + n30 eσY + n31 eσY 9407

ð64Þ



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Figure 2. Dimensionless temperature and velocity distributions in the porous medium. β = 0.001, σ = 3160, GR = 13, A = 1 107, τF = 280, τBr = 3 108.

Vr1 ¼ k1 + k2 Y + k3 Y 2 + k3 Y 3 + k5 coshðσY Þ + k6 sinhðσY Þ + k7 Y coshðσY Þ + k8 Y sinhðσY Þ + k9 Y 2 coshðσY Þ + k10 Y 2 sinhðσY Þ + k11 sinhð2σY Þ + k12 coshð2σY Þ + k13 Y coshð2σY Þ + k14 Y sinhð2σY Þ + k15 coshð3σY Þ + k16 sinhð3σY Þ

ð65Þ

Similarly, the velocity distribution of nanofluid for n = 0 and 1 can be expressed as Un0 ¼ p1 + p2 eσY + p3 e+σY + p4 eβY

ð66Þ

with Un1 ¼ q1 + q2 e+σY + q3 eσY + q4 eβY + q5 Y e+σY + q6 Y eσY + q7 e+2σY + q8 e2σY + q9 e2βY + q10 eð β + σÞY + q11 eðβ + σÞY ð67Þ Vn1 ¼ F1 + F2 e+σY + F3 eσY + F4 eβY + F5 Y eσY + F6 Y eσY + F7 e+2σY + F8 e2σY + F9 e2βY + F10 eð β + σÞY + F11 eð β σÞY + F12 Y + F13 Y eβY + F14 eð β + 2σÞY + F15 eð β 2σÞY + F16 eð 2β σÞY + F17 eð 2β + σÞY + F18 e3βY + F19 e3σY + F20 e+3σY

ð68Þ where Un ¼ Un0 + τF Un1 + τBr Vn1

ð69Þ

Finally, the temperature distributions of both the regular fluid and nanofluid can be evaluated by eq 16.

4. PARAMETERS The permeability of gas and oil reservoirs typically varies over the range of 10131011 m2. The values of the dimensionless parameters in a real reservoir for alumina nanoparticles in water at a volume fraction of ϕ = 5%, a temperature gradient of 1 (ΔT/H = 1 °C/m), and a porosity of 0.4 are β ¼ 0:001, σ ¼ 3160, GR ¼ 13, A ¼ 107 , τF ¼ 280, τBr ¼ 3 108 Considering these values, the dimensionless temperature and velocity distributions in the porous medium could be similar to those shown in Figure 2. As can be observed from this figure, the temperature distribution is almost linear in the medium, and the fluid velocity is almost the same at various points of the medium, with most of the variations occurring near the walls. In addition, our preliminary calculations showed that the effect of free convection is significant for GR/A > 3. In the present work, to investigate the effects of free convection, a porous medium with a porosity of K = 106 and a temperature gradient of ΔT/H = 45 °C/m was considered. Other dimensionless variables were as follows: β = 1.5, σ = 10, τF = 500, τBr = 107, GR = 300, and A = 100. Note that the effect of viscous dissipation cannot be studied well using these parameter values, because τBr is quite small; therefore, we had to increase this value to 1 (i.e.,τBr = 1) to study the effect of viscous dissipation at various conditions numerically.

5. NUMERICAL SOLUTION Note that the perturbation method applied for the determination of the velocity and temperature distributions can predict the 9408



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Figure 3. Dimensionless temperature and velocity distributions for regular fluids (β = 0) and nanofluids (β = 3). σ = 10, GR = 300, and A = 100.

appropriate results when the inertial force (i.e., τF) and the viscous dissipation term (i.e., τBr) are negligible (i.e., τBr, τF , 1). Therefore, a numerical method should be utilized to evaluate the velocity and temperature distributions for both the regular fluid and nanofluid in the porous medium when these forces are not negligible. In this regard, the momentum and thermal energy equations can be rewritten in dimensionless forms as d2 U GR τF θ σ2 U U 2 ¼ 0 A+ 2 dY M 2

ð70Þ

" # dθ τBr d2 U C2 dU 2 τF 3 d2 θ 2 2 β + U + 2 ¼0 M σ U C1 U 2 + + M dY 2 dY GR dY dY

ð71Þ where eq 70 is the dimensionless form of the momentum equation for both the regular fluid and nanofluid whereas eq 71 is the dimensionless form of thermal energy equation for the nanofluid. It is interesting to note that, by substituting β = 0 into eq 71, this equation reduces to the thermal energy equation for the regular fluid; therefore, the corresponding term, β(dθ/dY), accounts for the effect of the nanoparticles. In the present work, eqs 70 and 71 were solved numerically by the fourth-order RungeKutta method, and central difference schemes were used. Figure 3 shows the predictions of the perturbation and numerical methods for both the regular fluid and nanofluid when the inertial force and viscous dissipation are almost negligible (i.e., τF = τBr = 0.005). The values of the other dimensionless parameters for the regular fluid and nanofluid were as follows: β = 0 for the regular fluid, β = 3 for the nanofluid, σ = 10, GR = 300, and A = 100. The symbols shown in Figure 3 denote the predictions of the numerical method, whereas the solid lines show the predictions of the perturbation method. As can be observed from Figure 3, the predictions of the numerical method

are in good agreement with those of the perturbation method for both the regular fluid and the nanofluid.

6. RESULTS AND DISCUSSION In the following subsections, the effects of all of the dimensionless parameters on the velocity and temperature distributions and Nusselt number values for both the regular fluid and nanofluid are presented and discussed. 6.1. Effects of Nanoparticle Mass Flux. As mentioned earlier, the effect of nanoparticle mass flux (CBT) can be expressed by dimensionless parameter β, such that, for the regular fluid, the value of β is 0. In addition, eq 13 shows that the nanoparticle mass flux (CBT) is proportional to β, which means that, with an increase in the nanoparticle mass flux (CBT), the value of β increases. Moreover, it was also assumed that local thermal existed equilibrium between the nanoparticles, base fluid, and porous matrix. Furthermore, more nanoparticles move from the hot wall to the cold wall with increasing nanoparticle mass flux (CBT), which increases the nanofluid temperature at various points. Figure 4a shows the velocity and temperature distributions as versus the dimensionless width (Y) as a function of nanoparticle mass flux using the three models of viscous dissipation. As can be observed from this figure, the temperature and velocity profiles are approximately the same for the three models of viscous heating. Moreover, at large values of the nanoparticle mass flux (i.e., β > 1), the temperature of the nanofluid at various points approaches the hot wall temperature. Figure 4a also shows that, for small values of nanoparticle mass flux (β), particularly in the case of a regular fluid (β = 0), flow reversal occurs near the cold wall. This means that free convection plays an important role for small values of nanoparticle mass flux. However, with increasing nanoparticle mass flux, the temperature of the nanofluid at various points, especially near the cold wall, increases, which, in turn, decreases the density of the fluid near the cold 9409



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Figure 4. Effects of nanoparticle mass flux (β) on the (a) dimensionless temperature and velocity distributions and (b) Nusselt number values for both the regular fluid and nanofluid. σ = 10, τF = 500, τBr = 1, GR = 300, and A = 100.

Figure 5. Effects of viscous dissipation (τBr) on the (a) dimensionless temperature and velocity distributions and (b) Nusselt number values for both the regular fluid and nanofluid. σ = 10, GR = 300, A = 100, and τF = 500.

wall; hence, the fluid moves upward. Thus, the intensity of flow reversal decreases near the cold wall, which shows that, at high values of the nanoparticle mass flux, the effect of forced convection is dominant. The Nusselt number is a measure of the heat-transfer rate such that a positive value for Nu at the hot wall (Nu2 > 0) means that the heat-transfer direction at the hot wall is from the hot wall to the nanofluid, whereas a negative value of the Nusselt number (Nu2 < 0) means that the heat-transfer direction is from the nanofluid to the hot wall. Moreover, at the cold wall, a negative value of Nu (Nu1 < 0) means that the heat-transfer direction is from the cold wall to the nanofluid, whereas a positive value of Nu (Nu1 > 0) means that the heat-transfer direction is from the nanofluid to the cold wall. Figure 4b demonstrates the variations of the Nusselt number as a function of the nanoparticle mass flux using various types of viscous dissipation models. It can be seen that, at the cold wall, the Nusselt number values obtained using the three viscous dissipation models are almost the same, whereas at the hot wall, the Nusselt number values evaluated

by model 3 are larger than those obtained by models 1 and 2. Figure 4b also shows that the heat-transfer direction at the cold wall is always from the nanofluid to the cold wall. The heattransfer rate at the cold wall (Nu1) increases with an increase in the nanoparticle mass flux (β). As can be seen from Figure 4b, (1) the heat-transfer direction at the hot wall is from the hot wall to the nanofluid (Nu2 > 0) and (2) with increasing nanoparticle mass flux (β), the rate of heat transfer from the hot wall to the nanofluid decreases until it reaches 0 (Nu2 = 0), after which, with a further increase in β, the heat-transfer direction reverses and becomes from the nanofluid to the hot wall (Nu2 < 0). With increasing nanoparticle mass flux (β), the rate of heat transfer from the nanofluid to the hot wall increases as well. The change of heat-transfer direction with increasing nanoparticle mass flux (β) can be justified by the assumption that, for large values of β at regions near the hot wall, the temperature of the nanofluid rapidly becomes equal to the hot wall temperature. Viscous dissipation acts as a heat source in the fluid and increases the fluid temperature near the hot wall, so that the fluid temperature 9410



Figure 6. Effects of free and forced convection (i.e., GR/A) on the (a) dimensionless temperature and velocity distributions and (b) Nusselt number values for both the regular fluid and nanofluid. β = 1.5, σ = 10, τF = 500, and τBr =1.

becomes greater than the hot wall temperature. Therefore, the heat-transfer direction is reversed from the nanofluid to the hot wall. 6.2. Effects of Viscous Dissipation. Note that the dimensionless parameter τBr represents the effect of heat dissipation in the medium such that large values of τBr show that more heat dissipates in the medium. Heat dissipation can act as a heat source in the medium and increases the fluid temperature. This behavior can be seen in Figure 5a such that the fluid temperature increases at large values of τBr, which means that the dimensionless temperature increases with an increase in the heat dissipation. Figure 5a also shows that, with an increase in the heat dissipation, the fluid velocity increases as well. This is because, with an increase in the heat dissipation, the fluid temperature increases and its density decreases; consequently, the fluid moves upward. The effect of heat dissipation on the velocity distribution becomes significant when the free convection is not negligible in comparison with the forced convection.

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Figure 7. Effects of Forchheimer drag term (τF) on the (a) dimensionless temperature and velocity distributions and (b) Nusselt number value for both the regular fluid and nanofluid. σ = 10, GR = 300, A = 100, and τBr = 0.1.

Figure 5b shows that the heat-transfer direction at the cold wall for both the regular fluid and the nanofluid is from the nanofluid to the cold wall (Nu1 > 0). As can be observed from this figure, increasing the viscous dissipation does not have a profound influence on the heat-transfer rate at the cold wall for the regular fluid, but for the nanofluid, increasing the viscous dissipation (τBr) results in an increase in the heat-transfer rate (Nu1). For the nanofluid at the hot wall, the heat-transfer direction at low viscous dissipation values (τBr) is from the hot wall to the nanofluid (Nu2 > 0). In addition, with increasing viscous dissipation (τBr), the heat-transfer rate (Nu2) decreases until it reaches 0, after which a further increase in the viscous dissipation causes a reversal of the heat-transfer direction, that is, from the nanofluid to the hot wall (Nu2 < 0). However, for the regular fluid at the hot wall, heat transfer is always from the hot wall to the nanofluid, and with an increase in the viscous dissipation, the heat-transfer rate decreases. Moreover, the Nusselt number was evaluated using the three viscous dissipation models for both the regular fluid and nanofluid. It can be seen that, at the hot wall, model 3 9411



Figure 8. Effects of Darcy term (σ) on the (a) dimensionless temperature and velocity distributions and (b) Nusselt number values for both the regular fluid and nanofluid. β = 1.5, τF = 500, τBr = 1, GR = 300, and A = 100.

predicts larger values of the Nusselt number, whereas at the cold wall, the predictions of all of the models are the same. 6.3. Effects of Free and Forced Convection. The dimensionless parameter GR accounts for the ratio of free convection (Gr) to forced convection (Re). Moreover, the dimensionless parameter A represents the effect of the pressure drop, which has a direct relation to the forced convection. Figure 6a shows the temperature and velocity distributions as a function of GR/A as obtained using various types of viscous dissipation models for both the regular fluid and nanofluid. It can be seen that the predictions of all of the viscous dissipation models are the same. Moreover, as can be observed from Figure 6a, for GR/A > 1, free convection is dominant, and flow reversal occurs near the cold wall. However, with a decrease in the GR/A value, the contribution of free convection decreases, flow reversal near the cold wall does not occur, and forced convection becomes dominant. Note that, as the value of GR/ A is decreased, the Forchheimer drag term increases, and the fluid velocity decreases, but the values of the fluid velocity at different points are the same.

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Figure 6b shows that, for the cold wall at identical conditions, the predictions of the three models of viscous dissipation are the same, but for the hot wall, model 3 predicts larger Nusselt number values and the predictions of models 1 and 2 are almost the same. For both the regular fluid and the nanofluid, the heattransfer direction at the cold wall is from the nanofluid to the cold wall (Nu1 > 0). The rate of heat transfer decreases by increasing GR/A value, which can be easily justified because with an increase in the GR/A value, the contribution of forced convection decreases; therefore, the rate of heat-transfer decreases. For the nanofluid at the hot wall, the heat-transfer direction is always from the nanofluid to the hot wall (Nu2 < 0) and with an increase in the GR/A value, the rate of heat transfer from the nanofluid to the hot wall decreases as a result of decrease in the forced convection. 6.4. Effects of Forchheimer Drag Term. The effect of Forchheimer drag term was investigated by varying the dimensionless parameter τF. Note that there is a direct relation between Forchheimer drag term and τF, which means by increasing the Forchheimer drag term, the value of τF increases as well. Forchheimer drag term could be significant at high velocities of fluid in the porous medium and can be neglected at low velocities. Forchheimer drag term acts as the flow resistance, therefore, by increasing the Forchheimer drag term (τF), the fluid velocity decreases. As can be seen from Figure 7a, the dimensionless velocity decreases with an increase in the value of τF. In cases in which the effect of viscous dissipation is negligible, the Forchheimer drag term does not have a significant influence on the temperature profile. This is shown in Figure 7a, where the fluid temperature does not increase profoundly with increasing Forchheimer drag term. However, for cases in which the viscous dissipation is noticeable, with an increase in the Forchheimer drag term, the fluid temperature increases, and fluid velocity decreases. This figure also shows that the temperature and velocity profiles evaluated using three viscous dissipation models are almost the same. As can be seen from Figure 7b, for both the regular fluid and nanofluid, when the viscous dissipation is negligible, the heattransfer rate at both the hot and cold walls is independent of the Forchheimer drag term. The heat-transfer direction for both the regular fluid and the nanofluid at the hot wall is from the hot wall to the (nano)fluid, whereas at the cold wall, it is from the (nano)fluid to the cold wall. Figure 7b clearly demonstrates that the Nusselt number values predicted by various types of viscous dissipation models at both the hot and cold walls are the same. 6.5. Effects of Permeability or Darcy Term. The effects of the permeability of the medium, or the Darcy term, on the temperature and velocity distributions were investigated by changing the dimensionless parameter σ. With an increase in the permeability of the porous medium (K) at a fixed pressure drop, more fluid can pass through the porous medium; thus, the fluid velocity increases. Figure 8a shows that, with a decrease in the dimensionless parameter σ, the fluid velocity increases. The parameter σ is a measure of Darcy force in the porous medium, which is important at low velocities. Figure 8a also demonstrates that the temperature and velocity distributions at identical conditions are almost the same for various types of viscous dissipation models. Figure 8b shows the variations of the Nusselt number values for both the hot and cold walls as a function of the Darcy term as obtained using various types of viscous dissipation models. As can be observed from this figure, the heat-transfer rate (Nu1) at the cold wall for the regular fluid is almost constant for different values of σ. However, for the nanofluid, the heat-transfer rate 9412


Industrial & Engineering Chemistry Research decreases as the Darcy term increases. At the hot wall, the heattransfer direction is from the nanofluid to the hot wall at first (Nu2 < 0), and then the heat-transfer rate decreases with increasing σ until reaching 0, after which the heat-transfer direction changes and becomes from the hot wall to the nanofluid (Nu2 > 0). For the regular fluid at the hot wall, the heat-transfer direction is from the hot wall to the nanofluid, and its rate increases with decreasing permeability of the porous medium (increasing σ). Figure 8b also shows that, at the cold wall, the Nusselt number values are the same for three viscous dissipation models, but at the hot wall and for small values of the Darcy term, model 3 predicts larger values of the Nusselt number.

7. CONCLUSIONS The problem of combined forced and natural convection in a vertical porous channel saturated by both a regular fluid a and nanofluid was solved by perturbation and numerical methods, taking into account the influences of viscous heating and inertial force. Various types of viscous dissipation models were applied to account for the viscous heating. The velocity and temperature distributions of both the regular fluid and nanofluid and the Nusselt number values were obtained in terms of various dimensionless parameters. The findings of this work can be summarized as follows: (1) In the case of regular fluid, the temperature distribution is almost linear, but the presence of nanoparticles in the base fluid causes the temperature to approach the hotwall temperature, particularly at high nanoparticle fluxes. As the nanoparticle mass flux increases, the velocity profile becomes constant. The heat-transfer rate at the cold wall (Nu1) increases with increasing nanoparticle mass flux (β). The heat-transfer rate at the hot wall (Nu2) is from the hot wall to the nanofluid, and with an increase in the nanoparticle mass flux, the heat-transfer direction reverses. (2) The effect of nanoparticles on the temperature distribution becomes significant when the viscous dissipation term is non-negligible (i.e., τBr > 1). (3) With an increase in the viscous dissipation, the heattransfer rate from the hot wall to the nanofluid decreases until, at high viscous dissipations, the heat-transfer direction changes and becomes from the nanofluid to the hot wall. Moreover, at the cold wall, as the viscous dissipation (τBr) increases, the heat-transfer rate (Nu1) increases for both the regular fluid and the nanofluid. (4) The effect of nanoparticles on the velocity distribution is significant for GR/A > 3. (5) With increasing GR/A value, flow reversal is observed near the cold wall. (6) For both the regular fluid and the nanofluid, an increase in the GR/A value causes the heat-transfer rate from the nanofluid to the cold wall (Nu1 > 0) to decrease. (7) For the nanofluid at the hot wall, with an increase in the GR/A value, the rate of heat transfer from the nanofluid to the hot wall decreases as a result of the decrease in forced convection. (8) For the regular fluid at the hot wall, with a decrease in the forced convection (i.e., increasing GR/A value), the heattransfer rate from the nanofluid to the hot wall (Nu2 < 0) decreases until the heat-transfer direction changes and becomes from the hot wall to the nanofluid.

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’ AUTHOR INFORMATION Corresponding Author

*Tel.: +98-21-66165401. Fax: +98-21-66022853. E-mail: amolaeid@ sharif.edu.

’ ACKNOWLEDGMENT We acknowledge the financial support provided by Sharif University of Technology (Tehran, Iran). ’ NOMENCLATURE A = dimensionless pressure drop Br = Brinkman number C1, C2 = coefficients CF = Forchheimer coefficient Da = modified Darcy number F = modified Forchheimer drag term = Forchheimer drag force/ viscous force g = gravitational acceleration (m/s2) Gr = Grashof number GR = mixed-convection parameter (ratio of free to forced convection) k = thermal conductivity of fluid (W/m 3 K) keff = effective thermal conductivity of fluid (W/m 3 K) K = permeability of porous media (m2) Le = modified Lewis number Nu = Nusselt number = ∂θ/∂Y M = viscosity ratio p = pressure (Pa) P = dimensionless pressure Pr = Prandtl number Re = Reynolds number T = temperature (K) T0 = mean temperature (K) u = fluid velocity in the x direction, u = uD/ε (m/s) U = dimensionless fluid velocity uD = Darcy velocity in the x direction (m/s) um = average fluid velocity (m/s) x = Cartesian coordinate y = Cartesian coordinate Y = dimensionless coordinate (Y = y/H) Greek Symbols

β = dimensionless parameter for nanoparticle mass flux = nanoparticle mass diffusivity/thermal diffusivity γ = modified ratio of nanoparticle heat capacity to fluid heat capacity ε = porosity of porous media θ = dimensionless temperature, θ = (T T0)/(T2 T1) km = average thermal expansion coefficient (K1) μeff = effective viscosity (kg/m 3 s) μm = average viscosity (kg/m 3 s) Fm = average density of fluid (kg/m3) σ = dimensionless parameter for the Darcy term τBr = dimensionless parameter for viscous dissipation term = (buoyancy force/viscous force) (heat production by viscous dissipation/heat production by conduction) τF = dimensionless parameter for the Forchheimer drag term = (Forchheimer drag force/viscous force) (external pressure force/viscous force) 9413



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ϕ = nanoparticle volume fraction Φv = viscous heating term Subscripts

1 = properties at the cold wall 2 = properties at the hot wall f = fluid n = nanofluid p = particle r = regular fluid

’ REFERENCES (1) Choi, S. U. S. Enhancing Thermal Conductivity of Fluids with Nanoparticles. In Developments and Applications of Non-Newtonian Flows; Siginer, D. A., Wang, H. P., Eds.; American Society of Mechanical Engineers: New York, 1995; FED-Vol. 231/MD-Vol. 66, p 99. (2) Masuda, H.; Ebata, A.; Teramae, K.; Hishinuma, N. Alteration of Thermal Conductivity and Viscosity of Liquid by Dispersing Ultra-fine Particles (Dispersion of Al2O3, SiO2 and TiO2 Ultra-fine Particles). Netsu Bussei 1993, 7, 227 (in Japanese). (3) Buongiorno, J.; Hu, W. Nanofluid Coolants for Advanced Nuclear Power Plants. Presented at the International Congress on Advances in Nuclear Power Plants (ICAPP’05), Seoul, Korea, May 1519, 2005; Paper 5705. (4) Kleinstreuer, C.; Li, J.; Koo, J. Microfluidics of Nano-drug Delivery. Int. J. Heat Mass Transfer 2008, 51, 5590. (5) Prodanovi, M.; Ryoo, S.; Rahmani, A. R.; Kuranov R.; Kotsmar, C.; Milner, T. E.; Johnston, K. P.; Bryant, S. L.; Huh, C. Effect of Magnetic Field on the Motion of Multiphase Fluids Containing Paramagnetic Nanoparticles in Porous Media. Presented at the SPE Improved Oil Recovery Symposium, Tulsa, Oklahoma, Apr 2428, 2010; Paper 129850-MS. (6) Buongiorno, J. Convective Transport in Nanofluids. ASME J. Heat Transfer 2006, 128, 240. (7) Tzou, D. Y. Instability of Nanofluids in Natural Convection. ASME J. Heat Transfer 2008, 72401, 130. (8) Tzou, D. Y. Thermal Instability of Nanofluids in Natural Convection. Int. J. Heat Mass Transfer 2008, 51, 2967. (9) Vafai, K.; Tien, C. L. Boundary and Inertia Effects on Flow and Heat Transfer in Porous Media. Int. J. Heat Transfer 1981, 24, 195. (10) Umavathi, J. C.; Kumar, J. P.; Chamkha, A. J.; Pop, I. Mixed Convection in a Vertical Porous Channel. Transport Porous Media 2005, 61, 315. (11) Umavathi, J. C.; Kumar, J. P.; Chamkha, A. J.; Pop, I. Mixed Convection in a Vertical Porous Channel. Transport Porous Media 2008, 75, 129. (12) Nield, D. A.; Kuznetsov, A. V. Thermal Instability in a Porous Layer Saturated by a Nanofluid. Int. J. Heat Mass Transfer 2009, 52, 5796. (13) Ahmad, S.; Pop, I. Mixed Convection Boundary Layer Flow from a Vertical Flat Wall Embedded in a Porous Medium Filled with Nanofluids. Int. Commun. Heat Mass Transfer 2010, 37, 987. (14) Nield, D. A.; Kuznetsov, A. V. The ChengMinkowycz Problem for Natural Convective Boundary-Layer Flow in a Porous Medium Saturated by a Nanofluid. Int. J. Heat Mass Transfer 2009, 52, 5792. (15) Kuznetsov, A. V.; Nield, D. A. The Onset of Double-Diffusive Nanofluid Convection in a Layer of a Saturated Porous Medium. Transport Porous Media 2010, 85, 941. (16) Nield, D. A.; Bejan, A. Convection in Porous Media; Springer: New York, 2006. (17) Nield, D. A.; Kuznetsov, A. V.; Xiong, M. Effects of Viscous Dissipation and Flow Work on Forced Convection in a Channel Filled by a Saturated Porous Medium. Transport Porous Media 2004, 56, 351.

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