Transp Porous Med (2008) 73:173–186DOI 10.1007/s11242-007-9166-8

Non-Darcian Forced Convection Flow of ViscousDissipating Fluid over a Flat Plate Embedded ina Porous Medium

Orhan Aydın · Ahmet Kaya

Received: 11 July 2007 / Accepted: 11 September 2007 / Published online: 4 October 2007© Springer Science+Business Media B.V. 2007

Abstract In this study, laminar boundary layer flow over a flat plate embedded in a fluid-saturated porous medium in the presence of viscous dissipation, inertia effect and suc-tion/injection is analyzed using the Keller box finite difference method. The flat plate isassumed to be held at constant temperature. The non-Darcian effects of convection, bound-ary and inertia are considered. Results for the local heat transfer parameter and the localskin friction parameter as well as the velocity and temperature profiles are presented forvarious values of the governing parameters. The non-Darcian effects are shown to decreasethe velocity and to increase the temperature. It is also shown that the local heat transferparameter and the local skin friction parameter increase due to suction of fluid while injec-tion reverses this trend. It is disclosed that the effect of the viscous dissipation for negativevalues of Ec (Tw < T∞) is to enhance the heat transfer coefficient while the opposite is truefor positive values of Ec (Tw > T∞). The results are compared with those available in theexisting literature and an excellent agreement is obtained.

Keywords Porous medium · Boundary layer · Horizontal · Viscous dissipation · Eckertnumber · Porosity · Inertia effect · Suction/injection parameter

Nomenclaturecp Specific heat of the convective fluidEc Eckert numberf Dimensionless stream functionfw Suction/injection parameterF Inertial coefficientK Permeability of the porous mediumPr Prandtl numberRe Reynolds numberT Temperature

O. Aydın (B) · A. KayaDepartment of Mechanical Engineering, Karadeniz Technical University, Trabzon 61080, Turkeye-mail: oaydin@ktu.edu.tr

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174 O. Aydın, A. Kaya

u, υ Velocities in x and y directions, respectivelyx, y Coordinates in horizontal and vertical directions, respectively

Greek symbols

η Pseudo similarity variable, yRe1/2x /x

ε Porosityξ Non-similarity variable, vx/Ku∞γ Dimensionless inertia effect, FK1/2u∞/vρ Fluid densityµ Dynamic viscosityv Kinematic viscosityθ Dimensionless temperature profile in Eq. 5

Subscriptsw Wall∞ Free stream

1 Introduction

The study of convective flow through porous media has received a great deal of researchinterest over the last three decades due to its wide and important applications in environmen-tal, geophysical and energy related engineering problems. Prominent applications are theutilization of geothermal energy, the migration of moisture in fibrous insulation, drying ofporous solid, food processing, casting and welding in manufacturing processes, the disper-sion of chemical contaminants in different industrial processes, the design of nuclear reactors,chemical catalytic reactors, compact heat exchangers, solar power, etc. Forced convectionover a horizontal plate in a fluid-saturated porous medium is a good representative for manyof the areas listed above. The growing volume of research devoted to this subject can befound in the recent excellent reviews by Kaviany (1995), Pop and Ingham (2001), Inghamand Pop (1998, 2002) and Nield and Bejan (1999).

On the problem of forced convection boundary layer flow and heat and mass transferalong a flat plate embedded in a porous medium, many studies have been conducted to inves-tigate effects of variable viscosity, internal heat generation, the presence of an isotropic andsolid matrix, surface mass, etc. (Vafai and Tien 1980, 1981; Vafai 1984; Vafai et al. 1985;Chandrasekhara 1986; Beckerman and Viskanta 1987; Ling and Dybbs 1992; Postelnicuet al. 2001; Hady and Ibrahim 1997; Murthy and Singh 1997; Yih 1998; Gupta et al. 2003;Magyari et al. 2003; Nield 2000; Al-Hadhrami et al. 2003; Nield and Kuznetsov 2003). Effectof viscous dissipation has been generally neglected in the existing literature. However, thiseffect can be relevant if highly viscous fluids with a low thermal conductivity are consid-ered. Sonth et al. (2002) studied heat and mass transfer in a visco-elastic fluid flow over anaccelerating surface with heat source/sink and viscous dissipation using the similarity solu-tion. They determined the effect of various physical parameters like visco-elasticity, Eckertnumber, Prandtl number, heat source/sink, Schmidt number and suction/blowing parameteron temperature and concentration profiles. Mureithi and Mason (2002) studied the inviscidstability of an accelerating forced-free convection boundary layer with viscous dissipationusing the similarity solution. Israel-Cookey et al. (2003) investigated the influence of viscousdissipation and radiation on the problem of unsteady magneto-hydrodynamic free-convectionflow past an infinite vertical heated plate in an optically thin environment with time-dependentsuction. An increase in Eckert number was shown to increase the velocity and temperature

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Non-Darcian Forced Convection Flow of Viscous Dissipating Fluid 175

Table 1 Comparison of the values −θ ′ (0, 0) for various values Pr at ε = 1, ξ = 0, γ = 0, Ec = 0 andfw = 0

Pr Lin and Lin (1987) Yih (1999) Chamkha et al. (2003) Present study

0.01 0.051559 0.051589 0.051830 0.051437

0.1 0.140032 0.140034 0.142003 0.148053

1 0.332057 0.332057 0.332173 0.332000

10 0.728148 0.728141 0.728310 0.727801

100 1.571860 1.571831 1.572180 1.573141

x, u

Porous Medium

u ∞ , T∞y,υ

υ=V w (x)

Fig. 1 The schematic of the problem

profiles. El-Amin (2003) investigated the influence of viscous dissipation on buoyancy-induced flow over a horizontal or a vertical flat plate embedded in a non-Newtonian fluidsaturated porous medium under the action of transverse magnetic field using the second-levelnon-similarity method. The Eckert number was shown to enhance the temperature profile.Magyari et al. (2003) studied the uniform forced-convection flow in a fluid-saturated porousmedium adjacent to a plane surface with prescribed temperature distribution in the presence ofviscous dissipation. They obtained exact solutions by reducing the forced convection problemto some well-known heat conduction problems in a homogeneous semi-infinite solid. Aydinand Kaya (2006) studied influence of the viscous dissipation on mixed convection about onisothermal vertical plate embedded in a porous medium. Based on the thermal boundary layerconditions applied at the wall (wall heating/cooling) and the direction of free stream flow(upward/downward), four different mixed convection flow regimes were identified.

In a recent study, Aydin and Kaya (2005) studied the laminar boundary layer flow over apermeable flat plate with injection or suction imposed at the wall. This study is an extensionof that study. It is aimed at analyzing effect of the viscous dissipation on the momentum andheat transfer over a horizontal plate embedded in a fluid-saturated porous medium. The effectof viscous dissipation is studied for the wall heating and wall cooling cases. Interaction ofviscous dissipation with suction/injection at wall is included. The results have been obtainedfor a range of given parameters.

2 Analysis

Consider steady, incompressible, laminar, two-dimensional, boundary layer flow with viscousdissipation over a flat plate embedded in a Newtonian fluid-saturated porous medium. Theporous medium is considered to be homogeneous and isotropic (i.e. uniform with a constant

123

176 O. Aydın, A. Kaya

Table 2 Comparison of the values −θ ′ (0, 0) for various values Pr at ε = 1, ξ = 0, γ = 0, Ec = 0 andfw = 0

Pr Kuznetsov and Nield (2006) Present study

0.1 0.1580 0.1480

1 0.3320 0.3320

5 0.5700 0.5765

10 0.7300 0.7278

20 0.9100 0.9176

30 1.0500 1.0550

40 1.1500 1.1550

50 1.2450 1.2440

60 1.3200 1.3210

70 1.3900 1.3900

80 1.4500 1.4520

90 1.5100 1.5100

100 1.5700 1.5731

Table 3 Comparison of the values −θ ′ (ξ, 0) for various values ξ with ε = 1, Pr = 10, γ = 0, Ec = 0 andfw = 0

ξ Lloyd and Sparrow (1970) Chang (2006) Present study

0.00000 0.7281 0.7280 0.7278

0.00125 0.7313 0.7291 0.7291

0.00500 0.7404 0.7373 0.7328

0.01250 0.7574 0.7566 0.7556

0.05000 0.8259 0.8351 0.8351

0.12500 0.9212 0.9412 0.9432

0.25000 1.0290 1.0603 1.0603

porosity and permeability) and is saturated with a fluid which is in local thermodynamicequilibrium with the solid matrix. Far above the plate, the velocity and the temperatureof the uniform main stream are u∞ and T∞, respectively. The x-coordinate is measuredfrom the leading edge of the plate and y-coordinate is measured normal to the plate. Thecorresponding velocity components in the x and y directions are u and v, respectively. Theentire surface of the plate is maintained at a uniform temperature of Tw. The properties ofthe fluid and the porous media, such as viscosity, thermal conductivity, specific heat andpermeability, all are assumed to be constant. In order to study transport through non-Darcianmedia, the original Darcy model is improved by including the convective, viscous and inertiaeffects. Assuming that the Boussinesq approximation is valid, the boundary-layer form of thegoverning equations which are based on the balance laws of mass, momentum and energycan be written as

∂u

∂x+ ∂υ

∂y= 0 (1)

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Non-Darcian Forced Convection Flow of Viscous Dissipating Fluid 177

η0 1 2 3 4 5 6

η0 1 2 3 4 5 6

f'(ξ,

η)

0.0

0.2

0.4

0.6

0.8

1.0

ξ=0.08, 0.06, 0.04, 0.02, 0.0

Ec=0.0Pr=1.0γ=0.5ε=0.75

(a)

0.0

0.2

0.4

0.6

0.8

1.0(b)

θ(ξ,

η)

ξ=0.08, 0.06, 0.04, 0.02, 0.0

Ec=0.0Pr=1.0γ=0.5ε=0.75

Fig. 2 Dimensionless velocity (a) and temperature (b) profiles for different ξ atEc = 0.0,Pr = 1.0, γ = 0.0and ε = 1.0

1

ε2

[u∂u

∂x+ υ

∂u

∂y

]= v

ε

∂2u

∂y2 − v

K(u− u∞)− F

K1/2

(u2 − u2∞

)(2)

u∂T

∂x+ υ

∂T

∂y=

( v

P r

) ∂2T

∂y2 +(v

cP

)(∂u

∂y

)2

+ v

Kcpu2. (3)

The above equations are called Brinkman–Forchheimer-extended-Darcy equations (Lauriatand Ghafir 2000). Here u and υ are the velocity components parallel and perpendicular tothe plate, T is the temperature, v is the kinematic viscosity, ρ is the fluid density, cp is thespecific heat at constant pressure, ε is the porosity,K is the permeability and F is the inertialcoefficient which depend on permeability and microstructure of the porous matrix. In theenergy equation, Eq. 3, the last two terms represent the viscous dissipation effect. The firstterm is the classical expression of the viscous dissipation for a clear fluid, while the secondterm is the viscous dissipation in the Darcy limit (Al-Hadhrami et al. 2002).

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178 O. Aydın, A. Kaya

η0

0 1 2 3 4 5 6

2 4 6 8

f'(ξ,

η)

0.0

0.2

0.4

0.6

0.8

1.0

ε=0.25, 0.5, 0.75, 1.0

Ec=0.0fw=0.0Pr=1.0γ=0.5ξ=0.04

(a)

0.0

0.2

0.4

0.6

0.8

1.0(b)

η

θ(ξ,

η)

ε=0.25, 0.5, 0.75, 1.0

Ec=0.0fw=0.0Pr=1.0γ=0.5ξ=0.04

Fig. 3 Dimensionless velocity (a) and temperature (b) profiles for different ε atEc = 0.0,Pr = 1.0, γ = 0.5and ξ = 0.04

The appropriate boundary conditions for the velocity and temperature of this problem are:

x = 0 y > 0 T = T∞ u = u∞x > 0 y = 0 T = Tw u = 0 υ = Vw(x)

y → ∞ T → T∞ u → u∞

⎫⎬⎭ . (4)

Here, u∞ and T∞ are the free stream velocity and temperature, respectively.

ψ (x, y) = (vu∞x)1/2 f (ξ, η), η = y

xRe

1/2x , θ = T − T∞

Tw − T∞,

ξ (x) = vx

Ku∞, γ = FK1/2u∞

v(5)

where ψ(x, y) is the stream function that satisfies (1) with u = ∂ψ/∂y and υ = −∂ψ/∂x.

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Non-Darcian Forced Convection Flow of Viscous Dissipating Fluid 179

ξ0.00 0.02 0.04 0.06 0.08

f''(ξ

,η)

0.3

0.4

0.5

0.6

0.7

0.8

0.9ε=0.25ε=0.5ε=0.75ε=1.0

Pr=1.0Ec=0.0fw=0.0

γ=0.5

(a)

ξ0.00 0.02 0.04 0.06 0.08

−θ'(ξ

,η)

0.32

0.34

0.36

0.38

0.40

0.42

0.44ε=0.25ε=0.5ε=0.75ε=1.0

Pr=1.0Ec=0.0γ=0.5

(b)

Fig. 4 Effects of ε on the local skin friction parameter (a) and local heat transfer parameter (b)

In terms of these new variables, the velocity components can be expressed as

u = u∞f ′, (6)

υ = −{

1

2f

(u∞vx

)1/2 + (vu∞x)1/2[

v

Ku∞∂f

∂ξ− η

2xf ′

]}. (7)

The transformed momentum and energy equations together with the boundary conditions,Eqs. 2–4, can be written as

1

εf ′′′ + 1

2ε2 ff′′ − ξ

(f ′ − 1

) − γ ξ(f ′2 − 1

) = ξ

ε2

(f ′ ∂f ′

∂ξ− f ′′ ∂f

∂ξ

)(8)

1

Prθ ′′ + 1

2f θ ′ + Ec

[(f ′′)2 + ξ

(f ′)2

]= ξ

(f ′ ∂θ∂ξ

− θ ′ ∂f∂ξ

)(9)

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180 O. Aydın, A. Kaya

η0 2 4 6 8

f'(ξ,

η)

0.0

0.2

0.4

0.6

0.8

1.0

γ=3.5, 2.5, 1.5, 0.5

Ec=0.0Pr=1.0ε=0.75ξ=0.04

(a)

0.0

0.2

0.4

0.6

0.8

1.0(b)

η0 1 2 3 4 5 6

q(ξ,

η)

γ=3.5, 2.5, 1.5, 0.5

Ec=0.0Pr=1.0ε=0.75ξ=0.04

Fig. 5 Dimensionless velocity (a) and temperature (b) profiles for different γ at Ec = 0.0, Pr = 1.0,ε = 0.75 and ξ = 0.04

f (ξ, 0)+ 2ξ ∂f∂ξ

= −2fwf ′ (ξ, 0) = 0, θ (ξ, 0) = 1f ′ (ξ,∞) = 1, θ (ξ,∞) = 0

⎫⎬⎭ , (10)

where fw = − xvVwRe

1/2x , the case fw > 0 designates suction while fw < 0 indicates

injection or blowing, Pr , Ec and Re are the Prandtl, the Eckert and the Reynolds numbersdefined as:

Pr = µcp

k= v

α, Ec = u2∞

cp (Tw − T∞), Rex = u∞x

v(11)

In this study, the Keller’s box finite-difference method is used in the solution. The sys-tem of transformed equations together with the boundary conditions, Eqs. 8–10, is solvednumerically by an efficient and accurate finite-difference scheme similar to that described

123

Non-Darcian Forced Convection Flow of Viscous Dissipating Fluid 181

ξ0.00 0.02 0.04 0.06 0.08

0.35

0.40

0.45

0.50

0.55

0.60

0.65

0.70γ=0.5γ=1.5γ=2.5γ=3.5

Pr=1.0Ec=0.0fw=0.0

ε=0.75

(a)

(b)

ξ0.00 0.02 0.04 0.06 0.08

0.34

0.35

0.36

0.37

0.38

0.39

0.40γ=0.5γ=1.5γ=2.5γ=3.5

Pr=1.0Ec=0.0ε=0.75

f'(ξ,

η)−θ

'(ξ,η

)

Fig. 6 Effects of γ on the local skin friction parameter (a) and local heat transfer parameter (b)

η0 1 2 3 4 5 6

0.0

0.2

0.4

0.6

0.8

1.0Pr=1.0fw=0.0

ε=0.75γ=0.5ξ=0.04

Ec=-0.1, 0.0, 0.1

θ (ξ

,η)

Fig. 7 Dimensionless temperature profile for different Ec at Pr = 1.0, ε = 0.75, γ = 0.5 and ξ = 0.04

123

182 O. Aydın, A. Kaya

ξ0.00 0.02 0.04 0.06 0.08

−θ'(ξ

,η)

0.32

0.33

0.34

0.35

0.36

0.37

0.38

0.39

0.40Ec=0.1Ec=0.0Ec=-0.1

Pr=1.0fw=0.0

γ=0.5ε=0.75

Fig. 8 Effects of Ec on the local heat transfer parameter

in Cebeci and Bradshaw (1984). This numerical scheme with its very desirable features isvery appropriate for the solution of parabolic partial differential equations. A uniform gridwith a step size 0.01 in the η-direction and a non-uniform grid in the ξ -direction are used.Resolution of the grid sizes was kept satisfactory enough in terms of the convergence of thenumerical solution and the accuracy of the results.

3 Results and Discussion

At first, in order to assess the accuracy of the numerical results, the validity of the numericalcode developed has been checked for a limiting case. For ε = 1, γ = 0 and ξ = 0, wecompare our −θ ′ (0, 0) results with those given by Lin and Lin (1987), Yih (1999), Chamkhaet al. (2003) (Table 1) and Nield and Kuznetsov (2003) and Kuznetsov and Nield (2006)(Table 2) and for ε = 1, Pr = 10, γ = 0 and Ec = 0, compare our −θ ′ (ξ, 0) results at withthose given by Lloyd and Sparrow (1970) and Chang (2006) (Table 3). As it is seen fromTables 1–3 our results correspond very well with theirs.

Laminar boundary layer flow over a flat plate embedded in a porous medium in the presenceof viscous dissipation, inertia force and suction/injection effect is investigated numerically.The temperature of the plate is assumed to be constant (Fig. 1). In order to study the effectof the Eckert number, Ec, the range −0.1 ≤ Ec ≤ +0.1, the inertia force, γ , the range0.5 ≤ γ ≤ 3.5, the suction/injection parameter, fw , the range −0.1 ≤ fw ≤ +0.1 arechosen. Positive values of Eckert number are for the case that the wall temperature is greaterthan the free stream temperature, i.e. heat is transferred from the wall to the fluid. Oppositeis true for its negative values.

Figure 2 shows velocity (a) and temperature profiles (b) for different values of ξ at Ec =0.0, Pr = 1.0, γ = 0.5 and ε = 1.0. As expected, an increase in ξ increases velocity andtemperature profiles.

As seen from the Fig. 3, an increase at the porosity broadens velocity and temperatureprofiles, thereby thickening the momentum and thermal boundary layers. This will result inreduced momentum and heat transfer rates.

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Non-Darcian Forced Convection Flow of Viscous Dissipating Fluid 183

η

0.00 0 4 6 8

η0 1 2 3 4 5 6

0.2

0.4

0.6

0.8

1.0

fw=0.1, 0.0, -0.1

Ec=0.0Pr=1.0ε=0.75γ=0.5ξ=0.04

(a)

0.0

0.2

0.4

0.6

0.8

1.0(b)

fw=0.1, 0.0, -0.1

Ec=0.0Pr=1.0ε=0.75γ=0.5ξ=0.04

f'(ξ,

η)θ(

ξ,η)

Fig. 9 Dimensionless velocity (a) and temperature (b) profiles for different fw at Ec = 0.0, Pr = 1.0,ε = 0.75, γ = 0.5 and ξ = 0.04

Figure 4 shows the local skin friction parameter (a) and local heat transfer parameter (b)for different values of ε. It is seen that increasing porosity decreases local skin friction andlocal heat transfer parameter.

Figure 5 shows the velocity (a) and the temperature (b) profiles for various values of inertialforce, γ , at Ec = 0.0, Pr = 1.0, ξ = 0.04 and ε = 0.75. It is seen that the inertia effectincreases velocity and temperature profiles.

Figure 6 shows the local skin friction parameter (a) and local heat transfer parameter (b)for different values of γ at Ec = 0.0, Pr = 1.0, fw = 0.0 and ε = 0.75. It is seen thatincreasing inertia force increases local skin friction and local heat transfer parameter.

For the forced convection case, as seen, the Eckert number does not have any influence onvelocity profile since the momentum and energy equations are not coupled, however, it doeson the temperature profile. Figure 7 shows the effects of Ec on the temperature profile. For apositive value of Ec, since the temperature of the wall is greater than that of the free stream,there will be a heat transfer from the wall to the fluid. The viscous dissipation will cause to aheat generation inside the fluid, which results in increase in the temperature distribution in the

123

184 O. Aydın, A. Kaya

f''(ξ

,η)

(a)

−θ'(ξ

,η)

(b)

ξ0.00 0.02 0.04 0.06 0.08

0.30

0.35

0.40

0.45

0.50

0.55

0.60fw=0.1

fw=0.0

fw=-0.1

Pr=1.0Ec=0.0ε=0.75γ=0.5

ξ0.00 0.02 0.04 0.06 0.08

0.30

0.32

0.34

0.36

0.38

0.40

0.42

0.44fw=0.1

fw=0.0

fw=-0.1

Pr=1.0Ec=0.0ε=0.75γ=0.5

Fig. 10 Effects of fw on the local skin friction parameter (a) and local heat transfer parameter (b)

flow region. This is due to the fact that heat energy is stored in the fluid due to frictional heating.Due to the increased bulk fluid temperature, the temperature gradient will decrease. For anegative value of Ec, since Tw < T∞, the viscous dissipation will increase the temperaturedistribution in the flow region more. Finally, this leads to an increased temperature gradient,as will be shown later, which will result in increased heat transfer values. As it can be noticedfrom Fig. 7, the effect of Ec intensifies downstream. In addition, either for the wall coolingcase (Tw < T∞) or for the wall heating case (Tw > T∞), the porosity presents similar effectswhich thickens the temperature profile with an increase in the porosity.

Figure 8 shows the variation of local heat transfer parameter, −θ ′ (ξ, 0) with the Eckertnumber at Pr = 1.0, γ = 0.5 and ε = 0.75. As well known, viscous dissipation behaves likea heat generation source inside the fluid. For its positive values, the Eckert number presentsan opposing effect on the heat transfer, while representing an aiding effect for its negativevalues.

In order to study the effect of suction or injection, the following values of thesuction/injection parameter, fw are used: 0.1, 0.0, −0.1. Figure 9 shows velocity (a) and

123

Non-Darcian Forced Convection Flow of Viscous Dissipating Fluid 185

temperature (b) profiles for different values of fw. It should be noted that fw > 0 corre-sponds to suction, fw < 0 corresponds to injection and fw = 0 represents the flow over animpermeable surface. Injecting fluid into the boundary layer broadens the velocity distribu-tion and increases the hydrodynamic boundary layer thicknesses as shown in Fig. 9, while thesuction reverses this trend. Also, the wall shear stress would be increased with the applicationof suction whereas injection tends to decrease wall shear stress. This can be explained by thefact that the wall velocity gradient is increased with the increasing value of fw . Figure 9 alsoshows the effects of the injection parameter on the temperature profile. The effect of injectionis found to broaden the temperature distribution, decrease the wall temperature gradient, andhence reduce the heat transfer rate. On the other hand, the thermal boundary layer becomesthinner and the wall temperature gradient becomes larger when suction is applied.

Figure 10 shows the local skin friction parameter (a) and local heat transfer parameter (b)for different values of fw at Ec = 0.0, Pr = 1.0, γ = 0.5 and ε = 0.75. It is seen thatincreasing fw increases local skin friction and local heat transfer parameter.

4 Conclusions

The following main conclusions can be drawn from the study:

(i) Increasing porosity thickens the momentum and thermal boundary layers, and there-fore it decreases local skin friction and local heat transfer parameter.

(ii) Increasing inertia force increases local skin friction and local heat transfer parameter.(iii) The influence of the viscous dissipation on the heat transfer was shown to vary accord-

ing to the thermal boundary condition applied at the wall. For the wall heating case(Tw > T∞), the viscous dissipation was shown to decrease the heat transfer while itenhances it for the wall cooling case (Tw < T∞).

(iv) Employing suction at the wall increases the local skin friction and local heat transferparameter by making the momentum and thermal boundary layers thinner, whileinjection reverses this trend.

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