Meccanica (2009) 44: 133–144 DOI 10.1007/s11012-008-9156-0

Radiation effects on combined convection over a vertical flat plate embedded in a porous medium of variable porosity Dulal Pal · Hiranmoy Mondal

Received: 4 January 2008 / Accepted: 27 May 2008 / Published online: 16 July 2008 © Springer Science+Business Media B.V. 2008

Abstract The present paper is concerned with the study of radiation effects on the combined (forcedfree) convection flow of an optically dense viscous incompressible fluid over a vertical surface embedded in a fluid saturated porous medium of variable porosity with heat generation or absorption. The effects of radiation heat transfer from a porous wall on convection flow are very important in high temperature processes. The inclusion of radiation effects in the energy equation leads to a highly non-linear partial differential equations which are transformed to a system of ordinary differential equations using non-similarity transformation. These equations are then solved numerically using implicit finite-difference method subject to appropriate boundary and matching conditions. A parametric study of the physical parameters such as the particle diameter-based Reynolds number, the flow based Reynolds number, the Grashof number, the heat generation or absorption co-efficient and radiation parameter is conducted on temperature distribution. The D. Pal () Department of Mathematics, Siksha Bhavana, Visva-Bharati University, Santiniketan 731 235, West Bengal, India e-mail: [email protected] H. Mondal Department of Mathematics, Bengal Institute of Technology and Management, Santiniketan 731236, West Bengal, India e-mail: [email protected]

effects of radiation and other physical parameters on the local skin friction and on local Nusselt number are shown graphically. It is interesting to observe that the momentum and thermal boundary layer thickness increases with the radiation and decrease with increase in the Prandtl number. Keywords Convection · Non-Darcy · Thermal radiation · Boundary layer flow · Mechanics of fluids Nomenclature b, c C Cp Cf d F g Gr ke kf ks K L Nux Q0 Pr rk

empirical constants inertia co-efficient fluid heat capacity local skin-friction co-efficient particle diameter reduced stream function acceleration due to gravity Grashof number gβ(Tw − T∞ )L3 /ν 2 porous medium effective thermal conductivity fluid thermal conductivity thermal conductivity of the porous medium porous medium permeability characteristic plate length local Nusselt number dimensional heat generation or absorption coefficient Prandtl number μCp /kf ratio of ke and kf

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Re flow Reynolds number ρU∞ L/μ Rew Reynolds number based on the particle diameter, ρU∞ d/μ Rex local Reynolds number Nr thermal radiation parameter qr radiative heat flux k ∗ mean absorption co-efficient T fluid temperature Tw wall temperature T∞ free-stream temperature Da Darcy’s number u x component of fluid velocity v y component of fluid velocity U∞ free-stream velocity x vertical or tangential distance y normal distance Greek symbols β δ η ψ σ∗ ε ε0 μ ν ξ ρ θ λ

coefficient of thermal expansion heat-generation or absorption coefficient transformed normal coordinate stream function Stefan-Boltzmann constant porous medium porosity free-stream porosity fluid dynamic viscosity fluid kinematic viscosity, μ/ρ transformed tangential coordinate fluid density dimensionless temperature, (T − T∞ )/(Tw − T∞ ) assisting λ = 1 or opposing λ = −1 flow constant

1 Introduction The study of convection boundary layer flow in porous media have received considerable attention in recent years due to its important roles and wide applications in geophysics and thermal sciences, such as geothermal energy technology, petroleum recovery, building thermal insulation, packed bed reactors, underground disposal of chemical and nuclear waste. Cheng [1] presented a comprehensive review of heat transfer in geothermal system. Cheng and Minkowycz [2] presented the mathematical analysis for the natural convection flow about a heated surface embedded in a fluid saturated porous medium. In the study on porous

media, the Darcy’s law is used for slow flow and neglecting the effect of a solid boundary, inertia forces, variable porosity and thermal dispersion effects. These missing effects are very significant in most practical situations such as drying and metal processing. In the porous medium inertial effects have proven to be important for fast flows. Benenati and Brosilow [3], Vafai [4] and Vafai et al. [5] have shown that for an impermeable surface embedded in a porous medium, the porosity distribution exhibits a peak value close to the boundary and then decays asymptotically beyond that value. When the Reynolds number based on the pore diameter and local velocity is greater than order of unity, so in this situation, the well known Darcy’s law is inappropriate due to the pressure drop across the porous medium. Plumb and Huenefeld [6] studied the fundamental problem of natural convection from a heated vertical wall in a non-Darcy saturated porous medium. The inertial and viscous effects on mixed convection about a vertical surface have been studied by Ranganathan and Viskanta [7]. Their results show that the effects of inertia and boundary friction are quite significant and cannot be ignored. Takhar et al. [8] investigated mixed convection flow over a hot vertical plate in non-Darcian porous media. Nakayama and Pop [9] analyzed the inertia effects on mixed convection along a vertical wall using the Forchheimer flow model. Hsieh et al. [10] reported numerical solutions for mixed convection along a vertical flat plate embedded in a uniform porosity medium for the cases of variable wall temperature and variable wall heat flux. Chen et al. [11] studied non-Darcy mixed convection along non-isothermal vertical surfaces in porous media. In studies on spherical particles packed beds, the secondary flow effect caused by the mixing and recirculation of local fluid particles through tortuous paths formed by the spherical solid particles making up the porous medium is classified as thermal dispersion (Amiri and Vafai [12]). The study of thermal dispersion effects becomes prevalent in the porous media flow region. Fried and Combarnous [13] proposed a linear function to express the thermal dispersion. Cheng [14] investigated on flows and heat transfer in porous medium by taking thermal dispersion effects into consideration. Lai and Kulacki [15] studied thermal dispersion effect on non-Darcy convection from horizontal surface in saturated porous media. Murthy and Singh [16] investigated on mass flux effects on

Meccanica (2009) 44: 133–144

non-Darcy natural convection over a vertical flat plate in a fluid saturated porous medium. All the above mentioned studies treat permeability of the porous medium as constant. However, porosity measurements by Schwartz and Smith [17] show that porosity is not constant but varies from the wall to the interior due to variation in the permeability. Vafai and Tien [18] were the first to consider boundary and inertia effects on flow and heat transfer in a porous media of constant and variable porosity. Chandrasekhara and Namboodiri [19] studied the effects of variable permeability on combined, free and forced convection about inclined surfaces in porous media using a similarity solution approach. They have shown that the variation of porosity and permeability have greater influence on velocity and heat transfer. The effects of non-uniform porosity and thermal dispersion on vertical plate natural convection in porous media were analyzed by Hong et al. [20], and Hong and Tien [21] respectively. Boutros et al. [22] studied two-dimensional boundary layer stagnation point flow towards a heated stretching sheet placed in a porous medium applying Lie-group method of solution. All the above investigations were restricted to flow and heat transfer problems in porous medium. However, depending on the surface properties and solid geometry, the effect of radiation on flow and heat transfer in porous media has become more important in certain application, including waste heat storage in aquifers and gasification of oil shale which is of interest on combined convection since fluid is pumped into porous region. In addition, in the case of gasification, large temperature gradients exist in the neighborhood of the combustion hence radiation effects may become important. Radiation heat transfer in porous media has also been studied by many researchers. Whitaker [23] discussed radiative heat transfer in porous media. Plumb et al. [24] was the first to examine the effect of horizontal cross-flow and radiation on natural convection from vertical heated surface in saturated porous media. Rosseland diffusion approximation had been utilized in this investigation of convection flow with radiation. Ibrahim and Hady [25] have investigated mixed convection-radiation interaction in boundary layer flow over a horizontal surface. Ishak et al. [26] studied mixed convection boundary layers in the stagnation-point flow of an incompressible viscous fluid over a stretching vertical sheet. Gorla and Pop [27] studied the effects of radiation on mixedconvection flow over vertical cylinders. Hossain and

135

Takhar [28] have investigated the radiation effect on mixed convection boundary layer flow of an optically dense viscous incompressible fluid along a vertical plate with uniform surface temperature. Takhar et al. [29] studied the radiation effects on MHD free convection flow for a non gray-gas past a semi-infinite vertical plate whereas Bakier and Gorla [30] investigated the effect of the thermal radiation on mixed convection from horizontal surfaces in saturated porous media. Mansour [31] analyzed combined forced convection and radiation interaction heat transfer in boundary layer over flat plate immersed in porous medium of variable viscosity. Raptis [32] analyzed radiation and free convection flow through a porous medium using Rosseland approximation for the radiative heat flux. Chamkha [33, 34] studied solar radiation assisted free convection in the boundary layer adjacent to a vertical flat plate in a uniform porous medium considering a more general Darcy-Forchheimer-Brinkman flow model. The effect of radiation on the free convection heat transfer problem was studied by Hossain et al. [35] considering suction boundary condition. Mohammadein and Ei-Amin [36] studied the problem of thermal dispersion-radiation effects on non-Darcy natural convection in a fluid saturated porous medium. ElHakiem and El-Amin [37] investigated thermal radiation effect on non-Darcy natural convection with lateral mass transfer. Temperatures of the drying agent higher than 400◦ C, thermal radiation plays a substantial part in the drying process (Mezhericher et al. [38]). They utilized the model of drying of single droplet containing solids, which has been developed and validated their previous studies (Mezhericher et al. [39]). Drying models which describe the process use the continuity, motion and energy conservation equations (Dolinsky [40]). The model is enhanced by incorporating equations of heat transfer due to thermal radiation phenomenon. Consequently, the overall rate of heat flow to the dried droplet wet particle is determined as the sum of convection and radiation heat flow rates and, in similar manner, the total coefficient of heat transfer can be calculated as the sum of convection and radiation heat transfer coefficients. The combined effect of thermal radiation in nonDarcian convection flow over a vertical impermeable surface embedded in a porous medium having a variable porosity distribution in the presence of thermal dispersion and heat generation/absorption effects have not been studied by any investigators. It is, therefore,

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of great significance and interest to investigate combined effects of convective and radiative flow and heat transfer aspects in porous media. Thus in the present paper we have investigated the effects of thermal radiation and heat generation on the combined convection flow along an vertical flat plate embedded in a porous medium of variable porosity using Forchheimer-Darcy flow model (Chamkha et al. [41]) and utilizing the Rosseland diffusion approximation for thermal radiation flux. The basic equations governing the nonDarcy’s flow and heat transfer are expressed in terms of the local non-similarity boundary layer equation taking ξ as the non-similarity variable and integrated using numerical method described in Lawrence and Rao [42]. To our best knowledge this problem has not been studied before.

2 Formulation of the problem Consider the steady two-dimensional combined convection boundary-layer flow due to radiation over a vertical impermeable semi-infinite surface embedded in a variable porosity porous medium in the presence of heat generation or absorption effects. The Rosseland approximation is used to describe the radiative heat flux in the energy equation. The plate or surface is coincident with the half plate y > 0, x ≥ 0 and the flow far from the plate is a uniform stream in the x-direction parallel to the plate and y-axis is normal to x-axis (see Fig. 1). We assume that the plate is heated to a constant temperature, Tw , which is greater than T∞ , of the ambient fluid saturated porous medium temperature. Fluid is considered to be Newtonian in nature and thermal equilibrium is maintained in each phase. Further it is assumed that no phase change occurs in the fluid. Both the surroundings and the fluid are maintained at a constant temperature T∞ far away from

the plate surface. All physical properties of the fluid are assumed to be constant, except the density, in the buoyancy term of the momentum equation. Both the fluid and the porous medium are assumed to be in local thermal equilibrium. The original model of Darcy was improved by including the boundary and inertial effects using volume average principles by Vafai and Tien [18]. The governing equations for this investigation are based on the balance laws of mass, linear momentum, and energy which are modified to account for the presence of the porous medium, thermal dispersion and radiation effects in addition to the buoyancy and heat generation or absorption effects. Under these assumptions the boundary-layer form of these equations can be written as ∂u ∂v + = 0, ∂x ∂y   ρ ∂u μ ∂ 2u ∂u u = λρgβ(T − T∞ ) + + v 2 ∂x ∂y ε ∂y 2 ε μ u − ρC(y)u2 , − K(y)     ∂T ∂T ∂ ∂qr ∂T ρCp u +v = ke − ∂x ∂y ∂y ∂y ∂y + Q0 (T − T∞ )

(2)

(3)

where u, v are the velocity components along x- and y-axes, ρ is the density (= 0.5542 kg/m3 ), ε is the porosity of the porous medium, λ is assisting (λ = 1) or opposing (λ = −1) flow constant, g is the acceleration due to gravity, β is the co-efficient of thermal expansion (= 2.5 × 10−3 K), μ is the viscosity of fluid (= 13.44 × 10−6 kg/m s), Cp is the specific heat at constant pressure (= 12.014 K J/kg ◦ C), ke is the thermal conductivity of the porous medium, qr is the radiation heat flux and Q0 is the heat generation or absorption co-efficient. It has been shown by Vafai et al. [5] that the dependence of the porosity on the normal distance from the boundary can be well established by the following exponential relationship ε = ε0 (1 − b exp(−cy/d))

Fig. 1 Physical model and coordinate system

(1)

(4)

where ε0 is the porosity at the edge of the boundary layer. The values for ε0 , b and c were chosen to be 0.38, 1 and 2, respectively. For a particle diameter d = 5 mm these values were found to give good approximation to the variable porosity data (Benenati

Meccanica (2009) 44: 133–144

137

and Brosilow [3]). Nithiarasu et al. [43] used the same values on their work on natural convection heat transfer in a fluid saturated variable porosity medium inside a rectangular enclosure. The type of decay of porosity as the normal distance increases given by (4) is well established and has been extensively used by many investigators for studying flow in porous media with variable porosity. It is also established that K and C vary with porosity as follows: K(y) =

d 2 ε2 , 150(1 − ε)2

1.75 C(y) = √ 150εK(y)

(5)

where kf is the fluid thermal conductivity, ks is the thermal conductivity of the porous medium, Pr is the Prandtl number (= μCp /kf ), Rew is the Reynolds number based on the particle diameter (= ρU∞ d/μ), and U∞ is the free stream velocity. The porous medium is assumed to be nonmetallic (glass fibers), such that ks  kf and hence the last term of (5) is neglected. The physics of the problem suggests the following boundary conditions: u(x, 0) = 0, u(x, y) → U∞ ,

v(x, 0) = 0,

T (x, 0) = Tw , (6)

T (x, y) → T∞ ,

as y → ∞ (7)

where Tw is the constant wall or surface temperature. In addition, the radiation heat flux qr is employed according to Rosseland approximation such as qr = −

T 4 = 4T∞ 3 − 3T∞ 4 . The flow and heat transfer problem represented by (1) to (7) has no similar or exact solutions. To facilitate the solutions of the problem non-similarity transformations is used (Ramachandran et al. [44]): ξ=

where K(y) is the permeability of the porous medium, C(y) is the inertia co-efficient which vary with permeability of the porous medium. The effective thermal conductivity of the porous medium is of the following form (Amiri and Vafai [12]): ke = kf (ε + 0.1Pr Rew u/U∞ ) + (1 − ε)ks

about the free stream temperature T∞ and neglecting higher-order terms to yield.

4σ ∗ ∂T 4 3k ∗ ∂y

where σ ∗ and k ∗ are the mean absorption co-efficient and the Stefan-Boltzmann constant (= 5.669 × 10−8 W/m2 K4 ), respectively. According to Raptis [32], the fluid-phase temperature differences within the flow are assumed to be sufficiently small so that T 4 may be expressed as a linear function of temperature. This is done by expanding T 4 in a Taylor series

x , L

 η=

∂ψ , u= ∂y

U∞ νx

1/2 y, (8)

∂ψ v=− , ∂x

ψ(x, y) = (νU∞ x)1/2 F (ξ, η), T (x, y) − T∞ = (Tw − T∞ )θ (ξ, η)

(9)

where ψ is the stream functions satisfying the continuity equation, θ is the dimensionless temperature, ξ is the transformed tangential coordinate, η is the transformed normal coordinate and F is the reduced stream function. Substituting (8) and (9) into the governing equations result in the transformed equations: ξε F F 1.75ξ 2 − F − √ F 2ε DaRe 150ε 3 Da   Gr ξ ∂F ∂F F −F , + λξ ε 2 θ = ε ∂ξ ∂ξ Re

F +

(10)

rk + Nr 1 θ + 0.1Rew θ F + F θ Pr 2  √  ε0 bc  c ξ Re + ξ Re exp − η θ + ξ δθ Pr Rew Rew   ∂θ ∂F −θ . (11) =ξ F ∂ξ ∂ξ The transformed boundary conditions become F (ξ, η) = 0,

F (ξ, η) = 0,

θ (ξ, η) = 1 at η = 0, F (ξ, η) → 1,

θ (ξ, η) → 0 as η → ∞

(12) (13)

where a prime denotes partial differentiation with respect to η and other physical parameters are defined as follows

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Meccanica (2009) 44: 133–144

(10) and (11) for F = F0 and θ = θ0 at ξ = 0 become gβ(Tw − T∞ )L3 U∞ L , , Re = 2 ν ν    √ c ξ Re ε = ε0 1 + b exp − η , Rew

F0 +

Gr =

δ=

(14)

Q0 L , ρCp U∞

Da =

Rew =

ρU∞ d , μ

3 16σ ∗ T∞ , 3Kk ke rk = = ε + 0.1Pr Rew F kf

Nr =

(18)

(15)

−1/2

and Nux Rex

F1 F1 ξ ε 1.75ξ 2 − F1 − √ F 3 2ε DaRe 150ε Da Gr + λξ ε 2 θ1 Re   ξ F1 − F0 F1 − F0 F1 − F1 , = ε ξ ξ

F1 +

here, δ is the dimensionless heat generation/absorption parameter, Gr is Grashof number, Re is Reynolds number, Da is the Darcy’s number, Pr = μCp /kf is the Prandtl number and L is the characteristic length. The important physical quantities such as the local skin-friction co-efficient, Cf , and the local Nusselt number, Nux , are expressed in dimensionless form as follows: 1/2

1 rk + Nr θ0 + 0.1Rew θ0 F0 + F0 θ0 = 0. Pr 2

(17)

By using a central difference approximations for (10) and (11) for F = F1 and θ = θ1 at ξ = ξ become

Re2w ε3 , Re2 150(1 − ε)2

Cf Rex = 2F (ξ, 0)

F0 F0 = 0, 2ε

= −θ (ξ, 0) (16)

where Rex = U∞ x/ν is the local Reynolds number.

3 Numerical procedure The nonlinear boundary value problem (10)–(13) can not be solved in closed form, so we are required to solve this problem numerically to describe the physics of the problem. The equation governing the boundary layer flow are non-similar and a three-point backward finite-difference approximation similar to that discussed by Lawrence and Rao [42] is appropriate and the resulting nonlinear ordinary differential equations are integrated by fourth-order Runge-Kutta method with shooting technique. Assuming that a singular solution exist locally in the neighborhood of ξ = 0 and that frictional heating does not start until ξ > 0 along the sheet, the boundary layer equations

(19)

rk + Nr 1 θ + 0.1Rew θ1 F1 + F1 θ1 Pr 2  √   ε0 bc c ξ Re − ξ Re exp − η θ1 + ξ δθ1 Pr Rew Rew   θ1 − θ0 F1 − F0 − θ1 . (20) = ξ F1 ξ ξ Similarly, (10) and (11) for F = Fn and θ = θn at ξ = nξ for n ≥ 2 are approximated by using a threepoint backward finite -difference as follows: Fn Fn nξ ε 1.75ξ 2 − Fn − √ Fn 2ε DaRe 150ε 2 Da Gr + λξ ε 2 θn Re   3Fn − 4Fn−1 + Fn−2 nξ Fn = ε 2ξ    3Fn − 4Fn−1 + Fn−2 F , − 2ξ

Fn +

rk + Nr 1 θn + 0.1Rew θn Fn + Fn θn Pr 2  ε0 bc − nξ Re Pr Rew  √  −c nξ Re × exp θn + nξ δθn Rew

(21)

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139



3θn − 4θn−1 + θn−2 F2 2ξ  3Fn − 4Fn−1 + Fn−2 θn . − 2ξ

−θ (0, 0)

Table 1 Comparison of for different values of Pr at ε = 1, ξ = 0, and fw = 0 with Kuznetsov and Nield [45] and Aydin and Kaya [46] for λ = 1

= nξ

(22)

The boundary condition for (17)–(22) are Fn (ξ, η) = 0,

Fn (ξ, η) = 0,

θn (ξ, η) = 1 at η = 0, Fn (ξ, η) → 1,

θn (ξ, η) → 0,

(23)

as η → ∞ for n ≥ 0. The above system of equations is integrated using fourth-order Runge-Kutta method with shooting technique for ξ = 0.025 and η = 0.025. The convergence criterion employed is based on the difference between the current and the previous iterations result. If this difference reaches 10−6 for all points in the η directions, the numerical solution was assumed to have converged and the iteration process was terminated.

4 Results and discussions Extensive calculations have been performed to obtain the flow and temperature field with the parametric values of Nr, Gr, Re, Rew and Pr at ξ = 0.1. When the value of Nr is equal to zero, the present problem reduces to with those obtained by Chamka et al. [34]. In order to assess the accuracy of the numerical results, the validity of the numerical code developed has been checked for a limiting case. We have compared our results for −θ (0, 0) with those given by Kuznetsov and Nield [45] and Aydin & Kaya [46] in Table 1 for the values when ε = 1, λ = 1 and ξ = 0. For ε = 1, ξ = 0, and fw = 0, we have also compared the present results of −θ (0, 0) with those given by Lin and Lin [47], Yih [48], Chamkha et al. [49] and Aydin & Kaya [46] in Table 2 and a very good agreement has been obtained with their results. Table 3 shows the numerical values of local skinfriction coefficient, Cf Re1/2 , and Nusselt number, −1/2 Nux Rex , for two values of Rew = 100 and 300, against ξ in [0, 0.1], taking Nr = 0.1, 0.5, 1.0 and 5.0, Gr = 100, Re = 10, Pr = 0.7 and δ = 0. From this table it is noticed that increasing the value of Nr leads to rise in the value of skin-friction coefficient and fall in the value of local Nusselt number whereas reverse

Pr

Kuznetsov and

Aydin and

Nield [45]

Kaya [46]

Present result

0.1

0.1580

0.1480

0.1400

1

0.3320

0.3320

0.3320

5

0.5700

0.5765

0.5767

10

0.7300

0.7278

0.7281

20

0.9100

0.9176

0.9184

30

1.0500

1.0550

1.0517

40

1.1500

1.1550

1.1577

50

1.2450

1.2440

1.2473

60

1.3200

1.3210

1.3255

70

1.3900

1.3900

1.3955

80

1.4500

1.4520

1.4590

90

1.5700

1.5100

1.5175

100

1.5700

1.5731

1.5718

trend is seen by increasing the value of Rew . This is due to the fact that increase in the value of Nr implies more interaction of radiation with the momentum and less interaction with thermal boundary layer. Further, it is observed that as ξ increases there is also increase in the values of skin-friction co-efficient and local Nusselt number for all values of Nr. Similar results are seen by increasing the value of Rew . The numerical results computed for different values of physical parameters are shown in Fig. 1 to 9 for λ = 1. The variations of tangential velocity profiles for various values of Grashof number (i.e., buoyancy effects) in the boundary layer are depicted in Fig. 2 at ξ = 0.1. It is seen from this figure that the effects of Gr is to increase the tangential velocity in the boundary layer. This demonstrates the role of convection as to increase the tangential velocity component. The effect is more prominent around η = 2 and its effects diminishes as η → ∞. In addition, as Gr becomes greater than 100, an overshoot above the free-stream value in the velocity profiles occurs close to the boundary because buoyancy force act like a favorable pressure gradient and accelerates the fluid within the boundary layer which is similar to the result obtained by Takhar et al. [50] for viscous flows. Figure 3 illustrates the effect of the Grashof number Gr on the temperature profile at ξ = 0.1 in the boundary layer. It is clearly seen from this figure that increas-

140

Meccanica (2009) 44: 133–144 Table 2 Comparison of the value

Pr

−θ (0, 0)

for values of Pr at ε = 1, ξ = 0 for λ = 1

Lin & Lin [47]

Yih [48]

Chamkha et al. [49]

Aydin & Kaya [46]

Present result

0.01

0.051559

0.051589

0.051830

0.051437

0.051589

0.1

0.140032

0.140034

0.142003

0.148053

0.140029

1.0

0.332057

0.332057

0.332173

0.332000

0.332057

10.0

0.728148

0.728141

0.728310

0.727801

0.728142

1.571860

1.571831

1.572180

1.573141

0.571835

100

−1/2

Table 3 Variation of Cf Re1/2 and Nux Rex δ = 0, λ = 1 Nr

0.0

0.1

0.5

1.0

5.0

ξ

for various values of Nr and ξ for Gr = 100, Re = 10.0, Rew = 100, 300, Pr = 0.70, −1/2

1/2

Cf Rex

Nux Rex

Rew = 100

Rew = 300

Rew = 100

Rew = 300

0

0.76179178

0.76179178

0.95770484

2.00306894

0.25

0.79908050

0.79759554

0.97384590

2.04085068

0.05

0.83534416

0.83295673

0.98952008

2.07752259

0.075

0.87060721

0.86784486

1.00441195

2.11302564

0.1

0.90503135

0.90230481

1.01862460

2.14747562

0

0.76179178

0.76179178

0.86306863

1.81386007

0.025

0.80007751

0.79818958

0.87767852

1.84776277

0.05

0.83730019

0.83412476

0.89184505

1.88064446

0.075

0.87347909

0.86956429

0.90528730

1.91245154

0.1

0.90877847

0.90455414

0.91809882

1.94329064

0

0.76179178

0.76179178

0.65693220

1.34095358

0.025

0.80214230

0.80006622

0.66753824

1.36514060

0.05

0.84135195

0.83781180

0.67778742

1.38854257

0.075

0.87942950

0.87498701

0.68748081

1.41111927

0.1

0.91654472

0.91164167

0.69668929

1.43295346

0

0.76179178

0.76179178

0.52253792

1.03616331

0.025

0.80392010

0.80174221

0.53052930

1.05408914

0.05

0.84483887

0.84110127

0.53822737

1.07139565

0.075

0.88454701

0.87981924

0.54548546

1.08805114

0.1

0.92321917

0.91794983

0.55235954

1.10412194

0

0.76179178

0.76179178

0.25743222

0.44550573

0.025

0.80946420

0.80753714

0.26030976

0.45148716

0.05

0.85570842

0.85245702

0.26304507

0.45721740

0.075

0.90048832

0.89646836

0.26559157

0.46268286

0.1

0.94399390

0.93964063

0.26797517

0.46791311

ing the value of Gr there is a tendency of increasing in the thermal distribution close to the boundary layer because increasing the value of Gr (i.e. increasing the buoyancy effects) has the tendency to increase the coupling between the flow and thermal distributions.

Figure 4 is a plot of tangential velocity profile in the boundary layer for different values of Reynolds number, Re, for fixed value of Gr = 100 and radiation parameter Nr = 0.1. This figure shows that by increasing the Reynolds number Re, there is decrease in the tan-

Meccanica (2009) 44: 133–144

141

Fig. 2 Effects of Gr on the tangential velocity profiles in the boundary layer

Fig. 4 Effects of Re on the tangential velocity profiles in the boundary layer

Fig. 3 Effects of Gr on the temperature profiles in the boundary layer

Fig. 5 Effects of Re on the temperature profiles in the boundary layer

gential velocity profile in the boundary layer because the buoyancy dominated regime occurs for lower values of Re whereas higher values of Re correspond to the forced convection dominated regime. It is also observed that for Re = 15 and 20 the tangential velocity is almost same. Thus it is concluded that of higher values of Re on tangential velocity profile. In addition, it is bound that tangential velocity is almost constant after η = 5 but for small value of Re = 5, there is formation of peak in the value of f close to the leading edge of the plate. The effect of Re on velocity profile becomes negligibly small for higher values of Re. The effects of Re on temperature profiles in the boundary layer at ξ = 0.1 for various values of Re = 5, 10 and 20 for Gr = 100 which correspond to mixed convection is shown in Fig. 5. It is interesting to note that the effect of Re is to reduce the thermal distrib-

ution in the boundary layer and its effect diminishes for higher values of Reynolds number Re. It is due to the fact that lower values of Re corresponds to the buoyancy dominated regime whereas higher values of Re corresponds to the forced convection dominated regime. The effects of radiation parameter Nr on temperature profile are shown in Fig. 6 at ξ = 0.1. The effects of Nr are prominently seen throughout the boundary layer. It is interesting to see that the effect of Nr is to increase the temperature distribution. This is due to the fact that increase in the value of Nr implies more interacting of radiation in the thermal boundary layer and hence increases the value of temperature profile in the thermal boundary layer. Further, it is observed that the temperature decreases steadily with η for lower values of Nr whereas it decreases at a fast rate for higher values of Nr in the boundary layer.

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Fig. 6 Effects of Nr on the temperature profiles in the boundary layer

Fig. 7 Effects of η on the temperature profiles in the boundary layer

Figure 7 depicts the effects of heat generation/ absorption coefficient on temperature profiles for fixed value of radiation parameter. The direction of heat flow depends both on temperature difference (Tw − T∞ ) and the temperature gradient θ (0). We have considered both positive and negative values of δ. When δ is positive, we have a heat source in the boundary layer when Tw < T∞ and heat sink when Tw > T∞ . For the case of cooled wall Tw < T∞ , the heat transfer is from the fluid to the wall even without heat source. Hence the presence of heat source δ > 0 will further increase the heat flow to the wall. But when δ is negative, which indicates a heat source is for Tw > T∞ and a heat sink for Tw < T∞ . For the case of heated wall (Tw > T∞ ), the presence of a heat source δ < 0 creates a layer of hot fluid adjacent to the surface and therefore the heat from the wall decreases.

Meccanica (2009) 44: 133–144

Fig. 8 Effects of assisting and opposing flows on the tangential velocity profiles

For cooled wall Tw < T∞ , the presence of heat absorption δ < 0 blanket the surface with a layer of cooled fluid, and therefore heat flow along the plate decreases. This result coincides with those obtained by Acharya et al. [51]. It is well known that the heat generation causes the fluid temperature to increase which has a tendency to increase the thermal buoyancy effects. These effects are clearly shown in this figure. On the other hand heat absorption produces opposite effects i.e. it decreases the temperature distribution along the plate. The effects of opposing flow on tangential velocity profiles are shown in Fig. 8. It is seen that in assisting flow the effect of Gr is to increase the tangential velocity profiles and its effect reverses in the opposing flow. The increase in velocity profiles with increase in Gr for the case of assisting flow demonstrate the role of convection current to increase the velocity distribution, which indicates that the buoyancy force acts like a favorable pressure gradient and thereby enhances the fluid velocity in the boundary layer. Further, in the case of opposing flow, the velocity distribution decreases with increase in the Grashof number which acts like a negative buoyancy force and thereby retards the fluid velocity distribution in the boundary layer. Figure 9 depicts temperature profiles for various values of the Reynolds number based on the particle diameter Rew at ξ = 0.1 keeping other physical parameters fixed for aiding or assisting flow conditions. It is observed that Rew increases the value of temperature close to the wall whereas its effects get reversed thereafter till it attains the value θ = 0. Also, thermal boundary-layer thickness decreases as Rew increases.

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creases with the increase of Nr. Finally, it is observed that the effect of Gr in assisting flow is to increase the tangential velocity profile and this affect is reversed for opposing flow. Acknowledgements The authors express their sincere appreciation to the editor and the reviewers for their valuable comments and constructive suggestions which helped to improve the quality of the paper considerably. University Grants Commission (UGC), New Delhi, India financially supported the work of DP under SAP-DRS (Phase-I) Programme Grant No. F.510/ 8/ DRS/ 2004 (SAP-I) and HM is thankful to the Director, BITM for his encouragement.

Fig. 9 Effects of Rew on the temperature profiles in the boundary layer for Nr = 1.0

This behavior is believed to be related to or associated with the thermal dispersion effect. It may be of interest to some investigators to understand the effect of angle between flow and flat plate on convection flow. The increase in the inclination to the horizontal leads to increase in the values of skin friction coefficient, Nusselt number coefficient and boundary layer thickness (Hossain [52]). Further, both skin friction coefficient and Nusselt number coefficient may increase due to increase in the thermal radiation parameter Nr. When the plate is at a negative angle to the horizontal, the separation of the boundary layer would occur downstream from the leading edge of the plate as the opposed buoyancy force and induced pressure gradient are of comparable magnitude. The present work can be extended to analyze these effects due to change in the angle between flow and flat plate on convection flow.

5 Conclusions A comprehensive parametric study on combined convection heat transfer in a porous medium having variable porosity distribution by considering heat generation/absorption and radiation effects concludes that an increase in the radiation parameter Nr leads to increasing the skin-friction co-efficient and to decrease in the local Nusselt number. This is due to the fact that the increase of the value of Nr implies more interaction of radiation with the momentum boundary layer and less interaction of radiation thermal boundary layer. Further it was also found that the temperature profile in-

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