Meccanica (2012) 47:153–161DOI 10.1007/s11012-011-9423-3

Forced convective flow and heat transfer over a porous platein a Darcy-Forchheimer porous medium in presenceof radiation

Swati Mukhopadhyay · Prativa Ranjan De ·Krishnendu Bhattacharyya · G.C. Layek

Received: 17 May 2010 / Accepted: 11 January 2011 / Published online: 26 January 2011© Springer Science+Business Media B.V. 2011

Abstract A mathematical model is presented for an-alyzing the boundary layer forced convective flow andheat transfer of an incompressible fluid past a plateembedded in a Darcy-Forchheimer porous medium.Thermal radiation term is considered in the energyequation. The similarity solutions for the problem areobtained and the reduced nonlinear ordinary differen-tial equations are solved numerically. It is noticed thatthe boundary layer decreases with an increase in thevalue of inertial parameter and in this case the tem-perature profile is found to decrease smoothly withinthe boundary layer. In case of porous plate, fluid ve-locity increases whereas non-dimensional temperaturedecreases for increasing values of suction parameter.The rate of heat transfer increases with the increasingvalues of Prandtl number. The effect of thermal radia-tion on temperature field is also analyzed.

Keywords Forced convective flow ·Darcy-Forchheimer porous medium · Thermalradiation · Suction

Nomenclaturecp specific heat at constant pressure.F non-dimensional stream function.

S. Mukhopadhyay (�) · P.R. De · K. Bhattacharyya ·G.C. LayekDepartment of Mathematics, The University of Burdwan,Burdwan 713104, West Bengal, Indiae-mail: swati_bumath@yahoo.co.in

F ′ streamwise velocity.k Darcy permeability of the porous medium.k′ Forchheimer resistance factor.k1 parameter of the porous medium.k2 inertial parameter.k∗ absorption coefficient.N radiation parameter.Pr Prandtl number.qr radiative heat flux.S suction parameter.T temperature of the fluid.Tw temperature of the wall of the surface.T∞ free-stream temperature.u∞ free stream velocity.υw velocity of suction [υw > 0].Greek symbolsη similarity variable.κ coefficient of thermal conductivity.μ dynamic viscosity.ν kinematic viscosity.ψ stream function.σ Stefan-Boltzman constant.ρ density of the fluid.θ non-dimensional temperature.θr temperature ratio parameter.

1 Introduction

In recent years, considerable attention has been de-voted to the study of boundary layer flow behavior and

154 Meccanica (2012) 47:153–161

heat transfer characteristics of a Newtonian fluid pasta vertical plate embedded in a fluid saturated porousmedium because of its extensive applications in engi-neering processes, especially in the enhanced recov-ery of petroleum resources and packed bed reactors(Pal and Shivakumara [1]). A better understanding ofconvection through porous medium can benefit severalareas like insulation design, grain storage, geother-mal systems, heat exchangers, filtering devices, metalprocessing, catalytic reactors etc.

The radiative effects have important applications inphysics and engineering. The radiative heat transfereffects on different flows are very important in spacetechnology and high temperature processes. But verylittle is known about the effects of radiation on theboundary layer. At high temperatures, thermal radia-tion can significantly affect the heat transfer and thetemperature distribution in the boundary layer flow ofparticipating fluid. Thermal radiation effects may playan important role in controlling heat transfer in indus-try where the quality of the final product depends onthe heat controlling factors to some extent. High tem-perature plasmas, cooling of nuclear reactors, liquidmetal fluids, power generation systems are some im-portant applications of radiative heat transfer from avertical wall to conductive gray fluids. The effect ofradiation on heat transfer problems have studied byHossain and Takhar [2], Takhar et al. [3], Hossain etal. [4]. In most of the above mentioned studies, thelinearization process for the radiation term is used.

Cheng and Minkowycz [5] and Cheng [6] studiedthe free convective flow in a saturated porous medium.Wilks [7] had studied the combined forced and freeconvection flow along a semi-infinite plate extendingvertically upwards. Lai and Kulacki [8] discussed thecoupled heat and mass transfer by mixed convectionfrom a vertical plate in a saturated porous medium.Many problems of Darcian and non-Darcian mixedconvection about a vertical plate had been reported, asin Hsu and Cheng [9], Vafai and Tien [10]. Soundal-gekar et al. [11] investigated the combined free andforced convection flow past a semi-infinite plate withvariable surface temperature. Hong et al. [12] havestudied analytically the non-Darcian effects on a verti-cal plate natural convection in porous media. Kaviany[13] used the Darcy-Brinkman model to study the ef-fects of boundary and inertia forces on forced convec-tion over a fixed impermeable heated plate embeddedin a porous medium. Chen and Ho [14] have stud-

ied the effects of flow inertia on vertical, natural con-vection in saturated porous media. Kumari et al. [15]have investigated the non-Darcian effects on forcedconvection heat transfer over a flat plate in a highlyporous medium. Damesh et al. [16] has discussed themagnetic field and thermal radiation effects on forcedconvection flow. Duwairi and Damesh [17, 18] ana-lyzed thermophoresis effects on mixed convection heatand mass transfer from vertical surfaces embedded ina saturated porous media and the interaction of ther-mal radiation and thermophoresis particle depositionin mixed convection flow over a nonisothermal ver-tical flat plate embedded in a fluid-saturated porousmedium. Damesh and Duwairi [19] investigated thenatural convection heat and mass transfer by steadylaminar boundary layer flow over an isothermal verti-cal flat plate embedded in a porous medium. Recently,Mukhopadhyay and Layek [20] analyzed the effectsof porous media on forced convection flow. But inthis paper the linear Darcy term describing the distrib-uted body force exerted by the porous medium was re-tained whereas the non-linear Forchheimer term wasneglected. A new dimension is added to the abovementioned study by considering the effects of Darcy-Forchheimer porous media.

It is well known that Darcy’s law is an empiricalformula relating the pressure gradient, the bulk vis-cous fluid resistance and the gravitational force fora forced convective flow in a porous medium. De-viations from Darcy’s law occur when the Reynoldsnumber based on the pore diameter is within therange of 1 to 10 (Ishak et al. [21]). For flow througha porous medium with high permeability, Brinkman[22] as well as Chen et al. [23] argue that the momen-tum equation must reduce to the viscous flow limitand advocate that classical frictional terms be addedin Darcy’s law. The aim of this paper is to studythe steady forced convection flow and heat transferpast a porous plate placed in a fluid-saturated porousmedium using the Darcy-Forchheimer model.

Consideration of thermal radiation term in the en-ergy equation makes the equation highly non-linear.Using similarity variables, a third order and a secondorder differential equations corresponding to the mo-mentum and thermal boundary layer equations are ob-tained. These equations are solved numerically usingshooting method. The effects of the permeability of theporous medium, inertial parameter, suction parameter,temperature ratio parameter and the influence of ther-mal radiation on velocity and temperature fields are

Meccanica (2012) 47:153–161 155

investigated and analyzed with the help of their graph-ical representations.

2 Formulation of the problem

We consider a forced convective, two-dimensionalsteady laminar boundary-layer flow of an incompress-ible, viscous liquid over a flat plate of very small thick-ness and much larger breadth, embedded in a porousmedium. In the analysis of flow in porous media,the differential equation governing the fluid motion isbased on Darcy-Forchheimer model, which accountsfor the drag (represented by the Darcy term) exertedby the porous media as well as the inertia effect (rep-resented by the non-linear Forchheimer term). Takinginto account the thermal radiation term, the governingequations for boundary layer flows and heat transferare written as

∂u

∂x+ ∂υ

∂y= 0, (1)

u∂u

∂x+ υ

∂u

∂y= ν

∂2u

∂y2− ν

k(u − u∞)

− k′√

k(u2 − u2∞), (2)

u∂T

∂x+ υ

∂T

∂y= κ

ρcp

∂2T

∂y2− 1

ρcp

∂qr

∂y, (3)

where the last two terms on the right-hand side of (2)describe the non-linear Darcy-Forchheimer resistanceof the surrounding porous medium. At higher veloc-ities, inertial effects become appreciable, causing anincrease in the form drag. Experimental observationsdo indicate that the pressure drop in the bulk of aporous medium is proportional to a linear combina-tion of flow velocity and square of the flow velocity.The square term is caused by the inertial effects (Vafaiand Tien [10]).

Since the velocity of the fluid is low (laminar flow),the viscous dissipative heat is assumed to be negligi-ble here. The x co-ordinate is measured from the lead-ing edge of the plate, y co-ordinate is measured alongthe normal to the plate. Here u and υ are the compo-nents of velocity respectively in the x and y directions,μ is the coefficient of fluid viscosity, ρ is the fluid den-sity, ν = μ/ρ is the kinematic viscosity, k = k0x is theDarcy permeability of the porous medium, k0 is the

initial permeability, k′ = k′0√x

. Forchheimer resistance

factor, k′0 is the Forchheimer constant, which has been

experimentally measured for different porous media.Typical values of k′

0 range from 0.05 to 0.5, and a rep-resentative value for many materials is 0.1 (Wu et al.[24]). T is the temperature, κ is the thermal conduc-tivity of the fluid, u∞ is the free stream velocity, T∞ isthe temperature at infinity, qr is the radiative heat flux,cp is the specific heat at constant pressure.

Using Rosseland approximation for radiation

(Brewster [25]) we can write qr = − 4σ3k∗ ∂T 4

∂ywhere σ

is the Stefan-Boltzman constant, k∗ is the absorptioncoefficient.

The appropriate boundary conditions for the prob-lem are given by

u = 0, υ = 0, T = Tw at y = 0, (4)

u = u∞, T = T∞ as y → ∞. (5)

Here Tw is the wall temperature, T∞ is the free streamtemperature assumed constant with Tw > T∞.

For the consideration of porous plate, embedded ina porous medium, a suction/injection velocity is giventhrough the plate. The boundary condition for thevelocity component υ , then, becomes υ = ±υw(x),υw(x) = υ0√

xis the velocity of suction [υw(x) > 0] or

injection [υw(x) < 0] of the fluid, υ0 is the constantvalue of suction/injection.

2.1 Similarity analysis and solution procedure

We now introduce the following relations for u,υ as

u = ∂ψ

∂y, υ = −∂ψ

∂x(6)

where ψ is the stream function.We also introduce the following dimensionless

variables

θ = T − T∞Tw − T∞

, (7)

and

η = y

√u∞νx

, ψ = √u∞νxf (η). (8)

Using the relations (6) and (7) in the boundary layerequation (2) and in the energy equation (3) we get the

156 Meccanica (2012) 47:153–161

following equations

∂ψ

∂y

∂2ψ

∂x∂y− ∂ψ

∂x

∂2ψ

∂y2= ν

∂3ψ

∂y3− ν

k

(∂ψ

∂y− u∞

)

− k′√

k

{(∂ψ

∂y

)2

− u2∞}, (9)

and

∂ψ

∂y

∂θ

∂x− ∂ψ

∂x

∂θ

∂y

= κ

ρcp

∂2θ

∂y2+ 16σT 3∞

3ρcpk∗

[{1 + (θr − 1)θ}3 ∂2θ

∂y2

+ 3{1 + (θr − 1)θ}2(θr − 1)

(∂θ

∂y

)2](10)

where θr = Tw

T∞ is the temperature ratio of the medium.The boundary conditions (4) and (5) then become

∂ψ

∂y= 0,

∂ψ

∂x= υw, θ = 1 at y = 0, (11)

∂ψ

∂y= u∞, θ = 0 as y → ∞. (12)

Using (8), (9) and (10) finally can be put in the follow-ing form

f ′′′ + 1

2ff ′′ − 1

DaxRex

(f ′ − 1)

− k′x√k{(f ′)2 − 1} = 0, (13)

θ ′′

Pr

[1 + 4

3N{1 + (θr − 1)θ}3

]

+ 1

Pr

4

N{1 + (θr − 1)θ}2(θr − 1)θ ′2

+ 1

2f θ ′ = 0, (14)

where 1DaxRex

= νk0u∞ , Dax = k

x2 = k0x

is the localDarcy number, Rex = u∞x

νis the local Reynolds num-

ber, Pr = μcp

κis the Prandtl number, N = κk∗

4σT 3∞is the

radiation parameter.Equation (13) can be written as

f ′′′ + 1

2ff ′′ − k1(f

′ − 1) − k2(f′2 − 1) = 0, (15)

where k1 = 1DaxRex

= νk0u∞ is the parameter of the

porous medium (Damesh [26]) and k2 = k′0√k0

is the

inertial parameter. k−11 and k2 will reflect the effect of

Darcian and Forchheimer flows on the present prob-lem.

In view of these, the boundary conditions finallybecome

f ′ = 0, f = S, θ = 1 at η = 0, (16)

and

f ′ = 1, θ = 0 at η → ∞, (17)

where S = (−2υ0u∞

√x)(Rex)

12 = −2υ0√

u∞νis the suction

(υ0 < 0)/ injection (υ0 > 0) parameter.When k1 = 0 = k2, (15) reduces to the equation of

boundary layer flow on a flat plate at zero incidence(Batchelor [27]).

Equations (14) and (15) along with boundary con-ditions are solved by converting them to an initialvalue problem. We set⎧⎪⎨⎪⎩

f ′ = z,

z′ = p,

p′ = k1(z − 1) + k2(z2 − 1) − Fp/2,

(18)

⎧⎪⎪⎪⎨⎪⎪⎪⎩

θ ′ = q,

q ′ = ( 3N Pr3N+4{1+(θr−1)θ}3

)

× [− 12f q − 4

N Pr {1 + (θr − 1)θ}2(θr − 1)θ ′2],

(19)

with the boundary conditions

f (0) = S, f ′(0) = 0, θ(0) = 1. (20)

In order to integrate (18) and (19) as an initial valueproblem we require a value for p(0) i.e. f ′′(0) andq(0) i.e. θ ′(0) but no such values are given in theboundary. The suitable guess values for f ′′(0) andθ ′(0) are chosen and then integration is carried out. Wecompare the calculated values for f ′ and θ at η = 10(say) with the given boundary condition f ′(10) = 1and θ(10) = 0 and adjust the estimated values, f ′′(0)

and θ ′(0), to give a better approximation for the so-lution. We take the series of values for f ′′(0) andθ ′(0), and apply the fourth order classical Runge-Kutta method with step-size �η = 0.01. The aboveprocedure is repeated until we get the converged re-sults within a tolerance limit of 10−5.

Meccanica (2012) 47:153–161 157

(a)

(b)

Fig. 1 (a) Variation of horizontal velocity f ′(η) with η whenPr = 0.5. (b) Variation of temperature θ(η) with η whenPr = 0.5

3 Results and discussions

In order to get a clear insight of the physical problem,numerical computations have been carried out usingthe method described in the previous section for var-ious values of different parameters such as the para-meter of the porous medium (k1), inertial parameter(k2), thermal radiation parameter (N ), suction parame-ter (S), temperature ratio parameter (θr) and Prandtlnumber (Pr) encountered in this problem. For illustra-tions of the results, numerical values are plotted in thefigures Fig. 2(a) to Fig. 6(b).

In order to assess the accuracy of the method, theresults (in the absence of porous media and thermalradiation) are compared with those (in the absence ofmagnetic field and thermal radiation) of Damesh et al.[16] [Fig. 1(a) and (b)]. The results are found to agreewell.

Figure 2(a) and (b) illustrate the effects of the per-meability of the medium and the inertial effects re-spectively on velocity field in presence of suction

(a)

(b)

Fig. 2 (a) Velocity profiles for several values of the permeabil-ity parameter k1. (b) Velocity profiles for several values of theinertial parameter k2

(S = 1) and thermal radiation (N = 1). Both thesefigures show the variations of velocity profiles for in-creasing values of porosity parameter k1 (k1 = 0, 0.1,0.3, 0.6, 0.9) and inertial parameter k2 (k2 = 0, 0.1,0.3, 0.6, 0.9) when the other parameters (θr and Pr)are kept constant (i.e. θr = 1.1 and Pr = 0.7). Fig-ure 2(a) clearly indicates that the thickness of the ve-locity boundary layer decreases. In this case, horizon-tal velocity is found to increase with the increasingvalues of the permeability of the medium. With a risein permeability of the medium, the regime becomesmore porous. As a consequence, the Darcian bodyforce decreases in magnitude (as it is inversely pro-portional to the permeability). The Darcian resistanceacts to decelerate the fluid particles in continua. Thisresistance diminishes as permeability of the mediumincreases. So progressively less drag is experiencedby the flow and flow retardation is thereby decreased.Hence the velocity of the fluid increases as the para-meter (k1) increases in both cases. Here k1 = 0 repre-sents the effects of non-Darcy porous media whereas

158 Meccanica (2012) 47:153–161

(a)

(b)

Fig. 3 (a) Temperature profiles for several values of the perme-ability parameter k1. (b) Temperature profiles for several valuesof the inertial parameter k2

k2 = 0 represents the effects Darcy porous media. Itis also observed that an increase in the value of in-ertial parameter (k2) leads to an increase in the ve-locity profile within the boundary layer [Fig. 2(b)]. Itis also important to note that the boundary layer de-creases with an increase in the value of inertial para-meter. Thus, the non-Darcian term (i.e. Forchheimerterm) has a very significant effect on the velocity dis-tribution.

Next, we concentrate on the nature of tempera-ture profiles for several increasing values of perme-ability parameter k1 [Fig. 3(a)] and the inertial pa-rameter k2 [Fig. 3(b)] respectively by taking N = 1,θr = 1.1 and Pr = 0.7 in presence of suction (S = 1).From both figures, it is found that the temperature de-creases with the increase of k1 and also of k2 withinthe boundary layer. The rate of heat transfer (the ther-mal boundary layer thickness becomes thinner) is en-hanced when the velocity boundary layer thickness de-creases [Fig. 3(a)]. Figure 3(b) shows that the rate of

(a)

(b)

Fig. 4 (a) Velocity profiles for several values of suction para-meter S in case of non-porous media. (b) Velocity profiles forseveral values of suction parameter S in case of porous media

cooling is much faster for higher values of the inertialparameter k2.

We consider now, the case of porous flat plate em-bedded in both non-porous (k1 = 0 = k2) and porous(k1 = 0.1, k2 = 0.1) media. The velocity and tempera-ture distributions for various values of suction parame-ter (S) are shown in Figs. 4(a)–5(b) for the fixed valuesN = 1, θr = 1.1 and Pr = 0.7 in the absence of porousmedium [Figs. 4(a) & 5(a) for k1 = 0 = k2] as well asin presence of porous medium [Fig. 4(b) & Fig. 5(b)for k1 = 0.1 = k2]. With the increasing S (S > 0), thehorizontal velocity is found to increase [Fig. 4(a) &4(b)] i.e. suction causes to increase the velocity of thefluid. Since the effect of suction is to suck away thefluid near the wall, the momentum boundary layer isreduced due to suction (S > 0). Consequently the ve-locity increases. Hence the velocity gradient and so theskin friction increases with increasing S (S > 0).

In Fig. 4(b), velocity profiles are exhibited for thesame set of values of S (i.e. S = 0, 0.5, 1, 1.5, 2, 3)with k1 = 0.1 = k2, N = 1, θr = 1.1 and Pr = 0.7.

Meccanica (2012) 47:153–161 159

(a)

(b)

Fig. 5 (a) Temperature profiles for several values of suctionparameter S in case of non-porous media. (b) Temperature pro-files for several values of suction parameter S in case of porousmedia

The velocity profiles are fuller than that of the pre-vious case. The tendency is more pronounced towardsthe wall. Figure 5(a) demonstrates that the tempera-ture θ(η) decreases with the increasing suction para-meter S (S = 0, 0.5, 1, 1.5, 2, 3) when k1 = 0 = k2,N = 1, θr = 1.1 and Pr = 0.7. The thermal boundarylayer thickness decreases with the suction parameter S

which causes an increase in the rate of heat transfer.The explanation for such behavior is that the fluid isbrought closer to the surface and reduces the thermalboundary layer thickness. Figure 5(b) indicates that inpresence of porous medium (k1 = 0.1 = k2), the in-crease in the rate of heat transfer due to suction be-comes more pronounced. Here S = 0 means the non-porous plate.

In Fig. 6(a), the variation of temperature θ(η) withη for various values of the radiation parameter N

(N = 1, 1.5, 2, 3) with k1 = 0 = k2, S = 1, θr = 1.1and Pr = 0.7 is given. Figure 6(b) depicts the effectsof radiation parameter on the temperature field in pres-

(a)

(b)

Fig. 6 (a) Temperature profiles for several values of radiationN in case of non-porous media. (b) Temperature profiles forseveral values of radiation parameter N in case of porous media

ence of constant suction (S = 1) in porous media (i.e.k1 = 0.1 = k2). It is noticed from these two figures thatthe temperature decreases with the increasing valuesof the radiation parameter N . The effect of radiationparameter N is to reduce the temperature significantlyin the flow region. The increase in radiation parametermeans the release of heat energy from the flow regionand so the fluid temperature decreases as the thermalboundary layer thickness becomes thinner.

Figures 7(a) and (b) depict the temperature distrib-ution for various values of temperature ratio parameterθr in case of non-porous media (k1 = 0 = k2) as wellas in porous media (k1 = 0.1 = k2) with N = 1, S = 1and Pr = 0.7. From both the figures it is clearly ob-served that fluid temperature increases as θr increases.θr > 1 when Tw > T∞ and this is the usual case that isconsidered in this problem. θr ≤ 1 represents the casewhen Tw ≤ T∞ and this is possible if the porous plateis kept on ice pad.

160 Meccanica (2012) 47:153–161

(a)

(b)

Fig. 7 (a) Temperature profiles for several values of tempera-ture ratio parameter θr in case of non-porous media. (b) Temper-ature profiles for several values of temperature ratio parameterθr in case of porous media

Fig. 8 Variation of rate of heat transfer with radiation parame-ter for several values of Prandtl number Pr

Figure 8 exhibits the variations of the rate of heattransfer [−θ ′(0)] in terms of Nusselt number with ra-diation parameter (N ) for different values of Prandtlnumber [Pr = 0.3, 0.7, 1, 1.5]. From this figure it isnoticed that rate of heat transfer increases with the in-creasing values of Prandtl number.

4 Conclusions

The present study gives the numerical solutions forsteady forced convective boundary layer flow and heattransfer over a porous plate embedded in a Darcy-Forchheimer porous medium in presence of thermalradiation. The effect of permeability of the medium ona viscous incompressible fluid is to increase the fluidvelocity by reducing the drag on the flow which inturn causes a decrease in the temperature field. It isfound that the inertial parameter has a great influenceon decreasing the flow field, whereas its influence isreversed on the rate of heat transfer. The non-Darcianterm (i.e. Forchheimer term) has a very significant ef-fect on the velocity distribution.

The results pertaining to the present study indicatethat due to suction the skin-friction increases while therate of heat transfer increases due to suction. The tem-perature in the boundary layer decreases due to suc-tion. Due to thermal radiation, temperature is found todecrease. This effect is more pronounced in presenceof porous media. The combined effects of suction andthermal radiation can be used as means of cooling. Therate of heat transfer increases with the increasing val-ues of Prandtl number.

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