Transient and non-Darcian effects on natural convection flow in a vertical channel partially filled with porous medium: Analysis with Forchheimer–Brinkman extended Darcy model

International Journal of Heat and Mass Transfer 54 (2011) 1111–1120

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International Journal of Heat and Mass Transfer

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Transient and non-Darcian effects on natural convection flow in a vertical channelpartially filled with porous medium: Analysis with Forchheimer–Brinkmanextended Darcy model

Atul Kumar Singh a, Pratibha Agnihotri a, N.P. Singh b, Ajay Kumar Singh c,⇑a Department of Mathematics, V.S.S.D. College, Kanpur 208002, Indiab Department of Applied Sciences and Humanities, Rama Institute of Engineering and Technology, Mandhana, Kanpur 209217, Indiac Department of Mathematics, C.L. Jain College, Firozabad 283203, India

a r t i c l e i n f o a b s t r a c t

Article history:Received 3 May 2009Received in revised form 16 January 2010Accepted 13 February 2010Available online 15 December 2010

Keywords:Transient and non-Darcian effectsNatural convectionVertical channelPorous medium

0017-9310/$ - see front matter � 2010 Elsevier Ltd. Adoi:10.1016/j.ijheatmasstransfer.2010.11.011

⇑ Corresponding author. Address: 236, Durga NagTel.: +91 9411841345.

E-mail address: [email protected] (A

The present study addresses the transient as well as non-Darcian effects on laminar natural convectionflow in a vertical channel partially filled with porous medium. Forchheimer–Brinkman extended Darcymodel is assumed to simulate momentum transfer within the porous medium. Two regions are coupledby equating the velocity and shear stress in the case of momentum equation while matching of thetemperature and heat flux is taken for thermal energy equation. Approximate solutions are obtainedusing perturbation technique. Variations in velocity field with Darcy number, Grashof number, kinematicviscosity ratio, distance of interface and variations in temperature distribution with thermal conductivityratio, distance of interface are obtained and depicted graphically. The skin-friction and rate of heat trans-fer at the channel walls are also derived and the numerical values for various physical parameters aretabulated.

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1. Introduction

Considerable research efforts have been made on understandingthe convective flow between porous media and objects of differentshapes embedded in porous media. Interest in the study of thephenomena of convective flow through a fluid-saturated porousmedium stems both from fundamental considerations, such asthe development of better insight into the nature of the underlyingphysical processes, as well as from physical considerations such asthe fact that these idealized configurations serve as a launchingpad for modeling the analogous transfer processes in more realisticsystems. Such idealized geometries also provide a testing groundfor checking the validity of the theoretical analysis.

In the present stage of contemporary technologies, the practicalinterest on convective heat transfer in porous medium is expand-ing rapidly due to its wide range of applications. These applicationsinclude wide area such as geothermal energy utilization, insulationof high temperature gas–solid reaction vessels, petroleum reser-voirs, pollutants dispersion in aquifers, fiber and granular insula-tion including structures for high power density machines,combustion in situ in underground reservoirs for enhancement ofoil recovery, reduction of hazardous combustion products using

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ar, Firozabad 283203, India.

.K. Singh).

catalytic porous beds, ceramic radiant porous burners used byindustrial firms as efficient heat transfer devices, storage of grain,food and vegetables, industrial and agricultural water distribution,solar collectors with porous absorbers and porous bearings, bloodflow in lungs or in arteries, porous heat pipes, solidification of cast-ing, buried electrical cables, cores of chemical reactors, chemicalcatalytic connectors etc. These applications have attracted theattention of engineers and scientists from diversified disciplines,such as applied mathematics, geothermal physics, food sciences,nuclear, chemical, civil, mechanical and bio-engineering.

In the literature on convective heat transfer phenomena forfluid-saturated porous medium, a wealth of information is avail-able and satisfactory means have been evolved for the estimationof the velocity and temperature fields, as well as for the drag, heatand mass transfer coefficient involving porous media. Comprehen-sive literature on the subject is given in the monographs and booksof Ene and Polisevski [1], Nield and Bejan [2], Bejan [3], Nakayama[4] and Kaviany [5].

The earliest reference to fluid flowing through a porous mediadates back to Darcy [6]. However its importance and ramificationsin process design and operation have been recognized only duringthe last three/four decades. Irmay [7] presented an analysis on thetheoretical derivation of Darcy and Forchheimer models. The anal-ysis of Beavers and Joseph [8] is the first study on flow mechanismat fluid/porous surface for the flow field in a composite systemcontaining fluid and porous regions using Darcy’s law with

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Nomenclature

A non-dimensional constantC Forchheimer constantd0 distance of interface from the wall at y0 = 0d distance of interface in non-dimensional formDa Darcy numberGr Grashof numberg acceleration due to gravityH distance between vertical wallsK0 permeability of porous mediumk thermal conductivity of fluid layerkeff effective thermal conductivityNuf rate of heat transfer at the wall y = 0Nup rate of heat transfer at the wall y = 1t0 timet time in non-dimensional formT 0f temperature in the fluid regionT 0p temperature in the porous regionT 0c temperature of the cold wallT 0h temperature of the hot wall

u0f dimensional velocity in the fluid regionuf non-dimensional velocity in the fluid regionu0p dimensional velocity in the porous regionup non-dimensional velocity in the porous regionx0, y0 Cartesian coordinatesx, y Cartesian coordinates in non-dimensional form

Greek symbolse porosity of the mediuma thermal conductivity ratiob coefficient of thermal expansionc kinematic viscosity ratio#eff effective kinematic viscosity of fluid layer# kinematic viscosity of fluid layerhf temperature in non-dimensional form in fluid regionhp temperature in non-dimensional from in porous regionsf skin-friction at the wall y = 0sp skin-friction at the wall y = 1

1112 A.K. Singh et al. / International Journal of Heat and Mass Transfer 54 (2011) 1111–1120

slip- velocity condition at the interface between a Newtonian fluidand an isotropic porous medium. Theoretical support for theBJ-condition is provided by the results of Taylor [9] and Richardson[10]. Taylor [9] also observed that BJ-condition could be deduced asa consequence of the Brinkman equation. Consequently, Neale andNader [11] presented an important analysis for the flow in a chan-nel having fluid and porous regions and showed that the Darcymodel with BJ-condition gives the same results as that obtainedby using the Brinkman model by considering continuity of thevelocity and shear stress at the interface.

Somerton and Catton [12] used the Brinkman equation andobserved that for all values of Darcy number, the Nusselt numbergoes through a minimum as the relative thickness of the porous re-gion increases. Cheng and Pop [13] studied free convection flow ina porous medium and obtained results for high Darcy numbers(P37 � 10�4) and fluid layers (P0.43). They analysed the free con-vection boundary layer flow in the porous medium for the case of astep increase in wall temperature. Poulikakos and Kazmierczak[14] obtained closed form analytical solutions of the Brinkmanequation for parallel plates with constant heat flux on the walls,when there is a layer of porous medium adjacent to the wallsand the clear fluid interior. Renken and Poulikakos [15] performedan experimental investigation for the parallel plate configurationwith the walls maintained at constant temperature and also per-formed numerical simulations incorporating the effects of inertia,boundary friction and variable porosity. Their experimental andnumerical findings agreed on predicting an enhanced heat transferover that predicted earlier and that in the fully developed region,the effect of channeling was to produce a Nusselt number increaseusing Darcy model. Chandrasekhara and Nagaraju [16] studiedcomposite heat transfer in the case of a steady laminar flow of agray fluid with small optical density past a horizontal plate embed-ded in a saturated porous medium. Vafai and Kim [17] obtained anexact solution for forced convection in a channel filled with porousmedium using boundary layer approximation for fully developedflow subject to a constant-heat-flux boundary condition. Further,Vafai and Kim [18] presented a numerical study for forced convec-tion flow in a composite system containing fluid and porous layersbased on Darcy–Brinkman–Forchheimer model.

Nakayama et al. [19] have studied Forchheimer free convectionover a non-isothermal body of arbitrary shape in a saturated porousmedium. Consequently, a wealth of information on the mechanism

of convective heat transfer in a porous medium in a variety of pro-cesses and unit operations involving a range of geometric configu-rations has accrued over recent years. Indeed, several momentumtransfer models have been proposed and used during the last fewyears, such as the Forchheimer-extended Darcy model and thegeneral extended Darcy model which is Brinkman–Forchheimerextended Darcy model with an additional convectional inertialterm. With the widespread availabilities and power of computers,most of these models have been thoroughly investigated. Kladiasand Prasad [20] presented a numerical solution and observed theeffects of Darcy number, Prandtl number and conductivity ratioon flow transitions in buoyancy induced non-Darcy convection ina porous medium. Further, Kladias and Prasad [21] have experi-mentally verified the Darcy–Brinkman–Forchheimer model fornatural convection in porous media. Ruth and Ma [22] have derivedthe Forchheimer equation by means of averaging theorem andmade an integrated analysis of high Forchheimer number inporous media [23]. Shenoy [24] extended the work [23] for natural,forced and mixed convection heat transfer in non-Newtonianpower law fluid-saturated porous medium for Darcy–Forchheimermodel.

Vafai and Kim [25] have presented a numerical study based onthe Darcy–Brinkman–Forchheimer model for the forced convectionin a composite system containing fluid and porous regions. Knuppand Lage [26] have studied generalized Forchheimer-extendedDarcy flow model to the permeability case via a variation principle.Marpu [27] have studied the Forchheimer and Brinkman extendedDarcy flow model on natural convection in a vertical cylindricalporous medium. Singh and Thorpe [28] presented a comparativeanalysis of flow models on natural convection in a fluid, flow overa porous layer. Nield [29] obtained a closed-form solution of theBrinkman–Forchheimer equation for different values of Darcynumber using the Brinkman–Forchheimer model and stress jumpboundary condition at the porous interface. Kuznetsov [30]obtained an analytical solution for the steady fully developed flowin a composite region. Nakayama [31] has presented a unifiedtreatment of Darcy–Forchheimer boundary layer flows. Paul et al.[32] studied this problem for natural convection flow in a verticalchannel partially filled with a porous medium using Brinkman–Forchheimer extended Darcy model. Malashetty et al. [33] haveinvestigated two fluid flow and heat transfer in an inclined channelcontaining porous and fluid layers while stability of mixed

A.K. Singh et al. / International Journal of Heat and Mass Transfer 54 (2011) 1111–1120 1113

convection in a vertical channel filled with a porous medium haspresented by Chen [34]. Leong and Jin [35] have investigated heattransfer of oscillating and steady flows in a channel filled withporous medium. More recently, Singh and Takhar [36] have studiedheat transfer effects on flow of two immiscible viscous fluidsthrough parallel permeable beds using Brinkman model. Beji andGobin [37] numerically studied thermal dispersion effects on theBrinkman–Forchheimer model. These cause a significant increasein the overall heat transfer particularly when the thermal conduc-tivities of the fluid and the solid matrix are similar. Mishra andSarkar [38] have provided comparative numerical studies on aheated square cavity on Darcy, Brinkman and Brinkman–Forchhei-mer models and confirmed that boundary friction and quadraticdrag lead to a reduction in heat transfer. Whitaker [39] obtaineda momentum equation with a Forchheimer correction using themethod of volume averaging.

In the proposed investigation, an effort is made to present ananalysis on natural convective flow using Forchheimer–Brinkmanextended Darcy model in a vertical channel partially filled withporous medium. The model used to simulate momentum transferwithin the porous medium has some advances over studies re-ported by Vafai and Kim [25], Singh and Thorpe [28], Paul et al.[32], Beji and Gobin [37], Mishra and Sarkar [38] and Paul et al.[42]. In the present model the viscosity and the thermal conductiv-ity of the fluid are different from the effective viscosity and theeffective thermal conductivity of the porous medium envisagingthat the temperature of the hot isothermal wall decreases expo-nentially with time. The model may have possible application ingeothermal engineering.

Fig. 1a. Physical model of the problem.

2. Formulation of the problem

Let us consider unsteady, fully developed, natural convectionflow of a viscous incompressible fluid between two vertical iso-thermal walls consisting of fluid and porous regions. The walls sep-arated at a distance H apart, are such that the distance d0 occupiesthe fluid and the remaining distance (H � d0) occupies porous sub-tract. In Cartesian coordinate system, the x0-axis is taken along oneof the wall and y0-axis is normal to it as shown in physical model ofthe problem. Initially, when t0 6 0 the wall y0 = 0 is at rest and thewall temperature is T 0c . At time t0 > 0, the wall y0 = 0 suddenly startsto move with uniform velocity #/H in the direction of the flow andthe temperature of the wall y0 = 0 is instantaneously raised to T 0hcausing the phenomenon of natural convection in the verticalchannel walls. Since the walls are infinitely long, the flow charac-teristics depend on coordinate y0 and t0 only. The two regions arecoupled by equating the velocity and the shear stress for themomentum equation, while the matching of the temperature andheat flux are taken for energy equation. The fluid in the presentconfiguration is incompressible and the physical properties are as-sumed to be constant excluding density in the buoyancy terms. Forthermal convection to occur, the density of the fluid must be afunction of the temperature, as such the Boussinesq’s approxima-tion is taken into account. In addition, the analysis is based onthe following assumptions:

(i) The solid spheres are of uniform shape, non-conducting andincompressible.

(ii) The medium is isotropic. However, the dependency of thequantities such as effective thermal conductivity (keff) andeffective kinematic viscosity (veff) are accounted for.

(iii) The forced convection dominates the packed bed as suchnatural convection effects are negligible.

(iv) The inter-particle and inertia-particle heat transfer effectsare negligible.

(v) The medium considered is homogeneous and highly porous,as such for simplicity, the heat capacity ratio of the porouslayer to fluid layer is considered as unity [2], while the kine-matic viscosity ratio and the thermal conductivity ratio aretaken into account [5].

(vi) The porous medium is considered to be homogeneous, assuch the normalized porosity is taken as unity [42].

(vii) In the limit of very high Darcy number, the flow in porousmedium behaves as a pure fluid flow, as such the convectiveflow predicted by the present solution is comparable to thesolution reported in the convective fluid flow in a verticalchannel [41] (Fig. 1a).

Under the above stated assumptions, the equations governingthe flow in the fluid region and porous region [2,5] in non-dimen-sional form are:

Fluid Region:

@uf

@t¼ @

2uf

@y2 þ Grhf ; ð1Þ

@hf

@t¼ 1

Pr@2hf

@y2 : ð2Þ

Porous Region:

@up

@t¼ c

@2up

@y2 �up

Da� Cffiffiffiffiffiffi

Dap u2

p þ Grhp; ð3Þ

@hp

@t¼ a

Pr@2hp

@y2 : ð4Þ

The boundary and matching conditions [42] in non-dimensionalform are:

t 6 0 : uf ¼ 0; hf ¼ 0 for all y;

t > 0 : y ¼ 0; uf ¼ 1; hf ¼ 1þ ee�nt ;

y ¼ 1; up ¼ 0; hp ¼ 0;y ¼ d; uf ¼ up; hf ¼ hp;

@uf

@y¼ c

@up

@y;

@hf

@y¼ a

@hp

@y: ð5Þ

The above equations as well as the boundary and matching con-ditions have been rendered in non-dimensional form by using thefollowing non-dimensional variables:

d ¼ d0

H; y ¼ y0

H; t ¼ #t0

H2 ; n ¼ H2n0

#a ¼ KTeff

KT; c ¼ #eff

#;


Da ¼ K 0

H2 ; Pr ¼ lCp

KT; uf ¼

u0f H

#; up ¼

u0pH#

; hf ¼T 0f � T 0cT 0h � T 0c

;

hp ¼T 0p � T 0cT 0h � T 0c

and Gr ¼ gbH3 T 0h � T 0c� �

#2 :

The symbols are defined in the nomenclature.The inertia coefficient C, appearing in (3) of the model, is eval-

uated from the following empirical formula suggested by Ergun[40]:

C ¼ 1:75ffiffiffiffiffiffiffiffiffiffiffiffiffi175e3p :

3. Solution of the problem

To obtain the solution of Eqs. (1)–(4), we assume (e� 1):

uf ðy; tÞ ¼ u0f ðyÞ þ eu1f ðyÞ expð�ntÞ;upðy; tÞ ¼ u0pðyÞ þ eu1pðyÞ expð�ntÞ;hf ðy; tÞ ¼ h0f ðyÞ þ hu1f ðyÞ expð�ntÞ;hpðy; tÞ ¼ h0pðyÞ þ hu1pðyÞ expð�ntÞ: ð6Þ

Using (6) in (1)–(4), we get the following equations:Fluid Region:

@2u0f

@y2 ¼ �Grh0f ; ð7Þ

@2h0f

@y2 ¼ 0; ð8Þ

@2u1f

@y2 þ nu1f ¼ �Grh1f ; ð9Þ

@2h1f

@y2 þ nPrh1f ¼ 0: ð10Þ

Porous Region:

c@2u0p

@y2 �u0p

Da� Cffiffiffiffiffiffi

Dap u2

0p ¼ �Grh0p; ð11Þ

@2h0p

@y2 ¼ 0; ð12Þ

c@2u1p

@y2 �u1p

Da� 2Cffiffiffiffiffiffi

Dap u0pu1p þ nu1p ¼ �Grh1p; ð13Þ

@2h1p

@y2 þnPra

h1p ¼ 0: ð14Þ

conditions (5) are transformed to:

t 6 0 : u0f ¼ 0; u1f ¼ 0; h0f ¼ 0; h1f ¼ 0 for all y;t > 0 : y ¼ 0; u0f ¼ 1; u1f ¼ 0; h0f ¼ 1; h1f ¼ 1;

y ¼ 1 u0p ¼ 0; u1p ¼ 0; h0p ¼ 0 h1p ¼ 0;y ¼ d u0f ¼ u0p; u1f ¼ u1p; h0f ¼ h0p; h1f ¼ h1p;

@u0f

@y¼ c

@u0p

@y;

@u1f

@y¼ c

@u1p

@y;@h0f

@y¼ a

@h0p

@y;

@h1f

@y¼ a

@h1p

@y: ð15Þ

In order to get the solution of Eqs. (7) and (11), we further assume(since inertia coefficient C is small):

u0f ðyÞ ¼ u00f ðyÞ þ Cu01f ðyÞ þ oðC2Þ;u0pðyÞ ¼ u00pðyÞ þ Cu01pðyÞ þ oðC2Þ: ð16Þ

On introducing (16), we get the following equations:

@2u00f

@y2 ¼ �Grh0f ; ð17Þ

@2u01f

@y2 ¼ 0; ð18Þ

c@2u00p

@y2 �1

Dau00p ¼ �Grh0p; ð19Þ

c@2u01p

@y2 �1

Dau01p �

1ffiffiffiffiffiffiDap u2

00p ¼ 0: ð20Þ

The corresponding boundary conditions for u0f and u0p become:

at y ¼ 0; u00f ¼ 1; u01f ¼ 0;at y ¼ 1; u00p ¼ 0; u01p ¼ 0;at y ¼ d; u00f ¼ u00p; u01f ¼ u01p;

@u00f

@y¼ c

@u00p

@y;

@u01f

@y¼ c

@u01p

@y: ð21Þ

On using the boundary conditions (15) and (21), we get the solutionfor u0f, h0f, u0p and h0p.

Fluid Region:

u0f ¼ u00f þ Cu01f ¼A5

A4þ C

A19

A20

� �y� Gr

y2

2� GraA1

y3

6þ 1; ð22Þ

h0f ¼ aA1yþ 1: ð23Þ

Porous Region:

u0p ¼ u00p þ Cu01p

¼ ðA6 þ CA21ÞeAy þ ðA7 þ CA22Þe�Ay þ GrA1

cA2 ðy� 1Þ � 2CA6A7

cA2 ffiffiffiffiffiffiDap

þ CGrA1

c2A3 ffiffiffiffiffiffiDap A6eAy y2

2� y� y

2A

� �� A7e�Ay y2

2� yþ y

2A

� ��

� CGr2A21

c3A6 ffiffiffiffiffiffiDap ðy� 1Þ2 þ 2

A2

� �þ C

3A2cffiffiffiffiffiffiDap A2

6e2Ay þ A27e�2Ay

h i;

ð24Þ

h0p ¼ A1ðy� 1Þ: ð25Þ

In order to get the solution of Eqs. (9) and (13), we further assume:

u1f ðyÞ ¼ u10f ðyÞ þ Cu11f ðyÞ þ OðC2Þ;u1pðyÞ ¼ u10pðyÞ þ Cu11pðyÞ þ OðC2Þ: ð26Þ

On using (26), we get the following equations:

@2u10f

@y2 þ nu10f ¼ �Grh1f ; ð27Þ

@2u11f

@y2 þ nu11f ¼ 0; ð28Þ

c@2u10p

@y2 þ n� 1Da

� �u10p ¼ �Grh1p; ð29Þ

c@2u11p

@y2 þ n� 1Da

� �u11p �

2ffiffiffiffiffiffiDap u00pu10p ¼ 0: ð30Þ

The corresponding boundary conditions are:

at y ¼ 0; u10f ¼ 0; u11f ¼ 0;at y ¼ 1; u10p ¼ 0; u11p ¼ 0;at y ¼ d; u10f ¼ u10p; u11f ¼ u11p;

@u10f

@y¼ c

@u10p

@y;

@u11f

@y¼ c

@u11p

@y: ð31Þ

On using the boundary conditions shown in (15) and (31), we getthe solution for u1f, h1f, u1p and h1p.


Fluid Region:

u1f ¼ u10f þ Cu11f ¼ L8 cosffiffiffiffiffiffiffiffinPrp

y� cosffiffiffinp

yh i

þ L5L8 sinffiffiffiffiffiffiffiffinPrp

yþ L18

L19þ CL33

L34

� �sin

ffiffiffinp

y; ð32Þ

h1f ¼ cosffiffiffiffiffiffiffiffinPrp

yþ L5 sinffiffiffiffiffiffiffiffinPrp

y: ð33Þ

Porous Region:

u1p ¼ u10p þ Cu11p

¼ ðL20 þ CL35ÞeB0y þ ðL21 þ CL36Þe�B0y � L6 cos

ffiffiffiffiffiffiffiffinPra

ry

� L7 sin


ryþ C G1eðAþB0Þy þ G2eðA�B0Þy þ G3e�ðA�B0Þy

þ G4e�ðAþB0Þy

þ C G19eAy þ G21e�Ay þ G16

þ G17ð1� yÞ� sin


ryþ C G20eAy þ G22e�Ay þ G18

þ G15ð1� yÞ� cos


ryþ C G13eB0y y2

2� y� y

2B0

� ��

þ G14e�B0y y2

2� yþ y

2B0

� ��; ð34Þ

h1p ¼ ðL4 þ L3L5Þ cos


ryþ ðL2 þ L1L5Þ sin


ry: ð35Þ

4. Skin-friction and rate of heat transfer

The skin-friction (sf) at the wall y = 0 due to velocity in fluid/porous region and the skin-friction (sp) at the wall y = 1 due tovelocity in porous/fluid region are:

sf ¼@uf

@y

� �y¼0¼ A5

A4þ C

A19

A20þ e L5L8

ffiffiffiffiffiffiffiffinPrp

þffiffiffinp L18

L19þ C

L33

L34

� �� ; e�nt

ð36Þ

sp ¼@up

@y

� �y¼1¼ A ðA6þCA21ÞeA�ðA7þCA22Þe�A

� �þ L37

þ e½B0fðL20þCL35ÞeB0 �ðL21þCL36Þe�B0gþ L38�e�nt

ð37Þ

The rate of heat transfer (Nuf) at the wall y = 0 due to fluid/por-ous region and the rate of heat transfer (Nup) at the wall y = 1 dueto porous/fluid region are:

Nuf ¼@hf

@y

� �y¼0¼ aA1 þ e


L5e�nt; ð38Þ

Nup ¼@hp

@y

� �y¼1¼ A1 þ ee�nt


rðL2 þ L1L5Þ cos


r"

� ðL4 þ L3L5Þ sin


r #: ð39Þ

Fig. 1b. Effect of Prandtl number on temperature distribution.

5. Results and discussion

The non-dimensional Eqs. (1)–(4) are solved and the solutionsfor velocity and temperature fields are expressed in (22)–(24),(25), (33)–(35) and distributions plotted. The expressions for theskin-friction and the rate of heat transfer at both the isothermalwalls are derived and tabulated. The solutions obtained for velocityand temperature show that the velocities in clear fluid region andporous region are governed by Darcy number (Da), Grashof

number (Gr), kinematic viscosity ratio (c), thermal conductivity ra-tio (a) and distance of the fluid/porous interface (d) from the hotisothermal wall, while the temperature distribution is governedby the thermal conductivity ratio (a) and distance of fluid/ porousinterface (d). In describing the matching conditions at the fluid/porous interface in (5), the continuity of velocity and the shearstress are taken for velocity field following Vafai and Kim [18],while the continuity of temperature and heat flux are taken forthe temperature field following Paul et al. [32]. Consideration ofthe Brinkman–Forchheimer extended Darcy model gives rise toan addition empirical parameter C, which is called inertia coeffi-cient. The value of inertia coefficient (C) depends upon the porosityof the medium and is calculated considering that porous region,which consists of spherical beads (0.35 6 e 6 0.45). FollowingErgun [40], when e = 0.39, the value of C is found to be 0.543. Toexamine the effects of different parameters governing the convec-tive flow, the value of Prandtl number (Pr) is chosen to be Pr = 0.71corresponding to air. The values of Grashof number are chosen forcooling case (Gr > 0), a case of general interest in energy systemtechnologies. The remaining parameters are chosen arbitrarily,but authors of the field are followed [2,5].

Variations in the velocity of the fluid in fluid/porous region aredepicted for different numerical values of pairs (Da,d), (Gr,d), (c,d)and (a,d). Figs. 1b, 1c and 1d verify the accuracy of our analyticalsolution. Fig. 2 depicts the velocity profiles for different values ofDarcy number (Da) and distance of the fluid/ porous interface(d). Fig. 3 represents the velocity profiles for different values ofGrashof number (Gr) and distance of the fluid/ porous interface(d). Fig. 4 shows the velocity profiles for different values of kine-matic viscosity ratio (c) and distance of the fluid/ porous interface(d) while Fig. 5 represents the velocity profiles for different valuesof thermal conductivity ratio (a) and distance of the fluid/porousinterface (d). These variations are calculated for Da = 10�3, 10�4,10�5 and d = 0.5, 0.7 at Gr = 25, n = 1.0, t = 1.0, a = 0.5, c = 0.5(Fig. 2); Gr = 25, 50, 75 and d = 0.5, 0.7 at Da = 10�3, c = 0.5,n = 1.0, t = 1.0, a = 0.5 (Fig. 3); c = 0.5, 0.8, 1.0 and d = 0.5, 0.7 atDa = 10�3, a = 0.5, Gr = 25, n = 1.0, t = 1.0 (Fig. 4); a = 0.5, 0.8, 1.0and d = 0.5, 0.7 at Da = 10�3, Gr = 25, n = 1.0, t = 1.0, c = 0.5(Fig. 5). The variations of the temperature distribution in fluid/por-ous region showing effects of a = 0.5, 0.8, 1.0 and d = 0.5, 0.7 arerepresented in Fig. 6. The variations in skin-friction at the wallsy = 0 and y = 1 are numerically presented in Table 1 while rate ofheat transfer at the walls y = 0 andy = 1 are numerically presentedin Table 2.

Fig. 1c. Velocity distribution.

Fig. 1d. Velocity distribution.

Fig. 2. Effects of Da on velocity profiles Pr = 0.71, Gr = 25, c = 0.5, a = 0.5.

Fig. 3. Effects of Gr on velocity profiles Pr = 0.71, Da = 10�3, c = 0.5, a = 0.5.

Fig. 4. Effects of c on velocity profiles Pr = 0.71, Gr = 25, Da = 10�3, a = 0.5.


In order to examine the accuracy of our analytical solutions, thenumerical results are compared with most closely related analyti-cal and numerical solutions. This was achieved by making

necessary adjustments in our model to reduce it to a system equiv-alent to the simplified cases of Vafai and Thiyagaraja [43], Amiriand Vafai [44] and Amiri et al. [45]. These comparisons are shownin Figs. 1b, 1c and 1d, respectively. It is worth mentioning that allthe above comparisons of the present model are made in terms ofnon-dimensionl variables and are in good agreement with theabove stated studies.

Fig. 2 shows variations in the velocity (u) versus non-dimen-sional (y) for different values of the Darcy number. It is observedthat as Darcy number increases, the velocity increases in fluid re-gion as well as in porous region. The lower values of Darcy numbersignificantly affect the velocity in porous region. This verifies andsupports the fact that the Darcy equation is applicable for the flowthrough porous medium with low permeability [32]. An increase inthe distance of the fluid region increases the velocity in the porousregion more frequently. The physics behind this phenomena is thatan increase in velocity of the fluid in fluid-flow region exerts anadditional pressure on the velocity in porous region, which

Fig. 5. Effects of a on velocity profiles Pr = 0.71, Da = 10�3, c = 0.5, Gr = 25.

Fig. 6. Effects of a on temperature profiles Pr = 0.71.

Table 1Numerical values of skin-friction (Pr = 0.71).

Da Gr c a d = 0.5

sf sp

10�3 25 0.5 0.5 �14.3531 �25.2100910�4 25 0.5 0.5 �1.00376 �74.6373510�5 25 0.5 0.5 0.116606 �269.89510�3 50 0.5 0.5 �2.26182 �132.47910�3 75 0.5 0.5 �3.49145 �173.530110�3 25 0.8 0.5 �1.71041 �59.147110�3 25 1.0 0.5 �2.17793 �53.2134810�3 25 0.5 0.8 �0.80078 �63.065810�3 25 0.5 1.0 �0.69735 �57.14924

Table 2Numerical values of rate of heat transfer (Pr = 0.71).

a d = 0.5 d = 0.7

Nuf Nup Nuf Nup

0.5 �6.14895 �12.29998 �7.09509 �14.192240.8 �8.19897 �10.24989 �8.58036 �10.726621.0 �9.22397 �9.224862 �9.22397 �9.224862


accelerates the velocity in porous region. Also we note that forhigher values of Darcy number reverse flow is observed in porousregion.

Fig. 3 illustrates variations in the velocity (u) versus non-dimensional (y) for different values of Grashof number (Gr). Wenote that for higher values of Grashof number (Gr), the velocityis enhanced in comparison to lower values of Grashof number(Gr) in fluid region as well as in porous region. Besides, we ob-serve that the velocity in porous region increases significantlyfor the higher values of Grashof number. Hence, the buoyancyparameter Gr, has the dominant effect in escalating transientvelocities [41]. Also, an increase in the distance of the fluid/porousregion increases the velocity in the fluid region in comparison toporous region.

Fig. 4 indicates variations in the velocity (u) against y for differ-ent values of kinematic viscosity ratio (c). It is observed that forhigher values of kinematic viscosity ratio (c) the velocity (u) is lessthan lower values of kinematic viscosity ratio in fluid region andporous region. The higher values of kinematic viscosity ratio en-hance the velocity more efficiently in fluid/porous region. Mathe-matically, kinematic viscosity is defined as the ratio of viscosityand density. An increase in kinematic viscosity ratio results in a de-crease in viscosity, which in turn increases velocity in both the re-gions [17]. An increase in the distance of the fluid/porous regionincreases the velocity in the fluid region as well as in porousregion.

Fig. 5 indicates variations in the velocity (u) with respect to y fordifferent values of thermal conductivity ratio (a). It is observedthat for higher values of thermal conductivity ratio (a), the velocityis more sensitive in comparison to lower values of thermal conduc-tivity ratio in fluid region. But reverse effect is noted for lower val-ues of thermal conductivity ratio in porous region. An increase inthe distance of the fluid/porous interface increases the velocity inporous region as well as fluid region.

Fig. 6 illustrates variations in temperature versus y for differentvalues thermal conductivity ratio (a). We note that for higher val-ues of thermal conductivity ratio, the temperature decreases morerapidly in comparison to lower values of thermal conductivity ratioin fluid region as well as porous region. Hence, thermal conductiv-ity ratio boots the transient temperature. Higher thermal conduc-tivity ratio physically implies increase in the contribution of freeconvection currents. The regime becomes more sensitive towardsheat transfer, which in turn stabilizes the temperature for lowervalues of thermal conductivity ratio. An increase in the distanceof the fluid/porous interface increases the temperature in fluid re-gion as well as in porous region.

Table 1 represents numerical values of skin-friction sf at thewall y = 0 and sp at the wall y = 1, respectively for different valuesof Da, Gr, c or a. Table 2 presents the heat transfer Nuf at the wally = 0 and Nup at the wall y = 1, respectively for different values of aat d = 0.5 and d = 0.7. These tables are self-explanatory, as such theeffect of different parameter on skin-friction and heat transfer rateat y = 0 and y = 1 are obvious. Hence, any discussion about themseems to be redundant.

6. Conclusions

For steady flow, the results obtained for velocity and temper-ature field of the present study are in excellent agreement with


the results obtained by Paul et al. [42]. Further if there is noporous medium, the results are in agreement with those ofSingh et al. [41]. Brinkman–Forchheimer–Darcy model, used toanalyse the flow in porous region shows that the flow in theporous layer is not influenced by Brinkman term and transportphenomena are influenced with the width of the fluid/porouslayer. The findings of the study are applicable in the analysisin various branches of geothermal engineering and petroleumtechnology [46,47]. Main conclusions of the study are asfollows:

(i) An increase in the distance d, of the fluid region increases thevelocity in the porous region more frequently.

(ii) The buoyancy parameter Gr, has the dominant effect in esca-lating transient velocities in fluid region as well as porousregion.

(iii) An increase in kinematic viscosity ratio results in a decreasein viscosity, which in turn increases velocity in both theregions. Also an increase in the distance of the fluid/porousregion increases the velocity in the fluid region as well asin porous region.

(iv) The velocity is more sensitive for higher values of thermalconductivity ratio (a), in comparison to lower values of ther-mal conductivity ratio in fluid region.

(v) The temperature decreases more rapidly for higher values ofthermal conductivity ratio, in comparison to lower values ofthermal conductivity ratio in fluid region as well as porousregion. Hence, thermal conductivity ratio boots the transienttemperature.

(vi) An increase in Da decreases the skin-friction, while anincrease in Gr, c or a increases the skin-friction at the wally = 0, but reverse effect are noted at the wall y = 1. An impor-tant observation of the study is that an increase in ddecreases the skin-friction at the wall y = 0 as well as theskin-friction at the wall y = 1.

(vii) An increase in a increases the rate of heat transfer at the wally = 0 as well as the wall y = 1, while an increase in ddecreases the rate of heat transfer at the wall y = 0 and thewall y = 1.

Appendix A

A ¼ ½cDa��1=2; A1 ¼ ½dð1� aÞ � 1��1

;

A2 ¼ 1� GrA1

cA2 ðd� 1Þ � GraA1d3

6� Grd2

2;

A3 ¼ �GraA1d2

2cA� GrA1

cA3 �GrdcA

;

A4 ¼eA

2eAd

1þ cAdcA

� �þ e�A

2e�Ad

cAd� 1cA

� �;

A5 ¼ �ðA2 þ A3ÞeA

2eAd� ðA2 � A3Þe�A

2e�Ad;

A6 ¼A5

2A4eAd

1þ cAdcA

� �þ A2 þ A3

2eAd;

A7 ¼A5

2A4e�Ad

cAd� 1cA

� �þ A2 � A3

2e�Ad;

A8 ¼A2

6e2A þ A27e�2A

3cA2 ffiffiffiffiffiffiDap ;

A9 ¼ �2Gr2A2

1

c3A8 ffiffiffiffiffiffiDap � 2A6A7

c3A2 ffiffiffiffiffiffiDap ;

A10 ¼�GrA1

c2A3 ffiffiffiffiffiffiDap A6

Aþ 12A

� �eA þ A7

1� A2A

� �e�A

� �;

A11 ¼A2

6e2Ad þ A27e�2Ad

3cA2 ffiffiffiffiffiffiDap ;

A12 ¼GrA1

c2A3 ffiffiffiffiffiffiDap A6eAd d2

2� d

2A� d

!� A7e�Ad d2

2þ d

2A� d

!" #

� 2A6A7

cA2 ffiffiffiffiffiffiDap � Gr2A2

1

c3A6 ffiffiffiffiffiffiDap ðd� 1Þ2 þ 2

A2

� �;

A13 ¼2 A2

6e2Ad � A27e�2Ad

�3cA2 ffiffiffiffiffiffi

Dap ;

A14 ¼GrA1A6eAd

c2A4 ffiffiffiffiffiffiDap A

d2

2� d

2A� d

!þ d� 1� 1

2A

� �" #

� 2Gr2A21ðd� 1Þ

c3A7 ffiffiffiffiffiffiDap þ GrA1A7e�Ad

c2A4 ffiffiffiffiffiffiDap

� Ad2

2þ d

2A� d

!� d� 1þ 1

2A

� �" #;

A15 ¼1þ cAd

2cAeAd

!;A16 ¼ �

A11 þ A12 þ A13 þ A14

2eAd;

A17 ¼cAd� 1

2cAe�Ad; A18 ¼ �

A11 þ A12 � A13 � A14

2e�Ad;

A19 ¼ �A8 � A9 � A10 � A16eA � A18e�A;

A20 ¼ A15eA þ A17e�A; A21 ¼ A16 þA19A15

A20;

A22 ¼ A18 þA19A17

A20;

L1 ¼1ffiffiffiap cos


d: cos


rdþ sin


d � sin


rd;

L2 ¼ cosffiffiffiffiffiffiffiffinPrp

d � sin


rd� 1ffiffiffi

ap sin


d � cos


rd;

L3 ¼ sinffiffiffiffiffiffiffiffinPrp

d � cos


rd� 1ffiffiffi

ap cos


d � sin


rd;

L4 ¼ cosffiffiffiffiffiffiffiffinPrp

d � cos


rdþ 1ffiffiffi

ap sin


d � sin


rd;

L5 ¼ � L4 cos


rþ L2 sin


r" #� L3 cos


rþ L1 sin


r" #�1

;

L6 ¼GrðL4 þ L3L5Þc nPr þ aB2

0

� ; L7 ¼GrðL2 þ L1L5Þc nPr þ aB2

0

� ; L8 ¼Pr

nðPr � 1Þ ;

L9 ¼ L8 cosffiffiffiffiffiffiffiffinPrp

dþ L7 sinffiffiffinp

d �

þ L5L8 sinffiffiffiffiffiffiffiffinPrp

d

þ L6 cos


rdþ L7 sin


rd; L10 ¼ sin

ffiffiffinp

d;

L11 ¼ffiffiffinp

cB0cos

ffiffiffinp

d; L12 ¼1B0


rL6 sin


rd� L7 cos


rd

" #;

L13 ¼1

cB0

ffiffiffinp

L8 � sinffiffiffinp

d�ffiffiffiffiffiffiffiffinPrp

L8 sinffiffiffiffiffiffiffiffinPrp

dþ L5L8 cosffiffiffiffiffiffiffiffinPrp

d �h i

;

L14 ¼L10 þ L11

2eB0 d; L15 ¼

L9 � L12 þ L13

2eB0 d; L16 ¼

L10 � L11

2e�B0 d;

L17 ¼L9 þ L12 � L13

2e�B0 d;

L18 ¼ L6 cos


rþ L7 sin


r� L15eB0 � L17e�B0 ;

L19 ¼ L14eB0 þ L16e�B0 ;

L20 ¼L14L18

L19þ L15;

L21 ¼L16L18

L19þ L17;


L22 ¼ G1eAþB0 þ G2eA�B0 þ G3e�ðA�B0Þ þ G4e�ðAþB0Þ

þ G13ðB0 þ 1ÞeB0

2B0þ G14ðB0 � 1Þe�B0

2B0

þ ðG19eA þ G21e�A þ G16Þ sin


r

þ ðG20eA þ G22e�A þ G18Þ cos


r;

L23 ¼ G1eðAþB0Þd þ G2eðA�B0Þd þ G3e�ðA�B0Þd

þ G4e�ðAþB0Þd þ G13eB0d d2

2� d

2B0� d

!

þ ½G19eAd þ G21e�Ad þ G16 þ G17ð1� dÞ� sin


rd

þ G14e�B0d d2

2þ d

2B0� d

!

þ G20eAd þ G22e�Ad þ G18 þ G19ð1� dÞ

cos


rd;

L24 ¼ c½ðAþ B0ÞG1eðAþB0Þd þ ðA� B0ÞG2eðA�B0Þd

� ðA� B0ÞG3e�ðA�B0Þd � ðAþ B0ÞG4e�ðAþB0Þd�

þ cG13eB0d B0d2

2� d

2B0� d

!þ d� 1� 1

2B0

� �" #

� cG14e�B0d B0d2

2þ d

2B0� d

!� d� 1þ 1

2B0

� �" #

þ c AG19eAd � eAd


rG20 � AG21e�Ad

"

� e�Ad


rG22 � fG15ð1� dÞ þ G18g


r� G17

#sin


rd

þ c AG20eAd þ eAd


rG19 � AG22e�Ad

"

þ e�Ad


rG21 þ fG17ð1� dÞ þ G16g


r� G15

#cos


rd;

L25 ¼ L10 þ L11; L26 ¼ L23 þ L24; L27 ¼L10 þ L11

2eB0d;

L28 ¼L23 þ L24

2eB0d; L29 ¼ L10 � L11; L30 ¼ L23 � L24;

L31 ¼L10 � L11

2e�B0d; L32 ¼

L23 � L24

2e�B0d; L33 ¼ L22 þ L28eB0 þ L32e�B0 ;

L34 ¼ L27eB0 þ L31e�B0 ; L35 ¼L27L33

L34� L28; L36 ¼

L31L33

L34� L32;

L37 ¼GrA1

cA2 �GrCA1

c2A3 ffiffiffiffiffiffiDap AA6eAðAþ 1Þ

2Aþ A6eA

2Aþ AA7e�AðA� 1Þ

2Aþ A7e�A

2A

� �;

L38 ¼ L6


rsin


r� L7


rcos


r

þ fðAþ B0ÞG1eAþB0 þ ðA� B0ÞG2eA�B0� A� B0ÞG3e�ðA�B0Þ � ðAþ B0ÞG4e�ðAþB0Þg�

þ nPra

cos


rfG19eA þ G21e�A þ G16g

s

þ sin


rfAG19eA � AG21e�A � G17g

� nPra

sin


rfG20eA þ G22e�A þ G18g

s

þ cos


rfAG20eA � AG22e�A � G15g

#;

G1 ¼A6L20

cffiffiffiffiffiffiDap

ðA2 þ 2AB0Þ; G2 ¼

A6L21


ðA2 � 2AB0Þ;

G3 ¼A7L20


ðA2 � 2AB0Þ; G4 ¼

A7L21


ðA2 þ 2AB0Þ;

G5 ¼�2A6L6A

ffiffiffiffiffinPra

qcffiffiffiffiffiffiDap

B22 þ 4A2nPr

a

� ; G6 ¼�B2A6L6


B22 þ 4A2nPr

a

� ;

G7 ¼2A6L7A

ffiffiffiffiffinPra


B22 þ 4A2nPr

a

� ;G8 ¼�B2A6L7


B22 þ 4A2nPr

a

� ;

G9 ¼2AA7L6

ffiffiffiffiffinPra


B22 þ 4A2nPr

a

� G10 ¼�B2A7L6


B22 þ 4A2nPr

a

� ;

G11 ¼�2AA7L7

ffiffiffiffiffinPra


B22 þ 4A2nPr

a

� ; G12 ¼�B2A7L7


B22 þ 4A2nPr

a

� ;G13 ¼

GrA1L20

2c2A2 ffiffiffiffiffiffiDap

B0

; G14 ¼�GrA1L21

2c2A2 ffiffiffiffiffiffiDap

B0

;

G15 ¼�GrA1L6


B1

; G16 ¼�2GrA1L6


B21


r;

G17 ¼�GrA1L7


B1

; G18 ¼2GrA1L7


B21


r;

G19 ¼ G5 þ G8; G20 ¼ G6 þ G7; G21 ¼ G9 þ G12;

G22 ¼ G10 þ G11; B0 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� nDa

cDa

s;

B1 ¼ B20 þ

nPra

and B2 ¼ A2 � B20 þ

nPra

� �:

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Documents

Transient and non-Darcian effects on natural convection flow in a vertical channel partially filled with porous medium: Analysis with Forchheimer–Brinkman extended Darcy model