Mixed Convection in a Vertical Porous Channel | SpringerLink

DOI 10.1007/s11242-005-0260-5Transport in Porous Media (2005) 61: 315–335 © Springer 2005

Mixed Convection in a Vertical Porous Channel

J. C. UMAVATHI1, J. P. KUMAR1, A. J. CHAMKHA2 and I. POP3,�

1Department of Mathematics, Gulbarga University, Gulbarga-585 106 Karnataka, India2Manufacturing Engineering Department, The Public Authority for Applied Educationand Training, Shuweikh, 70654, Kuwait3University of Cluj, Faculty of Mathematics, R-3400 Cluj, CP 253, Romania

(Received: 14 June 2004; accepted in final form: 27 December 2004)

Abstract. A numerical study of mixed convection in a vertical channel filled with aporous medium including the effect of inertial forces is studied by taking into accountthe effect of viscous and Darcy dissipations. The flow is modeled using the Brinkman–Forchheimer-extended Darcy equations. The two boundaries are considered as isothermal–isothermal, isoflux–isothermal and isothermal–isoflux for the left and right walls of thechannel and kept either at equal or at different temperatures. The governing equations aresolved numerically by finite difference method with Southwell–Over–Relaxation techniquefor extended Darcy model and analytically using perturbation series method for Darcianmodel. The velocity and temperature fields are obtained for various porous parameter,inertia effect, product of Brinkman number and Grashof number and the ratio of Gras-hof number and Reynolds number for equal and different wall temperatures. Nusselt num-ber at the walls is also determined for three types of thermal boundary conditions. Theviscous dissipation enhances the flow reversal in the case of downward flow while it coun-ters the flow in the case of upward flow. The Darcy and inertial drag terms suppress theflow. It is found that analytical and numerical solutions agree very well for the Darcianmodel.

Key words: mixed convection, vertical channel, porous medium, non-Darcy model, analyt-ical and numerical solutions.

NomenclatureA constant defined in Equation (4).Br Brinkman number defined in Equation (10).Cp specific heat at constant pressure.CF porous media inertia coefficient.D =2L, hydraulic diameter.g acceleration due to gravity.GR mixed convection parameter (Gr/Re) defined in Equation (10).Gr Grashof number defined in Equation (10).I dimensionless porous medium inertia coefficient defined in Equation (10).K permeability of the porous media.k thermal conductivity.

�Author for correspondence: E-mail: [email protected]

316 J. C. UMAVATHI ET AL.

keff effective thermal conductivity.L channel width.p pressure.P =p +ρ0gX, difference between the pressure and the hydrostatic

pressure.Pr Prandtl number defined in Equation (10).Re Reynolds number defined in Equation (10).RT temperature difference ratio defined in Equation (10).T temperature.T1, T2 prescribed boundary temperatures.T0 reference temperature.u dimensionless velocity component in the X-direction.U velocity component in the X-direction.U0 reference velocity defined in Equation (11).V velocity component in the Y -direction.X,Y space coordinates.y dimensionless transverse coordinate.

Greek symbolsα =k/ρ0Cp, thermal diffusivity.β thermal expansion coefficient.�T reference temperature difference defined by Equation (12) or (13).ε dimensionless parameter defined by Equation (10).θ dimensionless temperature defined in Equation (10).µ fluid dynamic viscosity.µeff effective dynamic viscosity.ν =µ/ρ0, kinematic viscosity.σ =h/

√K, Darcy number defined in Equation (10).

1. Introduction

The problem of convective flow in fluid-saturated porous media has beenthe subject of several recent papers. Interest in understanding the con-vective transport processes in porous materials is increasing owing to thedevelopment of geothermal energy technology, high performance insulationfor building and cold storage, renewed interest in the energy efficient dry-ing processes and many other areas. It is also of interest in the nuclearindustry, particularly in the evaluation of heat removal from a hypotheti-cal accident in a nuclear reactor and to provide effective insulation. Com-prehensive literature surveys concerning the subject of porous media canbe found in the most recent books by Ingham and Pop (1998), Nieldand Bejan (1999), Vafai (2002), Pop and Ingham (2001), Bejan and Kraus(2003).

The present paper studies numerically the steady mixed convection flowin a vertical channel filled with a porous medium including the effect ofinertial forces and taking into account the effect of viscous and Darcydissipations. The main purpose is to reduce heat transfer by means of

MIXED CONVECTION IN A VERTICAL POROUS CHANNEL 317

open-pore insulators, such as glass fibers or rock fibrous material. In thistype of porous medium, the macroscopic velocity is not always small andhence the inertial force may not be negligible. Further, the distortion ofvelocity gives rise to shear stresses which in turn give rise to a viscousforce. Therefore, the study of convection in such a special type of coarseporous medium needs the generalized Darcy law which incorporates bothinertial and viscous forces in addition to the usual Darcy resistance. Itoffers a quantitative theory for the details of the transition from the con-duction regime to convection and a convenient means for demonstratingthe non-linear effects such as the preferred cell pattern, heat transport andso on.

In the literature about fluid flow in domains totally filled with porousmaterial, it has been realized that the local macroscopic inertial term isusually small compared to the microscopic Darcy drag term, and hence canbe neglected (Nield, 1991). However, the local inertial term may retain itsimportance in applications involving very thin porous substrates or at largeDarcy numbers. Most early works on porous media have used the Darcylaw which represents an empirical relation between the Darcian velocityand the pressure drop across the porous medium. Vafai and Tien (1981)have discussed the importance of boundary and inertia effects of porousmedia which are not accounted for by the Darcy law. Rudraiah (1984)studied non-linear convection in a porous medium with convective accel-eration and viscous force. Also, Vafai and Kim (Date) have studied andreported an exact solution for forced convection in a channel filled witha porous medium. Srinivasan and Vafai (Date) have reported a theoret-ical study for linear encroachment in two immiscible fluid systems in aporous medium taking into account the non-Darcian boundary and iner-tia effects. Chen and Vafai (1997) have analyzed free surface momentumand energy transport in porous media in the absence and presence of sur-face tension effects. Abu-Hijleh and Al-Nimr (2001) have investigated theimportance of the local inertial term in forced-convection fluid flow prob-lems in channels partially filled with porous material. Equations govern-ing transient flow and heat transfer aspects of a particulate suspensionwith a constant finite volume fraction in a porous channel were solvedby Chamkha (1996). Also, Chamkha (1997) studied the non-Darcy fullydeveloped mixed convection in a porous medium channel with heat genera-tion/absorption and hydromagnetic effect. Further, Chamkha (2000) studiedinertial effects for the flow of two-immiscible fluids in porous and nonpor-ous channel.

Keeping in view the importance of inertial effects for flow throughporous medium, it is the objective of this paper to consider the problemof laminar mixed convection flow in a vertical porous channel.


2. Mathematical Model

Consider a steady, laminar, fully developed mixed convection flow in anopen-ended vertical parallel plate channel filled with a porous material. Thefluid is assumed to be Newtonian and the porous medium is isotropic andhomogeneous. It is assumed that the thermal conductivity, the dynamic vis-cosity and the thermal expansion coefficient are considered constant. TheOberbeck–Boussinesq approximation is supposed to hold. It is worth men-tioning that Rajagopal et al. (1996) provided a systematic basis for theOberbeck–Boussinesq approximation. The X-axis is choosen parallel to thegravitational field, but with opposite direction and y-axis is transverse tothe plates. The origin is such that the channel walls are at positions Y =−L/2 and Y =L/2, respectively as shown in Figure 1.

By the condition of fully developed flow, the mass balance equation willbe ∂U

∂X=0 so that U depends only on Y , where U is the velocity along the

X-axis. The stream wise and the transverse momentum balance equationsincluding inertial effects for the porous matrix are (see Arpaci and Larsen,1984):

gβ (T −T0)− 1ρ0

∂P

∂X+ µeff

ρ0

d2U

dY 2− υ

KU + ρCF

ρ0U 2 =0, (1)

∂P

∂Y=0, (2)

where gβ (T −T0) is the buoyancy term, µeff/ρ0 d2U/dY 2 is the viscous

term, (ν/K)U is the Darcy term, (ρCF /ρ0)U2 accounts for the inertia

effect and P = p + ρ0gX is the difference between the pressure and the

X

g

0 Y

-L/2 L/2

Figure 1. Physical configuration.


hydrostatic pressure. In view of Equation (2), we can write Equation (1) as

gβ (T −T0)− 1ρ0

dP

dX+ µeff

ρ0

d2U

dY 2− υ

KU + ρCF

ρ0U 2 =0. (3)

We assume that the temperature of the boundary at Y = −L/2 is T1,while the temperature at Y =L/2 is T2 with T2 �T1. These boundary condi-tions are compatible with Equation (3) if and only if dP/dX is independentof X. Let there exists a constant A such that

dP

dX=A. (4)

Evaluating the derivative of Equation (3) with respect to X and usingEquation (4), one obtains

dT

dX=0 (5)

which implies that the temperature also depends only on the variable Y .The energy balance equation, which includes the effects of Darcy and

viscous dissipations is

αd2

T

dY 2+ ν

Cp

(dU

dY

)2

+ ν

CpkU 2 =0. (6)

The conservation equations for the porous layer are based on a non-Darcian model, incorporating the Brinkman and Forchheimer extensions.Beckermann et al. (1988) have shown in their experiments on naturalconvection in vertical enclosures with fluid and porous layers that, Brink-man’s extension is small compared to the Darcy term. However, Brink-man’s extension has been included in all the computations to ensurecontinuity of the velocities and stresses at the fluid/porous medium inter-face. Forchheimer’s extension serves to model the inertia and, hence, Pra-ndtl number effects on the flow in the porous medium. Although the effectof Forchheimer’s extension is insignificant mainly at low Prandtl numbers,it has been utilized in all the computations of the present study. It shouldbe noted that both extensions must be included simultaneously for a high-permeability porous medium (i.e., a high Darcy number). As a first approx-imation and owing to lack of conclusive information, µeff is taken equal tothe fluid viscosity µ in the present study. Equation (3) using Equation (6)become

d4U

dY 4= βg

αCp

(dU

dY

)2

+ 1K

d2U

dY 2+ βg

αCpKU 2 + CF

ν

d2U 2

dY 2. (7)


The no-slip boundary conditions on U are

U

(−L

2

)=U

(L

2

)=0. (8)

Using Equations (3) and (4), and using the boundary conditions on T oneobtains

(d2U

dY 2

)y=−L/2

= A

µ− βg (T1 −T0)

ν(d2U

dY 2

)y=L/2

= A

µ− βg (T2 −T0)

ν. (9)

Equations (7)–(9) can be written in a dimensionless form by employingthe dimensionless quantities

u= U

U0; θ = T −T0

�T;y = Y

D;Gr = gβ�T D3

ν2

Re= U0D

ν;Pr = ν

α;Br = µU 2

0

K�T(10)

σ = h√K

; I = CF U0D2

ν;GR = Gr

Re;RT = T2 −T1

�T,

where D =2L is the hydraulic diameter. The reference velocity U0 and thereference temperature T0 are given by

U0 =−AD2

48µ; T0 = T1 +T2

2. (11)

The physical meaning of other quantities defined in (10) are explained inthe Nomenclature. Moreover, the reference temperature difference �T isgiven by

�T =T2 −T1 if T1 <T2 (12)

�T = ν2

CpD2if T1 =T2. (13)

As a consequence, the dimensionless parameter RT can only take the val-ues 0 or 1. That is, RT is 1 for asymmetric heating i.e., T1 <T2, while RT

is 0 for symmetric heating i.e., T1 =T2, respectively.


Equation (4) implies that A can be either positive or negative. If A>0,then U0, Re and Gr are negative, i.e., the flow is downward. On the con-trary, if A < 0, the flow is upward, so that U0, Re and GR are positive.Using (10), Equations (6)–(9) become

d2θ

dy2=−σ 2Br u2 −Br

(du

dy

)2

(14)

d4u

dy4=σ 2 d2

u

dy2+ I

d2u2

dy2+σ 2GR Br u2 +GRBr

(du

dy

)2

(15)

subject to the boundary conditions

u

(−1

4

)=u

(14

)=0 (16)

(d2

u

dy2

)y=− 1

4

=−48+ RT GR

2;

(d2

u

dy2

)y= 1

4

=−48− RT GR

2. (17)

Further, Equation (3) using variables (10) and (11) can be written as

θ =− 1GR

(48−σ 2u− Iu2 + d2

u

dy2

). (18)

We notice that Equation (15) is highly nonlinear through inertia, buoy-ancy and viscous dissipations. However, they can be solved analytically ifthe inertial effects of the porous medium are neglected (i.e., I = 0). ThenEquation (15) becomes

d4u

dy4=σ 2 d2

u

dy2+σ 2GRBr u2 +GRBr

(du

dy

)2

(19)

The solution of this equation in the absence of the viscous dissipationterm, (i.e., (du/dy)2 =0 so that Br =0, as seen by Equation (14), with thisEquation (19) become linear) and using the boundary conditions (16) and(17) is

u= 48σ 2

(1− cosh(σy)

cosh(σ/4)

)+ 2GRRT

σ 2

(y − sinh(σy)

4 sinh(σ/4)

)(20)

Using Equation (19) in (18) we obtain

θ =2RT y. (21)


When buoyancy forces are dominating i.e., when GR→±∞, in the caseof asymmetric heating, Equation (20) gives

u

GR= 2

σ 2

(y − sinh(σy)

4 sinh(σ/4)

)(22)

Another simple solutions of Equations (19) and (14) can be obtainedwhen buoyancy forces are negligible and viscous dissipation is dominating,i.e., GR = 0, so that a purely forced convection occurs. For this conditionsolutions of Equations (19) and (18) are

u= 48σ 2

(1− cosh(σy)

cosh(σ/4)

)(23)

θ =A

(y2 − 1

16

)+B (cosh(2σy)− cosh (σ/2))

+C (cosh(σy)− cosh (σ/4))+2RT y (24)

where A, B and C are constants. For GR =0 the boundary conditions onθ are θ(−1/4)= θ(1/4)=RT /2.

3. Perturbation Method

In this section, Equation (19) along with boundary conditions (16) and (17)are solved analytically by a perturbation series method. Then temperaturefield is determined by employing Equation (18) in the absence of inertialeffects. The dimensionless perturbation parameter is defined by

ε =BrGR =ReP rβgD

Cp

. (25)

Thus, the solution of Equation (19) for a fixed value of GR �= 0 can beexpressed for small values of ε(<<1) in the form of series

u (y)=u0 (y)+ εu1 (y)+ ε2u2 (y)+· · ·=∞∑

n=0

εnun (y) . (26)

Case 1: Isothermal–Isothermal Walls

Substituting series (26) into Equations (16), (17) and (19), and equatingcoefficients of like powers of ε to zero, one obtains the boundary valueproblem for n=0 and 1 as

d4u0

dy4−σ 2 d2

u0

dy2=0, (27)


d4u1

dy4−M2 d2

u1

dy2=M2u2

0 +(

du0

dy

)2

, (28)

u0

(−1

4

)=u0

(14

)=0, (29)

(d2

u0

dy2

)y=− 1

4

= GRRT

2−48;

(d2

u0

dy2

)y= 1

4

=−GRRT

2−48, (30)

u1

(−1

4

)=u1

(14

)=0, (31)

(d2

u1

dy2

)y=− 1

4

=(

d2u1

dy2

)y= 1

4

=0 (32)

Equation (27), which represents the flow in the absence of viscous andDarcy dissipations can be solved in closed form. Thus its solution, whichsatisfies the boundary conditions (29) and (30) is

u0 = c1 + c2y + c3 cosh(σy)+ c4 sinh(σy), (33)

where ci(i = 1, . . . ,4) are constants. Equation (33) is also the solution ofEquation (15) in the case of Br = 0 and I = 0. On the other hand, Equa-tion (28) can be also solved exactly as u0 is known now and is given by

u1 =c5 +c6y+c7cosh(σy)+c8sinh(σy)+d1cosh(2σy)+d2sinh(2σy)

+d3y2cosh(σy)+d4y

2sinh(σy)+d5ycosh(σy)+d6y sinh(σy)

+d7y4 +d8y

3 +d9y2, (34)

where ci(i = 5, . . . ,8) and di(i = 1, . . . ,9) are constants. We truncate theseries in Equation (26) after n=1 and the solution is obtained up to O(ε)

as

u=u0 + εu1. (35)

The dimensionless temperature field is evaluated by Equation (18) in theabsence of inertial effects, which is given by

θ=− 1GR

48−σ 2C1 −σ 2C2y+

ε

(f1cosh(2σy)+f2 sinh(2σy)+f3ycosh(σy)+f4y sinh(σy)+f5cosh(σy)+f6sinh(σy)+f7y

4 +f8y3 +f9y

2 +f10y+f11

),

(36)

where C1, C2 and fi(i =1, . . . ,11) are constants.


Case 2: Isoflux–Isothermal Walls (q1 −T2)

The isoflux and isothermal boundary conditions for the channel walls aregiven by (see Chamkha, [22])

q1 =−k

(dT

dY

)Y= L

2

; T

(L

2

)=T2. (37)

In the non-dimensional form Equation (37) using Equation (10) with �T =q1D/k becomes

(dθ

dy

)y=−1/4

=−1; θ

(14

)=Rqt , (38)

where Rqt = (T2 −T0)/�T is the thermal ratio parameter.To solve Equation (15), apart from the no-slip conditions, two more

boundary conditions, are required. These are obtained from Equation (1)using Equation (37) as follows. Differentiating Equation (1) with respect toY and using Equation (4) we obtain

gβdT

dY+ µeff

ρ0

d3U

dY 3− µ

ρ0K

dU

dY+ ρCF

ρ0

dU 2

dY=0. (39)

Equation (39) can be written in non-dimensional form using Equation (10)with µeff =µ as

d3u

dy3+ Gr

Re

dθ

dy−σ 2 du

dy+ I

du2

dy=0. (40)

Evaluating Equation (40) at the left wall i.e., at y =−1/4 it yields(

d3u

dy3−σ 2 du

dy+ I

du2

dy

)y=−1/4

=GR. (41)

The boundary condition at the right wall is the same as that of isothermal–isothermal case replacing Rt by Rqt i.e.,

(d2

u

dy2

)y=1/4

=−48− Rqt

2GR. (42)

In the absence of inertia forces Equation (41) become(

d3u

dy3−σ 2 du

dy

)y=−1/4

=GR. (43)


Substituting Equation (26) in Equations (42) and (43) and equating coeffi-cients of like powers of ε to zero, we obtain the boundary conditions forzeroth and first order of ε as follows(

d2u0

dy2

)y=1/4

=−48− Rqt

2GR, (44)

(d3

u0

dy3−σ 2 du0

dy

)y=−1/4

=GR, (45)

(d2

u1

dy2

)y=1/4

=0, (46)

(d3

u1

dy3−σ 2 du1

dy

)y=−1/4

=GR. (47)

The solution of Equations (27) and (28) with the boundary conditionsgiven by Equations (29), (31) and (44)–(47) can be easily obtained and thesolutions are not presented.

On the other hand, in the absence of inertial force, the temperature fieldcan be obtained by Equation (18) using the solutions of Equations (27)and (28).

Case 3: Isothermal-Isoflux Walls (T1 −q2)

The thermal boundary conditions for this case are

T

(−L

2

)=T1; q2 =−K

(dT

dy

)L/2

, (48)

where q2 is a constant. Equation (48) in the non-dimensional form usingEquation (10) with �T =q2D/k becomes

θ

(−1

4

)=Rtq;

(dθ

dy

)y=1/4

=−1, (49)

where Rtq = (T1 −T0)/�T is the thermal ratio parameter for the isothermal–isoflux case.

Following the procedure used for isoflux–isothermal walls, the dimen-sionless boundary conditions obtained from using Equation (1) and apply-ing Equation (10) can be written as(

d2u

dy2

)y=−1/4

=−48+ Rtq

2GR, (50)


(d3

u

dy3−σ 2 du

dy

)y=1/4

=GR. (51)

Using Equation (26) in Equations (50) and (51) and equating like powersof ε to zero, one obtains the following boundary conditions for zeroth andfirst order as(

d2u0

dy2

)y=−1/4

=−48+ Rqt

2GR, (52)

(d3

u0

dy3−σ 2 du0

dy

)y=1/4

=GR, (53)

(d2

u1

dy2

)y=1/4

=0, (54)

(d3

u1

dy3−σ 2 du1

dy

)y=1/4

=GR. (55)

One can obtain easily the solution of Equations (27) and (28) usingboundary conditions (29), (31) and (52)–(55). The expression for temper-ature field in the absence of inertial forces using Equation (18) can beobtained using solutions of Equations (27) and (28).

4. Heat Transfer Aspects

We shall now calculate the heat transfer parameters on the wall expressesin terms of the Nusselt numbers.

1. Isothermal–Isothermal (T1 −T2) Walls

Nu1 = D�T

(dT

dY

)Y=−L/2

=(

dθ

dy

)y=−1/4

, (56)

Nu2 = D�T

(dT

dY

)Y=L/2

=(

dθ

dy

)y=1/4

, (57)

where Nu1 and Nu2 are the Nusselt numbers at the left and right walls,respectively.


2. Isoflux–Isothermal (q1 −T )2 Walls

Nu1 = h1D

K= Dq1

K (T1 −T0)= 1

θ(− 1

4

) , (58)

Nu2 = h2D

K= D

�T

(dT

dy

)=(

dθ

dy

)y=1/4

. (59)

3. Isothermal–Isoflux Walls (T1 −q2)

Nu1 = h1D

K= D

Dt

dT

dy=(

dθ

dy

)y=−1/4

, (60)

Nu2 = h2D

K= Dq2

K (T2 −T0)= 1

θ( 1

4

) . (61)

The expressions for Nusselt numbers for different wall temperatures areobtained and results are shown in Figure 7.

5. Numerical Method

The general Equation (15) including the inertial effects do not poses ananalytical solution. Therefore, a numerical procedure is employed usingfinite-difference method. Replacing the derivatives with corresponding cen-tral difference approximations, lead to m linear algebraic equations wherem is the number of divisions from y = −1/4 to 1/4. The solutions ofreduced algebraic equations are solved by the successive-over-relaxationmethod. The relaxation parameter is fixed by comparing the numerical val-ues with the analytical results for the case GR = 0. The convergence cri-teria is based on the step size and the previous iterations for the iterativedifference to the order 10−6. The comparison of numerical and analyticalresults is shown in Figure 2 and they agree very well for ε = 0 and varyas ε increases. Figures 3 and 4 display the effects of inertia on the velocityand temperature profiles for asymmetric wall temperatures.

6. Results and Discussion

An analytical solution for mixed convective flow and heat transfer in a ver-tical porous channel is obtained using regular perturbation method in theabsence of inertial forces. A numerical procedure is performed to obtain theflow solutions in the presence of inertial forces. The flow is modeled usingBrinkman–Forchhiemer extended Darcy equations. The viscous and Darcy


-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

Ec=0.1Pr=2.0P=-5.0=2.0=0.2

a=5.0

8 6 4 = 2

u

y

Figure 2. Dimensionless velocity profiles for different values of porous parameter σ .

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4

Ec=0.1Pr=2.0P=-5.0=2.0=0.2

a=5.0

8 6 4

= 2

y

Figure 3. Dimensionless temperature profiles for different values of porous para-meter σ .


-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

=2.0Ec=0.1Pr=2.0P=-5.0

=2.0a=5.0

0.8

0.6

0.4

= 0.2

u

y

Figure 4. Dimensionless velocity profiles for different values of ratio of viscosityparameter η.

dissipation terms are included in the energy equation. The analytical solu-tions has been determined using BrGr/Re as the perturbation parameter.The heat transfer coefficient have been tabulated for three different thermalboundary conditions.

The velocity and temperature fields in the case of asymmetric heating(RT = 1) are obtained and are depicted in Figures 2–4. For asymmetricheating the temperatures at the boundaries are different and hence velocityand temperature fields depend on perturbation parameter ε and GR. Whenthe flow is upward, ε and GR are positive. On the other hand, when theflow is downward, ε and GR are negative. The sign of ε and GR are equalbut their absolute values are different. It is also seen from Figure 2 thatanalytical and numerical solutions agree very well in the absence of iner-tial effects for ε =0 and vary largely for ε =8.

The effect of the mixed convection parameter GR(= Gr/Re) on thevelocity field with asymmetric isothermal–isothermal wall conditions isshown in Figure 2 in the absence of inertia effects. Figure 2 displays thevelocity profiles for GR=±500 for fixed σ . The curves with maximum val-ues are on the left for GR = −500 and on the right for GR = 500. Flowreversal occurs for large values of GR near the cold wall at y =−1/4 forpositive GR and near the hot wall at y = 1/4 for negative GR. Also it isinteresting to note that flow reversal is less for large ε. The influence of


the presence of the porous medium, Darcian and inertial effects on velocityand temperature distributions in the channel is illustrated for asymmetricwall temperatures in Figures 3 and 4. These results are obtained numer-ically. The presence of porous medium produces flow resistance. In addi-tion, the inertia effects adds on this resistance mechanism which furtherreduces the flow in the channel. Lai and Kulacki (1987) also proved thesimilar result that the inertial forces has no effect on the wall heat flux, butit does have the effect of flattening the dimensionless velocity profile. Theflow reversal in the absence of inertial forces (Figure 2) is same as in thepresence of inertial forces (Figure 3).

Figure 4 shows the plots of θ versus y for different values of ε, σ andI for the asymmetric heating of the walls. This graph shows that tempera-ture field increases with increase in the value of ε and it is suppressed inthe presence of inertia for both positive and negative values of GR.

Figure 5 illustrates the influence of porous parameter σ on temperaturewith isoflux–isothermal wall conditions. It is observed that the temperatureat the wall with constant heat flux decreases as σ increases for positiveand negative values of GR. However, the wall temperature is more influ-enced for positive GR than for the negative GR. Figure 6 displays the vari-ations on temperature profiles with isothermal–isoflux wall conditions forvalues of mixed convection parameter. The flow nature is similar to that for

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4

=2.0Ec=0.1Pr=2.0P=-5.0=2.0

a=5.0

0.80.6

0.4 = 0.2

y

Figure 5. Dimensionless temperature profiles for different values of ratio of viscosityparameter η.


-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

0.0 0.5 1.0 0.5 2.0 0.5 3.0 3.5 4.0

Ec=0.1Pr=2.0P=-5.0η=0.2a=5.0σ=2.0

864κ = 2

θ

y

Figure 6. Dimensionless temperature profiles for different values of ratio of thermalconductivity parameter κ.

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

0 3 4 5 6

Ec=0.1P=-5.0=2.0=0.2

a=5.0=2.0

6.04.02.0Pr= 0.01

y

1 2

Figure 7. Dimensionless temperature profiles for different values of Prandtl numberPr.


-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

0 2 3 4 5 6 8 10 11 12

y

θ

0.6

0.40.2

Pr=2.0P=-5.0κ=2.0η=0.2a=5.0σ=2.0

Ec =0.01

1 7 9

Figure 8. Dimensionless temperature profiles for different values of Eckert number Ec.

isoflux–isothermal wall conditions. Figure 7 displays the rate of heat trans-fer for R = 500 with σ for isothermal-isothermal, isothermal–isoflux andisoflux–isothermal wall temperatures. Rate of heat transfer is less at y = 1

4compared to at y =− 1

4 for different wall temperatures and the difference isvery large for isothermal–isoflux and isoflux–isothermal wall temperatures(Figures 8–10).

7. Conclusions

The problem of steady, laminar, mixed convective flow and heat trans-fer in a vertical channel embedded in a porous media with symmet-ric and asymmetric wall temperatures is studied by both analytical andnumerical methods. Three different combinations of thermal wall condi-tions such as isothermal–isothermal, isoflux–isothermal and isothermal–isoflux are considered. The nonlinear dimensionless equations are solvedanalytically using perturbation series using ε(= BrGr/Re) as the pertur-bation parameter. The mixed convection problem which includes the iner-tial effects was solved numerically using finite-difference technique. Thenumerical solutions are successfully validated by the analytical solutions inthe absence of inertial forces. Dimensionless Nusselt numbers at the leftand right walls for the above-mentioned three different wall conditions aresolved analytically and graphically depicted for different values of porous


1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0-8

-4

0

4

8

12

16

2

2

1

1

q B

q T

q B

q T

a =5.0

a =5.0

a =5.0

a =5.0

q

2. 42 2

> a2

1. 42 2

> a2

Figure 9. Dimensionless rate of heat transfer for different values of the parameter σ .

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0

-4

-2

0

2

4

6

8

10

12

14

2. 42 2

> a2

1. 42 2

> a2

q B

q B

q T

q T

q

a

Figure 10. Dimensionless rate of heat transfer for different values of couple stressparameter “a”.


parameter. The velocity and temperature are also evaluated as showngraphically for asymmetric boundary temperatures. Increase in the valuesof porous parameter and Forchhiemer drag term produces reduced flow inthe channel for unequal wall temperatures. Flow reversal near the walls isobtained for asymmetric wall temperatures and is found to increase in thepresence of porous and inertial effects. The viscous dissipation enhancesthe effect of flow reversal in the case of downward flow while it encoun-ter this effect in the case of upward flow.

Acknowledgement

The first two authors thank Prof. M.S. Malashetty for his suggestions andto UGC New Delhi for the financial support under the Special AssistanceProgramme, DRS.

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