Non-Darcy mixed convection in a vertical pipe filled with porous medium

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International Journal of Thermal Sciences 50 (2011) 725e735

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International Journal of Thermal Sciences

journal homepage: www.elsevier .com/locate/ i j ts

Non-Darcy mixed convection in a vertical pipe filled with porous medium

Ashok Kumar, P. Bera*, J. KumarDept. of Mathematics, Indian Institute of Technology, Roorkee 247667, India

a r t i c l e i n f o

Article history:Received 2 July 2010Received in revised form17 November 2010Accepted 23 November 2010Available online 8 January 2011

Keywords:Mixed convectionPorous mediumNon-Darcy modelChebyshev spectral-collocation methodDistortion

* Corresponding author. Tel.: þ91 1332 285071; faxE-mail address: [email protected] (P. Bera).

1290-0729/$ e see front matter � 2010 Elsevier Masdoi:10.1016/j.ijthermalsci.2010.11.018

a b s t r a c t

The present paper reports an analytical as well as numerical investigation of fully developed mixedconvective flow in a vertical pipe filled with porous medium. The motion in the pipe is induced byexternal pressure gradient and buoyancy force. The non-Darcy BrinkmaneForchheimer-extended modelhas been considered. The Chebyshev spectral-collocation method has been used to solve the coupleddifferential equations numerically. A comprehensive investigation on the dependency of mixedconvective flow on governing parameters indicates that depending on the values of other parameters thevelocity profile possess point of inflection beyond a threshold value of Rayleigh number (Ra). In the caseof buoyancy-opposed flow, the velocity profile may contain point of inflection in the center zone andpoint of separation at the vicinity of the wall. The appearance of point of separation causes the back flownear the wall. In contrast to buoyancy-opposed case, where enhancement of Ra increases the magnitudeof the center velocity as well as temperature till the distortion appears on both profiles, in buoyancy-assisted case both velocity as well as temperature decrease on increasing Ra at the center. The points ofseparation as well as inflection die out on reducing the media permeability or increasing magnitude ofthe form drag coefficient. Further, it was observed that for buoyancy-opposed flow the velocity as well astemperature profiles have a kind of distortion beyond a threshold value of Ra, which is also a function ofother governing parameters. In this situation, the heat transfer rate varies abruptly as a function of Ra.

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1. Introduction

The research works, in the area of convective heat transfer influid saturated porous media, have substantially increased duringrecent years due to its numerous practical applications encoun-tered in engineering and sciences. Among these works natural andforced convection studies occupy the majority of investigations.The inter-facial area of mixed convection which connects naturaland forced convection, in comparison, has not been given dueattention in porous media. Mixed convection problems in porousmedia occur very often in the nature, e.g., in studies of shallow-water and deep-sea hydrodynamics. One important example ofmixed convection in shallow-water seas is given by hydrothermalvents by which hot, mineral-rich water ejects through a permeablesea-bed [1]. This problem constitutes a very new research area, andtheoretical investigation of it has been largely overlooked. For fluidenvironments, however, there are many papers which deal withmixed convection, for example, in connection with fluid flow ina vertical pipe [2e8], in a vertical annulus [9,10], and in a verticalchannel [11,12].

:þ91 1332 273560.

son SAS. All rights reserved.

Few investigations in wall bounded mixed convection throughvertical annuli (e.g., [13,14]) and channels (e.g., [15e22]) filled withporous medium are reported. Parang and Keyhani [13] studiedthe fully developed buoyancy-added mixed convective flow ina vertical annulus employing DarcyeBrinkman model. They foundthat the Brinkman term has a negligible effect on the flow whenDarcy number (Da) is very small. According to Muralidhar [14], inthe case of buoyancy-assisted mixed convection, the Nusseltnumber (Nu) increases with increasing Rayleigh number (Ra).

In the vertical channel, when the walls are heated uniformly,Hadim [15] investigated the evolution of mixed convection in theentrance region for Darcy as well as non-Darcy cases. In the sameyear, Hadim along with Chen [16] have reported the extension ofthe above study for asymmetric heating of the walls. They havestudied the effect of media permeability on the buoyancy-assistedmixed convection in the entrance region of a vertical channel atfixed values of Reynolds number, Forchheimer number, Prandtlnumber, and Grashof number. It was found that distortions in thevelocity profile lead to increased heat transfer rates when Da isdecreased. While studying the same with discrete heat sources atthe walls, in place of asymmetric wall heating, they [17] found, thelocation of the flow separation from the cold wall does not changewhile reattachment moves further downstream. The Nusseltnumber increases with decreasing Da and the effect of Da is more

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A. Kumar et al. / International Journal of Thermal Sciences 50 (2011) 725e735726

pronounced over the first heat source and in non-Darcy regimes.A comprehensive review of laminar wall bounded forced or mixedconvection may be found in the book of Nield and Bejan [23]. Asindicated by Chen et al. [18] for non-Darcy mixed convection ina vertical channel filled with a porous medium, the buoyancy forcecan significantly affects Nu for higher values of Ra and Da and lowervalues of Forchheimer number. An extensive study of mixedconvective flow and its stability in vertical channel filled withporous medium has been reported elsewhere [19e22]. They haveshown that the fully developed one dimensional mixed convectiveflow in the vertical channel does not remain one dimensionalalways.

It is then natural to ask how these flow dynamics will bemodulated when vertical channel is replaced by vertical pipe. Tothe best knowledge of us, except the work of Chang and Chang [24]in which mixed convection in a vertical tube partially filled withporous medium is reported, mixed convection in vertical pipe filledwith a porous medium has not been considered yet. Therefore, anattempt has been made in this direction by investigating the flowdynamics of wall bounded fully developed mixed convection andits dependency on different controlling porous-media parameters.The aim of the article is precisely this.

An outline of the paper is as follows. In Section 2, the mathe-matical formulation and solution of the physical problem are given.Results and discussions along with validation of the numericalresults are reported in Section 3. Finally, some important features ofthe analysis are concluded in Section 4.

2. Mathematical formulation

We consider a fully developed mixed convection flow caused byan external pressure gradient and a buoyancy force in a semi-infinite vertical pipe [8], filled with porous medium (see Fig. 1). Thewall temperature is linearly varying with z* as Tw ¼ T0 þ C1R0z*,where C1 is a constant and T0 is upstream reference wall temper-ature and R0 is radius of the pipe. The gravitational force is alignedin the negative z*-direction.

The thermo-physical properties of the fluid are assumed to beconstant except for density dependency of the buoyancy term inthe momentum equation. The porous medium is saturated witha fluid that is in local thermodynamic equilibrium with the solid

IsotropicPorousMedium

T w=

T 0+

C1R

0z*

Flow

R0

ψr*

z*

g

Fig. 1. Schematic of the dimensional physical problem.

matrix. The medium is assumed to be isotropic in permeability aswell as thermal diffusivity. In expressing the equations for the flowin the porous medium, it should be noted that the Darcy modelpresents a linear relationship between velocity of discharge and thepressure gradient. As the Darcy model does not hold when the flowvelocity is not sufficiently small, or when the permeability is high[25e32], extensions to this model known as Brinkman-extended orForchheimer-extendedmodels exist [23,29]. In short, the Brinkmanterm is found to be needed for satisfying a no-slip boundarycondition at solid walls, whereas the Forchheimer term accountsfor the form drag. Also in analogy with the NaviereStokes equa-tions, the Darcymodel has been extended by including thematerialderivative. The necessity of the simultaneous inclusion of all orsome of these extensions has been discussed in the literature[28,30,33]. The objective of the paper is to understand the fluid flowas well as heat transfer mechanism of the steady, unidirectionalfully developed flow (basic flow). Therefore, it is assumed that flowis in vertical direction only i.e. the velocity vector is (0,0,w*). Fromthe continuity equation, it is clear that w* is function of r* only. Asa consequence of this, the governing differential equations formomentum and energy, in cylindrical coordinate, of the basic flowcan be written as

rfcF

K1=2

��w*��w* ¼ �dp*

dz*þ ~m

"d2w*

dr* 2 þ 1r*

dw*

dr*

#� mf

Kw*

þrf gbT�T* � Tw

�(1)

w*vT*

vz*¼ a

"v2T*

vr*2þ 1r*

vT*

vr*

#(2)

In the above equations, w*, K, p*, t*, T*, and rf are flow velocity,permeability, pressure, time, temperature, and fluid densityrespectively. Further, g, e, cF, ~m, bT, and s are the gravitationalacceleration, porosity, form drag coefficient, effective viscosity,volumetric thermal expansion coefficient, and ratio of heat capac-ities respectively.

Using the following non-dimensional quantities:

r ¼ r*

R0; W0 ¼ w*

W*c; z ¼ z*

R0;

p ¼ p*

rfW* 2c

; Q0 ¼ Tw � T*

C1R0RePr

the dimensionless momentum and energy equations are

d2W0

dr2þ 1

rdW0

dr� L

DaW0 � RaQ0 � Re

dpdz

� ReFjW0jW0 ¼ 0 (3)

d2Q0

dr2þ 1

rdQ0

dr¼ �W0: (4)

The corresponding boundary conditions are given by

dW0

dr¼ dQ0

dr¼ 0 at r ¼ 0; (5)

W0 ¼ Q0 ¼ 0 at r ¼ 1; (6)

with W0 and Q0 being the basic velocity and temperature,respectively.

In the above equations the dimensionless parameters are theRayleigh number (Ra), Forchheimer number (F), Darcy number

A. Kumar et al. / International Journal of Thermal Sciences 50 (2011) 725e735 727

(Da), viscosity ratio ðL ¼ ~m=mf Þ and Reynolds number (Re).Different values have been reported for ~m in the literature (e.g.,[34e36]) leading to aL other than unity. However, in the absence ofany specific measured value, in this study L ¼ 1 has been taken.

Above coupled equations (3) and (4) along with the boundaryconditions (5) and (6) are solved analytically, when the inducedform drag into the system is neglected, otherwise these are solvednumerically using the spectral Chebyshev collocation method [37].

2.1. Analytic solution

The analytic solution of basic flow is obtained when F ¼ 0. Thetemperature may be eliminated from the above equations (3) and(4) to give the following expression for the velocity field."�

V2 � 12Da

�2

þl2#W0 ¼ 0; (7)

where, l2 ¼ Ra� ð1=4Da2Þ and V2 ¼ ðd2=dr2Þ þ ð1=rÞðd=drÞ. Theaxial pressure gradient can be determined by requirement of globalmass conservation:

Z10

rW0ðrÞdr ¼ 12: (8)

The solution of equation (7), for the two different cases: (i)l2 > 0, (ii) l2 < 0, is given by

W0ðrÞ ¼ a1J0�P1=21 r

�þ a2I0

�P1=22 r

�; (9)

where,

a1 ¼h0:25P1=21 P1=22 I0

�P1=22

�ihP1=22 J1

�P1=21

�I0�P1=22

�� P1=21 J0

�P1=21

�I1�P1=22

�i;

a2 ¼ �a1J0�P1=21

�I0�P1=22

�;

P1 ¼8<:

i� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Ra� 14Da2

q �� 1

2Da for l2 > 0� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

4Da2 � Raq �

� 12Da for l2 < 0;

and

P2 ¼8<:

i� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Ra� 14Da2

q �þ 1

2Da for l2 > 0� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

4Da2 � Raq �

þ 12Da for l2 < 0:

J0 and I0 are zeroth order first and second kind of Bessel func-tions respectively. The corresponding temperature is given by

Q0ðrÞ¼a1P1

hJ0�P1=21 r

��J0�P1=21

�i�a2P2

hI0�P1=22 r

��I0�P1=22

�i(10)

2.2. Numerical solution

In order to solve the basic flow numerically, equations (3) and(4) are divided by the constant (Re(dp/dz)). As a consequence ofit, the above equations are modified as

d2Wdr2

þ 1rdWdr

� 1Da

W � Ra Q� 1� F 0jW jW ¼ 0 (11)

d2Qþ 1 dQ ¼ �W (12)
dr2 r dr
in which, W ¼ W0=ðReðdp=dzÞÞ;Q ¼ Q0=ðReðdp=dzÞÞ are themodified basic velocity as well as temperature respectively. Thecorresponding boundary conditions will remain unchanged.

To approximate the field variables, W and Q by Chebyshevpolynomials, the range, [0, 1], of the independent variable, r, ismapped into [�1, 1] by using the function x ¼ 1 � 2r.

The governing equations (11)e(12) and the correspondingboundary conditions in terms of Chebyshev variable x are

�F 0jW jW �1þ 4d2W

dx2� 21� x

dWdx

!� 1Da

W þRaQ ¼ 0; (13)

4d2Q

dx2� 21� x

dQdx

¼ �W; (14)

with boundary conditions

W ¼ Q ¼ 0 at x ¼ �1: anddWdx

¼ dQdx

¼ 0 at x ¼ 1:

The discretized governing equations (13)e(14) in terms of Cheby-shev variable x are

�F 0jW jWj�1þ4XNk¼0

BjkWk�

21�xj

!XNk¼0

AjkWk�1Da

Wj�RaQj¼0

(15)

Wj þ 4XNk¼0

BjkQk �

21� xj

!XNk¼0

AjkQk ¼ 0 (16)

where j¼1,2,3...N � 1,

Ajk ¼

8>>>>>><>>>>>>:

cjð�1Þkþj

ckðxj�xkÞ; jsk

xj

2�1�x

2j

; 1 � j ¼ k � N � 1

2N2þ16 ; j ¼ k ¼ 0

�ð2N2þ16 ; j ¼ k ¼ N

and

Bjk ¼ Ajm$Amk:

In the above

cj ¼2; j ¼ 0;N1; 1 � j � N � 1

and xj ¼ cosðpj=NÞ; for 0 � j � N are Chebyshev collocationpoints. With the help of boundary conditions the above equationsform the following system:

AX ¼ b (17)

where A, X and b are matrices of order (2N þ 2)�(2N þ 2),(2N þ 2) � 1 and (2N þ 2) � 1 respectively. The system of linearequations (17) was solved using the MATLAB command A\b.

To find W0 ¼ (Re(dp/dz))W and Q0 ¼ (Re(dp/dz))Q the axialpressure gradient, (Re(dp/dz)), is determined by the global massconservationZ10

r�Re

dpdz

�WðrÞdr ¼ 1

2: (18)

Table 2Relative error of numerical velocity (W0) at r ¼ 0, for different Ra when Da ¼ 10�2

and F0 ¼ 0.

Ra Analytic solution Numerical solution Absolute Relativeerror

(A) (B) ðjA�BA jÞ

100 1.1233138912334 1.1233133813621 4.5 � 10�7

500 0.7849727196176 0.7849722728651 5.7 � 10�7

1000 0.5145013287329 0.5145009738370 6.9 � 10�7

5000 0.0047390551543 0.0047390489666 1.3 � 10�6

10000 �0.0149183169018 �0.0149182901456 1.8 � 10�6

100000 �0.0001966019852 �0.0001966009225 5.4 � 10�6

�100 1.3577498985402 1.3577493730019 3.9 � 10�7

�500 2.0453854923506 2.0453850330507 2.2 � 10�7

�1000 3.7541981818232 3.7541984949787 8.3 � 10�8

�5000 �8.696336355505 �8.696339376125 3.4 � 10�7

�10000 42.5570092026544 42.5567057712656 7.1 � 10�6

�100000 48.3283417058817 48.3285388388905 4.1 � 10�6


This implies�Re

dpdz

�¼ 1Z1

0

rWðrÞdr:

The rate of heat transfer (Nusselt number, Nu) is determinedfrom the temperature field. Following Rogers and Yao [7], theaverage Nu is defined for this problem as follows:

Nu ¼ hR0k

¼dQ0ð1Þ

drZ10

W0Q0rdr

(19)

where h and k are convective heat transfer coefficient and fluidthermal conductivity. The integral in (19) is evaluated by GausseChebyshev quadratures.

3. Results and discussion

In this section a rigorous study is made to find the dependenceof basic velocity, temperature, and heat transfer rate i.e. Nusseltnumber, on different controlling parameters.

The heat source intensity in the pipe filled with a porousmedium is governed by Rayleigh number (Ra), whereas, mediapermeability and form drag are controlled by Darcy number (Da)and modified Forchheimer numbers (F0) respectively. Based, on thenon-dimensional analysis as well as available realistic data in theliterature [23], wide range of different parameters (10�1 � Da� 10�8, 0 � Ra � 1010, 0 � F0 � 1016) is chosen here. The discussionbelow has been split into three parts. The first two parts deal withthe influence of controlling parameters on velocity and tempera-ture for buoyancy assisted and opposed cases, whereas, the thirdpart deals with rate of heat transfer in both assisted as well asopposed cases. Before discussing the influence of controllingparameters on flowas well as heat transfer rate, a verification of thenumerical results is given.

3.1. Validation of numerical solution

In order to validate the results presented in this work, followingsteps were carried out. First, the response of solution to grid size bychanging the order of polynomials was tested. Second, the relativeerror of numerical solution was calculated as a special case F0 ¼ 0.Third, the numerical solutions in purely viscous medium werecompared with published one.

Table 1 illustrates the grid independence of the numericalsolution. We have taken different order of the Chebyshev

Table 1Dependence of the velocity of the assisted flow at r ¼ 0, onthe order of approximation polynomials at Da ¼ 10�2,F0 ¼ 102 and Ra ¼ 103.

N (terms) W0 at r ¼ 0

5 0.502559861954710 0.516204534142615 0.516410745984820 0.516442255283530 0.516453684312640 0.516455580880750 0.516456096492960 0.516456281265470 0.5164563603653

polynomials, ranging from 5 to 70, in spectral approximation to findthe basic velocity at r ¼ 0 when Da, Ra, and F0 are fixed at 10�2, 103,and 102 respectively. We have achieved 4-digit point accuracy up tothe 20 terms of the Chebyshev polynomials. As the number of termsincreases, the results remain consistent and the accuracy improvedup to 6 digits. The relative error, which is defined as the absolutevalue of the ratio of difference between analytical and numericalsolutions to analytical solution, is depicted in Table 2. As can beseen from the table, it is always less than 10�5. It implies that thesolution generated by the spectral-collocation method is excellentagreement with the analytic solution. The comparison betweenpublished [8] and present results in purely viscous medium is givenin Table 3.

3.2. Buoyancy-assisted flow

In order to understand the basic flow in vertical pipe, when pipeis filled with a homogeneous and isotropic porous medium, thevelocity as well as temperature are plotted as function of differentcontrolling parameters. It can be seen from Fig. 2(a) and (b) thatwhen Da ¼ 10�2 and F0 ¼ 102, the basic velocity profile containspoint of inflection for Ra� 200. For lower values of Ra (i.e. Ra< 200)flow is nearly flat in most of the domain of the flow field exceptnear the wall of the pipe. For Ra / 0 flow is isothermal in natureand due to this buoyancy force has no effect on the flow. As Raattains non-zero values (non-isothermal flow), fluid near the wallmoves faster due to heating and the fluid motion near the center ofthe pipe retards to satisfy the global mass conservation imposed bysome external pressure gradient. Its effect becomes significant onlyfor higher values of Ra. As a result of it, a point of inflection appearsfor Ra¼ 200, and it will be continued as Ra is increased further. Thepoint of inflection on the velocity profile moves from center to wallas Ra is increased. Also magnitude of the velocity of fluid, at the

Table 3Comparison between published [8] and present results (velocity (W0) at r ¼ 0, fordifferent Ra when Da ¼ 1012 and F0 ¼ 0).

Ra Su and Chung [8] Present work

0 1.0000000 0.999999868 0.7476530 0.7476528200 0.4259873 0.4259871800 �0.0593991 �0.05939911000 �0.1032193 �0.1032192�100 1.5715000 1.5714998�200 2.6388881 2.6388882�300 5.1883033 5.1883035

0 0.25 0.5 0.75 10

0.5

1

1.5

Ra = 0Ra = 50Ra = 100Ra = 200Ra = 500

r

W0

a

0 0.25 0.5 0.75 1-0.5

0

0.5

1

1.5

2

2.5

Ra = 103Ra = 2x103Ra = 3x103Ra = 5x103Ra = 8x103

r

W0

b

0 0.25 0.5 0.75 10

0.2

Ra = 0Ra = 50Ra = 100Ra = 200Ra = 500

r

Θ0

c

0 0.25 0.5 0.75 10

0.1

0.2

0.3Ra = 103Ra = 2x103Ra = 3x103Ra = 5x103Ra = 8x103

r

Θ0

d

Fig. 2. Effect of Rayleigh number (Ra) on velocity profiles: (a) and (b), and on the temperature profiles: (c) and (d) of buoyancy-assisted flow at Da ¼ 10�2 and F0 ¼ 102.

0 0.25 0.5 0.75 10

0.1

0.2

0.3

Da = 10-1Da = 10-2Da = 10-3Da = 10-4Da = 10-6Da = 10-8

r

b

Θ0

0 0.25 0.5 0.75 1-0.5

0

0.5

1

1.5

2

Da = 10-1Da = 10-2Da = 10-3Da = 10-4Da = 10-6Da = 10-8

r

a

W0

Fig. 3. Effect of Darcy number (Da) on the (a) velocity profiles and (b) temperature profiles for buoyancy-assisted flow at Ra ¼ 103 and F0 ¼ 102.


0 0.25 0.5 0.75 10

0.1

0.2

0.3

F’ = 102F’ = 103F’ = 104F’ = 105

r

Θ0

b

0 0.25 0.5 0.75 1

0

0.5

1

1.5

2

F’ = 102F’ = 103F’ = 104F’ = 105

r

W0

a

Fig. 4. Effect of modified Forchheimer number (F0) on the (a) velocity profiles and (b) temperature profiles for buoyancy-assisted flow at Ra ¼ 103 and Da ¼ 10�2.

0 0.25 0.5 0.75 1-2

2

6

10

Ra = -600Ra = -800Ra = -1000Ra = -1500

b

r

W0

0 0.25 0.5 0.75 10

0.5

1

1.5

2

2.5

Ra = -1Ra = -100Ra = -200Ra = -500

a

r

W0

0 0.25 0.5 0.75 10

0.1

0.2

0.3

0.4

0.5

Ra = -1Ra = -100Ra = -200Ra = -500

c

r

Θ0

0 0.25 0.5 0.75 10

0.2

0.4

0.6

0.8

1

Ra = -600Ra = -800Ra = -1000Ra = -1500

d

r

Θ0

Fig. 5. Effect of Rayleigh number (Ra) on velocity profiles: (a) and (b), and on the temperature profiles: (c) and (d) of buoyancy-opposed flow at Da ¼ 10�2 and F0 ¼ 102.


0 0.25 0.5 0.75 1-50

-30

-10

10

30

50

Ra = -5x103Ra = -104Ra = -5x104Ra = -105

a

r

W0

0 0.25 0.5 0.75 1-1

-0.5

0

0.5

1

Ra = -5x103Ra = -104Ra = -5x104Ra = -105

b

r

Θ0

Fig. 6. A kind of distortion in (a) velocity and (b) temperature profiles for buoyancy-opposed flow as a function of Rayleigh number (Ra) at Da ¼ 10�2 and F0 ¼ 102.


center point, is reduced on increasing of Ra. However, the same atthe vicinity of the wall, is enhanced on increasing of Ra, which isconsequence of the enhancement of buoyancy force. The corre-sponding temperature profile shows that the maximummagnitudeof the temperature decreases on the increasing of Ra and theparabolic profile becomes flat at the center zone of the pipe, whichcan be seen from Fig. 2(c) and (d). As the value of Ra is increasedbeyond a threshold value (e.g., for Da ¼ 10�2, it is around 5000)back flow starts (i.e. velocity becomes negative) near the center ofthe pipe (see Fig. 2(b)). Another important observation from thesame figure is that higher values of Ra indicate higher magnitude ofthe velocity at the vicinity of wall of the pipe.

It is to be noted that the appearance of point of inflection isa sufficient condition for instability of the basic flow [38]. Therefore,it is expected that forDa¼ 10�2 and F0 ¼ 102, there exist a minimumpressure gradient such that the basic flow will remain no longer asan one dimensional flow for Ra � 200.

The influence of media permeability on the velocity as well as ontemperature is shown in Fig. 3(a) and (b). Decreasing of the media

0 0.25 0.5 0.75 1-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4Da = 10-2Da = 10-3Da = 10-4Da = 10-6Da = 10-8

a

r

W0

Fig. 7. Effect of Darcy number (Da) on the (a) velocity profiles and (b) temp

permeability, by varyingDa from 10�1 to 10�8, disappears back flownear the center (see Fig. 3(a)) and also reduces themagnitude of thevelocity significantly. Whereas, the changes in the correspondingtemperature (profile as well as magnitude), are not significant incomparison to velocity, which can be observed from Fig. 3(b).Another important finding is that the point of inflection on thevelocity profile dies out on decreasing of Da (see Fig. 3(a)). Thequalitative explanation of the above results can be explained byrecalling the definition of the media permeability. When the valueof Da decreases, the penetrating capability of a porous mediumdiminishes and induces more drag force on fluid. As a consequenceof it above result is expected.

In order to understand the effect of inertia on the basic flow,both velocity as well as temperature as function of form drag i.e. F0,are plotted in Fig. 4(a) and (b). Since form drag in the mediumreduces the flow, as a result, the maximum magnitude of thevelocity is expected to decrease on the increasing of F0, which isshown in Fig. 4(a). It also changes the velocity profile drastically.However, the effects of F0 on the corresponding temperature profile

0 0.25 0.5 0.75 10

0.2

0.4

0.6Da = 10-2Da = 10-3Da = 10-4Da = 10-6Da = 10-8

b

Θ0

erature profiles of buoyancy-opposed flow at Ra ¼ �103 and F0 ¼ 102.

0 0.25 0.5 0.75 10

0.2

F’ = 102F’ = 104F’ = 106F’ = 108

b

r

Θ0

0 0.25 0.5 0.75 10

0.5

1

1.5

F’ = 102F’ = 104F’ = 106F’ = 108

a

r

W0

Fig. 8. Effect of modified Forchheimer number (F0) on the (a) velocity profiles and (b) temperature profiles of buoyancy-opposed flow at Ra ¼ �103 and Da ¼ 10�3.

10-1 101 103 105 107 1090

50

100

150 F’ = 108F’ = 1010F’ = 1012F = 1014

c

Da = 10-6

Ra

Nu

10-1 101 103 105 107 1090

5

10

15

20

25

F’ = 1010F’ = 1012F’ = 1014F’ = 1016

d

Da = 10-8

Nu

Ra

10-1 101 103 105 1070

20

40

60

80

100

F’= 102F’ = 104F’ = 106F’ = 108

a

Da = 10-2

Ra

Nu

10-1 101 103 105 1070

20

40

60

80

100

F’ = 104F’ = 106F’ = 108F’ = 1010

b

Da = 10-4

Ra

Nu

Fig. 9. The variation of the Nusselt number (Nu) with the Rayleigh number (Ra) for the buoyancy-assisted flow at (a) Da ¼ 10�2, (b) Da ¼ 10�4, (c) Da ¼ 10�6, and (d) Da¼ 10�8 alongwith the different modified Forchheimer number (F0).


102 103 104 105 106 107 1080

1

2

3

4

5c

Nu

|Ra|

100 102 104 106 1083

3.2

3.4

3.6

3.8

4b

Nu

F’

10-6 10-5 10-4 10-3 10-2 10-1

1

2

3

4

5a

Nu

Da

Fig. 10. The variation of the Nusselt number (Nu) on (a) Darcy number (Da) atRa ¼ �103 and F0 ¼ 102, (b) modified Forchheimer number (F0) at Ra ¼ �103 andDa ¼ 10�3 and (c) Rayleigh number (Ra) at Da ¼ 10�2 and F0 ¼ 102 for the buoyancy-opposed flow.


is not significant relative to its effects on the maximum magnitudeat center of the pipe (see Fig. 4(b)).

In addition of these, form drag effect is negligible whenF0 � 0.01/Da2. Similar relation is also reported by others [18] whilestudying the same in vertical channel.

3.3. Buoyancy-opposed flow

In order to investigate the flow dynamics in the pipe, whenbuoyancy force is against of the forced flow, similar study is alsomade here. The effect of heat source intensity i.e. Rayleigh number,Ra, on velocity as well as temperature, is shown in Fig. 5(a)e(d),when Da and F0 are 10�2 and 102 respectively. As can be observedfrom Fig. 5(a) and (b), that for jRaj � 650, the velocity profilecontains point of separation at the vicinity of the wall along withpoint of inflection at the center zone of the pipe. This point ofseparation is an indication of back flow near to the wall, and causesRayleigheTaylor type of instability of the flow (e.g., in pipe [8], inchannel [21]). At the same time, the corresponding temperatureprofile shows a drastic change in comparison to the same inassisted case. Profile wise, it is parabolic in nature (see Fig. 5(c)). ForjRaj � 1000 the concavity of the curve changes, and as a conse-quence of it point of inflection appears on the temperature profilewhich is shown in Fig. 5(d). An interesting question arises con-cerning variation of characteristic of the velocity as well astemperature for jRaj > 1500. Accordingly both velocity as well astemperature profiles are plotted in Fig. 6 (a) and (b) for differentvalues of Ra. An important finding is that both profiles possessunnatural deviation i.e. a kind of distortion for jRaj � 5000. It canalso be seen from the above figures that increasing of Ra increasesnumber of zero on the velocity as well as temperature profiles. It isalso expected that as Ra tends to N the number of zeroes on bothprofiles tends to N. Similar type of distortion on the basic flow ina horizontal pipe was also reported by other scientist [39] whilestudying the Poiseuille flow.

It is worth mentioning that under such situation the buoyancy-opposed flow will go through transition to another state in the realsituation. Since the flow is also assumed here to be a fully devel-oped, which in turn, gives such type of unnatural distortion on theboth temperature as well as velocity profiles. Therefore, theseresults have to be carefully used when it is applied to the realengineering applications.

Fig. 7(a) and (b) shows the influence of permeability on thevelocity as well as temperature. As can be seen from the abovefigures that decreasing of media permeability, by varying Da from10�2 to 10�8, decreases magnitude of velocity as well as tempera-ture and both point of separation as well as inflection die out fromthe velocity profile. The velocity profile becomes almost flat. AtDa ¼ 10�2, the temperature profile contains a point of inflection,which disappears on reducing media permeability by changing Dato 10�3 or less.

To characterize the effect of form drag on the appearance ofpoint of separation as well as point of inflection on flow profile,both velocity as well as temperature are plotted in Fig. 8(a) and (b)for different F0 at jRaj ¼ 2000,Da¼ 10�2. It can be observed from theabove figure that introducing higher values of drag force in themedium, makes the velocity profile smooth and flat, and alsoreduces its magnitude. The corresponding temperature variationshows that the maximum magnitude of temperature decreases onincreasing F0.

3.4. Variation of Nusselt number

In assisted flow, the variation of the Nusselt number, Nu, asa function of the Rayleigh number for different values of F0 is

plotted in Fig. 9(a)e(d). Four different values 10�2, 10�4, 10�6 and10�8 of Da are considered in Fig. 9(a)e9(d) respectively. From theabove figures, three facts can be pointed. First, in general the heattransfer rate increases with increasing Rayleigh number. Second,depending on the values of Da, the change in heat transfer rate isnegligible up to a certain threshold value, Ra*, of Ra and beyond it,Nu increases significantly. From our numerical experiments it hasbeen found that Ra* z 10/Da. Third, the effect of F0 on Nu isnegligible when F0 � 0.01/Da2. In order to understand the same indetail, a quantitative analysis is made at RaDa ¼ 102. It has beenfound that the Nusselt number at Da ¼ 10�2, for three values 104,106, and 108 of F0 are 8.16, 5.46, and 4.04 respectively. This indicatesthat on the enhancement of F0 by two order by changing from 104 to


106, the heat transfer rate decreases z33%. But as F0 is increasedfurther by twomore order, the heat transfer rate decreases toz50%of Nu at F0 ¼ 104. For Da ¼ 10�4 the heat transfer rate decreasesz14% andz63% ofNu at F0 ¼ 106 by changing F0 from 106 to 108 and108 to 1010 respectively. However, for Da equal to 10�6, the effect ofF0 on Nu is negligible when it is changed from 108 to 1010. Moreover,it decreases Nu up to z19.9% of the same at F0 ¼ 108, by increasingF0 further two more order. The variation of Nu as function of F0 isnegligible for Da ¼ 10�8.

So it can be concluded from the above analysis that the influenceof form drag on fluid flow as well as heat transfer mechanism issignificant only for highly permeable media.

To understand the heat transfer mechanism and its dependencyon controlling parameters, Da, F0, and Ra in case of buoyancy-opposed flow, Fig. 10(a)e(c) is plotted. It can be seen from Fig. 10(a)and (b) that the effect of Da and F0 is similar to assisted case. Forhigher values of Ra, when the flow profile possess a distortion, theNusselt number varies abruptly (see Fig. 10(c)), which is conse-quence of abrupt variation of the temperature profile.

4. Conclusion

We have attempted to understand the dynamics of mixedconvective flow in vertical pipe and its dependency on porousmedium parameters, where flow is induced by external pressuregradient and buoyancy force. Thewall temperature of the pipe varieslinearly with vertical coordinate. To this end, we have adopted thenon-Darcy BrinkmaneForchheimereWooding extendedmodel. Thefully developed, one dimensional, nonlinear coupled equations aresolved numerically by Chebyshev spectral-collocation method.Neglecting form drag effect, the same was solved analytically. Bymeans of rigorous numerical and analytical experiments, we wereable to extract detailed information on the flow and heat transfermechanisms in pipe filled with a porous medium. The followingconclusions can be drawn from this study.

� For buoyancy-assisted flow, the velocity profile possess point ofinflection beyond a threshold value of Ra, which depends alsoon other controlling parameters. Back flow at the center isfound when Ra is very large. As an example, it has beenobserved at Ra ¼ 5 � 103, when Da and F0 are 10�2 and 102

respectively (see Fig. 2(b)).� In case of buoyancy-opposed flow, the velocity profile maycontain point of inflection in the center zone and point ofseparation at the vicinity of thewall. The appearance of point ofseparation causes back flow at the vicinity of the wall (i.e.velocity becomes negative at the vicinity of the wall).

� Based on the values of other controlling parameters, there exista minimum value of Ra, beyond it, the temperature profilepossess point of inflection. A kind of distortion on the velocityas well as on the temperature is found on the furtherenhancement of Ra.

� In opposed flow, enhancement of absolute value of Ra increasesthe maximum magnitude of temperature as well as velocitywhich appear at the center of the pipe. Whereas, in case ofassisted flowmagnitude of both velocity as well as temperaturedecreases on increasing Ra at center of the pipe, and maximummagnitude of the velocity increases.

� The point of separation as well as inflection die out on reducingmedia permeability and enhancing form drag coefficient.However, the effect of F0 on flow profiles is negligible whenF0 � 0.01/Da2.

� In general for assisted flow, the heat transfer rate increaseswith increasing Rayleigh number as well as Darcy number anddecreases on increasing the value of F0. However, in case of

opposed flow, the characteristics ofNu as a function of differentcontrolling parameters are reverse. In case of distortion, theNusselt number profile varies abruptly.

It should be mentioned that stability analysis of the aboveproblem is essential to understand the dynamics of the problem,which is left for a future study.

Acknowledgment

One of the authors, A. Kumar, is grateful to Council of Scientificand Industrial Research (CSIR) India for providing financial supportduring the preparation of this manuscript. This work was alsopartially supported through the CSIR project grant: 22(0464)09/EMR-II provided to P.B. by the CSIR India.

Nomenclature

cF form drag constantC1 constant in wall temperatureDa Darcy number ¼ K

R20

F Forchheimer number ¼ R0CFK1=2

F0 modified Forchheimer number ¼ Re2Fdpdzg gravitational accelerationK permeabilityp* dimensional pressurep non-dimensional pressurePr Prandtl number ¼ n

a

R0 radius of the pipeRa Rayleigh number ¼ gbTC1R4

0na

Re Reynolds number ¼ W*c R0n

r*,j,z* dimensional coordinatesr,j,z non-dimensional coordinatest* dimensional timet non-dimensional timeT* dimensional temperature(u*,v*,w*)dimensional fluid velocity(u,v,w) non-dimensional fluid velocityW0 velocity at laminar base stateW*

c dimensional base velocity at center of the pipe

Greek symbolsbT thermal expansion coefficiente porosityL viscosity ratio ¼ mf

~m

a thermal diffusivityQ0 temperature at laminar base state~m effective viscositymf dynamic viscosityn kinematic viscosityr fluid densityrf fluid density at reference states heat capacity ratio

References

[1] P.R.S. Dando, D. Stuben, S.P. Varnavas, Hydrothermalism in the Mediterraneansea, Progress in Oceanography 44 (1999) 333e367.

[2] H.L. Weissberg, Laminar flow in the entrance region of a porous pipe, Physicsof Fluids 2 (1959) 510e516.

[3] G.F. Scheele, T.J. Hanratty, Effect of natural convection on stability of flow ina vertical pipe, Journal of Fluid Mechanics 14 (1962) 244e256.

[4] G.A. Kemeny, E.V. Somers, Combined free and forced convection on stabilityflow in vertical circular tubes-experiments with water and oil, Journal of HeatTransfer 84 (1962) 339e346.


[5] L.S. Yao, Is a fully-developed and non-isothermal flow possible in a verticalpipe, International Journal of Heat and Mass Transfer 30 (1987) 707e716.

[6] L.S. Yao, Linear stability analysis for opposed mixed convection in a verticalpipe, International Journal of Heat and Mass Transfer 30 (1987) 810e811.

[7] B.B. Rogers, L.S. Yao, Finite-amplitude instability mixed-convection in a heatedvertical pipe, International Journal of Heat and Mass Transfer 36 (1993)2305e2315.

[8] Y.C. Su, J.N. Chung, Linear stability analysis of mixed-convection flow ina vertical pipe, Journal of Fluid Mechanics 422 (2000) 141e166.

[9] L.S. Yao, B.B. Rogers, The linear stability of mixed convection in a verticalannulus, Journal of Fluid Mechanics 201 (1989a) 279e298.

[10] B.B. Rogers, L.S. Yao, The importance of Prandtl number in a mixed-convectioninstability, Journal of Heat Transfer 115 (1993) 482e486.

[11] S.A. Suslov, S. Paolucci, Stability of mixed-convection flow in a tall verticalchannel under non-Boussinesq conditions, Journal of Fluid Mechanics 302(1995) 91e115.

[12] Y.C. Chen, J.N. Chung, The linear stability of mixed convection in a verticalchannel flow, Journal of Fluid Mechanics 325 (1996) 29e51.

[13] M.Parang,M.Keyhani, Boundaryeffects in laminarmixedconvectionflowthroughan annular porous medium, Journal of Heat Transfer 109 (1987) 1039e1041.

[14] K. Murlidhar, Mixed convection flow in a saturated porous annulus, Interna-tional Journal of Heat and Mass Transfer 32 (1989) 881e888.

[15] A. Hadim, Numerical study of non-Darcy mixed convection in a vertical porouschannel, Journal of Thermophysics and Heat Transfer 8 (1994) 371e373.

[16] A. Hadim, G. Chen, Non-Darcy mixed convection in a vertical porous channelwith asymmetric wall heating, Journal of Thermophysics and Heat Transfer 8(1994) 805e808.

[17] A. Hadim, G. Chen, Non-Darcy mixed convection in a vertical porous channelwith discrete heat sources at the walls, International Communications in Heatand Mass Transfer 21 (1994) 377e387.

[18] Y.C. Chen, J.N. Chung, C.S. Wu, Y.F. Lue, Non-Darcy flow stability of mixedconvection in a vertical channel filled with a porous medium, InternationalJournal of Heat and Mass Transfer 43 (2000) 2421e2429.

[19] P. Bera, A. Khalili, Stability of mixed convection in an anisotropic porouschannel, Physics of Fluids 14 (2002) 1617e1630.

[20] Y.C. Chen, Non-Darcy flow stability of mixed convection in a vertical channelfilled with a porous medium, International Journal of Heat and Mass Transfer47 (2004) 1257e1266.

[21] P. Bera, A. Khalili, Influence of Prandtl number on stability of mixed convec-tive flow in a vertical channel filled with a porous medium, Physics of Fluids18 (2006) 124103e124111-10.

[22] P. Bera, A. Khalili, Stability of buoyancy opposed mixed convection ina vertical channel and its dependency on permeability, Advances in WaterResources 30 (2007) 2296e2308.

[23] D.A. Nield, A. Bejan, Convection in Porous Media. Springer-Verlag, New York,2006.

[24] W.J. Chang, W.L. Chang, Mixed convection in a vertical tube partially filledwith porous medium, Numerical Heat Transfer, Part A 28 (1995) 739e754.

[25] H.E. Rose, On the resistance coefficient-Reynolds number relationship for fluidflow through a bed of granular materials, Proceedings of the Institution ofMechanical Engineers 153 (1945) 154e161.

[26] A.E. Scheidegger, The Physics of Flow Through Porous Media. Univ. of TorontoPress, Toronto, 1974.

[27] A.A. Hannoura, F. Barends, Non-Darcy flow: a state of the art, in: Flow andTransport in Porous Media, Balkema, 1981.

[28] C.C. Mei, J.L. Auriault, The effect of weak inertia on flow through a porousmedium, Journal of Fluid Mechanics 222 (1991) 647e663.

[29] M. Kaviany, Principles of Heat Transfer in Porous Media. Springer-Verlag, NewYork, 1995.

[30] T. Giorgi, Derivation of the Forchheimer law via matched asymptotic expan-sions, Transport in Porous Media 29 (1997) 191e206.

[31] M. Firdaouss, J.L. Guermond, P.L. Quéré, Nonlinear corrections to Darcy’s lawat low Reynolds numbers, Journal of Fluid Mechanics 343 (1997) 331e350.

[32] E. Skjetne, J.L. Auriault, High-velocity laminar and turbulent flow in porousmedia, Transport in Porous Media 36 (1999) 131e147.

[33] J. Kumar, P. Bera, A. Khalili, Influence of inertia and drag terms on the stabilityof mixed convection in a vertical porous-medium channel, InternationalJournal of Heat and Mass Transfer 53 (2010) 5261e5273.

[34] R.C. Givler, S.A. Altobelli, A determination of the effective viscosity for theBrinkmaneForchheimer flow model, Journal of Fluid Mechanics 258 (1994)355e370.

[35] W.P. Breugem, The effective viscosity of a channel-type porous medium,Physics of Fluids 19 (2007) 103104e103111-8.

[36] A.A. Hill, B. Straughan, Poiseuille flow in a fluid overlying a porous medium,Journal of Fluid Mechanics 603 (2008) 137e149.

[37] C. Canuto, M.Y. Hussaine, A. Quarteroni, T.A. Zang, Spectral Method in FluidDynamics. Springer-Verlag, New York, Berlin, Heidelberg, 1988.

[38] P.G. Drazin, W.H. Reid, Hydrodynamic Stability. Cambridge University Press,Cambridge, 2004.

[39] Z. Zhe-wie, A kind of distortion of mean velocity profile in pipe Poiseuille flowand its stability behavior, Applied Mathematics and Mechanics 9 (1988)77e86.

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Non-Darcy mixed convection in a vertical pipe filled with porous medium