Chapter – II

23

2.1. INTRODUCTION

Flow through porous medium is abundantly prevalent in nature and therefore

the study of flow through porous medium has become of principal interest in many

scientific and engineering applications. In the theory of flow through a porous

medium, the role of momentum equation or force balance verified the numerous

experimental observations is summarized mathematically as the Darcy’s law. It is

observed that the Darcy’s law is applicable as long as the Reynolds number based on

average grain (pore) diameter does not exceed a value between 1 and 10. But in

general the speed of specific discharge increases, the convective forces get

developed and the internal stress generated in the fluid due to its viscous nature

produces distortion in the velocity field. Also in the case of highly porous media

such as fiber glass, Pappas dandelion etc., the viscous stress at the surface is able to

penetrate into medium and produce gradient. Thus the difference between the

specific discharge and hydraulic gradient is inadequate in describing high speed

flows or flows near surface which may be either permeable or not. Hence

consideration for non-Darcian description for the viscous flow through a porous

medium is warranted. Saffman (27) employing statistical method derived governing

equation for the flow in a porous medium which takes into account the viscous

stress. Later another modification has been suggested by Brinkman (4)

vvk

p 2)(0

in which v2 is intended to account for the distortion of the velocity profiles near

the boundary. The same equation was derived analytically by Tam (30) to describe

the viscous flow at low Reynolds number past a swam of small particles.

The process of free convection as a mode of heat transfer has wide

applications in the fields of Chemical Engineering, Aeronautical and Nuclear power

generation. It was shown by Gill and Casal (10) that the buoyancy significantly

affects the flow of low Prandtal number fluids which is highly sensitive to

gravitational force and the extent to which the buoyancy force influences a forced

flow is a topic of interest. Free convection flows between two long vertical plates

Chapter – II

24

have been studied for many years because of their applications in nuclear reactors,

heat exchangers, cooling appliances in electronic instruments. These flows were

studied by assuming the plates at two different constant temperatures or temperature

of the plates varying linearly along the plates etc. The study of fully developed free

convection flow between two parallel plates at constant temperature was initiated by

Ostrach (20). Combined natural and forced convection laminar flow with linear wall

temperature was also studied by Ostrach (21). The first exact solution for free

convection in a vertical parallel plate channel with asymmetric heating for a fluid of

constant properties was presented by Anug (2). Many of the early works on free

convection flows in open channels have been reviewed by Manca et. al., (12).

Recently, Campo et. al., (8) considered natural convection for heated iso-flux

boundaries of the channel containing a low-Prandtl number fluid. Pantokratoras (22)

studied the fully developed free convection flow between two asymmetrically heated

vertical parallel plates for a fluid of varying thermo physical properties. However,

all the above studies are restricted to fully developed steady state flows. Very few

papers deal with unsteady flow situations in vertical parallel plate channels.

Transient free convection flow between two long vertical parallel plates maintained

at constant but unequal temperatures was studied by Singh et. al., (29). Jha et. al.,

(11) extended the problem to consider symmetric heating of the channel walls.

Narahari et. al., (16) analyzed the transient free convection flow between two long

vertical parallel plates with constant heat flux at one boundary, the other being

maintained at a constant temperature. Singh and Paul (27) presented and analysis of

the transient free convective flow of a viscous incompressible fluid between two

parallel vertical walls occurring as a result of asymmetric heating/cooling of the

walls. Narahari (17) presented an exact solution to the problem of unsteady free

convective flow of a viscous incompressible fluid between two long vertical parallel

plates with the plate temperature linearly varying with time at one boundary, that at

the other boundary being held constant. There are many reasons for the flow to

become unsteady. When the current is periodic due to on-off control mechanisms or

due to partially rectified a-c voltage, there exist periodic heat inputs. Hence, it is

important to study the effects of periodic heat flux on the unsteady natural

convection, imposed on one of the plates of a channel formed by two long vertical

Chapter – II

25

parallel plates, the other being maintained at a constant initial fluid temperature.

Recently Narahari (18) has discussed the unsteady free convection flow of a

dissipative viscous incompressible fluid between two long vertical parallel plates in

which the temperature of one of the plates is oscillatory while the other plate is a

constant one.

In the context of space technology and in processes involving high

temperatures, the effects of radiation are of vital importance. Recent developments

in hypersonic flights, missile reentry, rocket combustion chambers, power plants for

inter planetary flight and gas-cooled nuclear reactors have focused attention on

thermal radiation as a mode of energy transfer and emphasize the need for improved

understanding of radiative transfer in these processes. Mansour (13) studied the

radiative and free convection effects on the oscillatory flow past a vertical plate.

Raptis and Perdikis (24) considered the problem of thermal radiation and free

convective flow past a moving plate. Das et. al., (9) analyzed the radiation effects on

the flow past an impulsively started infinite isothermal vertical plate. Prasad et .al.,

(23) considered the radiation and mass transfer effects on two dimensional flow past

an impulsively started isothermal vertical plate. Chamkha et. al., (5) studied the

radiation effects on the free convection flow past a semi-infinite vertical plate with

mass transfer. Raptis (26) analyzed the thermal radiation and free convection flow

through a porous medium by using perturbation technique. Bakier and Gorla (3)

investigated the effect of thermal radiation on mixed convection from horizontal

surfaces in saturated porous media. Satapathy et. al., (28) studied the natural

convection heat transfer in a Darcian porous regime with Rosseland radiative flux

effects. With regard to thermal radiation heat transfer flows in porous media,

Chamkha (6) studied the solar radiation effects on porous media supported by a

vertical plate. Forest fire spread also constitutes an important application of radiative

convective heat transfer as described by Meroney (14). More recently

Chitrphiromsri et. al., (7) have studied the influence of high-intensity radiation in

unsteady thermo-fluid transport in porous wet fabrics as a model of fire fighter

protective clothing under intensive flash fires. Impulsive flows with thermal

radiation effects and in porous media are important in chemical engineering systems,

Chapter – II

26

aerodynamic blowing processes and geophysical energy modeling. Such flows are

transient and therefore temporal velocity and temperature gradients have to be

included in the analysis. Raptis et. al., (24) studied numerically the natural

convection boundary layer flow past an impulsively started vertical plate in a

Darcian porous medium. The thermal radiation effects on heat transfer in magneto-

aerodynamic boundary layers has also received some attention, owing to its

applications in astronautically re-entry, plasma flows in astrophysics, the planetary

magneto-boundary layer and MHD propulsion systems. Mosa (15) discussed one of

the first models for combined radiative hydro magnetic heat transfer, considered the

case of free convective channel flows with an axial temperature gradient. Nath et.

al., (19) obtained a set of similarity solutions for radiative – MHD stellar point

explosion dynamics using shooting methods. Abd El- Naby et. al., (1) presented a

finite difference solution of radiation effects on MHD unsteady free convection flow

over a vertical porous plate.

In this chapter we make an attempt to analyse the unsteady convective heat

transfer of dissipative viscous fluid through a porous medium confined in a vertical

channel on whose walls an oscillatory temperature is prescribed. Approximate

solutions to coupled non-linear partial differential equations governing the flow and

heat transfer are solved by a perturbation technique. The velocity, temperature, skin

friction and rate of heat transfer are discussed for different variations of the

governing parameters G, D-1, M, N1, , γ, Ec, and P.

Chapter – II

27

2.2. FORMULATION AND SOLUTION OF THE PROBLEM

We consider the unsteady flow of a viscous incompressible fluid through a

porous medium in a vertical channel bounded by flat walls in the presence of heat

sources. The unsteadiness in the flow is due to the oscillatory temperature prescribed

on the boundaries. We choose a Cartesian coordinate system 0(x y) with walls at

y = 1 by using Boussinesq approximation we consider the density variation only

on the buoyancy term Also the kinematic viscosity ,the thermal conductivity k are

treated as constants. The equations governing the flow and heat transfer are

guHuky

uxp

tu oe

)()(

22

2

2

(2.1)

yqu

kuTTQ

yTK

tTC R

yfp

)(

)(2)( 2202

2

0 (2.2)

- 0 = -0(T – T0) (2.3)

where u is a velocity component in x-direction, T is a temperature, p is a

pressure, is a density, k is the permeability of the porous medium, is dynamic

viscosity, kf is coefficient of thermal conductivity, is coefficient of volume

expansion and Q is the strength of heat source .

By using the Rosseland approximation the radiative heat flux, qr is given by

yTq

Rr

)(4 4

(2.4)

Where and R are the Stefan-Boltzman constant and the mean absorption

coefficient, respectively. We assume that the temperature differences within the flow

are sufficiently small such that 4T may be expressed as a linear function of

temperature

434 34 ee TTTT (2.5)

Using (2.4) and (2.5) in the last term of equation (2.2) we get

Chapter – II

28

2

2

0

3

316)(

yT

CT

yq

pR

er

(2.6)

The boundary conditions are

u = 0, T = T1 at y = -L

u = 0, T = T2 + (T2 – T1) Cos t (2.7)

on introducing the non dimensional variables

Luu/

, y = y/L,12

1

TTTT

, t = t,

Equations 2.1 & 2.2 reduce to (dropping the dashes)

uMDyuG

tu )( 21

2

22

(2.8)

2122

22 )

341( uDEPuPE

yNtP cyc

(2.9)

Where

2123 )(

TTgLG (Grashoff number)

kLD

21 (Darcy parameter)

f

P

KCP

(Prandtl number)

fKLQ 2.

(Heat source parameter)

2

2

TLCEc

p (Eckert Number)

fR

e

kTN

34 3*

(Radiation parameter)

22 L (Womersley Number)

Chapter – II

29

4N3N3

4N3NP3P 11

1221

DMM

The transformed boundary conditions are

u = 0, = 0, at y = -1

u =0, = 1 + Cos (t) at y = +1 (2.10)

In view of the boundary conditions (2.10) we assume

(u, θ)= (u0, θ0) + є eit (u1,θ1) (2.11)

Substituting the series expansion (2.11) in equations (2.8) & (2.9) and

separating the steady and transient parts we get

002

120

2

GuMyu

(2.12)

1122

121

2

)( GuiMyu

(2.13)

020

112

02

10120

2

uDEP

yuEP

y cc

(2.14)

101

110

112

121

2

)(.)2()( uuEcDPyu

yuEcPiP

y

(2.15)

As the equations (2.12 – 2.15) are non-linear coupled equations, assuming Ec

<< 1 we take

u0 = u00 + Ec u01

u1 = u10 + Ec u11

0 = 00 + Ec 01

1 = 10 + Ec 11 (2.16)

Chapter – II

30

Substituting 2.16 in equations (2.12 – 2.15) and separating the like terms we

get

00002

11100 GuMu , u00 ( 1) = 0 (2.17)

00011100 , 00(-1) = 0, 00 (+ 1) = 1 (2.18)

01012

11101 GuMu , u01 ( 1) = 0 (2.19)

200

11

12001011

1101 uDPuP , 01 (- 1) = 0=01 (2.20)

101022

11110 )( GuiMu , u10 ( 1) = 0 (2.21)

0102

11110 iP , 10(-1) = 0, 10(+1) = 1 (2.22)

111122

11111 )( GuiMu , u11 ( 1) = 0 (2.23)

10001

1110

100111

211

1111 22)( uuDPuuPiP , 11( 1) = 0 (2.24)

The solutions of equations (2.17) - (2.24) are obtained as

))Ch(β

y)Ch(β)Sh(β

y)Sh(β0.5(

))Sh(β

y)Sh(β)Sh(βy)(Sh(βa))Ch(β

y)Ch(β)Ch(βy)(Ch(βaθ

2

2

2

2

2

2114

2

211300

Chapter – II

31

))()()()(()

)()()()((

))()()()(()

)()()()((

1

11114

1

11113

1

12212

1

1221100

MChyMShShySha

MChyMChChyCha

MChyMChChyCha

MChyMShShyShau

))()(

1(

))()(

)()(())()(

)()((

))()(

)()(())()(

)()((

))()(

)()(())()(

)()((

))()(

)()(())()(

)()((

))()(

)()(())()(

)()((

))()(

)()(())()(

)()((

))()(

)2()2(())()(

)2()2((

))()(

)2()2(())()(

)2()2((

))()(

)2()2(())()(

)2()2((

2

271

2

26666

2

25565

2

26668

2

25567

2

24464

2

23363

2

24462

2

23361

2

22260

2

21159

2

22258

2

21157

2

21156

2

21155

2

22254

2

22253

2

21152

2

2115101

ChyCha

ChyChkChykCha

ChyChkChykCha

ShyShkShykSha

ShyShkShykSha

ChyChkChykCha

ChyChkChykCha

ShyShkShykSha

ShyShkShykSha

ChyChkChykCha

ChyChkChykCha

ShyShkShykSha

ShyShkShykSha

ShyShShySha

ChyChChyCha

ShyShShySha

ChyChChyCha

ShyShMShyMSha

ChyChMChyMCha

Chapter – II

32

))()(

1(

)))()(

)()(())()(

)()((

)()(

)()(())()(

)()((

))()(

)()(())()(

)()((

))()(

)()(())()(

)()((

))()(

)()(())()(

)()((

))()(

)()(())()(

)()((

))()(

)2()2(())()(

)2()2((

))()(

)2()2(())()(

)2()2((

))()(

)2()2(())()(

)2()2((

))()(

)()(())()(

)()((

1

192

1

16691

1

15590

1

16689

1

153588

1

14487

1

13386

1

14485

1

13384

1

12283

1

11182

1

11180

1

12281

1

11179

1

11178

1

12277

1

12276

1

11175

1

11174

1

12273

1

1227201

MChyMCha

MShyMShkShykSha

MShyMShkShykSha

MChyMChkChykCha

MChyMChkChykCha

MChyMChkChykCha

MChyMChkChykCha

MShyMShkShykSha

MShyMShkShykSha

MChyMChkChykCha

MChyMChkChykCha

MShyMShkShykSha

MShyMShkShykSha

MChyMShShySha

MChyMChChyCha

MChyMShShySha

MChyMChChyCha

MChyMShMShyMSha

MChyMChMChyMCha

MShyMShShySha

MChyMChChyChau

)()(

)()(

(5.0

))()(

)()(())()(

)()((

4

4

4

4

4

433108

4

43310710

ShySh

ChyCh

ShyShShySha

ChyChChyCha

0

))()()()(()

)()()()((

))()()()(()

)()()()((

10

5

533112

5

533111

5

544110

5

54410910

C

ShyShShySha

ChyChChyCha

ShyShShySha

ChyChChyChau

where

,,,,, 221

25

21

24

21

23

22

21 iMiPiSc

,,,, 124213212211 kkMkMk

.. 116115 MkMk

Chapter – II

33

2.3. SHEAR STRESS AND NUSSELT NUMBER

The Shear stress at the boundariesLdy

du

in the non dimensional form

is given by1

2

2

ydydu

L

and the corresponding expressions are

y=-1 = b54 + Ec b56 + et (b58 + Ec b60)

y=+1 = b53 + Ec b55 + et (b57 + Ec b59)

The rate of heat transfer (Nusselt number) at y = L is given by

qw =1

)1(

ydydNu

and the corresponding expressions are

(Nu)y=-1 = b44 + Ec b46 + eit (b48 + Ec b52 )

(Nu)y=+1 = b43 + Ec b45 + eit (b47 + Ec b51 )

Where b1,b2,……………….,b60 are constants given in the Appendix.

Chapter – II

34

2.4. DISCUSSION OF THE NUMERICAL RESULTS

In this analysis we investigate the effect of radiation on the unsteady

convective heat transfer flow of viscous electrically conducting fluid through a

porous medium confined in a vertical channel whose walls are maintained at an

oscillatory temperature in the presence of heat generating sources.

The velocity (u) is shown in figures (1-5) for different values of the Grashoff

number ‘G’, the Hartmann number ‘M’, Darcy parameter D-1, Womersley number ,

Heat Source Parameter , radiation parameter N1 .Eckert Number Ec and Prandtl

number P. Fig. (1) represents the variation of ‘u’ with G. It found that the velocity

‘u’ is in the vertically downward direction for G > 0 and is vertically upward

direction for G < 0.

The magnitude of ‘u’ experiences an enhancement with increase in G

attaining its a maximum increase in max at y = -0.4. Fig. (2) represents the variation

of ‘u’ with ‘M’ and D-1. It is noticed that higher the Lorentz force/lesser the

permeability of the porous medium, smaller |u| in the region and for further lowering

of the permeability / enhancing the Lorentz force leads to an enhancement in |u|. The

variation of ‘u’ with Womersely number ‘’ shows for 4, ‘u’ is in the vertically

downward direction and for higher 6, u is in the vertically upward direction,

while u < 0 for any value of 1 |u| enhances with 4 except in a narrow region

adjacent to y = -1 and for higher = 6, |u| enhances. An increase in the strength of

the heat source leads to a depreciation in |u|. Thus the presence of the heat sources in

the flow region reduces |u|. (fig.(3)). An increase in the radiation parameter N1

results in a depreciation in |u| everywhere in the flow region. (fig.(4)). The variation

of u with Ec shows that higher the dissipative heat lesser the magnitude of u in the

left half and larger in the right half of the channel (fig.5).The variation of u with

Prandtl number ‘P’ is shown in fig. (6). An increase in P 10 depreciates |u| except

in a central region where it enhances and for higher P 30, |u| enhances in the left

half and reduces in right half.

Chapter – II

35

The non-dimensional temperature () is shown in figs. (7-12) for different

values of G, M, D-1, N1 , ,Ec and P. Fig. (7) illustrates variation of ‘’ with

Grashoff number G. It is noticed that the axial temperature enhances in the heating

case and reduces in the cooling of the channel walls. The variation of ‘’ versus M

shows that for higher Lorentz force larger the axial temperature in the left half and

smaller in the right half of the channel. The variation of ‘’ with D-1 shows that

lesser the permeability of the porous medium smaller the axial temperature in the

left half and larger in the right half and for further lowering of the permeability, the

axial temperature enhances in the left half and reduces in the right half (fig.(8)).

Fig.(9) represents the variation of ‘’ with ’ and ‘’. We notice that the axial

temperature increases in left half and reduces in right half with an increase in while

an increase in ‘’ results in a depreciation the axial temperature in the entire flow

region. From fig.(10) we notice that an increase in the radiation parameter N1 results

in a depreciation in the axial temperature. The variation of with Eckert number Ec

shows that higher the dissipative heat larger the actual temperature in the flow

region(fig.12) The variation of ‘’ with the Prandtl number ‘P’ shows that higher the

Prandtl number, larger the axial temperature in the left half and smaller in the right

half of the channel (fig.11).

The stress () at the boundary walls y = 1 are evaluated for different

parametric values is shown in tables 1-4. It is found that an increase in |G| enhances

the stress at both walls. The variation of ‘’ with M and D-1 shows that higher the

Lorentz force/lesser the permeability of porous medium, larger the stress at y = 1.

An increase in Womersely number reduces || at y = +1 and enhances at y = -1

(tables (1 and 4)). From tables (2 and 5) we observe that an increase in the radiation

parameter N1 or heat source parameter ‘’ leads to a depreciation in || at both the

walls. The variation of the stress with Ec shows that higher the dissipative heat

larger at y = 1 and smaller at y = -1 (Tables.3 and 6). The variation of ‘’ with

Prandtl number shows that the stress at y = +1 enhances while it reduces at y = -1

with an increase in the Pradntl number ‘P’ (Tables.1 and 5).

Chapter – II

36

The rate of heat transfer (Nusselt number) at y = 1 is shown in tables. (7-

12) for different values of G, M, D-1, N1,,, Ec and P. We notice that the rate of

heat transfer enhances with increase in G. Higher the Lorentz force/smaller the

permeability of the porous medium, larger |Nu| at y = 1. Also an increase in leads

to an enhancement in |Nu| at both walls (tables 7 and 8). From tables (10 and11) we

find that the rate of heat transfer enhances at y = 1 with increase in the radiation

parameter N1. Thus the inclusion of the radiative heat flux leads to an enhancement

in the rate of heat transfer. Also the Nusselt number enhances with increase in heat

source parameter (or) P. This indicates that the presence of the heat source results

in an enhancement in |Nu| at y = 1. From tables 9 and12 we find that the rate of

heat transfer depreciates with Ec at both the walls.

Chapter – II

37

2.5. APPENDIX

Cosh( )=Ch( ) Sinh( ) =Sh( )

221

1221 iPDMM

221

22 iM

213 M 214 M

116115 MM

21

1 2MGa

21

2 2MGa

21

41

3 2)1(

22

MG

MGa

1

134 ch

aaa

1

25 Msh

aa

11

1

36 M

aMch

Maa

a7 = 2a11

128 Msh

Maa

1

1

1

39 Mch

achM

aa1

210 Msh

aa

2)(

2)( 2

10291

22

28

26

11aaPDaaaPa

2)(

2)( 2

102916

28

12aaPDaaPa

a13 = [a6a8P + PD-1a9a10] a14 = 2PD-1a1a9

a15 = 2PD-1a2a9 + 2Pa7a8 a16 = 2PD-1a1a1

a17 =(2PD-1a2a10 – 2Pa6a7) a18 = 2P a2a8

a19 = 2P a2a722

12720 aPDaPa

211

21 2 aaPDa 21

122 aPDa

Chapter – II

38

211

23aa

617

24aa

1220

25aa

2021

26aa

3022

27aa

1

1428 M

aa

1

1629 M

aa 1

172

1

1430

2Ma

Maa

21

16

1

1531

2Ma

Maa 2

1

1831

162

1

1532

2Ma

Ma

Maa

31

142

1

222

1

1733

2Ma

Ma

Maa 2

1

1234 4M

aa

21

1335 4M

aa

a36 = ½ + a24 + a26 + a28 sh M1 – a30 ch M1 + a33 sh M1 + a35 sh (2M1)

a37 = ½ – a23 + a25 + a27 – a29 ch M1 – a31 sh M1 – a31 sh (2M1)

– a32 ch M1 – a34 ch(2M1)

1

3738 2M

Gaa 1

3639 4M

Gaa

21

3540 3

Gaa 21

3441 3

Gaa

1

.3321

3042 22

GaGaa 1

.3221

3143 22

GaGaa

1

.302

1

2844 44 M

GaMGaa

1

.312

1

2945 44 M

GaM

Gaa

Chapter – II

39

1

2846 6M

Gaa 1

2947 6M

Gaa

148 Mch2

)1()1( a

149 Msh2

)1()1( a

150 Msh2

2a

151 Mch2

2a

1

22

21

52 ch)(2 Ga

122

21

53 sh)(2

Ga

254 ch2

)1()1(a

255 sh2

)1()1(a

a56 = –a52 ch 1 a57 = –a53 sh 1

a58 = 1 a522

125359

shshaa

a60 = 1a53

2

58571461

aaMaa

2

59581462

aaMaa 68575914

63 2aaaMaa

57685914

64 2aaaMaa a68 = 2a1a58

a69 = 2a1a59 a70 = 2a1a60

a71 = a2a57 a72 = a2a58

a73 = a2a59 a74 = a2a60

2

1275

ch

chaa

2

35376

sh

shaa

Chapter – II

40

2754

77aaa

2754

78aaa

2)( 5376

479aaaa

)(2 5376

480 aaaa

)(2 537653

482 aaaaa )(

2 75525

83 aaaa

)(2 7552

584 aaaa a85 = a1a75

a86 = a2a70 a87 = a1a52

a88 = a1a53 a89 = a2a75

a90 = a2a76 a91 = a1a52

a92 = a2a53 a93 = a3a75

a94 = a3a76 a95 = a3a52

a96 = a3a553

21

23

77611

aab 21

24

78622

aab

21

23

79633

aab 21

24

80644

aab

21

25

68815

aab 21

26

82666

aab

21

25

807

ab 21

26

848

ab

1

879 6

ab 1

8810 6

ab

21

22

8511

ab 21

22

8612

ab

Chapter – II

41

21

926887113 4

)(

aaab 2

1

917088114 4

)(

aaab

221

22

90672

122852

15 )())((4

aaab 221

22

89692

122862

16 )())((4

aaab

21

74951689217 2

)( aaaab

21

7296917018 2

)( aaaab

221

22

93732

1229067285

19 )())(()(22

aaaaab

221

22

94712

1228969286

20 )())(()(2

aaaaab

1

22211 ch2

)1()1(

b 1

22222 sh2

)1()1(

b

22

23

121

Gbb 22

24

222

Gbb

22

23

323

Gbb 22

24

424

Gbb

22

25

525

Gbb 22

26

626

Gbb

22

25

727

Gbb 22

26

828

Gbb

22

21

929

Gbb 22

21

1030

Gbb

222

21

1422

21101

31 )()(6

GbGbb 22

22

1

139132 )(

6

GbGbb

2

1133 2

Gbb 2

1234 2

Gbb

Chapter – II

42

222

21

911335 )(

64

GbGbb 22

22

1

1014136 )(

64

GbGbb

22

161237 2

)(

bbGb 2

2

151138 2

)(

bbGb

22

1539 2

Gbb 22

1640 2

Gbb

1

3341 ch2

)1()1(

b 1

3342 sh2

)1()1(

b

b43 = – + 0.5 b44 = + 0.5

b45 = 2a23 – 2a24 – 4a25 – 4a26 – 6a27 – a28 (sh M1 + M1 ch M1) + a29 (2 ch M1 + M1 sh

M1)+ a30 M1 sh M1 + a31 (2ch M1 + M1 sh M1) + a30 M1 sh M1 + a31 (sh(M1 )++M1

ch(M1) – a32 M1 sh(M1 )– a33 (M1 ch(M1 )– M1 chM1 –sh M1) + 2M1 a34 sh (2M1) +

a35 (2M1 ch 2 M1 – sh 2M1) + 0.5

b46 = –2a23 – 2a24 + 4a25 – 4a26 + 6a27 – a28 (sh(M1 )+ M1 ch(M1)) – a29 (2ch(M1 )+

M1 sh(M1)) + a30 M1 sh M1 – a31 (sh(M1 )+ M1 ch(M1)) + a32 M1 sh (M1 )–a33 (M1 ch

(M1+) sh(M1)) – 2 M1 a34 sh (2M1) + a35 (2M1 ch 2M1 –sh 2M1)+ 0.5

b47 = 0.5 (1 Th 1 + 1 ct h 1)

b48 = 0.5 (–1 Th 1 + 1 ct h 1)

b49 = b13 sh 3 + b24 sh 4 + b33 ch 3 + b44 ch4 – b55 sh5 +b64 sh 6

+ b75 ch 5 + b86 ch 6 – b9 (3 ch 1 + 1 sh 1)–b10 (3sh 1 + 1 ch 1)

+b11 (2 ch 2 + 2 sh 2) + b12 (2 sh 2 + 2 ch 2) + b13 (2 sh 1+1 ch 1) + + b14 (2

ch 1 + 1 sh 1) + b15 (sh 2 + 2 ch 2) + b16 (ch 2 + 2 sh 2)

+ b17 (ch 1 + 1 sh 1) + b18 (sh 1 + 1 ch 1) + 2 b19 sh 2 + b20 2 ch 2.

Chapter – II

43

b50 = –b13 sh 3 – b24 sh 4 + b33 ch 3 + b44 ch 4 + b55 sh 5 + b66 sh 6 +

b75 ch 5 + b86 ch 6 – b9 (3 ch 1 + 1 sh 1) + b10 (3 sh 1 + ch 1) – b11 (2 ch

2 + 2 sh 2) + b12 (2 sh 2 + 2 ch 2) + b13 (2 sh 1 + 1 ch 1)– b14 (2 ch 1 + 1 sh

1) – b15 (sh 2 + 2 ch 2) + b16 (ch 2 + 2 sh 2)

+ b17 (ch 1 + 1 sh 1) – b18 (sh 1 + 1 ch 1) – 2 b19 sh 2 + b20 2 ch 2.

b51 = 1b21 sh 1 + 1b22 ch 1 + 1(+1)

b52 = –1b21 sh 1 + 1b22 ch 1 + 1(–1)

b53 = a3 M1 Th M1 + a1 (2+M1 Th M1) + a2 (1–M1 cth M1)

b54 = –a3 M1 Th M1 + a1 (2+M1 Th M1) + a2 (1–M1 cth M1)

b55 = –a39 (2M1 –Th M1) – a41 (2M1 sh 2M1 – M1 ch M1 Th M1)

+ a43 (sh M1 + M1 ch M1 – M1 sh M1Th M1) + 2a45 ch M1 – a48 (1 – M1 cth M1)

– a40 (2M1 ch 2M1 – M1 sh 2M1 cth M1) + a42 (ch M1 + M1 sh M1 – M1 ch M1 cth

M1)+ 2a44 sh M1 + a46 (3ch M1 + M1 sh M1 – M1 ch M1 cth M1).

b56 = a39 (2 – M1Th M1) + a41 (2M1 sh 2M1 – M1 ch 2 M1 Th M1)

– a43 (sh M1 + M1 ch M1 – M1 sh M1Th M1) – 2a45 ch M1 – a38 (1 + M1 cth M1)

– a40 (2M1 ch 2M1 – M1 sh 2M1 cth M1) + a42 (ch M1 + M1 sh M1 – M1 ch M1 chM1)

+ 2a44 sh M1 + a46 (3ch M1 + M1 sh M1 – M1 ch M1 cth M1)

b57 = a52 (–2 Th 2 ch 1 + 1 sh 1) + a53 (2 cth 2 sh 1 – 1 ch 1)

b58 = a52 (–2 Th 2 ch 1 + 1 sh 1) + a53 (2 cth 2 sh 1 – 1 ch 1)

b59 = 2b41 sh 2 + 2b42 ch 2 + 13 (+1)

b60 = –2b41 sh 2 + 2b42 ch 2 + 13 (–1)

Chapter – II

44

2.6. REFERENCES

1. Abd El-Naby M. A, Elsayed M. E., Elberabary and Nader Y. A.: Finite

difference solution of radiation effects on MHD free convection flow over a

vertical porous plate, Appl. Maths Comp. Vol. 151, pp. 327-346(2004).

2. Anug, W.: Fully developed laminar free convection between vertical plates

heated asymmetrically. Int. J. Heat and Mass Transfer, 15, pp.1577-

1580(1972).

3. Bakier, A.Y. and Gorla, R. S. R.: Thermal radiation effects on mixed

convection from horizontal surfaces in porous media, Transport in porous

media, Vo. 23 pp. 357-362(1996).

4. Brinkman, H. C: A calculation of the viscous force external by a flowing fluid

on a dense swam of particles: Application Science Research. Ala. pp.81 (1948)

5. Chamkha, A. J. Takhar, H. S. and Soundalgekar, V. M.: Radiation effects on

free convection flow past a semi-infinite vertical plate with mass transfer,

Chem. Engg. J, Vol. 84, pp. 335-342(2001).

6. Chamkha, A. J.: Solar Radiation Assisted natural convection in a uniform

porous medium supported by a vertical heat plate, ASME Journal of heat

transfer, V.19, pp.89-96(1997).

7. Chitrphiromsri, P. and Kuznetsov, A. V.: Porous medium model for

investigating transient heat and moisture transport in firefighter protective

clothing under high intensity thermal exposure, J. Porous media, Vo. 85, pp.

10-26(2005).

8. Campo, A. Manca, O. and Marrone, B.: Numerical investigation of the natural

convection flows for low-Prandtl fluid in vertical parallel-plates channels.

ASME Journal of Applied Mechanics, 73, pp. 6-107(2006).

9. Das, U.N. Deka, R.K. and Soundalgekar, V.M.: Radiation effects on flow past

an impulsively started vertical plate – an exact solution, J. Theo. Appl. Third

Mech., Vol.1, pp. 111-115 (1996).

Chapter – II

45

10. Gill, W. M and Casal, A. D.: A theoretical investigation of natural convectioneffects in forced horizontal flows, Amer. Inst.Chem.Engg. Jour,V. 8, pp. 513-520 (1962).

11. Jha, B .K. Singh, A.K. and Takhar, H. S.: Transient free convection flow in avertical channel due to symmetric heating. International Journal of AppliedMechanics and Engineering, 8(3), pp.497-502(2003).

12. Manca, O. Marrone, B. Nardini, S. and Naso, V.: Narural convection in openchannels. In: Sunden B, Comini G. Computational Analysis of Convection HeatTransfer. WIT press, Southempton, pp.235-278(2000)

13. Mansour, M. H.: Radiative and free convection effects on the oscillatory flowpast a vertical plate, astrophysics and space science, Vol. 166, pp. 26-75(1990).

14. Meroney, R. N.: Fires in porous media, May 5-15, Kiev, Ukraine (2004).

15. Mosa, M. F.: Radiative heat transfer in horizontal MHD channel flow withbuoyancy effects and axial temp. gradient, Ph.D. thesis, Mathematics Dept,Brodford University, England, UK (1979).

16. Narahari, M. Sreenadh, S. and Soundalgekar, V. M.: Transient free convectionflow between long vertical parallel plates with constant heat flux at oneboundary. Thermophysics and Aeromechanics, 9(2), pp.287-293(2002).

17. Narahari, M.: Free convection flow between two long vertical parallel plateswith variable temperature at one boundary. Proceedings of InternationalConference on Mechanical & Manufacturing Engineering (ICME 2008), JohorBahru, Malaysia (2008).

18. Narahari, M.: Oscillatory plate temperature effects of free convection flow ofdissipative fluid between long vertical parallel plates. Int. J. of Appl. Math. AndMech. 5(3). pp 30-46(2009).

19. Nath, Ojha, S. N. and Takhar, H. S.: A study of stellar point explosion in aradiative MHD medium s and space science, V.183, pp. 135-145(1991).

20. Ostrach, S.: Laminar natural-convection flow and heat transfer of fluids withand without heat sources in channels with constant wall temperatures.Technical Report 2863, NASA, USA (1952).

Chapter – II

46

21. Ostrach, S.: Combined natural and forced convection laminar flow and heattransfer of fluids with and without heat sources in channels with linearlyvarying wall temperature. Technical Report 3141, NASA, USA(1954).

22. Pantokratoras, A.: Fully developed laminar free convection with variablethermo physical properties between two open-ended vertical parallel platesheated asymmetrically with large temperature differences. ASME Journal ofHeat Transfer, 128, pp.405-408(2006).

23. Prasad, P.M.V. Raveendranath, P. Prasada Rao, D.R.V.: Transienthydromagnetic convective heat and mass transfer through a porous mediuminduced by travelling thermal waves in a vertical channel. Acta Ciencia Indica,Vol. 34M, No.2 pp 677-692. (2008).

24. Raptis, A. A. and Singh, A. K.: Free convection flow past an impulsivelystarted vertical plate in a porous medium by finite difference method,Astrophysics space science J, Vol. 112, pp. 259-265(1985).

25. Raptis, A. A. and Perdikis, C.: Radiation and free convection flow past amoving plate, Appl. Mech. Eng. Vol. 4, pp. 817-821(1999).

26. Raptis, A. A.: Radiation and free convection flow through a porous medium,Int. Commun. Heat mass transfer, Vol. 25,pp.289-295(1998).

27. Saffman, P. G: On the boundary conditions at the free surface of a porousmedium, Stud .Application Maths, V. 2, pp.93(1971)

28. Satapathy, S. Behford, A. and Raptis, A.: Radiation and free convection flowthrough a porous medium, Int. Comm. Heat mass transfer, Vol. 125, 2, pp. 289-297(1998).

29. Singh, A.K. and Paul, T.: Transient natural convection between two verticalwalls heated/cooled asymmetrically. International Journal of AppliedMechanics and Engineering. 11(1), pp. 143-154(2006).

30. Tam. C. K. W.: The drag on a cloud of spherical particle in a low Reynoldsnumber flow, J. Fluid Mech.V.51.pp.273(1969).

-5

-4

-3

-2

-1

0

1

2

3

4

5

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

y

u

I

II

III

IV

V

VI

-2

-1

0

1

2

3

4

5

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

y

u

I

II

III

IV

V

Fig. 1 : Variation of u with G Fig. 2 : Variation of u with D-1

, M

I II III IV V VI I II III IV V

G 103 3x10

3 5x10

3 -10

3 -3x10

3 -5x10

3 D

-1 10

2 10

2 10

2 2x10

2 3x10

2

M 2 4 6 2 2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

y

u

I

II

III

IV

V

-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

y

u

I

II

III

IV

V

VI

Fig. 3 : Variation of u with , Fig. 4 : Variation of u with N1

I II III IV V I II III IV V VI

2 4 6 2 2 N1 0.5 1.5 2.5 5.0 10 100

2 2 2 4 6

-0.25

-0.2

-0.15

-0.1

-0.05

0

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

y

u

I

II

III

IV

-1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

y

u

I

II

III

Fig. 5 : Variation of u with Ec Fig. 6: Variation of u with P

I II III IV I II III

Ec 0.001 0.003 0.005 0.007 P 0.71 10 30

0

0.2

0.4

0.6

0.8

1

1.2

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

y

q

I

II

III

IV

V

VI

0

0.2

0.4

0.6

0.8

1

1.2

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

y

q

I

II

III

IV

V

Fig. 7 : Variation of q with G Fig. 8 : Variation of q with D-1

, M

I II III IV V VI I II III IV V

G 103 3x10

3 5x10

3 -10

3 -3x10

3 -5x10

3 D

-1 10

2 10

2 10

2 2x10

2 3x10

2

M 2 4 6 2 2

0

0.2

0.4

0.6

0.8

1

1.2

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

y

q

I

II

III

IV

V

0

0.2

0.4

0.6

0.8

1

1.2

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

y

q

I

II

III

IV

V

VI

Fig. 9 : Variation of q with , Fig. 10 : Variation of q with N1

I II III IV V I II III IV V VI

2 4 6 2 2 N1 0.5 1.5 2.5 5.0 10 100

2 2 2 4 6

0

0.2

0.4

0.6

0.8

1

1.2

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

y

q

I

II

III

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

y

q

I

II

III

IV

Fig. 11 : Variation of q with P Fig. 12 : Variation of q with Ec

I II III I II III IV

P 0.71 10 30 Ec 0.001 0.003 0.005 0.007

Table-1

Shear stress () at y = 1

P = 071

G I II III IV V VI VII

103

-34.2899 -38.2509 -47.1085 -35.6022 -36.3559 -33.2899 -32.0899

3X103

-117.6777 -147.6176 -289.5542 -120.6293 -124.3313 -116.2777 -115.0777

-103

34.2899 38.2509 47.1085 35.6022 36.3559 33.2099 32.0899

-3X103

117.6777 147.6176 2893.5542 120.6293 124.3313 116.27777 117.5077

M 2 4 6 2 2 2 2

D-1

102

102 10

2 2x10

2 3x10

2 10

2 10

2

2 2 2 2 2 4 6

Table-2

Shear stress () at y = 1

G I II III IV V VI VII VIII IX

103

-0.279 -0.218 -0.138 -0.12011 -0.103 -0.2569 -0.1140 -34.2999 -34.3092

3X103

-0.698 -0.541 -0.345 -0.3003 -0.257 -0.6417 -0.2850 -118.6777 -118.6777

-103

0.279 0.218 0.138 0.12011 0.103 0.2569 0.1140 34.2999 34.2899

-3X103

0.698 0.541 0.345 0.3003 0.2575 0.6417 0.2850 1118.6777 118.3277

N1 1.5 1.5 3.5 5 10 0.5 0.5 0.5 0.5

2

2 2 2 2 4 6 2 2

P 0.71 0.71 0.71 0.71 0.71 0.71 0.71 7 10

Table – 3

Shear stress at y = 1

G I II III IV

1x103

0.46187 0.46722 0.47223 0.49623

3x103 1.43906 1.56199 1.68387 1.80576

-1x103 -0.46187 -0.46722 -0.47223 -0.49623

-3x103 -1.43906 -1.56199 -1.68387 -1.80576

Ec 0.001 0.003 0.005 0.007

Table-4

Shear stress () at y = -1

G I II III IV V VI VII

103

145.1947 308.5397 477.5152 179.6205 310.1173 146.0947 147.2947

3X103

275.4981 659.7590 972.3858 355.8984 428.9495 276.2981 277.4981

-103

-145.1947 -308.5397 -477.5152 -179.6205 -210.1173 -146.0947 -147.2947

-3X103

-275.4981 -659.7590 -972.3858 -355.8984 -428.9494 -276.2981 -277.4981

M 2 4 6 2 2 2 2

D-1

102

102 10

2 2x10

2 3x10

2 10

2 10

2

2 2 2 2 2 4 6

Table-5

Shear stress () at y = -1

G I II III IV V VI VII VIII IX

103

0.4711 0.251 0.0510 0.0136 -0.0150 0.395 0.0024 144.1947 143.0947

3X103

1.1756 0.626 0.127 0.337 -0.0378 0.986 0.0058 273.0981 272.0981

-103

-0.4711 -0.251 -0.051 -0.0136 0.0150 -0.395 -0.0024 144.1947 -143.0947

-3X103

-1.17561 0.626 -0.127 -0.0337 0.0378 -0.986 -0.0058 -273.0981 -272.0981

N1 1.5 1.5 3.5 5 10 0.5 0.5 0.5 0.5

2

2 2 2 2 4 6 2 2

P 0.71 0.71 0.71 0.71 0.71 0.71 0.71 7 10

Table – 6

Shear stress at y = -1

G I II III IV

1x103

-1.04128 -1.03828 -1.03494 -1.03161

3x103 -3.08744 -3.00381 -2.91980 -2.83579

-1x103 1.04148 1.03828 1.03494 1.03161

-3x103 3.08744 3.00381 2.91980 2.83579

Ec 0.001 0.003 0.005 0.007

Table-7

Nusselt number (Nu) at y = 1

P=0.71

G I II III IV V VI VII

103

229.763 1878.747 429.1209 379.8239 565.845 230.743 232.703

3X103

1436.902 42.5026 530.9426 474.7840 656.166 361.902 365.922

M 2 4 6 2 2 2 2

D-1

102

102 10

2 2x10

2 3x10

2 10

2 10

2

2 2 2 2 2 4 6

Table-8

Nusselt number (Nu) at y = -1

P=0.71

G I II III IV V VI VII

103

343.284 440.108 509.125 489.842 675.538 346.284 349.284

3X103

440.108 573.295 693.249 556.093 763.694 440.108 468.108

M 2 4 6 2 2 2 2

D-1

102

102 10

2 2x10

2 3x10

2 10

2 10

2

2 2 2 2 2 4 6

Table – 9

Nusselt number (Nu) at y = 1

G I II III IV

1X103

-0.55027 -0.53007 -0.51297 -0.49587

3x103 -0.48588 -0.33687 -0.19097 -0.04507

Ec 0.001 0.003 0.005 0.007

Table-10

Nusselt number (Nu) at y = 1

P=0.71

G I II III IV V VI VII VIII

103

-0.2085 -0.23198 -0.23747 -0.24417 -0.21132 -0.23967 229.763 303.763

3X103

-0.19613 -0.23253 -0.23807 -0.24466 -0.20283 -0.24025 146.902 149.902

N1 1.5 1.5 3.5 5 10 0.5 0.5 0.5

2

2 2 2 2 4 6 2

P 0.71 0.71 0.71 0.71 0.71 0.71 0.71 7

Table-11

Nusselt number (Nu) at y = -1

P=0.71

G I II III IV V VI VII VIII

103

1.24185 1.37796 1.42197 1.48252 1.25284 1.44093 34.2843 43.2843

3X103

1.25798 1.37809 1.42234 1.48332 1.26325 1.4144 140.108 240.208

N1 1.5 1.5 3.5 5 10 0.5 0.5 0.5

2

2 2 2 2 4 6 2

P 0.71 0.71 0.71 0.71 0.71 0.71 0.71 7

Table – 12

Nusselt number (Nu) at y = -1

G I II III IV

103

1.55250 1.54583 1.53466 1.52350

3X103

1.50385 1.39987 1.29140 1.18294

Ec 0.001 0.003 0.005 0.007