chapter ii dusty flow and heat transfer through a porous...

CHAPTER II

DUSTY FLOW AND HEAT TRANSFER THROUGH A POROUS

MEDIUM BOUNDED BY A POROUS VERTICAL SURFACE

WITH VARIABLE PERMEABILITY AND HEAT SOURCE�

� Published in the International Journal of Mathematics, Computer Science and

Information Technology, 1,2,137-156,2008.

2.1 INTRODUCTION

The study of flow and heat transfer in porous medium has attracted

the attention of researchers in recent times, as it has wide applications in nature

and engineering practice such as thermal insulation, catalytic reactor, drying

technology etc. In particular, the dusty fluid flows phenomena are important in

sedimentation, pipe flows, fluidized beds, gas purification and transport

processes. In view of these applications, a series of investigations have been

made by many authors. Sharma R C and Sharma K N (1982) have described the

thermal instability of fluids through a porous medium in the presence of

suspended particles. Rao U S and Ramamurthy V (1987) have discussed the

laminar flow of a dusty fluid under pressure gradient in channels in porous walls.

Hamdan and Barron (1990) have cast the model equations in vorticity-stream

function forms and applied them to study the flow in a rectangular cavity,

obtained a numerical solution for the flow considered and compared the results

with the solution for a clean fluid flow in porous media. Pankaj Mathur and

Sharma (1995) have investigated the steady laminar free convective flow of

23

electrically conducting fluid along a porous plate in the presence of heat

source/sink.

Dusty gas flow through porous media has been analyzed by Marwann

Awartani and M H Hamdan (1999) in an attempt to study the effect of

permeability on the flow geometries that are possible for the two dimensional

flow at hand, considering the particle number density as constant as well as

variable. Fathi M allan et. al (2004) have studied an overview of the equations

governing the flow of a dusty gas in various type media including that in

naturally occurring porous media , using numerical techniques to solve the set of

partial differential equations describing the flow for different values of the

porous media parameters namely, Forchheimr drag coefficient, permeability and

the Reynolds Number. Adam Z Weber and John Newman (2005) have

presented modeling for gas phase flow in porous media. Allan and Hamdan

(2006) have studied Fluid- Particle model of flow through porous media

assuming the case of uniform particle distribution and parallel velocity fields.

Hani I Siyyam and Hamdan (2008) have obtained three exact solutions for flow

through porous media as governed by Darcy-Lapwood-Brinkman model for a

given velocity distribution and they found that the resulting flow fields are

identified as reversing flows , stagnation point flows and flows over a porous flat

plate with blowing or suction. Madhura, Gireesha and Bagewadi (2009) have

presented solutions for the problem of the motion of dusty gas through porous

media in an open rectangular channel. Three dimensional flow and heat transfer

through porous medium bounded by a porous vertical surface with variable

permeability and heat source was studied by Sharma and Yadav (2005).

Motivated by these papers on the flow of fluid in Porous medium, this chapter

has been presented on dusty fluid flow and heat transfer through porous medium

bounded by a porous vertical surface with variable permeability and heat source.

This chapter is an extension of [111] for dusty fluids.

24

2.2 MATHEMATICAL FORMULATION

The steady free convection flow of a viscous incompressible dusty

fluid through a porous medium bounded by an infinite vertical porous surface

with variable suction in the presence of heat source is considered. The surface is

lying vertically on the x*z* - plane with x* - axis taken along the surface in the

upward direction. The y* - axis is taken normal to the plane surface and directed

into the dusty fluid flowing laminarly with a uniform free stream velocity U.

The permeability of the porous medium is assumed to be of the form 1*

*0

* zcos1k)z(k�

��

��

��

�

��

��

� ��

� (2.1)

where is the mean permeability of the medium, is the wavelength of

permeability distribution, �(<<1) is the amplitude of permeability variation. All

the physical quantities are independent of x*.The flow remains three dimensional

due to periodic suction velocity

*0k �

��

�

��

��

� ��

�

** zcos1Vv (2.2)

where V is the mean suction velocity.

Denoting in nondimensional form the velocity components of fluid by

u, v, w and that of dust particle by along x, y, z - directions,

respectively, and temperature of fluid by

ppp wvu ,,,

� , and that of dust particle by p� , the

flow of the dusty fluid through a porous medium is governed by the following

equations.

2.3 EQUATIONS OF MOTION FOR FLUID PHASE

0zw

yv

��

�� (2.3)

25

� � � �� 1uzcos1kRe

1uufzu

yu

Re1ReGr

zuw

yuv

0p2

2

2

2

��

��

��

��

��

��

��

(2.4)

� �Rek

)zcos1(vvvfzv

yv

Re1

yp

zvw

yvv

0p2

2

2

2 ��

��

�

��

��

��

��

��

�� (2.5)

� � � �zcos1kRe

wwwfzw

yw

Re1

zp

zww

ywv

0p2

2

2

2

��

��

��

��

��

��

��

��

(2.6)

��

�

��

��

��

��

��

��

!��

��

��

��

��

��

�� 22

22

2

2

2

zw

yvEcPr2

zw

yvPrRe

zy

��

�

��

��

��

��

��

��

��

��

��

��

!�222

2

zu

yuEcPr

zv

ywEcPr

� ��

��

��

��

��

!� p

2

2 Ref32S

zw

yvEcPr

32 � (2.7)

2.4 EQUATIONS OF MOTION FOR PARTICLE PHASE

0z

wyv pp �

��

��

(2.8)

� uu1z

uw

yu

v pp

pp

p ��

��

��

�� (2.9)

� vv1zv

wyv

v pp

pp

p ��

��

��

�� (2.10)

� ww1z

ww

yw

v pp

pp

p ��

��

��

�� (2.11)

�"��

��

��

Pr32

zw

yv pp

pp

p (2.12)

after non dimensionalising the parameters using the following transformations,

�

*y y � , �

*z z � , Uu u

*

� , Uu

u*p

p � , Vv v

*� ,

Vw w

*�

26

Vv

v*p

p � , V

p*

p

w w � , 2

*

Vp p

#� , 2

*0

0k k�

� , $

$

��

��TTTT

w

*

$

$

�

��

TTTT

w

*p

p , � �2

w

UVTTg Gr $�%&

� '

(� pC

Pr

&�

�V Re , '

�2*Q S � ,

UV �! , � �$�

�TTC

U Ecwp

2

Mass concentration parameter #

#�

#� pNm f

Relaxation time parameter ,Km

��K

mV�� ,

p

s

CC

�"

where the stared quantities represent the respective dimensional parameters.

Boundary conditions of the problem in nondimensional form are

given by,

y = 0 u = 0 v = - � �zcos1 �� w = 0 � = 1

up =0 vp = - � �zcos1 �� p = 1 (2.13)

y) $ u )1 w) 0 p)p$ � )0

up)1 wp)0 �p ) 0

2.5 SOLUTION OF THE PROBLEM

The partial differential equations (2.3) to (2.12) are solved by using

perturbation method. Since � and Ec are small parameters, the velocities and

temperatures are expressed as a power series in � and Ec. By comparing the

zeroth and first order terms in the series, the partial differential equations are

converted into third order ordinary differential equations. Descarte’s rule is

applied to find the number of positive and negative roots. It is found that all the

differential equations have only real roots. The positive roots are neglected in

accordance with the boundary conditions as y) $. We express all the physical

quantities in powers of � as,

F(y,z) = F0(y) + �F1(y,z) + O(�2) (2.14)

where F0 is any one of u,v,w,p,up, vp, wp, � or �p.

27

Substituting equation (2.14) in equations (2.3) to (2.12) and

comparing the zeroth order term, we get the following differential equations.

��

�

��

��

��

��

��

��

0

00

000 k

uuRek1Refu1Reu

= ��

��

��0

0

2

02

k1ReGrReGr (2.15)

��

��

��

��

��

�

��

��

��

Refk1ReGru

k1RefuReuu

00

20

0000p (2.16)

��"�

��

��

��"

��

��

��

��"

�� Pr3S2

3Re2

3Ref2S

Pr32PrRe 0

000

= ��"�

��3

uEc2uuEcPr22

000 (2.17)

��

��

��

��

��

��

��

Ref23uEcPr

3Ref2SPrRe 2

00000p (2.18)

The corresponding boundary conditions are

y = 0 u0 = 0 v0 = -1 w0 = 0 � = 1

up0=0 vp0 = -1 �p0 = 1 (2.19)

y) $ u0)1 w0 )0 p0 )p$

�0 )0 up0 )1 �p0) 0

Integrating 0v0 �� and using the boundary conditions (2.19), we get, 1v0 ��

Similarly, integrating and using the boundary conditions (2.19), we get,

0v 0p ��

1v 0p ��

Since the Eckert number Ec is small for compressible fluid flows,

u0(y), up0(y), �0(y) and �p0(y) can be expanded in the powers of Ec as given

below.

28

u0(y) = u00(y) + Ec u01(y) + O � �2Ec (2.20)

�0(y) = �00 (y) + Ec �01(y) + O � �2Ec (2.21)

up0(y) = up00 (y) + Ec up01(y) + O � �2Ec (2.22)

�p0(y) = �p00 (y) + Ec �p01(y) + O � �2Ec (2.23)


y = 0 u00 = 0 �00 = 1 up00 = 0 �p00 = 1

u01 = 0 �01 = 0 up01 = 0 (2.24)

y) $ u00 )1 �00 )0 up00 )1 �p00) 0

u01 ) 0 up01 )0 �01 )0 �p01 ) 0

Substituting (2.21) and (2.23) in equations (2.17) and (2.18) and


0Pr3S2

3Re2

3Ref2S

Pr32PrRe 00

000000 ��"

��

�

��

��"

��

��

��

��"

�� (2.25)

��

��

��

��

��

�

��

��

Ref23

3Ref2SPrRe 00000000p (2.26)

Equation (2.25) is a third order ordinary differential equation and is

solved directly using the above boundary conditions (2.24) to obtain . It has

two negative roots and one positive root. The positive root is neglected using the

boundary condition

00�

$))� yas000 . Substituting 00� in (2.26), �p00 is

obtained.

Substituting (2.20) and (2.22) in (2.15) and (2.16) respectively, and


��

�

��

��

��

��

��

��

0

0000

00000 k

uuRek1Refu1Reu

= ��

��

��0

00

2

002

k1ReGrReGr (2.27)

29

��

��

��

��

��

�

��

��

��

Refk1ReGru

k1RefuReuu

000

200

0000000p (2.28)

Equation (2.27) is a third order differential equation and is solved

directly using the boundary condition (2.24) to obtain u00. It has one negative

root and two positive roots. The positive roots are neglected using the boundary

condition. Substituting in (2.28), up00 is obtained. Thus the solutions are, 00u

ym2

ym100

21 eAeA ��

ym4

ym300p

21 eAeA ��

1eAeAeAu ym7

ym6

ym500

213 ��

1eAeAeAu ym10

ym9

ym800p

213 ��

Comparing the coefficient of Ec in equations (2.15) and (2.17), we get

��"�

��

��

��"

��

��

��

��"

�� Pr3S2

3Re2

3Ref2S

Pr32PrRe 01

010101

= 2000000 u

32uuPr2 ��"

�� (2.29)

��

��

��

��

��

��

��

Ref23uPr

3Ref2SPrRe 2

0001010101p (2.30)

01

2

012

0

0101

00101

ReGrReGrkuuRe

k1Refu1Reu �

��

��

�

��

��

��

��

��

��

(2.31)

��

��

��

��

��

�

��

��

��

RefReGru

k1RefuReuu 01

201

0010101p (2.32)

Equation (2.29) is third order differential equations and is solved

directly using the boundary condition (2.24) to obtain 01� . It has two negative

roots and one positive root. The positive root is neglected using the boundary

condition . Equation (2.31) has one negative root and two $))� yas001

30

positive roots. The positive roots are neglected using the boundary condition

.Substituting in (2.30), up01 is obtained as follows. $)) yas0u 01

01 A��

A�

01p A��

A�

01 Au �

A�

01p Au �

A�

01u

ym2e�

m( 1e�

m2e�

m( 1e�

m1e�

m2 2e�

m37 e�

43A

ym214

ym21312

ym11

131 eAeAAe ��

y)mm(18

y)mm(17

y)m16

ym215

313222 eAeAAe ��

ym222

ym221

y20

ym19

131 eAeAAe ��

y)mm(26

y)mm(25

y)m24

ym223

313222 eAeAAe ��

ym230

ym29

y28

ym27

323 eAeAAe ��

y)mm(35

y)mm(34

y)mm(33

y32

ym231

3132211 eAeAeAAe ��

ym241

ym240

ym239

ym38

yym36

213213 eAeAeAeAAe �� y)mm(

44y)mm(y)mm(

42313221 eAee ��

When � * 0, substituting (2.14) in equations (2.3) to (2.12) and

equating the coefficients of �, we get

0z

wyv 11 �

��

�� (2.33)

0z

wy

v 1p1p ��

��

��

(2.34)

� �11p21

2

21

2

1Re�01

1 uufzu

yu

Re1Gr

yuv

yu

��

��

��

��

��

��

��

�

� �+ ,100

u1uzcos�kRe

1�� (2.35)

� 11 �0p1p

1p uy

uv

yu

��

��

��

pu1�

(2.36)

� �0

1011p2

12

21

2

yv

��11

kRe)vzcosv(vvf

zv

Re1

yp

yv ��

��

��

��

��

��

��

� (2.37)

� 11p1p vv1

yv

��

�� (2.38)

31

� � � �0

1011p2

12

21

211

kRewzcoswwwf

zw

yw

Re1

zp

yw ��

��

��

��

��

��

��

��

�

(2.39)

� 11p1p ww1

yw

��

��

� � (2.40)

� �11p110

1012

12

21

2

3Ref2S

yu

yuEcPr2

yPrRe

yvPrRe

zy

��

��

��

�

��

��

��

��

(2.41)

� �11p0

1p1

Pr32

ypv

yp

��"

��

��

� (2.42)


y = 0 u1 = 0 v1 = zcos�� w1 = 0 �1 = 0

up1 = 0 vp1 = zcos�� wp1 = 0 �p1=1

(2.43)

y) $ u1)0 w1) 0 p1)0 �1) 0

up1)0 wp1)0 �p1) 0

Equations (2.37) – (2.40) are independent of the temperature field.

So, in view of the boundary conditions, applying separation of variables method,

it is assumed that

v1 (y,z) = -v11(y) cos�z vp1(y,z) = -vp11 (y) cos�z

zsin)y(v1)z,y(w 111 ��

� zsin)y(v1)z,y(w 11p1p ��

� (2.44)

p1 (y,z) = p11(y) cos�z

The equations (2.44) are so chosen such that the continuity equations

(2.33) and (2.34) are satisfied. Substituting (2.44) in equations (2.37) - (2.40)

and simplifying, we get

32

110

02

110

21111 v

k1kvRe

k1Refv1Rev ��

�

��

��

��

��

��

��

��

��

��

��

��

��0

1111 k1pRepRe (2.45)

(2.46) 0pp 112

11 ��

��

��

��

��

��

�

��

��

��

Refk1pRev

k1RefvRevv

01111

0

2111111p (2.47)


y = 0 v11 = 1 011 ��v vp11 = 1 011 ��pv (2.48)

y) $ ) 011v� 11pv� )0 p11) 0

Equation (2.45) and (2.46) are ordinary differential equations and are

solved directly using the boundary condition (2.47). The solutions are given by y

111 eBp ��

4y

3yA

211 BeBeBv ��

7y

6yA

511p BeBeBv ��

In view of the boundary conditions, separation of variables method

can be applied for u and �. It is assumed that

u1(y,z) = u11(y) cos�z up1(y,z) = up11(y) cos�z

�1(y,z) = �11(y) cos�z �p1(y,z) = �p11(y) cos�z (2.49)

Substituting (2.49) in (2.35), (2.36), (2.41) and (2.42) and

simplifying, we get

110

02

110

21111 u

k1kuRe

k1Refu1Reu ��

�

��

��

��

��

��

��

��

��

��

= 11p0p0

011

2110110 vuRef

kuReGrvuRevuRe �

��

��

)1u(k

1ReGrvuRe

00

11

2

110 ��

��

��

� (2.50)

33

��

��

��

�

�

��

�

��

��

��

��

��

�Ref

)1u(k1ReGrvuRe

uk1RefuReu

u

00

112

110

110

21111

11p (2.51)

112

1111 S3

Re23

Ref2Pr3

2PrRe ��

��

��

��"�

��

�

��

��"

��

� �11011011p0p11

2

uu3

Ec4v3

Re2v3

Ref2Pr3

S2 ��"

��"

��

��"

�� (2.52)

� � � �110110011011 uuuuEcPr2vvPrRe ��

��

��

��

�

�

��

�

��

��

��

��

��Ref23

SuuEcPr2vPrRe3

Ref2PrRe

11110011

112

111111p (2.53)


y = 0 u11 = 0 �11 =0 up11 = 0 �p11 = 0

y ) $ u11 )0 �11 )0 up11 )0 �p11 ) 0 (2.54)

Now expanding u11 and �11 in powers of Ec as a series, we get,

u11 = u110 +Ec u111 + O( ) 2Ec

up11 = up110 +Ec up111 + O( ) 2Ec

�11 = �110 + Ec �111 + O( ) 2Ec

�p11= �p110 + Ec �p111 + O( ) (2.55) 2Ec

Substituting (2.55) in equations (2.50) – (2.53) and comparing zeroth

order term, we get,

1102

110110 S3

Re23

Ref2Pr3

2PrRe ��

��

��

��"�

��

�

��

��"

��

)vv(PrRev3

Re2v3

Ref2Pr3

)S(200110011001100p11p110

2

��"�

��

��"

�� (2.56)

34

��

��

��

�

��

��

��

Ref23vPrReSRefPrRe 0011110

2110110110p (2.57)

1100

02

1100

2110110 u

k1kuRe

k1Refu1Reu ��

�

��

��

��

��

��

��

��

��

��

00p11p0

00110

200110011 uvRef

kuReGruvReuvRe �

��

��

� 1uk

1ReGruvRe

000

110

2

0011 ��

��

��

� � (2.58)

� ��

��

��

��

�

��

��

��

��

��

��

��

�Ref

1uk1ReGruvRe

uk1RefuReu

u

000

1102

0011

1100

2110110

110p (2.59)


y = 0 u110 = 0 �110 =0 up110 = 0 �p110 = 0 (2.60)

y) $ u110)0 �110 )0 up110 ) 0 �p110 ) 0

Equation (2.56) and (2.58) are third order coupled ordinary

differential equations, and solved using the boundary conditions (2.60) to get

�110 and u110. Using them in (2.57) and (2.59), �p110 and up110 are obtained as

follows. y)m(

12y)Am(

11y)Am(

10ym

9ym

811012154 eBeBeBeBeB ��

ym15

ym14

y)m(13

212 eBeBeB ��

y)m(20

y)Am(19

y)Am(18

ym17

ym16110p

12154 eBeBeBeBeB ��

ym23

ym22

y)m(21

212 eBeBeB ��

y)Am(27

y)Am(26

y)Am(25

ym24110

2136 eBeBeBeBu ��

31y)m(

30y)m(

29y)m(

28213 BeBeBeB ��

ym35

ym34

ym33

ym32

5421 eBeBeBeB ��

ym3e�

35

y)Am(39

y)Am(38

y)Am(37

ym36110p

2136 eBeBeBeBu ��

ym43

y)m(42

y)m(41

y)m(40


ym47

ym46

ym45

ym44


Comparing the coefficient of Ec in equations (2.50) – (2.53), we get

1112

111111 S3

Re23

Ref2Pr3

2PrRe ��

��

��

��"�

��

�

��

��"

��

11001011101p11p111

2

uu3

4v3

Re2v3

Ref2Pr3

)S(2 ��"

��"

��

��"

�� (2.61)

)uuuu(Pr2)vv(PrRe 110001100001110111 ��

��

��

��

� ��

�

��

��

Ref23uuPr2vPrReS

3Ref2PrRe 110000111111

2111111111p

(2.62)

1110

02

1110

2111111 u

k1kuRe

k1Refu1Reu ��

�

��

��

��

��

��

��

��

��

��

= 01p11p0

01111

201110111 uvRef

kuReGruvReuvRe �

��

��

��

��

��

0

01111

2

0111 kuReGr

uvRe (2.63)

��

��

��

�

��

��

��

��

RefkuReGruvReu

k1RefuReuu

0

01111

20111111

0

2111111111p

(2.64)


y = 0 u111 = 0 up111 =0 �111 = 0 �p111 = 0

y) $ u111) 0 up111 )0 �111 ) 0 �p111 ) 0 (2.65)

Equation (2.61) and (2.63) are third order coupled ordinary

differential equations, and solved using the boundary conditions (2.65) ) to get

36

�111 and u111. Using them in (2.62) and (2.64), �p111 and up11 are obtained. The

solutions are

y)mm(82

y)mm(81

y)mm(80

y)mm(79

y)mm(78

y)mm(77

y)mm(76

y)mm(75

y)mm(74

y)mm(73

y)mm(72

y)mm(71

ym270

ym269

ym268

ym67

ym66

y)mm(65

y)mm(64

y)mm(63

y)m2(62

y)m2(61

y)m2(60

y)m(59

y)m(58

y)Amm(57

y)Amm(56

y)Amm(55

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where the constants are not given for the sake of brevity.

2.5.1 Skin – friction coefficient

The skin friction coefficient at the surface in the x* - direction is given

by

zcos)0()0(UV 10x

*

��#�

��

Where 0y

*

*

x*y

u

��

��

��

(�� is the shear stress.

2.5.2 Nusselt Number

The rate of heat transfer in terms of nusselt number Nu at the surface

is given by zcos)0(Nu)0(Nu)TT(C

qNu 10wP

��$�#&

��

Where 0y

*

*

*yTkq

��

��

��

��

2.6 RESULTS AND DISCUSSION

As the exact solutions are very lengthy, the velocity and temperature

of the fluid and the dust particles are shown through graphs, by incorporating the

exact solutions in the graph directly.

38

Some results for the fluid phase temperature profile and the particle

phase temperature profile

�

p� , the fluid velocity u and the particle velocity

based on the analytical solutions reported above are presented in figures 2.1

through 2.9. These results presented illustrate the influence of some of the

physical parameters on the solutions. The values of some of the physical

parameters employed to obtain the graphical result may or may not represent

actual conditions of applications.

pu

In the case of a clean fluid, the steady state for the fluid velocity and

temperature are attained around y=1.5. But in the case of the dusty fluid the

steady state is attained around y=100. This huge difference in the steady state

may be attributed to the presence of dust particles in the dusty fluid.

Figure 2.1. Temperature of the fluid for various values of Re and Pr

Fixing S=0.5, Gr=1, Ec=0.01, �=0.25,

z=0.25, f=0.2, "=0.2, ��=0.2, k0=0.25

39

2.6.1 Temperature profiles of fluid phase for various values of Re and Pr

A close look at figure.2.1 shows that the profiles of temperature of

the fluid increase with increase in both the Prandtl Number Pr and Reynolds

Number Re. They all start from 1 at the lower end of the plate and decrease

gradually to zero as y increases for all values of Re and Pr.

Figure 2.2. Temperature of the dust for various values of Re and Pr

Fixing S=0.5, Gr=1, Ec=0.01, �=0.25,

z=0.25, f =0.2, "=0.2, ��=0.2, k0=0.25

2.6.2 Temperature profiles of particle phase for various values of Re and

Pr

The temperature profiles of dust particles are shown in figure.2.2.

They increase with increase in both the Prandtl Number Pr and Reynolds

Number Re. It starts from 1 at the lower end of the plate and approaches steadily

zero away from the plate, as y increases. Both the fluid and dust particles behave

in the same manner. But the profiles of dust particles are slightly higher

40

compared with that of the fluid. In the case of clean fluid, the temperature

profiles decrease with increase in Reynolds Number and Prandtl Number

whereas in the case of a dusty fluid the profiles decrease with increase in these

two parameters. This phenomenon may be due to the influence of dust particles.

Figure 2.3. Velocity of the fluid for various values of Re and Pr

Fixing S=0.5, Gr=1, Ec=0.01, �=0.25,

z=0.25, f =0.2, "=0.2, ��=0.2, k0=0.25

2.6.3 Velocity profiles of the fluid phase for various values of Re and Pr

From figure 2.3, it can be concluded that increase in Reynolds

Number and Prandtl Number increases the velocity profile of the fluid for some

fixed values of other parameters. The velocity profiles maintain an increasing

trend near the lower end of the plate and attain their maximum very near the

lower end and thereafter they decrease steadily and reach the value 1 as y

approaches 100. In the case of clean fluid, the velocity profiles increase steadily

from zero and reach the value 1 as y increases.

41

In the case of clean fluid, increase in Reynolds Number decreases the

velocity profile of the fluid whereas increase in Reynolds number increases the

velocity of the dusty fluid. This shows the influence of dust particles on the flow

of the dusty fluid.

Figure 2.4. Comparison of temperature of the fluid and the dust for various

values of f Fixing Re=1, Pr=5, S=0.5, Gr=1, Ec=0.01, �=0.25,

z=0.25, "=0.2, ��=0.2, k0=0.25

2.6.4 Comparison of temperature of the fluid phase and the particle phase

for various values of mass concentration parameter f

It is noted from figure 2.4 that the temperature profiles of both the

fluid phase and the dust particles phase increase with increase in the mass

concentration parameter f. But they both decrease steadily to zero as y increases

for any value of f. It can be seen that the temperature of the fluid is at a greater

height than that of the dust particles. This shows the influence of the dust

particles on the motion of the fluid.

42

Figure 2.5. Velocity of the fluid for various values of f

Fixing Re=1, Pr=5, S=0.5, Gr=1, Ec=0.01, �=0.25,

z=0.25, " =0.2, ��=0.2, k0=0.25

2.6.5 Velocity of the fluid phase for various values of mass concentration

parameter f

It is observed from figure 2.5 that the velocity profiles of the fluid

phase increase steadily for increase in the mass concentration parameter f. As y

increases the velocity profiles of the fluid phase goes on increasing upto 1.4

nearly and then maintain a decreasing trend and reach the value 1. As the

profiles of the velocity of the fluid phase are very significant upto y=3, the

graphs are drawn upto y=3.

43


values of " Fixing Re=1, Pr=5, S=0.5, Gr=1, Ec=0.01, �=0.25,

z=0.25, f =0.2, ��=0.2, k0=0.25

2.6.6 Comparison of temperature of the fluid phase and the dust phase for

various values of "

The temperature of the fluid phase and the particle phase are

compared in figure 2.6 for various values of " which is ratio of specific heat of

dust to that of the fluid. Both the temperature profiles of the fluid phase and

particle phase increase with an increase in the values of ". It is also seen that the

temperature of the fluid is slightly higher than that of the particle phase.

44

Figure 2.7. Velocity of the fluid for various values of "


z=0.25, f =0.2, ��=0.2, k0=0.25

2.6.7 Velocity of the fluid phase for various values of "

The velocity profiles of the fluid phase are shown in figure 2.7 for

various values of" . It is noted from figure 2.7. that the velocity profiles of the

fluid phase decrease with an increase in the value of " which is ratio of specific

heat of the dust to that of the fluid. As the profiles show a significant difference

very near the lower end of the plate, the graphs is drawn upto y=3. As y

increases the velocity of the fluid phase increases till y=3 and then they will

decrease to reach the value 1. The cases I and II in figures 2.6 and 2.7 are as

given in the following table 2.1.

45

Table 2.1 Cases I and II represented in figure 2.7 and 2.8

CASES " I

II

0.2

0.4


values of �� Fixing Re=1, Pr=5, S=0.5, Gr=1, Ec=0.01, �=0.25,

" =0.2, z=0.25, f =0.2, k0=0.25

2.6.8 Comparison of temperature of the fluid phase and the dust phase for

various values of� � , the time relaxation parameter.

The temperature profiles of the fluid phase and the particle phase are

compared in figure 2.8 for various values of� � , the time relaxation parameter. It

is seen that the temperature profiles of both the fluid phase and the particle phase

increase as � � increases. Also it can be noted that the fluid temperature is

slightly higher than the particle temperature for any value of� � . The temperature

profiles of both phases maintain a decreasing trend as y increases.

46

Figure 2.9.Velocity of the fluid for various values of ��


" =0.2, z=0.25, f =0.2, k0=0.25

2.6.9 Velocity of the fluid phase for various values of � � , the time

relaxation parameter

The velocity profiles of the fluid phase are shown in figure in 2.9. It

is observed that the velocity profiles of the fluid phase increase with an increase

in the values of time relaxation parameter. As y increases, the velocity profiles

increase up to 1.4 nearly and thereafter will decrease steadily and reach the value

1. As the curves show significant difference near the lower end, the curves have

been drawn up to y=3.

47

Table 2.2 Values of Skin –Friction Coefficient x� in the x* - direction for various

values of the Prandtl Number Pr and f

Pr f Re

5 6 7 0.2 0.4 0.6

0.1 32.1506 32.2494 32.379 32.1506 33.5517 34.9407

0.15 21.4337 21.4996 21.586 21.4337 22.3678 23.2938

0.2 16.0753 16.1257 16.1895 16.0753 16.7759 17.4704

Table 2.3 Values of Skin –Friction Coefficient x� in the x* -

direction for various values of "

" Re

0.2 0.4 0.6

0.1 32.1506 32.7994 32.8448

0.15 21.4337 21.8663 21.8966

0.2 16.0753 16.3997 16.4224

The variation of the skin friction coefficient x� in the main flow

direction x* are shown in table 2.2 and table 2.3.

From table 2.2, it can be concluded that increase in both Prandtl

Number Pr and mass concentration parameter f increase the skin friction

coefficient for a given Reynolds Number Re. Also for a given value of Prandtl

Number Pr and mass concentration parameter f, the skin friction coefficient

decreases for increase in Re. Even for a small increase in the Reynolds Number

Re, the skin friction coefficient shows an appreciable decrease.

It is noted from table 2.3 that increase in " increases the skin friction

coefficient in the x* direction for a fixed value of Re. Also for given" , an

appreciable decreasing trend is observed even for a small change in the value of

Reynolds Number Re.

48

Table 2.4 Values of Nusselt Number Nu in the x* - direction for various

values of Pr and� �

Pr � � Re

5 6 7 0.2 0.4 0.6

0.1 0.4308 0.4396 0.4621 0.4308 0.7942 1.1458

0.15 0.2872 0.2930 0.3081 0.2872 0.5294 0.7639

0.2 0.2154 0.2198 0.4621 0.2154 0.3971 0.5729

Values of heat transfer coefficient i.e. the Nusselt Number along the

x* direction is shown in the table 2.4. It is noted that when the Prandtl Number

Pr and mass concentration parameter f are increased there is an appreciable

increase in the values of Nusselt Number for a given Re. Also it is observed that

when Pr and f are fixed, increase in Reynolds Number Re decreases the Nusselt

Number values.

In the case of a clean fluid the skin friction coefficient and the Nusselt

Number decrease with an increase in the Prandtl Number Pr whereas a reverse

trend is seen in the dusty fluid.

If the mass of the dust particle are negligibly small, their

influence on the fluid flow is reduced and in the limit as m)0, the fluid

becomes ordinary viscous and we get the problem studied in [111].

2.7 CONCLUSION

1. Both the fluid and the particles behave in the same pattern for various

values of the physical parameters

2. Skin friction coefficient and Nusselt Number decrease with increase in

the values of the Reynolds Number Re.

3. It is hoped that the exact solutions reported in this chapter will serve as

stimulus for experimental work.

chapter ii dusty flow and heat transfer through a porous...

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