23

CHAPTER II

THREE DIMENSIONAL COUETTE DUSTY FLOW

WITH TRANSPIRATION COOLING*

* Published in the Journal of Zhejiang University Science (A), China, Vol.8, No.2, 2007 Feb, pp: 313-322 also available at on line : ISSN 1862 – 1775 (on

line). www.zju.edu.cn/jzus; www.springerlink.com

2.1 INTRODUCTION

Transpiration cooling is a very effective process to protect certain

structural elements like combustion chamber walls, exhaust nozzles and gas

turbine blades from the influence of hot gases in turbojet and rocket engines.

In view of this, Eckert (1958) obtained an exact solution of the plane Couette

24

flow with transpiration cooling. The problem remained two-dimensional due

to the uniform injection and suction applied at the porous plates. Flow and

heat transfer along a plane wall with periodic suction velocity was studied by

Gersten and Gross (1974). Effects of such a suction velocity on various flow

and heat transfer problems along flat and vertical porous plates were also

studied extensively by Singh et al (1978a) and Singh (1990,1978b,1991,1993).

Nag, et al (1979) discussed the couette flow of a dusty gas

between two parallel infinite plates for impulsive start as well as uniformly

accelerated start of one of the plates. Saffman (1962) formulated the equation

of motion of dust laden gas in a simplified form by making some assumptions

on dust particles. Michael and Miller (1966) investigated various aspects of

hydro dynamic and hydro magnetic two-phase fluid flows in a non-rotating

system. Debnath and Ghosh (1988) analyzed unsteady hydromagnetic flow

of a dusty fluid between two oscillating plates. Zung (1969) investigated the

flow induced in fluid particle suspension by an infinite rotating disc. Datta

and Dalal (1992) analyzed the generalized couette flow of a dusty gas due to

an impulsive pressure gradient as well as due to impulsive start of the lower

plate. Ahmed and Sharma (1997) investigated three dimensional free

convective flow and heat transfer through a porous medium. Raptis (1983)

analyzed the unsteady free convective flow through a porous medium bounded

by an infinite vertical plate. Raptis and Perdikis (1985) studied the oscillatory

flow through a porous medium in the presence of free convective flow. Singh

and Verma (1995) investigated the three dimensional oscillatory flow through

a porous medium with periodic permeability. Ahmed and Ahmed (2004)

studied two dimensional MHD oscillatory flow along a uniformly moving

infinite vertical porous plate bounded by porous medium.

Through the present chapter an attempt has been made to study the

effect of dust parameters on the three dimensional couette flow with

transpiration cooling. The object of the present chapter is to study the effect

of the injection / suction parameter, Prandtl number and dust parameters on

25

the velocity field, temperature field, skin friction and Nusselt number in the

three dimensional couette dusty flow with transpiration cooling using two

phase fluid. The present chapter is an extension of Singh (1999) for dusty

fluid.

2.2 FORMULATION OF THE PROBLEM

Consider the couette flow of a viscous incompressible dusty fluid

between two parallel flat porous plates as shown in figure 2.1. A co-ordinate

system is introduced with its origin on the lower stationary plate lying

horizontally on the x* - z* plane. The upper plate in uniform motion U along

the x* - axis is subjected to a constant suction V0. The lower plate is subjected

to a transverse sinusoidal injection velocity distribution of the form plate is

subjected.

V* (z*) = V0 [1+ � cos ���

���dz * ] , (2.1)

where � is a positive quantity (<<1).

Without any loss of generality, the distance ‘d’ between the plates

is taken to be equal to the wave length of the injection velocity. The lower

and upper plates are assumed to be at constant temperatures T0 and T1

respectively, where T1 > T0. All physical quantities are independent of x* for

this problem of fully developed laminar flow, however, the flow remains three

dimensional due to the injection velocity (2.1)

The exact form of particle-phase boundary condition at a surface

is currently unknown and many investigators in the field have used the no-slip

condition as for the fluid phase. This is also done in this thesis and the

conditions are not written separately for the particle phase so that what ever

conditions are valid for the fluid phase are assumed for the particle phase as

well.

26

Fig 2.1 Couette dusty flow with periodic injection and constant suction at t

Figure 2.1 Couette dusty flow with periodic injection and constant suction at the porous plates.

Denoting the velocity components of the fluid by u, v, w in the x,

y, z directions and that of the dust particles by up, vp, wp respectively, and the

temperature of the fluid by and that of the dust by p, the problem is

governed by the following non-dimensional equations.

2.3 EQUATIONS OF MOTION FOR FLUID PHASE

,0���

��

zw

yv (2.2)

),(12

2

2

2

uufzu

yu

zuw

yuv p ��

���

����

���

��

���

��

� (2.3)

27

),(12

2

2

2

vvfzv

yv

yp

zvw

yvv p ��

���

����

���

��

��

����

��

� (2.4)

),(12

2

2

2

wwfzw

yw

zp

zww

ywv p ��

���

����

���

��

��

����

��

� (2.5)

)(Pr3

2Pr1

2

2

2

2

�

�

� ��

�

����

���

��

���

��

pf

zyzw

yv , (2.6)

2.4 EQUATIONS OF MOTION FOR PARTICLE PHASE

0��

�

�

�

zw

yv pp , (2.7)

)(1p

pp

pp uu

zu

wyu

v ��

��

�

�

�, (2.8)

)(1p

pp

pp vv

zv

wyv

v ��

��

�

�

�, (2.9)

)(1p

pp

pp ww

zw

wyw

v ��

��

�

�

�, (2.10)

�

���

�

�

�

�

�

Pr3)(2 pp

pp

p zw

yv , (2.11)

Where

00

*,*,*,*,*Vww

Vvv

Uuu

dzz

dyy �����

2

*,*

,*,*

VopP

Vow

wVovv

Uu

u ppp

pp �

����

,,,*

,)()*( 0

01

0

01

0

dVmN

fTTTT

TTTT opp

p

��

����

��

��

�Km

p �� , s

p

CC

��

The ‘*’ Stands for dimensional quantities

The boundary conditions for the problem in the dimension less form are:

y = 0: u = 0, v (z) = 1+� cos z, w = 0, = 0,

up = 0, vp (z) = 1+� cos z, wp = 0, p = 0,

28

y = 1: u = 1, v = 1, w = 0, = 1,

up = 1, vp = 1, wp = 0, p = 1, (2.12)

The exact form of particle – phase boundary condition at a

surface is currently unknown and many investigators in the field have used the

no-slip condition as for the fluid phase. Whatever conditions are valued for

fluid phase are assumed for the particle phase as well.

2.5 Solution of the Problem

In order to solve these differential equations, we assume the

solution of the following form because the amplitude of the injection velocity

� (<<1) is very small.

h (y,z) = h0 (y) + � h1 (y,z) + �2 h2 (y,z) + . . . . . (2.13)

where h stands for any of u, up, v, vp, w, wp, p,, p. When � = 0 the problem

reduces to the two dimensional flow with constant injection and suction at

both plates. The same results for clean fluid when � = 0 is given by Eckert

(1958). The solution of corresponding the two dimensional problem for the

dusty fluid is

uo (y) = Aem1

y + Be m2y +C,

uPo (y) = 11

1

�mAe ym

+ 12

2

�mBe ym

+ C + D )(

��y

e ,

o = A1em3

y + B1 em4y + C1,

po = D1e- b1

y + 112

11

11

1143

CbmbeB

bmeAb ymym

, w0 = 0, wPo = 0, v0 =1 vPo = 1,

p0 = constant, where (2.14)

m1 =��

�

�1

)1( f , ,112 mm �

��� �

m3= 2)(4)()( 1

211 bprdbprbpr ��� � �

,

m4= 2

)(4)()( 12

11 bprdbprbpr ��� ���,

29

when � � 0, substituting (2.13) in equations in (2.2) to (2.11) and comparing

the coefficients of identical powers of �, neglecting those of �2, �3 etc., the

following first order equations are obtained with the help of (2.14). The

expressions of A, B, C, D, A1, B1, C1, and D1 are given.

FLUID PHASE:

,011 ���

��

zw

yv (2.15)

),(112

12

21

210

1 1uuf

zu

yu

yu

yu

v p ��

���

����

���

��

���

��

� (2.16)

),(1112

12

21

211 vvf

zv

yv

yp

yv

p ��

���

����

���

��

��

����

� (2.17)

),(1112

12

21

211 wwf

zw

yw

zp

yw

p ��

���

����

���

��

��

����

� (2.18)

)(Pr32

Pr1

1121

2

21

210

1

�

��

���

����

���

��

���

��

pf

zyyyv , (2.19)

PARTICLE PHASE

011 ��

�

�

�

zw

yv pp , (2.20)

)(111

101 p

ppp uu

yu

yu

v ��

��

�

�

�, (2.21)

)(111

1p

p vvyv

��

��

�, (2.22)

)(111

1p

p wwyw

��

��

�, (2.23)

)( 11110

1

����

�

�

�p

ppp b

yyv , (2.24)

The corresponding boundary conditions reduce to

y = 0: u1 = 0, v1 = cos z, w1 = 0, 1 = 0,

up1 = 0, vp1 = cos z, wp1 = 0, p 1 = 0,

30

y = 0: u1 = 0, v1, w1 = 0, 1 = 0,

up1 = 0, vp1, wp1 = 0, p 1 = 0, (2.25)

These are the linear partial differential equations describing the

three – dimensional cross flow and main flow. In order to solve these

equations, let us first consider the equations (2.15), (2.17), (2.18), (2.20),

(2.22) and (2.23) for cross flow, being independent of the main flow

component u1, up1, and the temperature field 1 and p1.

We assume v1, vp1, w1, wp1, p1 of the following form:

v1 (y,z) = v11 (y) cos z, (2.26)

w1 (y,z) = 1� {v1

11 (y) sin z}, ( 2.27)

vp1 (y,z) = vp11 (y) cos z, (2.28)

wp1 (y,z) = 1� {vp1

11 (y) sin z}, (2.29)

p1 (y,z) = p11 (y) cos z, (2.30)

where dash denotes differentiation with respect to y. Equations

(2.26), (2.27), (2.28) and (2.29) have been so chosen that the continuity

equation (2.15) and (2.20) are satisfied. Substituting these equations into

equations (2.17) and (2.18) and (2.22) and (2.23) and applying the

corresponding transformed boundary conditions, we get the solutions of v1,

vp1, w1, wp1, and p1 as

v1 (y,z) = {C3 er1

y + C4 e r2y + C5 r3y – d1c1 ey –d2 c2 e-y} cos z, (2.31)

W1 (y,z) = 1� {C3 r1 er

1y + C4 r2 e r2y + C5 r3 r3y – d1c1 ey +d2 c2 e-y} sin z,

(2.32)

,cos)1)1()1(

),( 2211

3

5

2

4

1

3)(

611

321

zebcde

acd

rec

rec

rec

eczyv yyyryryry

p

���

���

�� �

�

�

� ���

(2.33)

zebcde

acd

rerc

rerc

rerceczyw yy

yryryry

p

sin

)1()1()1(1),( 2211

3

35

2

24

1

13)(611

321

���

���

�� �

�

�

�

��� ��

�

(2.34)

31

p1 (y,z) = (c1 ey + c2 e-y) cos z, (2.35)

Now assuming u1, up1, 1, p1 in the form:

u1 (y,z) = u11 (y) cos z, (2.36)

up1 (y,z) = up11 (y) cos z, (2.37)

1 (y,z) = 11 (y) cos z, (2.38)

p1 (y,z) = p11 (y) cos z, (2.39)

and substituting in equations (2.16), (2.19), (2.21) & (2.24) we obtain the

following equations:

[�D3 + (1-� �) D2 – ((1+f)� + �2) D-2] u11 =

� [v11 u10 + f u1

p0 Vp11+ � (v111 u1

0 + v11 u110)], (2.40)

(D�+1) up11 = u11 - � vp11 u1p0, (2.41)

[D3 + (b1-�pr) D2 – D (2 + �3

2 �f +�prb1) -2b1] 11 =

� Pr [v11 110 + v1

11 10 + b1 V11 1

0] + �3

2 �f vp11 1p0, (2.42)

(D+b1) p11 – b1 11 = -vp11 1p0, (2.43)

with corresponding boundary conditions

y = 0: u11 = 0, 11 = 0, up11 = 0, p11 = 0,

y = 1: u11 = 0, 11 = 0, up11 = 0, p11 = 0, (2.44)

where primes denote differentiation with respect to y. Solving equations

(2.40), (2.41), (2.42) and (2.43) under the boundary conditions (2.44) and

using equations (2.36), (2.37), (2.38) and (2.39) we get

u1 (y,z) = [L er1y + M er

2y + N er

3y + K1 e(r

1+m

1) y + K2 e(r

2+m

1) y + K3 e(m

1+r

3) y

+ K4 e(m1

+) y +

K5 e (m1

- ) y + K6 e(m2

+r1) y + K7 e(m

2+ r

2) y + K8 e(m

2+r

3) y + K9 e(m

2+) y + K10

e(m2

-) y +

K11 e(m1

- �1

) y + K12 e(m2

- �1

) y + K13 e �� y2

+

K14 e( r1

- �1

) y + K15 e ( r2 - �1

) y + K16 e( r3

- �1

) y +

32

K17 e ( - �1

) y + K18 e - ( +�1

) y ] cos z, (2.45)

up11 (y, z) =

���

�

�

�

�

�

��

1)(1)(111 12

)(2

11

)(1)(

321

12113211

mreK

mreKeR

rNe

rMe

rLe ymrymryyryryr

+ �

�

��

�

�

�

1)(1)(1)(1)(1)( 22

)(7

12

)(6

1

)(5

1

)(4

31

)(3

22121113

rmeK

rmeK

meK

meK

rmeK yrmyrmymymymr

� �

��

�

��

��

�

�

���

��

�� )2(

13

2

)1(

12

1

)1(

11

2

)(10

2

)(9

32

)(8

1)1(1)1(1)(1)(1)(

212232 yymymymymyrm

eKm

eK

m

eKm

eKm

eKrm

eK

��

�

��

�

��

�

��

�

�� �

��

��

��

��

yyyryryreKeK

r

eK

r

eK

r

eK)1(

18

)1(

17

3

)1(

16

2

)1(

15

1

)1(

141321

1)1(1)1(1)1(

- � ��

��

�

� � �

��

��

]1))[()(1()1( 11111

)(31

11

)1(

6111

1

rmrmrecAm

mmecAm yrmym

+ � � �

� � �

]1))[(1)(1(]1))[(1)(1( 3131

)(51

2112

)(41

3121

rmrrmecAm

rmmrecAm yrmyrm

- �� �

� � �

�

]1)(()1(]1))[(1( 11

)(221

11

)(111

11

mbmecdAm

mmaecdAm ymym

+ � � �

�

�� �

��

]1))[(1)(1(1)1)[(1( 1211

)(32

22

)1(

6212

2

rmmrecBm

mm

ecBm yrmym

� � �

� � �

]1)()[1)(1(]1)()[1)[(1( 3232

)(52

2222

)(42

3222

rmrmecBm

rmmrecBm yrmyrm

�� �

� � �

� �

]1))[(1(1))[(1( 22

)(222

22

)(112

22

mmbecdBm

mmaecdBm ymym

212

)1(

4

12

)1(

3

2

6

)1(]1[

2

2

1

rrecD

rrecDecD

yryry

���

���

�

��

��

��

33

2

)1

(

11

33

)1(

5

)1(2

3

�

��� �

��

�

aeDcd

rrecD

yyr

����

��

� ye

bDdc )1(

222

cos z, (2.46)

where K1, K2 ….. K18 are known constants which are not presented

here due to the sake of brevity. But these constants were taken in to account

while drawing the velocity profiles of the dusty fluid u and dust up.

1 (y, z) = [R1es1

y + R2e s2 y + R3 es

3y + t1e (m

1 + r

1) y + t2 e (m

1+ r

2) y + t3e (m1

+ r 3

) y + t4

e (m1

+ ) y + t5 e (m1

- ) y + t6 e( m2

+ r1) y + t7 e (m

2+ r

2) y + t8 e ( m

2+ r

3) y + t9 e (m

2+ ) y +

t10 e (m2 - ) y –

)1

(

111 � � b

et - t12 e (r1 – b

1) y – t13 e( r

2 – b

1) y – t14 e( r

3 – b

1) y + t15 e( - b

1) y

+ t16 e – ( + b1) y +

ymet

)1(

171 ��

+ ym

et)1(

182 ��

] cos z, (2.47)

p1 (y, z) =

� ybeeys

eRbs

Rbbs

RbbseRb

ysys

1

321

413

31

12

21

11

11 b1

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

112

)(6

11

)(5

11

)(4

131

)(3

121

)(2

11

)(1

1211312111

brmet

bmet

bmet

brmet

brmet

brmet yrmymymyrmyrmyrm

�

�

3

)(14

2

)(13

1

)(12

)1

(

11

12

)(10

12

)(9

132

)(8

122

)(7

1312111

223222

)1()()()()( ret

ret

retet

bmet

bmet

brmet

brmet ybrybrybrybymymyrmyrm ����

��

����

� �

+

�

��

��

��

��

� ��

��

� ��

)1(]

11 11

)(13)1(

16

12

)12(

18

11

)1(

17)(

16)(

1511

11

11

rreDbCebDC

bm

et

bm

etetet ybrybymymybyb

� � �

�

����

ybybybrybr

ebDbcde

aDbcd

rrecDb

rreDbC )(122)(111

33

)(51

22

)(14 11

1312

)1()1(

� �

� �

�

��

��

))(1)(())(1)(()1)(( 122211

)(411

111111

)(311

1111

)1(

6112111

1

brmrbmecAmb

bmrrbmecAmb

bmbm

ecAmb yrmymrym

34

� �

�

�

))(())(())(1)(( 1111

)(1122

1111

)(1111

131311

)(115

1121

bmbmbAembcd

bmbmaeAmbcd

brmrbmeAmbc ymymyrm

))(1)(())(1)(()1)(( 122212

)(214

121112

)(321

1212

)12(

6212221

bmrrbmBembc

bmrrbmeBcmb

bmbm

eBcmb ymrymrym

� �

� �

�

�

��

-

]))(())(())(1)(( 1212

)(2122

1212

)(2111

132312

)(521

2232

bmbmbBembcd

bmbmaBembcd

brmrbmeBcmb ymymyrm

�

�

�

cosz, (2.48)

where A, B, C, D satisfy the equations

A+B+C =0, Aem1 + Bem

2 + C = 1,

11 2

1

1

1

�

� mB

mA bb + C + D = 0,

11 2

12

1

11

�

� mbBe

mbAe mm

+c+ D 11�

�e and b1 =

��

� 32,

Pr32 fd�

,

A1, B1, C1, D1 satisfy the equations

A1 + B1 + C1 = 0, A1em

1 + B1em2 + C1 =1, 011

12

11

11

11 �

DCbmBb

bmAb ,

114

112

11

11

311 �

�b

mm

DeCbmeBb

bmeAb , c1 = A3 / A0 ; c2 = A4 / A0,

A3 = (r1 - r2) er1+ r

2 + (r2 - ) e r1- - (r1 - ) er2-, A4 = (r1 - r2) er

1+r

2 + (r2 +

) e r1+ - (r1 + ) er2+,

A0 = 2 (r2 - r1) {1 + er1+r

2 } - {(r2 - r1) + 2} {e r1+ + er2-} - {er

1- + er

2+}

(r2 – r1) - 2),

,)1(2

)1(4})1{())1(( 222

1

2

���� � �

��

��� ffr

,)1(2

)1(4})1{())1(( 222

2

2

���� � ��

��

��� ffr

fbbd

faadbarrr

�

����� ���

��� 21323 ,,1,1,1 � ,

c3, c4, & c5 satisfy the following equations.

35

(1-er1) c3 + (1er2) c4 + (1-e r3) c5 = X1 – X2 , r1 c3 + r2 c4 + r3 c5 = X3 , r1 er1

c3 + r2 er2 c4 + r3 er3 c5 = X4 ,

X1 = 1+ d1c1 + d2c2 ; X2 = d1c1 e + d2 c2 e-, X3 = (d1c1 – d2c2) ; X4 =

(d1c1 e - d2c2 e-),

c3, c4, c5 & c6 satisfy the following equations

c6 + 53

5

2

4

1

3

)1()1()1(X

rc

rc

rc

� �

�

�

, c6

63

35

2

24

1

13

1

)1()1()1(X

rec

rec

rec

errr

� �

�

�

��

,

73

35

2

24

1

136

)1()1()1(X

rrc

rrc

rrcc

� �

�

�

�

� ,

83

335

2

224

1

113

16

)1()1()1(X

rerc

rerc

rerc

ec rrr

� �

�

�

�

� ��

,

X5 = bcd

acd 2211 , X6 = b

ecdaecd �

2211 , X7 = bcd

acd 2211 ,

X8 = becde

acd

�

� 2211 ,

L, M, N, R satisfies the following equations

L + M + N = X 9, L er1 + Mer2 + Ner3 = X10,

11

)1(

3

3

2

2

1

1

Re111

XrNe

rMe

rLe rrr

� �

�

�

��

, 12321 111

XRrN

rM

rL

� �

�

�

,

where X9 = -(K1+K2+K3+K4+K5+K6+K7+K8+K9+K10+K11+K12+

K13+K14+K15+K16+K17+K18)

X10 = - [K1 e(m1

+r1) + K2 e(r

2+m

1) + K3 e(m

1+r

3) + K4 e(m

1+) + K5 e(m

1-) + K6

e(m2

+r1) + K7 e(m

2+ r

2) K8 e(m

2+ r

3) +

K9 e(m2

+) + K10 e(m2

+) + )1(

14

2

13

)1(

12

)1(

111

21 �

����

��

�

r�

mmeKeKeKeK

])1(

18

)1(

17

)1(

16

)1(

1532 �

� ��

���

���

��

eKeKeKeK

rr

R1, R2, R3, R4 satisfy the following equations.

R1 + R2 + R3 = X13, R1es1 + R2 e

s2 + R3e

s3 = X 14

36

1513

31

12

21

11

114 X

bsRb

bsRb

bsRbR �

, 16

13

31

12

21

11

114

3211 X

bseRb

bseRb

bseRbeR

sssb �

�

2

)Pr32(4)Pr()Pr( 1

211

1

bfbbs

���� �

� �� ,

2

)Pr32(4)Pr()Pr( 1

211

2

bfbbs

���� �

���� ,

s3 = (�Pr-b1) –s1 – s2.

Now, after knowing the velocity field, the skin-friction

components Tx and Tz in the main flow and transverse directions are give by

zdydu

dydu

UTxdT

yyx �

�cos*.

0

11

0

0

�����

����

� ��

�

����

��� , (2.49)

= (Am1+Bm2) + � [Lr1+Mr2 + Nr3 + K1 (m1+r1) + K2 (m1+r2) + K3 (m1+r3) +K4

(m1+) + K5 (m1-) + K6 (m2 + r1) + K7 (m2+r2) + K8 (m2 + r3) + K9 (m2+) +

K10 (m2-)

+ )1(2)1()1( 11413

212111 ��

��

��

�� rKKmKmK

)1()1()1( 17316215 ��

��

�� KrKrK

zK cos)]1(18 � � , (2.50)

zdydw

VTzdTz

y

��

sin*.

0

11

0 ����

����

���

= - � (C3 r12 + c4 r2

2 +C5r23- 2D - 2E) Sin z, (2.51)

From the temperature field, the heat transfer co-efficient interms of the Nusselt

number can be obtained as

zdyd

dyd

TTkqwdNu

yy

�

cos)(

*.

0

11

0

0

01 �����

����

� ��

�

����

��

�� , (2.52)

= (Am1+Bm2) + � [{R1s1 + R2s2 + R3s3 + t1 (m1+r1) + t2 (m1+r2) + t3 (m1+r3) +

t4 (m1+)

37

+ t5 (m1-) + t6 (m2+r1) + t7(m2+r2) + t8 (m2+r3) + t9 (m2+) + t10(m2-) + t11

(b1+ )1�

-

t12 (r1-b1) – t13 (r2-b1) - t14 (r3-b1) +

.cos)]1()1()()( 218117116115 zmtmtbtbt �

� �

� � (2.53)

From expressions (2.50) and (2.53) for skin friction and heat transfer

coefficient, it can be shown easily that Nu = Tx for Pr = 1. This means that

the Reynolds analogy holds when Pr=1.

2.6 RESULTS AND DISCUSSION

2.6.1 Main flow velocity profiles of the fluid phase

From figure 2.2 we see that, irrespective of any value of �, the

velocity profiles decrease with an increase in the mass concentration of the

dust particles. Also, the profiles decrease with an increase in the injection

parameter for any value of the mass concentration of the dust particles. The

velocity profiles maintain an increasing trend near the lower plate and attain

their maximum value very near the lower plate and thereafter they decrease

steadily and reach the value one at the other plate.

2.6.2 Main flow velocity profiles of the particle phase

From figure 2.3 we conclude that the velocity profiles of the dust

particles decrease with an increase of either the mass concentration of the dust

particles (or) injection parameter. The velocity profiles increase steadily near

the lower plate and reach the maximum value a little away from the lower

plate and thereafter decrease steadily and reach the value one at the other

plate.

38

Figure 2.2 Main flow velocity profiles of the fluid with z = 0, �= 0.15 and for

various values of mass concentration parameter (f) and injection / suction parameter �.

Figure 2.3 Main flow velocity profiles of the dust particles with z = 0, �=

0.15 and for various values of mass concentration parameter (f) and injection / suction parameter �.

39

Both the fluid and dust particles behave in the same manner. But

the profiles of the dust are at a lower height as compared with the fluid.

In the case of clean fluid, the velocity profiles increase steadily

where as in the case of the dusty fluid the profiles increase steadily near the

lower plate and a decreasing trend is seen as they approach the other plate and

attain the value one at the other plate. This phenomenon can be attributed to

the presence of dust. So, we can say that the presence of dust has an influence

of accelerating the motion of the fluid.

2.6.3 Cross flow velocity profiles of the fluid phase

The cross flow velocity component w1 is due to the transverse

sinusoidal injection velocity distribution applied through the porous plate at

rest. This secondary flow component is shown in figure 2.4 and figure 2.5. It

is interesting to note from figure 2.4 and figure 2.7 that w1 the cross flow

velocity, increases with an increase in the concentration of the dust particles at

a point which is located a little away from the mid way between the two plates

and thereafter reverses its trend.

40

Figure 2.4 Cross flow velocity profiles of the fluid with z = 0.5, �= 0.15, injection / suction parameter � = 0.2, and various values of mass concentration

parameter (f).

Figure 2.5 Cross flow velocity profiles of the fluid with z = 0.5, �= 0.15,

mass concentration parameter f = 0.2, and various values of injection / suction parameter �.

41

All the profiles increase steadily near the lower plate and reach the

maximum value at a point a little away from the lower plate and thereafter a

reverse trend is followed. From figure 2.4 it is clear that the profiles increase

with an increase in the injection parameter up to a point which is located at a

distance little away from the mid way between the two plates and thereafter

the profiles reverse their trend. All the profiles increase steadily near the

lower plate and reach the maximum value very near the lower plate and

decrease steadily and become zero at the other plate. This is due to the fact

that there is injection at the stationary plate and suction at the plate, which is in

motion. These two are exactly opposite processes.

2.6.4 Cross flow velocity profiles of the particle phase:-

Figure 2.6 Cross flow velocity profiles of the dust particles with z = 0.5, �=

0.15, f = 0.2 and various values of injection / suction parameter �.

From figure 2.6, irrespective of any value of �, the velocity

profiles of the dust particles in the cross flow direction increase with an

increase in the mass concentration of the dust particles up to a point which is

located a little away from the mid way between the 2 plates and thereafter a

42

reverse trend is seen. All the profiles increase steadily near the lower plate

and reach the maximum value very near the lower plate and decrease steadily

and become zero at the other plate.

Figure 2.7 Cross flow velocity profiles of the dust particles with z = 0.5, �=

0.15, injection / suction parameter � = 0.2, and various values of mass concentration parameter f.

From figure 2.4 and figure 2.7 it is noted that the cross flow

velocity profiles of the fluid and dust behave in the same way. It is interesting

to point out from them that the cross flow velocity profiles of both fluid and

dust increase with an increase in mass concentration of the dust particles at a

point which is located a little away from the mid way between the two plates

and thereafter they reverse their trend. Profiles increase steady near the lower

plate and reach the maximum value at a point little away from the lower plate

and thereafter a reverse trend is followed. The profiles of the dust are at

greater height than the fluid.

43

Figure 2.8 Main flow velocity profiles of the fluid with z = 0, f = 0.2, injection

/ suction parameter � = 0.5 and for various values of the relaxation time parameter �

Figure 2.9 Main flow velocity profiles of the dust particles z = 0, f = 0.2,

injection / suction parameter � = 0.5 and for various values of the relaxation time parameter �

44

From figure 2.8 and figure 2.9 it is clear that both main flow

velocity profiles of fluid (figure 2.8) and dust (figure 2.9) decrease when there

is an increase in the relaxation time parameter.

Figure 2.10 Cross flow velocity profiles of the fluid for z = 0.5, f = 0.2,

injection / suction parameter � = 0.5 and for various values of the relaxation time parameter �

Figure 2.11 Cross flow velocity profiles of the dust particles for z = 0.5, f = 0.2, injection / suction parameter � = 0.5 and various values of the relaxation

time parameter �

45

From figure 2.10 and figure 2.11, it is clear that the cross flow

velocity profiles of the fluid and dust decrease when the relaxation time

increases. All the profiles of the fluid and dust increase steadily very near the

lower plate thereafter they change their trend and become zero at the other

plate.

Table – 2.1: Variations of Tx , Tz and Nu for Pr = 0.7 and Pr = 7.0 with

various values of � (injection parameter) f = 0.4, � = 0.2

� Tx Tz Nu for Pr=0.7 Nu for Pr=7.00.2 0.4 0.6

5.8771 5.8905 5.9047

0.9657 1.0163 1.0446

9.6605 9.6586 9.6568

1.4321 1.1387 0.7801

f = 0.5, � = 0.2 0.2 0.4 0.6

5.8900 5.9157 5.9417

0.9897 1.0347 1.0602

9.6939 9.6916 9.6893

1.5610 1.3239 1.1290

f = 0.6, � = 0.20.2 0.4 0.6

5.9028 5.9408 5.9783

1.0098 1.0505 1.0738

9.7271 9.7243 9.7216

1.6526 1.4255 1.2651

Table – 2.2: Variations of the skin friction Tx, Tz and Nu for Pr=0.7 and Pr=7.0 with various values of � (relaxation time)

f = 0.2, � = 0.5� Tx Tz Nu for Pr=0.7 Nu for Pr=7.0

0.10 0.11 0.12

10.1276 9.2472 8.5301

1.5290 1.4235 1.3367

19.1161 17.3844 15.9412

3.0722 2.9277 2.8048

f = 0.4, � = 0.50.10 0.11 0.12

10.2186 9.3361 8.6166

1.5840 1.4765 1.3880

19.1841 17.4522 16.0089

2.2546 2.0452 1.8673

f = 0.6, � = 0.5 0.10 0.11 0.12

10.3081 9.4236 8.7016

1.6241 1.5148 1.4250

19.2517 17.5195 16.0759

2.5167 2.3082 2.1337

46

2.6.5 Skin friction Tx and Tz

The variations of the skin friction components Tx and Tz in the

main flow and transverse directions and the Nusselt number Nu are shown in

Table 2.1 and Table 2.2.

For clean fluid, Singh (1999) showed that the skin friction Tx

decreases with the increase of the injection parameter �.

From the above table 2.1 and table 2.2, we conclude that, for a

given value of the mass concentration and relaxation time of the dust particles,

the skin friction Tx increases with an increase in the injection parameter.

Also, for a given value of the injection parameter, the skin friction Tx

increases with an increase in the mass concentration of the dust particles

whereas a reverse trend is seen when the relaxation time of the dust particle

increases. We find an appreciable increase in Tx value even for a slight

increase in the relaxation time of the dust particles.

The skin friction component in transverse direction Tz increases

with an increase in the injection parameter for fixed values of the mass

concentration and relaxation time of the dust particles. Also, for a given value

of the injection parameter, the skin friction Tz increases with an increase in the

mass concentration of the dust particles. Tz decreases with an increase in the

relaxation time of the dust particles whereas for the clean fluid, the skin

friction Tz decreases with an increase of the injection parameter.

2.6.6 Nusselt number

From table 2.1 and table 2.2 we see that the Nusselt number Nu

for both air (Pr=0.7) and water (Pr=7.0) decrease with an increase in the

injection parameter for a given value of the mass concentration of the dust

particles whereas in the case of air and water it increases with an increase of

47

the mass concentration of the dust particles for any value of the injection

parameter. Nu for both air and water decreases with an increase in the

relaxation time of the dust particles for a fixed value of the mass concentration

and injection parameters. It is to be noted that, for clean fluid, the heat

transfer coefficient Nu decreases with increase of the injection parameter � for

both air and water. It is also noted that the heat transfer coefficient is much

lower in the case of water (Pr=7.0) than in the case of air (Pr=0.7) for both

dusty fluid and clean fluid.

When the concentration parameter of the dust particles is neglected it is

found that our results are in perfect agreement with the results obtained

by Singh (1999) for clean fluid.

2.7 CONCLUSION

1. The effect of mass concentration parameter on u, up are similar. But

this effect is opposite to Tx, Tz and Nu.

2. The effect of injection parameter on u, up and Nu are same. But this

effect is opposite to Tx and Tz.

3. The effect of relaxation time parameter on Tx, Tz, and Nu are same.