Laminar flow of couple stress fluids for Vogel's model

Scientific Research and Essays Vol. 7(33), pp. 2936-2961, 23 August, 2012 Available online at http://www.academicjournals.org/SRE DOI: 10.5897/SRE11.1822 ISSN 1992-2248 ©2012 Academic Journals

Full Length Research Paper

Laminar flow of couple stress fluids for Vogel's model

M. Farooq1, S. Islam2*, M. T. Rahim1 and A. M. Siddiqui3

1Department of Mathematics, National University of Computer and Emerging Sciences, Peshawar, Pakistan.

2Department of Mathematics, Abdul Wali Khan University, Mardan, Khyber Pakhtunkhwa, Pakistan.

3Department of Mathematics, Pennsylvania State University, York Campus, 1031 Edgecomb Avenue, York, PA 17403,

USA.

Accepted 7 June, 2012

The coupled nonlinear equations for heat transfer flow of variable viscosity couple stress fluids between two parallel plates are derived for four different problems, namely plane Couette flow, plug flow, plane Poiseuille flow and generalized plane Couette flow. These equations are made dimensionless with the help of non-dimensional parameters and solved by using regular perturbation technique. The effect of various emerging parameters embedded in the problem is discussed graphically. Key words: Couple stress fluids, vogel's viscosity model, perturbation technique, heat transfer.

INTRODUCTION Theoretical research on the flow of non-Newtonian fluids has got substantial attention because of their applications in the process industry (Harris, 1977; Rajagopal, 1982; Erdogan, 1981; Fetecau and Fetecau, 2002, 2005; Tan and Xu, 2002; Tan and Masuoka, 2005; Chen et al., 2004). The non-Newtonian fluids cannot fit into a single constitutive model because of their complexity and hence several constitutive models have been suggested for different categories of these fluids. The flow behavior of such fluids cannot be properly explained on the basis of the classical linearly viscous model. Several constitutive equations that have been proposed try to characterize the deviation of relevant non-Newtonian behavior from the classical theory.

Among the many models which have been used to describe the non-Newtonian behavior showed by certain fluids, the couple stress fluids have received considerable attention (Rajagopal and Na, 1983; Asghar et al., 2003; Erdogan, 1975; Siddiqui et al., 2005, 2006; Stokes, 1966). They represent those fluids which consist of rigid and randomly oriented particles suspended in a viscous

*Corresponding author. E-mail: [email protected]. Tel: +92-333-9844540. PACS: 44.15.+a, 45.10.Hj, 47.50.-d.

medium. In these fluids, the stress tensor is antisymmetric, so their exact flow behavior cannot be predicted by the classical Newtonian theory. Therefore various micro-continuum models and theories were offered (Ariman and Sylvester, 1973; Massoudi and Christie, 1995). To include the effect of the couple stresses, Stokes (1966) generalized the classical model. This model has been extensively used for its relative mathematical simplicity as compared with other models developed for the fluids under consideration.

The study of heat transfer flow has importance in various engineering applications, examples include for drag reduction and thermal recovery of oil, the design of thrust bearings and radial diffusers transpiration cooling. Heat transfer plays an important role in handling and processing of non-Newtonian mixtures (Tsai et al., 1988; Makinde, 2008; Chinyoka and Makinde, 2010).

This paper focuses on the study of flow of couple stress fluids with temperature dependent viscosity between two parallel plates, as in the study of El-Dabe and El-

Mohandis (1995), kept at two different temperatures 0

and 1 . Vogel's model was used for the temperature

dependent viscosity. Four different flow problems, that is, plane Couette flow, plug flow, plane Poiseuille flow and the generalized plane Couette flow are investigated. Perturbation solutions are obtained for the coupled non-linear ordinary differential equations and are discussed

graphically. Basic equations The basic equations governing the flow of a couple stress fluid given in Cartesian tensor notation are (Stokes, 1966; Ariman and Sylvester, 1973; Massoudi and Christie, 1995; Islam et al., 2009; El-Dabe and El-Mohandis, 1995; El-Dabe et al., 2003; Aksoy and Pakdemirli, 2010): Continuity equation

0,=,rrv (1)

Cauchy's first law of motion

,= , irrii fa (2)

Cauchy's second law of motion

0,=, rsirsirri elm (3)

Energy equation

,= )( hqkmD rrrs

D

rsrsrs (4)

where is the constant density, iv are the velocity

components, ia are the components of acceleration, ji

is the second order stress tensor, if is the body force

vector per unit mass, jim is the second order couple

stress tensor, il is the body moment per unit mass, h is

an energy source density per unit mass, jiD is the rate

of deformation tensor which is the symmetric part of the velocity gradient and is given by:

),(2

1= ,, ijjiji uuD (5)

where Kji

is the curvature twist rate tensor defined to be

the gradient of the vorticity field, is the thermal energy

given by c= , c is the specific heat of the fluid,

assumed to be constant, is the absolute temperature,

iq is the influx of energy per unit area given by

ri kq ,= , when only thermal flux of energy is

considered. The superimposed "." denotes the material

derivative and ijse is the third order alternating pseudo

tensor, which is defined as:

Farooq et al. 2937

.equalare,,indicestheofmoreortwoif0,

3,2,1ofnpermutatiooddanis,j,iif1,

3,2,1ofnpermutatioeven an is,j,iif1,

=

sji

s

s

e sji

Also

,44= '

jiij

D

ij kkm

(6)

,2= ijijrrij

S

ij DDp (7)

where ,, and ' are material constants and

.jif0,

=if1,=

i

jiij

.

Formulation of the problems and their solutions Plane Couette flow Assuming that the couple stress fluid is flowing between

two infinite parallel plates which are d2 apart. And the

upper plate is moving steadily with a constant velocity

U . We take the origin of Cartesian coordinates to be on

the plane of symmetry of the flow. Both the lower and

upper plates are placed in the plane at dy = and

dy = and their temperature is maintained at 0 and

1 respectively. For the steady one dimensional flow of

an incompressible fluid, we let:

.=,=0,=0,=,= 321 yyvvyuv

The continuity Equation 1 is satisfied identically. In the absence of body forces, body moment and pressure gradient, the Equations of motion (Equations 2 and 3) and the energy Equation (Equation 4) becomes:

(3.1)0,=2

2

4

4

dy

du

dy

d

dy

ud

dy

ud

(8)

(3.2)0.=

2

2

22

2

2

dy

ud

kdy

du

kdy

d

(9)

The corresponding boundary conditions are:

(10)0,=''='',=0,= duduUdudu (10)

(11).=,= 10 dd (11)

2938 Sci. Res. Essays Let us introduce the following non-dimensional parameters:

.)(

=,=,=,=,=,=01

2

00

0

*

01

0***

k

UdB

d

yy

U

uu

Then Equations 8, 9, 10 and 11 is reduced to the following dimensionless form, by omitting the asterisks:

(3.5)0,=2

2

22

4

4

dy

du

dy

dB

dy

udB

dy

ud

(12)

(3.6)0,=

2

2

2

2

2

2

2

dy

ud

Bdy

du

dy

d

(13)

14)(0,=1''0,=1''1,=10,=1 uuuu (14)

15)(1.=10,=1 (15) The Vogel's viscosity model (Massoudi and Christie, 1995; Makinde, 2007) in the non-dimensional form is:

(3.9).exp=0

0*

B

A

(16) Applying Taylor series expansion, we get

(3.10),1=2

0

02

B

A

(17)

where

0

0*

2 exp=B

A and 00 , BA are viscosity

parameters related to Vogel's model. Let dA =0 ,

where is a small parameter. Using perturbation

technique the approximate velocity and temperature profiles are:

(3.11)= 2

2

10 uuuu (18)

(3.12)= 2

2

10 (19)

Substituting Equations 17, 18 and 19 into Equations 12, 13 14 and 15, and separating terms at each order of

yielding finally

Order 0

0,='''''' 0

22

0 uBu

0,=)''()'('' 2

02

2

0

2

0 uB

u

1.=(1)0,=10,=1''=1''1,=10,=1 000000 uuuu

Order 1

0,=)''''('''''' 00002

0

22

1

22

1 uuB

dBuBu

0,=''''2

)'(''2'' 102

2

002

0

2

10

2

1 uuB

uB

duu

0.=(1)=1)(0,=1''=1''0,=1=1 111111 uuuu

Order 2

0,=''''''''('''''' 100101102

0

22

2

22

2

uuuu

B

dBuBu

0,=''''2'''''2''2''' 20

2

1

22

011002

0

2

20

2

1

2

2 uuuB

uuuB

duuu

0,=(1)=1)(0,=1''=1''0,=1=1 222222 uuuu where primes denote derivative with respect to y.

0O rder solutions satisfying the boundary conditions

are

(20),2

1=0

yyu

(20)

(21).82

1= 2

2

10 yyLy

(21)

Substituting these solutions to order 1 equations, we

obtained: (22),coshsinh= 65

3

4

2

3211 ByKByKyKyKyKKyu (22) (23),coshsinh= 87

4

6

3

5

2

4321 ByLByLyLyLyLyLLy (23)

Using these solutions in order 2 equations, we get

:

,cosh

sinh=)(

3

20

2

191817

3

16

2

151413

5

12

4

11

3

10

2

9872

ByyKyKyKK

ByyKyKyKKyKyKyKyKyKKyu

(24)

Farooq et al. 2939

.2cosh2sinhcosh

sinh=)(

2524

3

23

2

222120

3

19

2

181716

6

15

5

14

4

13

3

12

2

111092

ByLByLByyLyLyLL

ByyLyLyLLyLyLyLyLyLyLLy

(25)

Using Equations 20 to 25 in Equations 18 and 19, we have:

,cosh

sinh

coshsinh2

1=)(

3

20

2

191817

3

16

2

151413

5

12

4

11

3

10

2

987

2

65

3

4

2

321

ByyKyKyKK

ByyKyKyKKyKyKyKyKyKK

ByKByKyKyKyKKy

yu

(26)

.2cosh2sinhcosh

sinh

coshsinh82

1=)(

2524

3

23

2

222120

3

19

2

181716

6

15

5

14

4

13

3

12

2

11109

2

87

4

6

3

5

2

432

22

1

ByLByLByyLyLyLL

ByyLyLyLLyLyLyLyLyLyLL

ByLByLyLyLyLyLLyyLy

(27)

The constants used in this solution are given in Appendix A.

Plug flow Here, we assume that the flow is because of the motion of both the plates which move with the same velocity U and pressure gradient is absent. Rest of the assumptions and conditions on the velocity and temperature fields remain the same. Again the governing equations are Equations 12 and 13 with boundary conditions as follows: (3.21),0=1''0,=1''1,=11,=1 uuuu (28) (3.22)1.=10,=1 (29) Making use of Equations 17 to 19 in Equations (12), (13) and (28), (29), and then separating at each order of approximation gives:

Order 0

0,='''''' 0

22

0 uBu

0,=)''()'('' 2

02

2

0

2

0 uB

u

1.=(1)0,=1)(0,=(1)''=1)(''1,=(1)1,=1)( 000000 uuuu

Order 1

0,='''''''''' 00002

0

22

1

22

1 uuB

dBuBu

0,=''''2

)'('')(2'' 102

2

002

0

2

10

2

1 uuB

uB

duu

0.=(1)=1)(0,=(1)''=1)(''0,=(1)=1)( 111111 uuuu

Order 2

0,=)''''''''('''''' 100101102

0

22

2

22

2

uuuu

B

dBuBu

0,=''''2'''''2''2'' 20

2

12

2

011002

0

2

20

2

1

2

2 uuuB

uuuB

duuu

0.=(1)=1)(0,=(1)''=1)(''0,=(1)=1)( 222222 uuuu

0O rder solutions are

3.23)(1,=0 yu (30)

(3.24).2

1=0

yy

(31)

Substituting these solutions in order 1 equations and

applying boundary conditions one obtains:

3.25)(0,=1 yu (32)

3.26)(0.=1 y (33)

2940 Sci. Res. Essays

Using these solutions in order 2 equations and applying

boundary conditions, we get: 3.27)(0,=2 yu (34) 3.28)(0.=2 y (35) Putting Equations 30 to 35 into Equations 18 and 19, we have

3.29)(1,=yu (36)

(3.30).

2

1=y

y

(37) Fully developed plane Poiseuille flow Let the couple stress fluid be flowing between two infinite parallel plates which are placed at a distance 2d from each other. Let both the plates are stationary and motion of the fluid is due to the external pressure gradient. All other conditions and assumptions remain the same. In this case Equations 2, 3 and 4 reduce to

3.31)(0,=2

2

4

4

x

p

dy

du

dy

d

dy

ud

dy

ud

(38)

(3.32),=0=

z

p

y

p

(39)

3.33)(0.=

2

2

22

2

2

dy

ud

kdy

du

kdy

d

(40)

Equation 39, gives Adx

dp= , where A is considered to be a

negative constant. Then;

3.34)(0,=2

2

4

4

Ady

du

dy

d

dy

ud

dy

ud

(41)

3.35)(0.=

2

2

22

2

2

dy

ud

kdy

du

kdy

d

(42) Taking x-axis in the midway between both the plates so that the boundary conditions are

3.36)(0,=''=''0,=0,= dudududu (43)

(3.37).=,= 10 dd (44)

We introduced the following non-dimensional variables and quantities:

.=,)(

=,=,=,=,=,=4

*

01

2

00

0

*

01

0***

U

AdA

k

UdB

d

yy

U

uu

By dropping the `*' for convenience, Equations 41 to 44 become

3.38)(0,=2

2

22

4

4

Ady

du

dy

dB

dy

udB

dy

ud

(45)

3.39)(0,=)()( 2

2

2

2

2

2

2

dy

ud

Bdy

du

dy

d

(46) 3.40)(0,=1''0,=1''0,=10,=1 uuuu (47) 3.41)(1.=10,=1 (48) Inserting Equations 17 to 19 into Equations 45 to 48, and separating at each order of approximation yields:

Order 0

0,='''''' 0

22

0 AuBu

0,=)''()'('' 2

02

2

0

2

0 uB

u

1.=(1)0,=1)(0,=(1)''=1)(''0,=(1)=1)( 000000 uuuu

Order 1

0,=)''''('''''' 00002

0

22

1

22

1 uuB

dBuBu

0,=''''2

'''2'' 102

2

002

0

2

10

2

1 uuB

uB

duu

0.=(1)=1)(0,=(1)''=1)(''0,=(1)=1)( 111111 uuuu

Order 0O rder

solutions, applying the boundary conditions are:

,cos= 3

2

210 ByMyMMyu

(49)

Farooq et al. 2941

.2coshscosh

2

1= 654

4

3

2

210 ByNByinhyNByNyNyNyNy (50)

Substituting these solutions to order 1 equations we

obtain:

,3cosh2cosh

coshs

=)(

2019

4

18

2

171615

5

14

3

13

2

121110

6

9

4

8

3

7

2

6541

ByMByM

ByyMyMyMMByinhyM

yMyMyMMyMyMyMyMyMMyu

(51)

).(4)3(c)(3i)2(c)

()2(s)(

)()()()

(=)(

373635

4

34

2

333231

5

30

3

29

2

282726

6

25

4

24

3

23

2

222120

5

19

3

18

2

171615

8

14

6

13

5

12

4

11

3

10

2

9871

BycoshNByoshNBynhysNByoshyN

yNyNNByinhyNyNyNyNN

BycoshyNyNyNyNyNNBysinhyNyN

yNyNNyNyNyNyNyNyNyNNy

(52)

Substituting Equations 49 to 52 into Equations 18 and 19, we obtain:

,3cosh2coshcosh

s

cos=)(

2019

4

18

2

171615

5

14

3

13

2

121110

6

9

4

8

3

7

2

6543

2

21

ByMByMByyM

yMyMMByinhyMyMyMyMM

yMyMyMyMyMMByMyMMyu

(53)

)].(4)(3)(3)(2)

()(2)(

)()()()

([

2coshscosh2

1=)(

373635

4

34

2

333231

5

30

3

29

2

282726

6

25

4

24

3

23

2

222120

5

19

3

18

2

171615

8

14

6

13

5

12

4

11

3

10

2

987

654

4

3

2

21

BycoshNBycoshNByysinhNBycoshyN

yNyNNBysinhyNyNyNyNN

BycoshyNyNyNyNyNNBysinhyNyN

yNyNNyNyNyNyNyNyNyNN

ByNByinhyNByNyNyNyNy

(54)

The constants used in this solution are given in Appendix B.

Generalized plane Couette flow

Finally we suppose that the motion of the fluid is maintained due to the constant pressure gradient and the movement of the upper plate. Assume that the upper plate is moving with a constant velocity U. All the other conditions and assumptions on the velocity and temperature fields are the same. In this case the governing Equations are 46 and 47 with the corresponding boundary conditions

55)(0,=1''0,=1''1,=10,=1 uuuu

(55)

(55)

56)(1.=10,=1 (56)

( Equations 17, 18 and 19 are substituted in Equations 46,

47, 55 and 56, and terms are separated at each order of yielding finally:

Order 0

0,='''''' 0

22

0 AuBu

0,=)''()'('' 2

02

2

0

2

0 uB

u

1.=(1)0,=1)(0,=(1)''0,=1)(''1,=(1)0,=1)( 000000 uuuu

Order 1

0,=)''''('''''' 00002

0

22

1

22

1 uuB

dBuBu

0,=''''2

'''2'' 102

2

002

0

2

10

2

1 uuB

uB

duu


0.=(1)=1)(0,=(1)''=1)(''0,=(1)=1)( 111111 uuuu

0Order solutions satisfying the given boundary

conditions are

,cosh2

1= 3

2

210 ByEyEyEyu (57)

(57)

)58(.2coshcoshsinh= 9876

4

5

3

4

2

3210 ByFByFByyFFyFyFyFyFFy (58)

Substituting this solution in order 1 equation, we have:

),(3c)(2c)(2s)(

)(c)()(s)

(=)(

25242322

4

21

3

20

2

191817

5

16

4

15

3

14

2

131211

6

10

5

9

4

8

3

7

2

6541

ByoshEBoshEByinhyEE

ByoshyEyEyEyEEByinhyEyE

yEyEyEEyEyEyEyEyEyEEyu

(59)

).(4c)(3c

)(3s)()(2c)(

)(2s)()(c)

()(s)

(=)(

4645

4443

4

42

3

41

2

403938

5

37

4

36

3

35

2

343332

6

31

5

30

4

29

3

28

2

272625

5

24

4

23

3

22

2

21

2019

8

18

7

17

6

16

5

15

4

14

3

13

2

1211101

ByoshFByoshF

ByinhyFFByoshyFyFyFyFF

ByinhyFyFyFyFyFFByoshyFyF

yFyFyFyFFByinhyFyFyFyF

yFFyFyFyFyFyFyFyFyFFy

(60)

Making use of Equations 57 to 60 into Equations 18 and 19, we get

,3c2c2sc

s

cosh2

1=)(

25242322

4

21

3

20

2

191817

5

16

4

15

3

14

2

131211

6

10

5

9

4

8

3

7

2

6543

2

21

ByoshEByoshEByinhyEEByoshyE

yEyEyEEByinhyEyEyEyEyEE

yEyEyEyEyEyEEByEyEyEyu

(61)

)].(4c)(3c)(3s)(

)(2c)()(2s)

()(c)

()(s)(

[)2(c

)(c)(s)(=)(

46454443

4

42

3

41

2

403938

5

37

4

36

3

35

2

343332

6

31

5

30

4

29

3

28

2

27

2625

5

24

4

23

3

22

2

212019

8

18

7

17

6

16

5

15

4

14

3

13

2

1211109

876

4

5

3

4

2

321

ByoshFByoshFByinhyFF

ByoshyFyFyFyFFByinhyFyF

yFyFyFFByoshyFyFyFyFyF

yFFByinhyFyFyFyFyFFyF

yFyFyFyFyFyFyFFByoshF

ByoshFByinhyFFyFyFyFyFFy

(62)

The constants involving in this solution are given in Appendix C.

DISCUSSION

The effect of the properties of various parameters

especially A, B and on the velocity field and

temperature distribution is illustrated graphically through Figures 1 to 12. Figure 1 shows a plot of u against y for the plane Couette flow. Figure 2 shows that the fluid

temperature increases with increase in the value of

for the plane Couette flow case. Figures 3 and 4 are plotted for the velocity profile in the case of plane Poiseuille flow which illustrate that there is an inverse relation between the velocity u and parameters B and A, respectively. Figures 5 to 7 are plotted to show the

behavior of the temperature in the plane Poiseuille flow

while varying the parameters B, A and respectively. In

Figures 5 and 7, it is found that the phase of velocity decreases with increase in B and A while it increases with decrease in both B and A, respectively. The

temperature of the fluid increases as the value of

increases (Figure 6). The impact of parameters B and A on the velocity profile in the case of generalized Couette flow is demonstrated in Figures 8 and 9, respectively. In Figure 8, it is noted that the velocity decreases as B increases. It is clear from Figure 9 that as the value of A increases, the velocity profile also increases and vice versa. Figures 10 to 12 are plotted for the temperature distribution of the generalized Couette flow. From Figures 10 and 12, we show that the temperature increases with decreasing the parameters B and A, respectively.

Farooq et al. 2943

1.0 0.5 0.0 0.5 1.0

0.0

0.2

0.4

0.6

0.8

1.0

y

uy

-1.0 -0.5 0.0 0.5 1.0

u(y

)

Figure 1. Velocity profile for plane Couette for fixed values

1.0 0.5 0.0 0.5 1.0

0.0

0.5

1.0

1.5

2.0

2.5

y

y

= 4 = 3 = 2

= 1

-1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0

y

Θ(y)

Figure 2. Effect of λ on Θ(y) for plane Couette flow keeping .

aa

1.0 0.5 0.0 0.5 1.0

0.00

0.01

0.02

0.03

0.04

0.05

0.06

y

uy

B = 0.8B = 1.0

B = 1.2 B = 1.4

-1.0 -0.5 0.0 0.5 1.0

y

-1.0 -0.5 0.0 0.5 1.0

y

u(y

)

Figure 3. Effect of B on u(y) for plane Poiseuille flow keeping


1.0 0.5 0.0 0.5 1.0

0.00

0.05

0.10

0.15

y

uy

A = -4 A = -3 A = -2 A = -1

-1.0 -0.5 0.0 0.5 1.0

y

-1.0 -0.5 0.0 0.5 1.0

y

u(y

)

Figure 4. Effect of A on u(y) for plane Poiseuille flow keeping

1.0 0.5 0.0 0.5 1.0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

y

y

B-1.4B=1.2

B=0.8 B=1.0

-1.0 -0.5 0.0 0.5 1.0

y

Θ

(y)

Figure 5. Effect of B on Θ(y) for plane Poiseuille flow when

1.0 0.5 0.0 0.5 1.0

0.0

0.2

0.4

0.6

0.8

1.0

y

y

= 1 = 2

= 4 = 3

-1.0 -0.5 0.0 0.5 1.0

y

Θ

(y)

Figure 6. Effect of on Θ(y) for plane Poiseuille flow when

Farooq et al. 2945

1.0 0.5 0.0 0.5 1.0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

y

y

A = -1

A = -2

A = -3A = -4

-1.0 -0.5 0.0 0.5 1.0

y

Θ

(y)

Figure 7. Effect of A on Θ(y) for plane Poiseuille flow when

1.0 0.5 0.0 0.5 1.0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

y

uy

B = 1.5B = 1.0B = 0.5

B = 0.1

-1.0 -0.5 0.0 0.5 1.0

y

u

(y)

Figure 8. Effect of B on u(y) for generalized plane Couette flow keeping

1.0 0.5 0.0 0.5 1.0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

y

uy

A = -1A = -2

A = -4

A = -3

-1.0 -0.5 0.0 0.5 1.0

y

u

(y)

Figure 9. Effect of A on u(y) for generalized plane Couette flow keeping


1.0 0.5 0.0 0.5 1.0

0

2

4

6

8

10

12

y

y

B = 1.0

B = 0.8

B = 0.6B = 0.4

-1.0 -0.5 0.0 0.5 1.0

y

Θ(y)

Figure 10. Effect of B on Θ(y) for generalized plane Couette flow when

1.0 0.5 0.0 0.5 1.0

0.0

0.5

1.0

1.5

2.0

2.5

3.0

y

y

= 1 = 2 = 3 = 4

-1.0 -0.5 0.0 0.5 1.0

y

Θ(y)

Figure 11. Effect of on Θ(y) for generalized plane Couette flow

when

1.0 0.5 0.0 0.5 1.0

0

1

2

3

4

5

y

y

A = -1A = -3A = -5A = -7

-1.0 -0.5 0.0 0.5 1.0

y

Θ(y)

Figure 12. Effect of A on Θ(y) for generalized plane Couette flow when

Figure11 illustrates that increasing the values of , the

temperature profile also increases.

Conclusion In this paper, we have studied the flow of couple stress fluids between two parallel plates for Vogel's model. Four different flow problems, that is, plane Couette flow, plug flow, plane Poiseuille flow and generalized plane Couette flow are discussed in detail. The nonlinear coupled differential equations are solved for fluid velocity and temperature in each case by using the perturbation technique. It was found from the graphs that both the velocity field and temperature distribution are strongly

dependent on the non-dimensional parameters A, ,

and B . REFERENCES Aksoy Y, Pakdemirli M (2010). Approximate analytical solutions for flow

of a third-grade fluid through a parallel-plate channel filled with a porous medium. Transp. Porous Med. 83: 75-395.

Ariman TT, Sylvester ND (1973). Microcontinum fluid mechanics. A review. Int. J. Eng. Sci. 11: 905-930.

Asghar S, Mohyuddin MR, Hayat T (2003). Unsteady flow of a third-grade fluid in the case of suction. Math. Comp. Model. 38(1-2):201-208.

Chen CI, Chen CK, Yang YT (2004). Unsteady unidirectional flow of an Oldroyd-B fluid in a circular duct with different given volume flow rate. Int. J. Heat Mass Transfer. 40:203-209.

Chinyoka T, Makinde OD (2010). Computational dynamics of unsteady flow of a variable viscosity reactive fluid in a porous pipe. Mech. Res. Commun. 37:347-353.

El-Dabe NTM, El-Mohandis SMG (1995). Effect of couple on pulsatile hydromagnetic poiseuille flow. Fluid Dyn. Res. 15:313-324.

El-Dabe NTM, Hassan AA, Mohamed MAA (2003). Effect of couple stresses on pulsatile hydromagnetic Poiseuille flow. Z. Naturforsch. 58a:204-210.

Erdogan E (1981). Steady pipe flow of fluid of fourth grade. ZAMM 61:466-469.

Erdogan ME (1975). On the flow of a non-Newtonian fluid past a porous flat plate. Z. Angew. Math. Mech. 55:99-103.

Fetecau C, Fetecau C (2002). The Rayleigh-Stokes problem for heated second grade fluids. Int. J. Non-Linear Mech. 37:1011-1015.

Fetecau C, Fetecau C (2005). Decay of a potential vortex in an Oldroyd-B fluid. Int. J. Non-Linear Mech. 43:340-351.

Harris J (1977). Rheology and non-Newtonian flow. London-New York: Longman.

Islam S, Ishtiaq Ali, Shah A, Ran XJ, Siddiqui AM (2009). Effects of couple stresses on flow of third grade fluid between two parallel plates using Homotopy perturbation method, Int. J. Non-linear Sci. Numeric. Simul. 10(1):99-112.

Farooq et al. 2947 Makinde OD (2007). Hermite-Padé approximation approach to thermal

criticality for a reactive third-grade liquid in a channel with isothermal walls. Int. Commun. Heat Mass Transfer. 34(7):870-877.

Makinde OD (2008). Thermal criticality in viscous reactive flows through channels with a sliding wall: An exploitation of Hermite-Padé approximation method. Math. Comp. Model. 47:312-317.

Massoudi M, Christie I (1995). Effects of variable viscosity and viscous dissipation on the flow of a third grade fluid in a pipe. Int. J. Nonlinear Mech. 30(5):687-699.

Rajagopal KR (1982). A note on unsteady unidirectional flows of a non-Newtonian fluid. Int. J. Non-Linear Mech. 17:369-373.

Rajagopal KR, Na TY (1983). On Stokes problem for a non-Newtonian fluid, Acta Mech. 48:233-239.

Siddiqui AM, Ahmed M, Islam S, Ghori QK (2005). Homotopy analysis of Couette and Poiseuille flows for fourth grade fluids. Acta Mech. 180(1-4):117-132.

Siddiqui AM, Ahmed M, Ghori QK (2006). Couette and Poiseuille flows for non-Newtonian fluids. Int. J. Non-Linear Sci. Numer. Simul. 7(1):15-26.

Stokes VK (1966). Couple Stresses in fluid. The physics of fluids. 9:1709-1715.

Tan WC, Xu MY (2002). The impulsive motion of a flat plate in a generalized second grade fluid. Mech. Res. Comm. 29:3-9.

Tan WC, Masuoka T (2005). Stoke first problem for a second grade fluid in a porous half space with heated boundary. Int. J. Non-Linear Mech. 40:515-522.

Tsai CY, Novack M, Roffle G (1988). Rheological and heat transfer characteristics of flowing coal-water mixtures. Final report. DOE/MC/23255-2763.

2948 Sci. Res. Essays APPENDIX Appendix A

,

8

][Csch=,

48=,

8=,

48

6=,

8

2=

2

0

252

0

2

42

0

32

0

2

22

22

0

22

22

1BB

BdK

B

dK

B

dK

BB

BdK

BB

BdK

,4

][Sech=

2

0

226BB

BdK

,][Cosh824848][Cosh16

1632][Sinh4][Cosh2

][Cosh4][Sinh8][Cosh86

1442448(16

=

7

33

5

44

53

44

3

22

16

44

13

44

13

22

6

53

6

64

6

42

5

44

5

33

4

44

43

64

3

2

2

44

2

22

2

0

447

LBBLBLLBLBLKBB

LKBLKBKBBKBB

KBBKBBKBBKB

KKBKKBKBBB

dK

,][Sinh60

12144020120][Sinh120

120720][Sinh60][Cosh60

][Sinh15][Sinh30][Cosh309

10804024053060

=

8

33

6

44

64

44

4

22

15

44

14

44

14

22

6

33

6

44

5

64

5

42

5

53

4

64

4

2

3

44

3

22

2

64

2

42

2

0

448

LBB

LBLLBLBLKBB

LKBLKBKBBKBB

KBBKBBKBBKB

KKBKBKBKBBB

dK

,641834

= 53

22

13

22

43

2

2

22

2

0

229 LLBLKBKKKBBB

dK

,4842436824

= 64

22

14

22

4

2

3

22

2

42

2

0

2210 LLBLKBKKBKBBB

dK

,4340

=,2616

= 64

2

2

0

12543

2

2

0

11 LKB

dKLKK

B

dK

,][Sinh48][Cosh24192

115296][Sinh96][Cosh48576

][Sinh12][Sinh48][Cosh12][Sinh3

][Sinh24][Cosh2][Cosh3

1448641922496

][Csch=

8

33

8

44

6

22

64

22

15

44

15

55

14

22

6

55

6

33

6

44

5

64

5

42

5

75

5

53

4

42

4

2

3

22

2

42

2

0

4413

LBBLBBLB

LLBLKBBLKBBLKB

KBBKBBKBBKBB

KBBKBBKBB

KBKKBKBBB

BdK

,81671232

= 716

2

652

0

14 BLLKBKBKBB

dK

,48

=,4332

=2

0

6

3

16652

0

15B

KBdKBKK

B

dK

,)][Sinh24][Cosh4814428848

][Sinh48][Cosh96192][Sinh2



4328647214448192

][Sech=

7

44

7

33

5

22

53

22

16

55

16

44

13

22

6

75

6

53

6

64

6

42

5

44

5

55

5

33

4

22

43

42

3

2

2

22

2

0

4417

LBBLBBLBLLB

LKBBLKBBLKBKBB

KBBKBBKBB

KBBKBBKBB

KBKKBKKBBB

BdK

Farooq et al. 2949

,81612732

= 815

2

652

0

18 BLLKBBKKBB

dK

,48

1=,

48=,34

32= 2

12

0

5

3

20652

0

19

LB

KBdKKBK

B

dK

,48][Cosh38496192384

= 15

2

04

2

02

2

0

3

2

0

2 LBdKBBKBBKBBBdBB

L

,48

=,][Sinh481648

1= 12

2

02

0

2

46

2

03

2

02

0

3 dLKBB

LKBBKBBBdBB

L

,=,=,96384

=,1648

= 5

8

6

74

2

0

2

2

0

2

63

2

02

0

2

5B

KL

B

KLKBd

BLKBd

BL

,286

361663

2848][Cosh602530

236202031072

68020248485

2402628

24(][Sinh240][2Cosh120480

=

8

22

1

2222

42

22

4

2

052020181513

15

2

063

2

0

22

642

12108

2

0

22

1

2

4

222

4

2

0

22

2

3

2

0

22

3

22

14

2

0

2

2

22

2

2

2

0

2222

20181520

2

063

2

0

22

54

2

0

22

6

2

5

2

0

44

2

0

249

LdBLBBd

KKBKBKBKKKKKBB

KBBKKBdBBLLLd

KKKBBLKdBKBB

KBBKdBdLKBdKB

KBBBKKBKBKBKKBdB

KKBdBBBKKBBBBB

L

,286

3616

326848][Sinh30

310183516803

320482

62248][Cosh120

][2Sinh120820240

=

7

22

1

2222

42

22

4

2

061917

16141916

2

053

2

0

22

534119

2

0

44

1

244

4

2

0

2222

3191614

16

2

064

2

0

2

53

2

0

22

65

2

0

44

3

2

02

44

2

0

2410

LdBLBBd

KKBKBKBKKB

KKBKBKBKKBdBB

LLKdKKBBLdB

KBBBKKKKBB

KBKKBdBKKBdBBB

KKBBBKBdKBBB

L

,441648

= 2

22

12

22

8

2

0

222

3

2

0

2

2

2

0

22

2

0

211 LdBLKdBKBBKBKBBBB

L

,889616224

= 3

22

13

22

9

2

0

22

43

2

032

2

0

22

2

22

2

0

212 LdBLKdBKBBKKBKKBBKdBBB

L

,22424

2884832896

=

4

22

14

22

10

2

0

22

2

4

2

042

2

0

222

3

2

0

22

3

22

2

42

2

0

213

LdBLKdBKBB

KBKKBBKBBKdBKdBBB

L


,1648680

= 511

2

043

2

043

2

2

0

2

14 dLKBKKBdKKdB

L

,240723240

= 612

2

0

2

4

2

04

2

2

0

2

15 dLKBKBKdB

L

64

2

062

2

0

33

6

3

53

2

0

22

5

22

2

0

2416 48831644

= KKBBKKBBKBdKKBBKdBBB

L

,122

48242

=

20

2

018

2

0

22

15

2

063

2

0

22

6

22

54

2

05

3

2

0

317

KBKBB

KBBKKBBKdBKKBBKBdBB

L

,=,824488

= 20

1919

2

016

2

064

2

06

3

2

0

218B

KLKBBKBKKBBKBd

BBL

,424484

16448834

=

8

22

15

33

20

2

018

2

0

22

15

2

013

2

0

33

63

2

0

22

6

22

54

2

052

2

0

33

5

3

2

0

2420

LdBLKdBKBKBBKBBKBB

KKBBKdBKKBBKKBBKBdBB

L

19

2

016

2

0

14

2

0

22

64

2

06

3

53

2

0

22

5

22

321

412

22482

=

KBBKB

KBBKKBBKBdKKBBKdBB

L

,248488

= 20

2

015

2

054

2

05

3

2

0

222 KBKBBKKBBKBdBB

L

,4

1=,

2

1=,= 2

6

2

5

2

2565

2

24

16

23 KKLKKLB

KL

Appendix B

,][Sech

=,2

=,22

=443222

22

441

B

BAM

B

AMB

B

AM

,][Sinh3][Cosh32126

1= 33

22

2

22

22

0

225 MBBdMBdBMdBdMBB

M

,][2Sinh6][2Cosh3

2][Cosh122][Sinh12

23267214404

62248][Cosh12

][3Cosh42][2Cosh3][2Sinh6

26212][Sinh4824

=

6

22

6

5

22

454

32

33

1

22

2

22

32

6

44

32321

2222

3

6

44

54

33

5

44

5

2233

32

22

33

22

2

0

664

NBBNBB

NBNBBNNBBB

NNBNBBBNBNM

NBNNNNNBBNB

NBBNNBBBNBB

NBBNNBNBBMBBB

dM

Farooq et al. 2951

,4802444

= 53

55

3222

22

12

44

2

0

446 NMBNMNMBNMBBB

dM

,3

=,202

=,3

=2

0

32

932

22

2

0

22

282

0

27

B

NdMMNNB

BB

dMM

B

dMM

,][Coth4][Csch168

1= 3

22

3

33

322

0

2210 MBdBMdBMBdMBdBB

M

,68284

36558421224

1=

63

44

62

22

53

33

52

42

22

3323

22

13

44

2

0

3311

NMdBNMdeBNMdeBNMBd

NMdBNdMNMdBNMdBBB

M

ByBy

,10

=,2216

=,8

=2

0

3314523323

22

2

0

132

0

22

3

33

12B

NMBdMNMBNMNMB

BB

dM

BB

MdBM

,][2Sinh24][2Cosh20

233][Sinh38

4][Cosh9533610216

432040][3Cosh13524

3896][Cosh1801561030

21545][Sinh12][2Sinh240

36056][2Cosh401360

=

6

22

6

5

22

45

22

4

22

32132

22

326

44

6321

22

32

22

36321

22

32

22

35

44

5

33

54

33

3

22

2

0

62615

NBBNBB

NBNBBBNBNB

BBNNNBBNNB

NMNBBNNNNB

NNBNBNNNNB

NNBNBBNBB

NBNNBBBMBBeB

deM

B

B

,1249068

=,24

9= 5242

22

3323

22

2

0

22172

0

33

3

33

16 NMBNMBNMNMBBB

dM

BB

MdBM

,48

=,4411672

=,4

3=

2

0

63

20625343

22

2

0

22192

0

33

18B

NdMMNMNMBNMB

BB

dM

B

NdMM

,][2Cosh3

][Sinh48][Cosh48424612

1=

2

3

22

3232

2

2

222

2

2

21

MBB

MMBBMMBMBMBB

N

,4

1=,

4=,

4=,

3

1=,

2= 2

3

2

6

32

52

32

4

2

2

2

32

2

2

2 MNB

MMN

B

MMNMN

B

MN


,601821200][Sinh216][3Sinh24

][3Cosh1610621024

720][Cosh432232][2Cosh27

182][2Sinh54560][2Sinh420

][2Cosh23105184][Sinh840

466][Cosh8401528

70288][4Cosh13515253072

26315][2Cosh540

23][3Cosh80][3Sinh240][Sinh64806

][Cosh216026][2Sinh21607

3253

62103615][2Cosh3

][4Cosh1815615122

720][Cosh48][3Cosh162

2103615][2Sinh6

122362512

32360][Sinh96105])[3Sinh3

][3Cosh(7140][2Sinh630][2Cosh630

642281030

762161320][Sinh1260812

5340184

600482401440][Cosh12604556

420670382460480

=

6

44

321

22

32

22

36

55

6

44

6321

22

32

22

35

22

4

33

5

77

54

44

326

22

6

22

5

22

4

5

22

4

22

32

1

552

26

44

6321

66

6321

22

32

22

3

54

33

5

44

5

44

5

4

33

32

22

3

2

3

22

20181715141311

181714131814

20

66

19

66

986

22

98

22

919

66

181714

131114131814

22

1817141311

66

98

986

2222

9320

55

19

66

19

55

181715141311181714

1318141817

1817151413141311

22

1413181498

22

986

2255

2

2

02

0

577

NBNNNB

NNBNBBNBB

NBBNNNNBNNB

NBNBNBBB

NBNNBBBMMNBB

NBBBNBNBBB

NBNBBBNN

NBMBNBBNNNNB

NNNNBNNBNB

NNBBBNBBNBBN

NBBBNNBNBBMBBd

MMMMBMMMB

MMBMMBMBMBB

MBBMBMMMBMMB

MBMBBMMM

MMBBMMBMBMBB

MMMMMBBMM

MMMBBMBBMBMBB

BBMBBMBB

MMMBMMMBMMBM

MBMBMBBMM

MMMMMBBMMMBBB

MMBMBMBMMB

MMMBBMBBB

N

,][Sinh4][Cosh2][Sinh43

][Cosh10])[Sinh][Cosh(20

320310][Sinh6

][Cosh43016])[2Sinh][2Cosh(23

][2Sinh63][2Cosh64][2Sinh6

][Sinh2][Cosh272][Sinh244

][2Sinh3][2Cosh32524240

=

16

22

12

22

22

10

22

7

2222

5

442

03

22

216

12

2233

10

22

7

22

5

222

0

3

3322

3

2

2

44

2

0

248

MBBBBBMBB

BBBMBBBB

MBBMBBMBB

BBBdMMBBBB

MBBBBBMBBB

MBBBBMBBB

MBBBBdBMBMdBBB

N

,284328

= 6

2

3

44

1

2

3

44

173

2

0

22

113

2

0

33

62

2

02

0

29 NMdBNMdBMMBBMMBBMMBBB

N

,8961624

= 123

2

0

33

72

2

052

2

0

222

3

44

2

0

210 MMBBMMBMMBBMdBBB

N

,48

246961624

=

532

33

2

2

3

44

1

2

2

22

183

2

0

22

133

2

0

33

82

2

062

2

0

22

2

0

211

NMMdBNMdBNMdB

MMBBMMBBMMBMMBBBB

N

Farooq et al. 2953

,36288604860480

1= 72

2

0

772

2

77

2

0

5712 MMBBMdBBB

N

,8102403260

= 3

2

3

44

2

2

2

22

143

2

0

33

92

2

082

2

0

22

2

0

213 NMdBNMdBMMBBMMBMMBBBB

N

,614

= 329

2

02

0

2

2

14 NdMMBB

MN

,824

2662

=

162

2

0122

2

0

102

2

0

22

73

2

053

2

0

33

322

0

2415

MMBBMMB

MMBBMMBBMMBBMMBdBB

N

,8328864

9600288163456

1601621120672

325760192164

=

632

55

5

2

3

66

5

2

2

22

4

2

2

33

332232

33

132

55

182

2

0

172

2

0

33

152

2

0

55

142

2

0132

2

0

22

112

2

0

44

93

2

083

2

0

33

63

2

0

55

2

0

4616

NMMdBNMdBNMdBNMdB

NMMBdNMMdBNMMdBMMBB

MMBBMMBBMMBMMBB

MMBBMMBBMMBBMMBBBB

N

,2632

= 162

2

0122

2

073

2

0322

0

217 MMBBMMBMMBBMMBdBB

N

,6028

18046024

=

5

2

2

22

332232

33

182

2

0172

2

0

33

142

2

0132

2

0

22

93

2

083

2

0

33

2

0

2418

NMdBNMMBdNMMdBMMBBMMBB

MMBMMBBMMBBMMBBBB

N

,634

= 332182

2

0142

2

093

2

02

0

219 NMMBdMMBBMMBMMBBBB

N

,16

63849611520

3843283840

1921623040768

645760192164

=

632

55

5

2

3

66

5

2

2

22

4

2

3

77

4

2

2

33

332

232

33

132

55

193

2

0

77

182

2

0

172

2

0

33

152

2

0

55

142

2

0132

2

0

22

112

2

0

44

93

2

083

2

0

33

63

2

0

55

2

0

5720

NMMdB

NMdBNMdBNMdBNMdBNMMBd

NMMdBNMMdBMMBBMMBB

MMBBMMBBMMBMMBB

MMBBMMBBMMBBMMBBBB

N

,210324

= 162

2

0122

2

0102

2

0

22

73

2

0322

0

321 MMBBMMBMMBBMMBBMMBdBB

N

,)6240

6963600

1818064

=

5

2

2

22

4

2

2

33

332

232

33

182

2

0172

2

0

33

142

2

0

132

2

0

22

112

2

0

44

93

2

083

2

0

33

2

0

3522

NMdBNMdBNMMBd

NMMdBMMBBMMBBMMB

MMBBMMBBMMBBMMBBBB

N


,4

= 12223

B

MMN

,10540154

= 332182

2

0142

2

0132

2

0

22

93

2

02

0

324 NMMBdMMBBMMBMMBBMMBBBB

N

,4

= 14225

B

MMN

,44816

= 16

2

012

2

010

2

0

22

32

0

2

326 MBBMBMBBMBd

BB

MN

,84

2686

215324

=

6

2

2532

22

432

33

3

2

32

2

3

33

192

2

0

33

183

2

0

173

2

0

33

143

2

0133

2

0

22

113

2

0

44

2

0

2427

NMBdNMMdB

NMMdBNMBdNMdBMMBBMMBB

MMBBMMBMMBBMMBBBB

N

,252

=,2

1= 3318

2

014

2

013

2

0

22

2

0

2

3

29123

2

28 NMBdMBBMBMBBBB

MNMMN

,2

1= 143

2

30 MMN

,224

1281532

2432124

8306416

=

6

2

3

55

6

2

2

532

22

432

33

3

2

32

2

3

33

1

2

3

55

203

2

0

55

192

2

0

33

183

2

0173

2

0

33

153

2

0

55

143

2

0133

2

0

22

113

2

0

44

2

0

3531

NMdBNMBd

NMMdBNMMdBNMBdNMdBNMdB

MMBBMMBBMMBBMMBB

MMBBMMBMMBBMMBBBB

N

,8816

= 16

2

012

2

032

0

3

32 MBBMBMBdBB

MN

,849

1243068

=

6

2

2532

22

3

2

32

2

3

33

183

2

0173

2

0

33

143

2

0133

2

0

22

2

0

333

NMBdNMMdBNMBdNMdB

MMBBMMBBMMBMMBBBB

N

,4108

= 3318

2

014

2

02

0

3

34 NMBdMBBMBBB

MN

,84836

= 6325

2

3202

2

02

0

35 NMdMNMBdMMBBB

N

,162333672108

= 6325

2

34

2

3

22

202

2

0193

2

0

22

2

0

236 NMdMNMBdNMdBMMBMMBBBB

N

Farooq et al. 2955

.4864

= 6320

2

02

0

3

2

37 NdMMBB

MN

Appendix C

,][Sech

=,2

=,22

1=

443222

4422

441

B

BAE

B

AEBABA

BE

,][2Sinh6][2Cosh32][Sinh12

][Cosh122326][Cosh24

7214404622

48][Cosh12][3Cosh42][2Cosh3

][2Sinh626212][Sinh48

][Cosh42237224

=

9987

8753

22

1

22

7

3

22

529

44

53531

2222

59

44

87

33

7

44

7

2233

53

22

53

642

2222

4

22

2

0

664

FBBFBFBFB

FFBBFFBFBBFBB

FBFEFBFFFFFBB

FBFBBFBFBB

FBBFBBFFBFBBE

FBFBBFBBFBBB

dE

,][2Sinh15

4362][Sinh30

462][Cosh301206

120365861060

=

963

33

876242

22

43

76242

22

43

22

5

44

4

44

2

2222

23

2222

2

0

445

FFEBBB

FBFFEBFFBFEBB

FFEBFFBFFBBFB

FBFBBEFBBBB

dE

,48062444

= 73

55

524

22

32

22

2

44

12

44

2

0

446 FEBFEFBFEBFBFEBBB

dE

,124846

= 5423

22

22

22

2

0

227 FFEFBFEBBB

dE

,3

=,410

=,8048

=2

0

52105422

0

9524

22

32

22

2

0

228B

FdEEFFE

B

dEFEFBFEB

BB

dE

,][2Sinh8444

34248][Sinh3

2423][Cosh3

21221242424

][Csch=

963

33

8762

22

4

22

2

2222

43

876242

22

43

22

53

22

542

22

422

0

4411

FFEBBBFBFFEBB

FBFBBFEBB

FBFFEBFFBFEBB

FFBFFFBFEBB

BdE

,61493724

= 93

44

82

22

726

33

5333

22

13

44

2

0

3312 FEBFEBFEBFBFEFEBFEBBB

dE

,42128

= 7624323

22

2

0

13 FBFEBFEFEBBB

dE


,10

=,8

=,2216

=2

0

53162

0

4315725333

22

2

0

14B

FEBdE

B

FEBdEFEBFEFEB

BB

dE

,][2Sinh24][2Cosh204][Cosh9

323][Sinh35336

][Cosh7210216432040][3Cosh135

243896][Cosh180

156103021545][Sinh12

65][2Cosh40][2Sinh240360

])[Sinh][Cosh(23212180720

][Sech=

99878

87

22

531

753

22

529

44

9531

22

53

22

5

9531

22

53

22

5

87

33

7

44

7

33

3

642

22

4

22

2

0

6617

FBBFBFBFBFB

FBFBBFFFB

FBBFFBFEFBB

FFFFBFFBFB

FFFFBFFBFBB

FBFBBFBBFBE

FBBBBFFBFBBB

BdE

,23124568

= 8

22

7624323

22

2

0

2218 FBFBLEBFEFEBBB

dE

,4

3=,26453

4=

2

0

432082

22

725333

22

2

0

2219B

FdEEFEBFEBFEFEB

BB

dE

,412

=,12

=,4

3= 92732

0

239632

0

222

0

5321 FEFEB

BB

dEFFEB

BB

dE

B

FdEE

,48

=,4461172

=2

0

93259283

22

732

0

2224B

FdEEFEFEBFEB

BB

dE

,][2Cosh6

][Sinh96][Cosh9684831224

1=

2

3

22

3232

2

2

222

2

222

21

EBB

EEBBEEBEBEBBB

F

,3

1=,16

8

1=,][Sinh623

6

1= 2

2

4

2

2

22

2332

2

2 EFEBB

FEBBEBB

F

,4

1=,

4=,

4=,=,

3

1= 2

3

2

92

32

8

32

7

3

6

2

2

2

5 EFB

EEF

B

EEF

B

EFEF

Farooq et al. 2957

,601821200

84181631

184

2012066422

64228103076216

13204426

244][Sinh15120238

2631544

62325

362103615

4442][2Cosh37808

222624

221036

15][2Sinh15120163228

612][3Cosh56082

24][3Sinh168048][4Cosh94515

253075252815284

68335225712

23623354556420

67042104361062

1024720166

231286

4616260244

154814405962404

3612484

41602400576012

244721922

4164][Cosh1512060480

=

9

44

531

22

53

22

53

6

33

287

222

27

2

3

22

42

22

43

22

23

66

10

44

22

8

2233

6

55

3211917

1614122119161421

1622018151320

15

222

09

2

2

2222

95

31

22

53

22

5

2

382726

7232521191716

14122119161421163

2423222232

222

0

22

9

2

2

873253

22

5

2

324232

22

2119161412161421

163

2

0

33

928733252

24233

2

0

55

92733252

233

2

0

66

9325

2

03

77

9

533637342253

2

2

1

2

3

222

253

22

2119

161412397

22

108

22

108

6

22

25

222

0

55

9531

22

53

22

536

33

287

2

3

33

8742

22

43

22

8

22

7

222

224

66

231086

22

108

22

103212

4422

20

55

20

33

192

55

192

33

18

55

172

55

162

66

162

44

162

22

16215

66

15

44

15

22

142

66

142

44

142

22

13

66

13

44

122

66

122

44

11

662

02

0

5710

FBFFFBFFBFE

FBEFBFBEBFEBB

FFBFEBBdEBEB

BBEBBEBEEEEB

EEEBEEBEEBEB

EEEEBEEBEB

EBBBBFEBBFF

FFBFFBFEFEFEFBB

FEEBBdEEEEBE

EEBEEBEEBEBEE

EEBEEBBEEBBBBFE

FBFEEBFFBFEBdEBEEB

EEEEEBBEEBEB

EEBBBFEFBFEBdEEE

EBEEBBBBFEFEBdEEE

EEBBBBFdEEBEBBF

FFEBFEBFEBFFEFFE

FEBEFFdBEE

EEEBEEEBEEBEE

EBEEBBBFFFFB

FFBFEFBEFBFEB

FBFBFFBFEBFBB

FBEBBdEBEBEEEB

EEBEEBEEBBB

EBEBEEBEEBEBEEB

EEBEEBEEBEEEBEB

EBEEBEEBEEBEBEB

EEBEEBEBBBBB

F


,361202361208

46222268

6163

12245232

536210624

53184812

4824041204][Sinh12604

433282

32623

6][2Sinh6308422

94222

364][2Cosh3152

212][3Sinh14022

24224224422

2062422

2810422164

12061034][Cosh5040140

7010217081021163102

1715357152135

5063704201635040

=

9

44

5

4422

42

4422

322

2222

1

44

387

22

2

6

2

2

2222

6

2

3

55

22

66

9

44

22

7

2233

5

55

3211917

1614122119161421

201815131120181513

2015216

2

09287

62342

22

4

2

32423222

22

2018151311151320

153

2

0

22

9276242

22

43323222

22

20181513

20153

2

0

33

963325

223

2

0

55

82

33

742

22

4326

2

253

22

532119

1614121614212018

151311151320152

169

22

7

33

3

2

073

63531242

2

242

2

3

22

22

20151331086

22

975

22

972

2

0

44

2

0

4611

FBFBBFEBB

FFEBBFBEFBFBE

FEBBBFEBBdEBEB

BBEBBEBEEEEB

EEEBEEBEEBEB

EEBEEEBEEBEEB

EBEEEBBFEFBF

FEEBFFBFEBdEBEEEB

EEEEEBBEEBEB

EEBBBFEFFEBFFB

FEEBdEEEBEEBEEB

EBEEBBBFFEBdEE

EEBBBBFEdBFBFFB

FEEFEBFFBFEBdEE

EEEBBEEBEBEE

EEEBBEEBEBEE

EEBBEBEBBBFEB

FFBFFFEFFEFFEB

dBEEBEBEEEEB

EEEBEEEBBBB

F

,22

843248

=

9

2

3

44

63

33

1

2

3

44

1

22

193

2

0

22

123

2

0

33

62

2

05

2

0

22

2

0

212

FEdBFEdBFEdB

FdBEEBBEEBBEEBEBBBB

F

,2828

2489681624

=

73

33

632

33

2

2

3

44

2

22

12

22

203

2

0

22

133

2

0

33

72

2

06

2

0

22

52

2

0

22

2

0

213

FEdBFEEdBFEdBFdBFEdB

EEBBEEBBEEBEBBEEBBBB

F

,82816

4812192123248

=

732

33

3

2

3

44

3

22

22

22

1

2

2

22

213

2

0

22

143

2

0

33

82

2

07

2

0

22

62

2

0

22

2

0

214

FEEdBFEdBFdBFEdBFEdB

EEBBEEBBEEBFBBEEBBBB

F

,2816

16320164880

1=

4

2

3

44

4

22

32

22

2

2

2

22

153

2

0

33

92

2

08

2

0

22

72

2

0

22

2

0

215

FEdBFdBFEdBFEdB

EEBBEEBEBBEEBBBB

F

Farooq et al. 2959

,2816

204802064120

1=

5

2

3

44

5

22

42

22

3

2

2

22

163

2

0

33

102

2

09

2

0

22

82

2

0

22

2

0

216

FEdBFdBFEdBFEdB

EEBBEEBEBBEEBBBB

F

,614

=,231021

= 5210

2

02

0

2

2

18524

2

210

2

092

2

02

0

2

17 FdEEBB

EFFdEFdEEBEEB

BF

,208041222

316963120

622426

48412240412044

=

9

44

542322

22

1

44

3

8726

2

2

22

6

2

3

55

22

66

9

7

33

5

55

32119171214

2018111315216

2

02

0

4619

FBFFEFFEBFBE

FBFEFEBBFEBBdEBEB

EBEBEEEEBEBBEBB

EEEBBEBBEEEBBB

F

,1200

362829323

18472024

2221610242

13204622444

=

9

44

5

3

22

1

44

36

33

287

2

27

2

3

33

742

22

3

22

23

66

108

33

6

55

32119171214

16220181315

222

02

0

4620

FBF

FBFBEFBFFBFEBFEB

FBFFBEBBdEBEBEB

EBEEEEBEBBEBB

EEEEBEBBEBBBB

F

,3622436

123183964

60206=

742

22

326

2

253

22

3

211914201813152

1697

33

3

2

02

0

2421

FBFFBEEFEBFFBEBd

EEBEBBEEBEBBEE

EEBEBEBBB

F

,42404284

18044308=

7

2

252432

22

3211914

1622015

22

108

33

3

2

02

0

2422

FEBFEFFEBEBdEEBEBB

EEEBEBEBEBEBBB

F

,)

4452010=

53

43221

2

0202

2

016

2

0152

2

093

2

02

0

223

FEBd

FEEBdEBBEEBBEBEEBEEBBBB

F

,634

= 532212

2

0162

2

0103

2

02

0

224 FEEBdEEBBEEBEEBBBB

F


,720242166

2128496

21440484

96064845760

2419221644

=

9

44

53

22

1

44

36

33

287

2

3

33

8742

22

3

22

87

2

2

24

77

23

55

108

22

6

44

3

21220

33

192

33

18

55

172

55

162

15

22

142

22

13

44

122

44

11

662

02

0

5725

FBFFBFBEFBEFBFEB

FBFBFFBEBFBFEBBd

EBEBEEBEBBE

EEBEBEEBEBEEBEE

EBEEBEBEEBEBBBB

F

,424882422

2426422

1021641202012=

8742

22

326

2

253

22

3

21191214201811

131521697

33

3

2

02

0

3526

FBFBFFBEEFEBFFBEBd

EEEBBEBBEEEBB

EBBEEEEBEBEBBB

F

,4416046

963186004

3123024=

8

2

2

33

7

2

252432

22

3

21191214162

201315

22

108

33

3

2

02

0

3527

FEdBFEBFEFFEBEBd

EEEBBEBBEE

EEBBEBEBEBEBBB

F

,832416

20112440=

5343221

2

0202

2

0

16

2

0152

2

014

2

0

22

132

2

0

22

93

2

02

0

328

FEBdFEEBdEBBEEBB

EBEEBEBBEEBBEEBBBB

F

,40

20160460=

532

2

212

2

0162

2

015

2

0

22

142

2

0

22

103

2

02

0

329

EEEBd

EEBBEEBEBBEEBBEEBBBB

F

,4

=,4= 162

311615230B

EEFEEE

BF

,4

43282

32268

=

928

762342

222

32423222

22

20181113153

2

02

0

2432

FEFB

FFEFBFFBEBdEBEEEB

EEEBBEBBEEBBB

F

,8226)24

623154

=

9

2

2873253

222

324232

22

21191214163

2

02

0

2433

FEFBFEEBFFBEBdEBEEB

EEEBBEBBEEBBB

F

,361248

= 4320

2

015

2

013

2

0

22

2

0

2

3

34 FEBdEBBEBEBBBB

EF

,2

1=,

2

1=,25

2= 163

2

37153

2

365321

2

016

2

014

2

0

22

2

0

2

3

35

EEFEEFFEBdEBBEBEBB

BB

EF

Farooq et al. 2961

,2153

224462

3223615

44216

=

9

44

53

22

1

442

39

2

2

22

826723

251712191421163

2423222

22222

02

0

3538

FBFFB

FBEFEBFEFBBFEEBBd

EEBEBEBEBEBEE

EBEEEBBBBB

F

,84292

42326416

=

92762342

222

3

23222

22

201813153

2

02

0

339

FEFFEEBFFBEBd

EEEBEEBEBBEEBBB

F

,84916

1243068

=

9

2

2732

22

5

2

33

2

3

33

232

2

0

22

213

2

0193

2

0

33

163

2

0143

2

0

22

2

0

340

FEBdFEEdBFEBdFEdBEEBB

EEBBEEBBEEBEEBBBB

F

,1048

=,488

= 5316

2

0

2

02

0

3424320

2

015

2

02

0

341 FEBdEBBB

BB

EFFEBdEBBEB

BB

EF

,2122436

= 936

2

325

2

0223

2

02

0

43 FdEFEBdEBEEBBBB

F

,8482436

= 9327

2

3252

2

0233

2

02

0

44 FEdEFEBdEEBEEBBBB

F

,163

23367212108

=

9328

2

3

22

7

2

3252

2

0243

2

0

22

233

2

02

0

245

FEdEFEdB

FEBdEEBEEBBEEBBBB

F

.4864

= 9325

2

02

0

3

2

46 FdEEBB

EF

Documents

Laminar flow of couple stress fluids for Vogel's model