laminar flow of couple stress fluids for vogel's model
TRANSCRIPT
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
2942 Sci. Res. Essays
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
2944 Sci. Res. Essays
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
2946 Sci. Res. Essays
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
][Sinh3][Cosh3][Cosh24
][Sinh12][Cosh12][Cosh48
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
2950 Sci. Res. Essays
,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
2952 Sci. Res. Essays
,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
2954 Sci. Res. Essays
,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
2956 Sci. Res. Essays
,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
2958 Sci. Res. Essays
,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
2960 Sci. Res. Essays
,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