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STEADY PLANE COUETTE FLOW OF
VISCOUS INCOMPRESSIBLE FLUID
BETWEEN TWO POROUS PARALLEL
PLATES THROUGH POROUS MEDIUM WITH
MAGNETIC FIELD
Dr. Anand Swrup Sharma (D.Sc. Scholar)
Professor, Department of Applied Sciences (Mathematics)
Future University, Bareilly (UP), India
ABSTRACT: In this paper, we have investigated the Steady Plane Couette Flow of Viscous
Incompressible Fluid between two Porous Parallel Plates through Porous Medium with Magnetic Field.
We have studied the velocity, average Velocity, Shear stress, Skin frictions, volumetric flow, Drag
Coefficients & Streamlines.
KEY WORDS: Steady Couette Flow, Viscous Parallel Plates, Incompressible Fluid, Porous Medium &
Magnetic Field
NOMENCLATURE
u Velocity component along
x–axis
v Velocity component along
y–axis
t The Time
The Density of Fluid
p The Fluid Pressure
k The Thermal Conductivity
Coefficient of Viscosity
Kinematic Viscosity
Q The Volumetric Flow
INTRODUCTION
We have investigated the Steady Plane Couette Flow of Viscous Incompressible Fluid between two
Porous Parallel Plates through Porous Medium with Magnetic Field. Attempts have been made by
several researchers i.e. O. R. Burggraf [1] investigated Analytical and Numerical Studies of the Structure
of Steady Separated Flows. O. R. Burggraf [2] investigated Computational Study of Supersonic Flow
over Backward–Facing Steps at High Reynolds Number. G. I. Busws [3] investigated the Construction of
Special Explicit Solution of the Boundary Layer Equations Steady Flows. K. Butler & B. F. Farell [4]
investigated Three–Dimensional Optimal Perturbations in Viscous Shear Flow. Canadam & T. Mulnke
[5] investigated Forced Vibrations of a Piezoelectric Layer of Six (mm) Crystal Class. V. C. Carey & J. C.
Mollendorf [6] investigated Variable Viscosity Effects in Several Natural Convection Flows. W.
Cazemier, R.W.C.P. Verstappen, & A. E. P. Veldman [7] investigated Proper Orthogonal Decomposition
and Low–Dimensional Models for the Driven Cavity Flows. T. Cebeci & P. Bradshaw [8] investigated
Momentum Transfer in Boundary Layers. T. Cebeci & K. C. Chang [9] investigated a General Method
for Calculating Momentum and Heat Transfer in Laminar and Turbulent Duct Flows. T. Cebeci, F.
Thiele, P. G. Williams & K. Stewartson [10] investigated on the Calculation of Symmetric Waves–I, Two–
Dimensional Flow. I. Celik & Z. Wei–Mung [11] investigated Calculation of Numerical Uncertainty
Using Richardson Extrapolating Application to Some Simple Turbulent Flow Calculations. Chamkha
and H. Al–Naser [12] investigated Hydro Magnetic Double–Diffusive Convection in a Rectangular
Enclosure with Uniform Side Heart and Mass Fluxes and Opposing Temperature and Concentration
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IJCRT2012009 International Journal of Creative Research Thoughts (IJCRT) www.ijcrt.org 51
Gradients. O. P. Chandna & E. O. Oku–Ukpong [13] investigated Some Solutions of Second Grade Fluid
Flow Von Misses Co-ordinates Transformations. D. S. Chauhan & R. Agarwal [14] investigated MHD
through a Porous Medium Adjacent to a Stretching Sheet Numerical and an Approximation Solution. G.
Chavepeyer, J. K. Plat Ten, & M. B. Bada [15] investigated Laminar Thermal Convection in a Vertical
Slot. A. Gulhan, T. Thiele, F. Siebe, B. Kronen & T. Schleutker [16] investigated Aero Thermal
Measurements from the ExoMars Schiaparelli Capsule Entry. I. Hashem & M. H. Mohamed [17]
investigated Aerodynamic Performance Enhancements of H-Rotor Darrieus Wind Turbine. M. Jafari, A.
Razavi & M. Mirhosseini [18] investigated Affect of Airfoil Profile on Aerodynamic Performance and
Economic Assessment of H-Rotor Vertical Axis Wind Turbines. In this paper, we have investigated the
Velocity, average Velocity, Shear stress, Skin frictions, volumetric flow, Drag Coefficients & Streamlines.
FORMULATION OF THE PROBLEM
Let us consider two infinite Porous plates AB & CD separated by a distance 2h. The fluid enters in y-
direction. The velocity component along x-axis is a function of y only. The motion of incompressible
fluid is in two dimension and is steady then
( ), 0 & 0u u y wt
The equation of continuity for incompressible fluid
0 0 & 0 0u v w u v
put wx y z x y
Figure-1
v is independent of y but motion is along y-axis. So we can say that v is constant velocity 0
. .i e v v or
the fluid enters in flow region through one plate at the same constant velocity 0
v .
Also Navier-Stoke's equations for incompressible fluid in the absence of body force when flow is
steady
22
00 2
1 1 1(1) & 0 (2)
Bd u p d u pv u
d y x d y k y
SOLUTION OF THE PROBLEM
Equation (2) shows that the pressure does not depend on y hence p is a function of x only & so
equation (1) reduces to 2 22 2
0 0 002 2
1B u v Bd p d u d u u d u du P dpv u where P
d x d y d y k dy dy k dx
2 2
0 0 0
22 22 0 0 0 0 0
14
1 1. . 0
2 2 2
v v B
kv B v v BA E m m m
k k
2 22 2 2
0 0 0 0 01 1 1, &
2 2
v B B v BLet A B
k k k
1 2 1 2
0 02 2. . & . . ( )
v vy yP P
C F e C Cosh Ay C Sinh Ay P I u y e C Cosh Ay C Sinh AyB B
Using boundary conditions 0 &u at y h u U at y h
1 2 1 2
0 0
2 20........ (3) & ........ (4)
v vh h
C C C S C C C SP P
e osh Ah inh Ah U e osh Ah inh AhB B
0 0
2 21 2 1 2&
v vh hP P
e C Cosh Ah C Sinh Ah U e C Cosh Ah C Sinh AhB B
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1 2
0 0 0 02 2 2 21 1
&2 2
v v v vh h h hP P P P
C U e e C U e eCosh Ah B B Sinh Ah B B
00 0
00 0 2
2 22
2 2( )2 2
vy v v
h h
vy v v
h he Cosh Ay P P e Sinh Ay P Pu y U e e U e e
Cosh Ah B B Sinh Ah B B B
0 0( ) ( )2 2( ) ( )
( )2 2
v vy h y h
P e Sinh A y h P e Sinh A y h Pu y U
B Sinh Ah Cosh Ah B Sinh Ah Cosh Ah B
0 0( ) ( )2 21
( ) ( ) ( ) .......... (5)2
v vy h y hP P P
u y U e SinhA y h e SinhA y hSinh Ah B B B
FOR PLANE COUETTE FLOW In this case put 0P in equation (5)
0 ( )
21( ) ( ) 6
2
y hv
u y U e Sinh A y hSinh Ah
THE SHEAR STRESS AT ANY POINT
00 0( ) ( )
2 2( ) ( )2 2
y h y h
x y
v vvd u U
e Sinh A y h Ae Cosh A y hd y Sinh Ah
0
0 ( )2
( ) ( ) 72 2x y
y hv
vU eSinh A y h ACosh A y h
Sinh Ah
THE SKIN FRICTIONS AT LOWER AND UPPER PLATE
0 02 2 2 82 2 2
x y hy
USinh Ah ACosh Ah U A Coth Ah
Sinh Ah
v v
0 0
92 2
h h
x y y h
v v
U e U AeA
Sinh Ah Sinh Ah
THE AVERAGE VELOCITY DISTRIBUTION IN PLANE COUETTE FLOW
00 ( ) ( )( )( ) 221( ) ( )
2 2 2 2 2av
A y h A y hy hy h h
h
h
h
vvU U e e
u e Sinh A y h dy e dyh Sinh Ah h Sinh Ah
0 00 0
2
20
2 2
2 2
4 2
2
h hAh Ah
av
v vv v
A e e A e eU
uh S i nh Ah
Av
2 22 220 0 0 01 1
Since ,2 2
v B B vA B A B
k k
00 0 0 0
2 2 2 2( )4 2 2
vh h h hAh Ah Ah Ah
av
v v vvU
u e e e e A e e e eBh Sinh Ah
00
2 2 102 2 2av
hv
vUu Sinh Ah A Cosh Ah e
Bh Sinh Ah
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THE VOLUMETRIC FLOW
00
2 2 2 11 2 2av
hv
vUQ h u Sinh Ah A Cosh Ah e
B Sinh Ah
THE DRAG COEFFICIENTS
0
22
02 2 2
02
2 2 2 2
112 2 22 24 2
f y hav
xy y h
hv
vUSinh Ah ACosh Ah
Sinh Ah
u vUSinh Ah ACosh Ah Ae
B h Sinh Ah
C
02 2
0
20
8 2 2 22
12
2 22
y hf
hv
vB h Sinh Ah Sinh Ah A Cosh Ah
vU Sinh Ah A Cosh Ah A e
C
2 2 2
/
020
2
0
8 2
2
2 22
f y hh
hv
v
U Ae B h Sinh Ah
Sinh Ahv
U Sinh Ah ACosh Ah Ae
C
2
/
0
0
02
28 213
2 22
f y h
h
h
v
v
B h A e Sinh AhC
vU Sinh Ah ACosh Ah Ae
THE STREAM LINE IN THE PLANE COUETTE FLOW
00 ( )
2
ˆˆ ˆ0
( ) 2
y hv
d x d y d z d x d y d zw h e r e q u i v j w k
u v w vUe S i n h A y h
S i n h A h
Taking first two equations 1
0
0( )
2 2 ( )
y hv
CSinh Ah
dx e Sinh A y h dyU
v
1
0 ( ) ( )( )0 2
2
2
A y h A y hy hC
vSinh Ah e e
x e dyU
v
01
0 0( ) ( ) ( ) ( )2 21
2 2
y h A y h y h A y hC
v v
Sinh Ah x e e dyU
v
0 0
0 0
1
( ) ( ) ( ) ( )2 2
0 1 2
2
2 2
y h A y h y h A y hv
C
v
e eSinh Ah x
U v vA A
v
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0 0
01
( ) ( )0( )2 2 ( )01
2 2 2 2
y h y hA y h
v v
A y hC
vSinh Ah x A e e A e e
U B
vv
0
1
( )2
0 ( ) ( ) ( ) ( )0 2 2 2
y hA y h A y h A y h A y h
C
vve
Sinh Ah x e e A e eU B
v
1
0
0
( )2
0 2 ( ) 2 ( )2
y h
C
vve
Sinh Ah x Sinh A y h A Cosh A y hU B
v
1
0
0
( )2
0 2 ( ) ( ) 142
y h
C
v
veSinh Ah x Sinh A y h A Cosh A y h
U B
v
2& 15CSecond stream line is given by z
0
0 ( )2
ˆ ˆˆ ˆ ˆ ˆ
( )0
2
y hv
i j k i j k
Now Curl qx y z x y z
u v wU e Sinh A y h
vSinh Ah
0
0 ( )2
( ) ( ) 0 2 2
y hv
vU eCurl q Sinh A y h ACosh A y h k
Sinh Ah
.Motion of the Fluid is Rotational
Comparison between Porous medium, Magnetic Field & Porous medium with Magnetic
Field
Tables for velocity and skin friction 2
2
0 00 12
26, 5, 6 &
2
v BA
k
vLet U h all are fixed
2 22 20 0 0 011
& & .2 2k
B v v BLet are vary are also vary
k
2 2 22 2 2
0 0 0 0 0 01 1 116 20 2
2 2:
21 Let
k k kC
vase
B v v B B
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Table-1 (for velocity)
y 0 .1 .2 .3 .4 .5 .6
2
0 120
2
v
k
u(y) .032 .091 .259 .738 2.11 6 17.09
2 20 0 20
2
v B
u(y) .032 .091 .259 .738 2.11 6 17.09
2 20 01
22 k
v B
u(y) .097 .227 .52 1.184 2.67 6 13.44
Graph of table-1
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Table-2 (for skin friction)
y 0 .1 .2 .3 .4 .5 .6
2
0 120
2
v
k
xy .167 .476 1.36 3.87 11.02 31.42 89.54
2 20 0 20
2
v B
xy .167 .476 1.36 3.87 11.02 31.42 89.54
2 20 01
22 k
v B
xy .417 .95 2.15 4.835 10.84 24.22 54.08
Graph of table-2
Tables for velocity and skin friction 2
2
0 00 12
26, 5, 6 &
2
v BA
k
vLet U h all are fixed
2 22 20 0 0 011
& & .2 2k
B v v BLet are vary are also vary
k
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2 22 2 2
0 0 0 0 01 1 111, 21 5 & 15
2 22 :
B B v v BLCa et
k kse
k
Table-3 (for velocity)
y 0 .1 .2 .3 .4 .5 .6
2
0 15
2
v
k
u(y) .024 .073 .221 .665 1.997 6 18.025
2 20 0 15
2
v B
u(y) .042 .115 .309 .832 2.23 6 16.11
2 20 01
22 k
v B
u(y) .097 .227 .52 1.184 2.67 6 13.44
Graph of table-3
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Table-4 (for skin friction)
y 0 .1 .2 .3 .4 .5 .6
2
0 15
2
v
k
xy .135 .405 1.22 3.66 10.98 33 99.14
2 20 0 15
2
v B
xy .212 .57 1.53 4.112 9.21 29.63 79.53
2 20 01
22 k
v B
xy .417 .95 2.15 4.835 10.84 24.22 54.08
Graph of table-4
Tables for velocity and skin friction 2
2
0 00 12
26, 5, 6 &
2
v BA
k
vLet U h all are fixed
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2 22 20 0 0 011
& & .2 2k
B v v BLet are vary are also vary
k
2 22 2 2
0 0 0 0 031 1 1
21, 11 15 & 52
:2
CB B v v B
Letk k k
ase
Table-5 (for velocity)
y 0 .1 .2 .3 .4 .5 .6
2
0 115
2
v
k
u(y) .042 .115 .309 .832 2.23 6 16.11
2 20 0 5
2
v B
u(y) .024 .073 .221 .665 1.997 6 18.025
2 20 01
22 k
v B
u(y) .097 .227 .52 1.184 2.67 6 13.44
Graph of table-5
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Table-6 (for skin friction)
y 0 .1 .2 .3 .4 .5 .6
2
0 115
2
v
k
xy .212 .57 1.53 4.112 9.21 29.63 79.53
2 20 0 5
2
v B
xy .135 .405 1.22 3.66 10.98 33 99.14
2 20 01
22 k
v B
xy .417 .95 2.15 4.835 10.84 24.22 54.08
Graph of table-6
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CONCLUSION AND DISCUSSION
In this paper, we have investigated the velocity by the table-1 of equation (6). The velocity in Porous
medium and Magnetic field at
2 2 20 0 01
202 2k
v v B
is less than the corresponding
value of velocity in Porous with Magnetic field at
2 20 01
22 k
v B
in the interval 0 4y
and equal 6u y in all medium at 5.y But the value of velocity in Porous medium and Magnetic
field at
2 2 20 0 01
202 2k
v v B
is greater than the corresponding value of velocity in
Porous with Magnetic field at
2 20 01
22 k
v B
at 6.y
Again by the table-3, the value of the velocity in Porous medium at
2
0 15
2
v
k
is less than the
corresponding value of velocity in Magnetic field at
2 20 0 15
2
v B
and also is less than the
corresponding value of velocity in Porous medium with Magnetic field at
2 20 01
22 k
v B
in
the interval 0 4.y The Velocity is equal 6u y in all medium at 5y and is greater than the
corresponding value of velocity in Magnetic field and Porous with Magnetic field at 6.y
Again by the table-5, the value of the velocity in Magnetic field at
2 20 0 5
2
v B
is less than the
corresponding value of velocity in Porous medium at
2
0 115
2
v
k
and also is less than the
corresponding value of velocity in Porous medium with Magnetic field at
2 20 01
22 k
v B
in
the interval 0 4.y The velocity is equal 6u y in all medium at 5y and is greater than the
corresponding value of velocity in Porous medium and Porous with Magnetic field at 6.y
Again we have investigated the skin friction by the table-2, of equation (7). The skin friction in Porous
medium and Magnetic field at
2 2 20 0 01
202 2k
v v B
is less than the corresponding
value of Skin friction in Porous with Magnetic field at
2 20 01
22 k
v B
in the interval
0 3y and the Skin friction in Porous medium and Magnetic field at
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2 2 20 0 01
202 2k
v v B
is greater than the corresponding value of Skin friction in
Porous with Magnetic field at
2 20 01
22 k
v B
in the interval 4 6.y
Again by the table-4, the Skin friction in Porous medium at
2
0 15
2
v
k
is less than the
corresponding value of Skin friction in Magnetic field at
2 20 0 15
2
v B
and also is less than
the corresponding value of Skin friction in Porous with Magnetic field at
2 20 01
22 k
v B
in
the interval 0 3y and Skin friction in Porous medium at
2
0 15
2
v
k
is greater than the
corresponding value of Skin friction in Magnetic field at
2 20 0 15
2
v B
and is also greater
than the corresponding value of Skin friction in Porous with Magnetic field at
2 20 01
22 k
v B
in the interval 4 6.y
Again by the table-6, the Skin friction in Magnetic field at
2 20 0 5
2
v B
is less than the
corresponding value of Skin friction in Porous medium at
2
0 115
2
v
k
and also is less than the
corresponding value of Skin friction in Porous with Magnetic field at
2 20 01
22 k
v B
in the
interval 0 3y and Skin friction in Magnetic field at
2 20 0 5
2
v B
is greater than the
corresponding value of Skin friction in Porous medium at
2
0 115
2
v
k
and is also greater than
the corresponding value of Skin friction in Porous with Magnetic field at
2 20 01
22 k
v B
in
the interval 4 6.y Also we have investigated the Skin frictions, average Velocity, the Volumetric
flow, Drag Coefficients & Stream lines by the equations (8), (9), (10), (11), (12), (13), (14) & (15)
respectively.
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REFERENCES
[1] O. R. Burggraf (1966), Analytical and Numerical Studies of the Structure of Steady Separated Flows.
Journal of Fluid Mechanics, Vol. 24, pp. 113–151.
[2] O. R. Burggraf (1970), Computational Study of Supersonic Flow over Backward-Facing Steps at
High Reynolds Number. Journal of Aero. Res. Lab. Rept., Vol. 2, pp. 270–275.
[3] G. I. Busws (1994), The Construction of Special Explicit Solution of the Boundary Layer Equations
Steady Flows. Journal of Mechanics & Applied Math., Vol. 17, No. 2, pp. 242–268.
[4] K. Butler & B. F. Farell (1992), Three-Dimensional Optimal Perturbations in Viscous Shear Flow.
Journal of Physics of Fluids, Vol. 4(A), pp. 1637–1642.
[5] Canadam & T. Mulnke (1994), Forced Vibrations of a Piezoelectric Layer of 6 (mm) Crystal Class.
Journal of Proc. Indian National Sci. Acad., Vol. 60(A), No. 3. pp. 551–561.
[6] V. C. Carey & J. C. Mollendorf (1980), Variable Viscosity Effects in Several Natural Convection
Flows. Journal of Heat & Mass Transfer, Vol. 23, pp. 95–109.
[7] W. Cazemier, R.W.C.P. Verstappen & A. E. P. Veldman (1998), Proper Orthogonal Decomposition &
Low-Dimensional Models for the Driven Cavity Flow. Journal of Physics of Fluids, Vol. 10, pp.
1685–1699.
[8] T. Cebeci & P. Bradshaw (1977), Momentum Transfer in Boundary Layers. Journal of Washington,
Hemisphere Publishing Corporation, Vol. 8, pp. 391–402.
[9] T. Cebeci & K. C. Chang (1977), A General Method for Calculating Momentum and Heat Transfer in
Laminar and Turbulent Duct Flows. Journal of Numerical & Heat Transfer, Vol.1, pp. 39–68.
[10] T. Cebeci, F. Thiele, P. G. Williams & K. Stewartson (1979), On the Calculation of Symmetric
Waves–I, Two–Dimensional Flow. Journal of Numerical & Heat Transfer, Vol. 2, pp. 35–60.
[11] I. Celik & Z. Wei–Mung (1995), Calculation of Numerical Uncertainty Using Richardson
Extrapolating Application to Some Simple Turbulent Flow Calculations. Journal of Fluids
Engineering, Vol. 117, No. 3, pp. 439–446.
[12] Chamkha & H. Al–Naser (2002), Hydro Magnetic Double–Diffusive Convection in a Rectangular
Enclosure with Uniform Side Heart and Mass Fluxes and Opposing Temperature and
Concentration Gradients. Journal of Thermal Sciences, Vol. 41, No. 10, pp. 936–948.
[13] O. P. Chandna & E. O. Oku–Ukpong (1994), Some Solutions of Second Grade Fluid Flow Von
Misses Co–Ordinates Transformations. Journal of Engineering Sci., Vol. 32, No. 2, pp. 257–266.
[14] D. S. Chauhan & R. Agarwal (2011), MHD through a Porous Medium Adjacent to a Stretching
Sheet Numerical and an Approximation Solution. The European Physical Journal, Vol. 126, pp. 43–
49.
[15] G. Chavepeyer, J. K. Plat Ten, & M. B. Bada (1995), Laminar Thermal Convection in a Vertical Slot.
Applied Scientific Research, Clawer Academic Publishers Printed in Netherlands, Vol. 55, pp. 1–29.
[16] A. Gulhan, T. Thiele, F. Siebe, B. Kronen & T. Schleutker (2019), Aero Thermal Measurements
from the ExoMars Schiaparelli Capsule Entry. Spacecraft Rockets, Vol. 56, No.1, pp. 68–81.
[17] I. Hashem & M. H. Mohamed (2018), Aerodynamic Performance Enhancements of H-Rotor
Darrieus Wind Turbine. Journal of Energy, Vol. 142, pp. 531–545.
[18] M. Jafari, A. Razavi & M. Mirhosseini (2018), Affect of Airfoil Profile on Aerodynamic Performance
and Economic Assessment of H-Rotor Vertical Axis Wind Turbines. Journal of Energy, Vol. 165, pp.
792–810.
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