steady plane couette flow of viscous incompressible …

www.ijcrt.org © 2020 IJCRT | Volume 8, Issue 12 December 2020 | ISSN: 2320-2882

IJCRT2012009 International Journal of Creative Research Thoughts (IJCRT) www.ijcrt.org 50

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

http://www.ijcrt.org/



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




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




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




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




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




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


2

0 00 12

26, 5, 6 &

2

v BA

k


2 22 20 0 0 011

& & .2 2k


k




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


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





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


2

0 00 12

26, 5, 6 &

2

v BA

k





2 22 20 0 0 011

& & .2 2k


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


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





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




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


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




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


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.




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.


steady plane couette flow of viscous incompressible …

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