Steady Mhd Couette Flow Of An Incompressible Viscous Fluid Through A Porous Medium Between

The International Journal Of Engineering And Science (IJES) || Volume || 3 || Issue || 9 || Pages || 18-37 || 2014 || ISSN (e): 2319 – 1813 ISSN (p): 2319 – 1805

Steady MHD Couette Flow of An Incompressible Viscous Fluid Through A Porous Medium Between Two Infinite Parallel Plates Under Effect Of Inclined Magnetic Field 1,

L.Sreekala , 2,E.Kesava Reddy

1, 2,

Department of Mathamatics, CRIT,Anantapur.

Department of Mathamatics, JNTUA,,Anantapur

-------------------------------------------------------ABSTRACT--------------------------------------------------In this paper, we discuss the steady hydro magnetic three dimensional couette flow of an incompressible viscous flow through a porous medium between two infinite parallel plates under the influence of effect of inclined magnetic filed, and taking viscous dissipation into account. The stationary plate is subjected to a slightly sinusoidal surface temperature with transverse sinusoidal suction velocity while moving plate is isothermal with injection velocity. The velocity and the temperature are evaluated by using perturbation technique. The expressions for the components of skin friction at the stationery plate and the insulated plate and the rate of heat transfer are also obtained and its behaviour computationally discussed with reference to the various governing parameters in detail.

KEYWORDS: viscous fluid, steady flow, porous medium infinite parallel plates, MHD couette flow --------------------------------------------------------------------------------------------------------------------------------------Date of Submission: 19 September 2014 Date of Publication: 30 September 2014 ---------------------------------------------------------------------------------------------------------------------------------------

I.

INTRODUCTION:

Heat transfer in porous media has received considerable attention and has been the field of a number of investigations during the last two decade. The need for fundamental studies in porous media heat transfer stems from the fact that a better understanding of a host of thermal engineering applications in which porous materials are present is required. Some of examples of thermal engineering disciplines which stand to benefit from a better understanding of heat and fluid flow processes through porous media are geothermal systems, thermal insulations, grain storage, solid matrix heat exchangers, oil extraction and manufacturing numerous products in the chemical industry. The application of electromagnetic fields in controlling the heat transfer as in aerodynamic heating leads to the study of magneto hydrodynamic heat transfer. This MHD heat transfer has gained significance owing to recent advancement of space technology. The MHD heat transfer can be divided in two parts. One contains problems in which the heating is an incidental by product of electro magnetic fields as in MHD generators and pumps etc, and the second consists of problems in which the primary use of electromagnetic fields is to control the heat transfer (Toshivo Tagawa [39]). Heat transfer in channels partially filled with porous media has gained considerable attention in recent years because of its various applications in contemporary technology. These applications include porous journal bearing, blood flow in lungs or in arteries, nuclear reactors, porous flat plate collectors, packed bed thermal storage solidification of concentrated alloys, fibrous and granular insulation, grain storage and drying, paper drying, and food storage. Besides the use of porous subtracts to improve heat transfer in channels, which is considered as porous layers, finds applications in heat exchangers, electronic cooling, heat pipes, filtration and chemical reactors etc. In these applications engineers avoid filling entire channel with a solid matrix to reduce the pressure drop.

The flow between parallel plates is a classical problem that has important applications in magneto hydro dynamic power generators and pumps, accelerators, aerodynamic heating, electrostatic precipitation, polymer technology, petroleum industry, purification of crude oil and fluid droplets and spays. Hastmann and lazarus [13] studied the influence of a transverse uniform magnetic field on the flow of viscous incompressible electrically conducting fluid between two infinite parallel stationary and insulating plates. Then the problem was extended in numerous ways. Closed form solutions for the velocity fields and temperature distributions were obtained [1, 8, 36 and 38] under different physical effects. Some exact and numerical solutions for the heat transfer problem are found by Nigam (20) and Attia and Kotb (5). Attia and Kotb (5) Studied the steady MHD fully developed flow and heat transfer between two parallel plates with temperature dependent viscosity in the presence of uniform magnetic field.. Later Attia [4] extended the problem to the transient state.

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Steady Mhd Couette Flow Of An Incompressible…

The heat transfer in a viscous incompressible fluid neglecting the buoyancy force has been studied by several authors notably Alpher(1), Roming(24), Perimutten et al(21) Sielgel(25), Viskanta(40), Jain and Bansal(16), Gopalan and Sundaram(11), Hwang and Hong(15), Balaram(6), Hahn(12), Radha Krishnamacharya et al (22). But it was show by Gill and Casal (10) that viscosity variations and temperature differences on the horizontal forced convection flow between two parallel infinite plates can induce such flows as might significantly increase or decrease the tendency towards instability. Many authors have shown that the fluids with low Prandtl numbers are to a good deal more sensitive to gravitational field effects than the fluids with high Prandl number. Mehta and Shobha Sood (19) investigated the effect of temperature dependent viscosity on the heat transfer rate for a transient free convection flow along a non-isothermal vertical surface. They observed that a decrease in viscosity increases the heat transfer rate and reduces the time to reach steady state. Hazem A. Attia (14) studied the unsteady flow and heat transfer of a dusty conducting fluid between two parallel plates with variable viscosity and electric conductivity. The effect of the variation in the viscosity and electric conductivity of the fluid and the uniform magnetic field on the velocity and temperature fields for both the fluid and dust particles was discussed. O. Jumbul et al. (17) studied laminar heat transfer in parallel plates and circular ducts subject to uniform wall temperature by taking into account both viscous dissipation and fluid axial heat conductions in an infinite region. And extension of this work was done by O.Jambal et al. (18). Later M. Guria et al (12) studied the unsteady couette flow of a viscous incompressible fluid confined between parallel plates, rotating with a uniform angular velocity about an axis normal to the plates, here the flow was induced by the motion of the upper plate and the fluid and plates rotate in unison with the same angular velocity.

Channel flows of a Newtonian fluid with heat transfer have been studied by many authors [3, 4 and 8].

Gersten and Gross [9] imposed slightly sinusoidal transverse suction velocity on two dimensional flows over a plane wall and studied the resulting three dimensional flow and heat transfer. Since then, many authors have used this idea in different two dimensional flow problems. Singh et al [26, 31, 33 and 35] studied three dimensional free convection flow past an infinite vertical plate with periodic suction. The effect of viscous dissipation on three dimensional flow along a porous plate was studied by Singh [28]. Hydro magnetic effects and porous medium effects on these flows have also been studied by Singh [29, 30]. The three dimensional couette flow with transpiration cooling was studied by Singh [27], which was further extended to include MHD effects by Singh and Sharma [32, 34]. All papers quoted above, have been concerned with transverse suction velocity which causes the flow to be three dimensional. Rees [23] has considered the effect of a sinusoidal surface temperature distribution in a span wise direction, on the steady laminar free convection boundary layer flow a long the vertical surface, which also causes the flow to be three dimensional. Later Tak and Vyas [37] studied three dimensional couette flow and beat transfer in porous medium, taking viscous dissipation into account and assuming that the stationery plate exhibits a sinusoidal temperature and sinusoidal suction velocity, and the moving plate is isothermal with uniform injection velocity.

II.

FORMULATION AND SOLUTION OF THE PROBLEM:

We consider the steady hydro magnetic three dimensional couette flow of an incompressible viscous fluid through a porous medium between two infinite parallel plates under the effect of inclined magnetic field with an angle of inclination  . The stationary plate is being on xz-plane and the other plate at a distance h, moving with uniform velocity U along x-direction. The stationery plate is subjected to sinusoidal surface temperature and sinusoidal suction velocity both, varying in z-direction and the moving plate is assumed to be isothermal with a uniform injection velocity and taking viscous dissipation into account. The steady hydro magnetic equations through a porous medium with inclined magnetic field are governed by the following equations. v y



w z

0

(2.1)

 u  u v  w     2 2 y z z  y u

v

v y

u

 w

v z

2

 

2

2

  2v    y 2  y  1 p



w

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1 p

2

2



u 



u

2

(2.2)

k

2 2 2 2  v  σ μ e H 0 Sin    v  v 2  z   k

 w  w v  w       2 2 y z  z z  y w

2

σ μ e H 0 Sin     

 σ μ e H 0 Sin     w  w  k  2

2

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(2.3)

2

(2.4)

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Steady Mhd Couette Flow Of An Incompressible…  T  T T   T    K1     C p  v  w  2 2    z  z   y  y Where  is the viscous dissipation function given by 2

 v  w   2    2 2 z  y 2

2

u , v and w

2

  u         y  

   

2

 w w    z  y

(2.5)

   

2

2   u     ,   z   

are the velocity components in the directions x, y and z respectively.  is the density of

the fluid, p is the pressure, 

is the conductivity of the medium, μ e is the magnetic permeability,  is the

coefficient of viscosity , k is the permeability of the medium, H 0 is the applied magnetic field,  is the kinematic viscosity, T is the temperature, K 1 is the thermal conductivity and C

p

is the specific heat of the

fluid at constant pressure.

The boundary conditions are,  z   v   V  1   cos ( ) h  

u  0,

 z  w  0 , T  T 1  1   cos ( h 

 ) 

 u  U , v  V , w  0  T  T 2 , T 2  T1   V  h  p  k 

     

, p  0

y  0

at

(2.6)

y  h

at

(2.7)

Where   1 , U and V are constants with dimension of velocity, h and T 1 dimension of length and temperature respectively.

We introduce the non-dimensional variables are z  *

z

y

,y  *

h

,u

u



*

h

v

,v  *

U

, w  *

U

w

,

p

U

*



p ρU

2

,  

T - T1 T 2  T1

are constants with

.

Using the non-dimensional variables the governing equation reduces to v y



w z

0

(2.8)

1  u  u  v  w   2 2  y z R  y z u

v

v

v

v y

w y  y

u

 w

 w

 w

v z

w z  z

2

p

 

y

 

p z



2

2 1  v  2 R   y



2

2

1       2 2  PR   y z

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2

2

Sin

2

 D 1

 u 

RD

2  v  M  2   z  

1  w  w   2 2  R  y z

 2



 M      

2

Sin

2

 D 1

 v 

RD

 M      

2

Sin

(2.9)

2

 D 1

RD

 E    R 

 w 

(2.10)

(2.11)

(2.12)

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Steady Mhd Couette Flow Of An Incompressible…

Where the viscous dissipation function given by 2    v    w        2        z      y  2

 u    y

2

  w w      z   y

2

  u        z  

2

(2.13)

Corresponding boundary conditions are u  0 , v    1   cos (  z )    w  0 ,   a  cos (  z )   p  0 

y  0

at

(2.14)

and u  1, v    ,

  w  0,   1, p  d 

y 1

at

(2.15)

Where the non-dimensional parameters are R 

D 

Uh



E 

C

p

is the Prandtl number

K1 2

k h

P 

is the Reynolds number

is the Darcy parameter

2

U

M

2



σ μe H



2

C p (T 2  T 1 )

is the Eckert number

2 0

 

V

h

2

is the Hartmann number d 

is the Suction parameter

U

 R D

In order to solve the above partial differential equations, we assumed to that u ( y , z )  u 0 ( y )   u1 ( y , z ) v ( y , z )  v 0 ( y )   v1 ( y , z ) w ( y , z )   w1 ( y , z )

(2.16)

p ( y , z )  p 0 ( y )   p1 ( y , z )

 ( y, z)   0 ( y)   1 ( y, z)

This is valid, since it is assumed that amplitude  (  1 ) of sinusoidal suction velocity is small compare to its wave length. Substituting equation (2.16) in (2.8) to (2.15) and taking   0 . The unperturbed quantities satisfies the following equations dv

 0

0

(2.17)

dy 2

d u0

D

dy dp

2

 R D

du

0

 (M

2

2

Sin



dy

D  1) u 0  0

 d

0

(2.18) (2.19)

dy

d 0 2

dy

2

 du 0  PR   EP  dy  dy du

0

   

2

 0

(2.20)

With the boundary conditions u 0  0, v0       0  0 , p0  0 

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at

y  0

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(2.21)

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Steady Mhd Couette Flow Of An Incompressible… u 0  1,v0       0  1 , p0  d 

y 1

at

(2.22)

The solutions of the equations (2.17) to (2.20) making use of the boundary conditions (2.21) and (2.22) (2.23) v 0 ( y )   e

u0 ( y) 

m1 y

 e

m2y

m1

 e

m1

e

(2.24)

p0 ( y)  d y

(2.25)  ( PR  ) y

 0 ( y)  C1  C 2e

 m 1e 1 m 2e 2   m 2   e 1 )  2 ( 2 m 1  PR  ) 2 ( 2 m 2  PR  ) 2m y

EP

 (e

m1

2m

y

( m1  m 2 ) y

  ( m 1  m 2 )  ( PR  )( m 1  m 2 )  2 m 1m 2e



(2.26)

2

Likewise the perturbed quantities in equations (2.17) to (2.20) satisfies the following equations v1 y

v1

w1



z

u 0







y

v1 y

y

y

p1 y

p



2

2 1   v1  2 R   y



 0 y

 M      

2

 M      

 1 1   1   2 2  PR   y z 2



2

2  v1   M  2   z  

 w1 1   w1   2 2  R  y z 2



z

 v1

 u1 1   u1   2 2  R  y z 2



y

 

 1

(2.27)

u1

 

w1

 v

0

2

Sin

2

 D 1

 u 1 

RD 2

Sin

2

 D 1

 v 1 

RD 2

Sin

2

 D 1

 w 1 

RD

 2 E u 0 u1   R y y 

(2.28)

(2.29)

(2.30)

(2.31)

Corresponding the boundary conditions u 1  0 , v 1    cos (  z ) w 1  0 ,  1  a cos (  z ) p1  0

    

y  0

at

(2.32)

and u 1  0 , v1    w1  0,  1  0 p1  0

    

y 1

at

(2.33)

The equations (2.27) to (2.31) are linear partial differential equations which describes perturbed three dimensional flow due to variation of the suction velocity stationery surface temperature along three directions.

The form of the suction velocity stationery surface temperature suggests the following forms u 1  u 2 ( y ) cos (  z ) (2.34) v 1  v 2 ( y ) cos (  z )

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(2.35)

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Steady Mhd Couette Flow Of An Incompressible… w1  

1



v 2 ( y ) cos (  z ) '

(2.36)

p 1  p 2 ( y ) cos (  z )

(2.37)

 1   2 ( y ) cos (  z )

(2.38)

The expressions for v 1 and w 1 have been chosen so that the equation of continuity (2.27) is trivially satisfied. The equations (2.29) and (2.30) being independent of the main flow and temperature field can be solved initially. Therefore substituting equations (2.35), (2.36) and (2.37) are in equations (2.29) and (2.30).Hence the following simultaneous ordinary differential equations are obtained.

3

2

d v2

D

dy

 RD 

3

2

dy

2

dy

d v2

D

d v2

 RD 

2

dv

 (D

 (D

2

 M

2

 M

2

2

2

Sin

Sin

2

dy

2

  1 ) v 2   RD p 2 2

dp

  1 ) v 2  RD

2

(2.39) (2.40)

dy

Corresponding the boundary conditions

v 2   dv

and

 0

2

at

y  0

at

y 1

(2.41)

dy v2  0 dv

and

 0

2

(2.42)

dy

Solving the equations (2.39) and (2.4.) using the equations (2.41) and (2.42), which is straight forward and is substituted in equations (2.35) to (2.37). There fore we obtain the following expressions. v 1  C 3 e

w1   p1 

1



C

1

 RD 2

 C 4e

m3 y

3

e

7

 C 5e

 C 4e

m3 y

C

m4 y

e

m3 y

 C 6e

 C 5e

m4 y

 C 8e

 y

m4 y

 y

 y

 cos

 C 6e

 C 9e

 y

( z )

 cos

 y

 C 10 e

 y

(2.43)

( z )

 cos

(2.44)

( z )

(2.45)

Now for the main flow and the temperature field, the expressions (2.34) and (2.38) are substituted in equations (2.28) and(2.31) to give the following ordinary differential equations 2

d u2

D

dy

2

d 2

 RD 

du

 (D

2

2

 M

dy

2

dy

2

 PR 

2

   2  v 2 PR 2

2

Sin   1 ) u 2  RD v 2

d 0

2

 2 PE

du

dy

dy

0

du

2

du

0

(2.46)

dy

(2.47)

dy

Corresponding the boundary conditions u2  0

and

(2.48) 2  0

at

y  0

and

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Steady Mhd Couette Flow Of An Incompressible… u2  0

and

(2.49) 2  0

y 1

at

Solving equations (2.46) and (2.47) with boundary conditions (2.48) and (2.49) and substituting the solutions in the equations (2.34) and (2.35), the expressions for u 1 and  1 can be written as u1 



C 11 e

m3 y

 C 12 e

R



m4 y

e

m1

 e

m2

( B1e

( m1  m 3 ) y

 B2e

( m1  m 4 ) y

 B3e

( m1   ) y

 (2.50)

  G  e a 2 ( F  a )  a y  G  e a1 ( F  a )  a y   e 1    e 2  f ( y )  cos (  z ) a2 a1 a1 a2 e e e e     

 1   

(2.51)

The skin friction components  x and  z along main flow and transverse direction and the non dimensional rate of surface heat transfer (Nusselt number) at both the plates can be calculated as 

x



d

 du 0   U  dy x

     y0

 du 2    dy

 m1  m 2    m   m2 1 e e 

R

 (e

m1

 e

m2





  cos(  z )   y0 R m 5 ( B  Ae

 (e

e

m6

m5

)( e

m6

)

e

m1

m2

R m 6 ( B  Ae

 )

(e

m6

e

m5

)( e

z



d

)

e

m2

 )

B1 ( m 1  m 3 )  B 2 ( m 1  m 4 )  B 3 ( m 1   )  B 4 ( m 1   ) 

)

 B1 ( m 1  m 3 )  B 2 ( m 1  m 4 )  B 3 ( m 1   )  B 4 ( m 1   )



m1

m6

  cos(

 z)

 dw 1       U  dy  y  0 z

 



C



m 3  C 4 m 4  C 5 2

3

2

2

 C 6

2



sin(  z )

Nusselt number at the stationery plate Nu  

 T      T 2  T1   y  y  0 h

   m1 m2 2m1m 2 EP     PR  C 2      m1 m2 2  2 m 2  PR  m 1  m 2  PR   (e  e )  2 m 1  PR   2

 

 G  e a2 ( F  a )   a a (e 2  e 1 ) 

2

  G  e 1 (F  a)  a1   a a   (e 1  e 2 )   a

   '  a 2  f ( 0 )  cos(  z )     

Nusselt number at the moving plate Nu  

 T      T 2  T 1   y  y 1 h

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Page 24

Steady Mhd Couette Flow Of An Incompressible… m m

  m1 e 1 m2e 2 2 m 1m 2e 1 2  EP  PR      PR  C 2 e       m m 2 ( e 1  e 2 )  2 m 1  PR  2 m 2  PR  m 1  m 2  PR    2

 G  e a2 ( F  a )   a a (e 2  e 1 ) 



III.

2m

2

2m

  G  e 1 (F  a) a  a1e 1   a a   (e 1  e 2 )   a

   a '  a 2 e 2  f (1 )  cos(  z )     

RESULTS AND DISCUSSION:

The computational analysis is carried out to discuss the behaviour of the velocity and temperature for different variations in the governing parameters Reynolds number R, Darcy parameter D, Hartmann number M,  the suction parameter, E Eckert number, P Prendtl number. In this chapter we discuss the steady hydro magnetic three dimensional couette flow of an incompressible viscous flow through a porous medium between two infinite parallel plates under the influence of transverse magnetic field and taking viscous dissipation into account. The stationary plate is subjected to a slightly sinusoidal surface temperature with transverse sinusoidal suction velocity while moving plate is isothermal with injection velocity. The velocity, temperature and the expressions for the components of skin friction and also the rate of heat transfer at the stationary plate and recovery factor at the insulated plate are obtained. The figures 1-4, 5-7, 8-11 represent the velocity profiles for u, v and w and for z  0 ,   0 . 1 . The temperature profiles for  displayed in the figure 12-17 for z = 0 level and for  = 0.1, a = 0.78,  



.

3

At z = 0 level, the velocity components u, v and w enhances with increasing the Reynolds number R, suction parameter  and Darcy parameter D, while it decreases with increasing the Hartmann number M fixing the other parameters Fig (1-11). The temperature  enhances with increasing the Reynolds number R, suction parameter  , Darcy parameter D, Prandtl number P, Eckert number E while it reduces with increasing the intensity of the magnetic field (Hartmann number M ) at z = 0 level (Fig 12-17).

The absolute values of the skin friction components  x in plane z = 0, z = ½ and  z in the plane z = 0 and z = ½ are evaluated and tabulated in the tables (I-IV). The magnitude of  x enhances with increasing Reynolds number R, suction parameter  , Hartmann number M and Darcy parameter D for both z = 0 and z = ½ (tables I & II). Like wise the magnitude of  z enhances with increasing Reynolds number R, suction parameter  , and Darcy parameter D while it reduces with increasing intensity of the magnetic field M for both z = 0 and z = ½ (tables III & IV).

Tables (V-VIII) represent the variation of Nusselt number Nu for z= 0 and z = ½ and at y = 0 and 1.

The rate of surface Heat transfer (Nusselt number) enhances with Reynolds number R, suction parameter  , Darcy parameter D, Prandtl number P and Eckert number E and reduces with increasing the magnetic parameter M. This can be seen that for all z and both the plates.

IV.

CONCLUSIONS:

1. The velocity components u, v and w enhances with increasing the Reynolds number R, suction parameter  and Darcy parameter D, while it decreases with increasing the Hartmann number M fixing the other parameters at z = 0 level and  



.

3

2. The temperature  enhances with increasing the Reynolds number R, suction parameter  , Darcy parameter D, Prandtl number P, Eckert number E while it reduces with increasing the intensity of the magnetic field (Hartmann number M) at z = 0 level.

3. The magnitude of 

x

enhances with increasing Reynolds number R, suction parameter  , Hartmann number

M and Darcy parameter D for both z = 0 and z = ½. Like wise the magnitude of  z enhances with increasing Reynolds number R, suction parameter  , and Darcy parameter D while it reduces with increasing intensity of the magnetic field M for both z = 0 and z = ½ at  



.

3

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Page 25

Steady Mhd Couette Flow Of An Incompressible…

5.The rate of surface Heat transfer (Nusselt number) enhances with Reynolds number R, suction parameter  , Darcy parameter D, Prandtl number P and Eckert number E and reduces with increasing the magnetic parameter M. This can be seen that for all z and both the plates.

GRAPHS AND TABLES The velocity profiles in the plane z=0: 1

1

0.8

0.8 R=50

0.6

D=500 0.6

R=200

u 0.4

D=1000

u

R=400

D=3000

0.4

R=1000

D=5000

0.2

0.2

0

0 0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

y

0.6

0.8

1

y

Fig. 1: The velocity component u with R M=2, D=500,  =0.01,   0 . 1

Fig. 2: The velocity component u with D M=2, R=50,  =0.01,   0 . 1

1

1

0.8

0.8 α=0.01

M=2 0.6

0.6

M=5

u 0.4

α=0.03

u

M=8

α=0.04

0.4

α=0.05

M=10 0.2

0.2

0

0 0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

y

0.8

1

y

Fig. 4: The velocity component u with  R=50, D=500, M=2,   0 . 1

Fig. 3: The velocity component u with M R=50, D=500,  =0.01,   0 . 1

-0.198

-0.198 0

0.2

0.4

0.6

0.8

1

-0.2

-0.2 -0.202

R=200 R=1000

0.6

0.8

1

D=500

-0.204

D=1000

-0.206

D=3000 D=5000

-0.208

-0.208

-0.21

-0.21

-0.212 y

y

Fig. 5: The velocity component v with R   0 . 05 ,   0 . 2 , D=500, M=2

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0.4

v

R=400

-0.206

0.2

-0.202

R=50

v -0.204

0

Fig. 6: The velocity component v with D   0 . 05 ,   0 . 2 , R=50, M=2.

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Page 26

Steady Mhd Couette Flow Of An Incompressible…

-0.198 0

0.2

0.4

0.6

0.8

0.008

1

-0.2

0.007

-0.202

0.006

M=2

v -0.204

R=200

w 0.004

M=8

R=400

0.003

M=10

-0.206

R=50

0.005

M=5

R=1000

0.002 0.001

-0.208

0

-0.21

0

0.2

0.4

y

0.6

0.8

1

y

Fig. 7: The velocity component v with M   0 . 05 ,   0 . 2 , R=50, D=500

Fig. 8: The velocity component w with R   0 . 1 , M=2, D=500,  =0.01

0.007

0.01

0.006

0.008

0.005

M=2

0.004

M=5

D=3000

0.003

M=8

D=5000

0.002

M=10

D=500 0.006

D=1000

w 0.004

w

0.002

0.001 0

0 0

0.2

0.4

0.6

0.8

0

1

0.2

0.4

0.6

0.8

1

y

y

Fig. 9: The velocity component w with D

.

10: The velocity component w with M   0 . 1 , R=50, D=500,  =0.01

  0 . 1 , M=2, R=50,  =0.01

0.01 0.009 0.008 0.007 0.006 w 0.005 0.004 0.003 0.002 0.001 0

α=0.01 α=0.03 α=0.04 α=0.05

0

0.2

0.4

0.6

0.8

1

y

Fig 11: The velocity component w with    0 . 1 , R=50, D=500, M=2

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Steady Mhd Couette Flow Of An Incompressible…

The temperature profiles in the plane z = 0: 1 0.9 0.8 0.7 0.6 θ 0.5 0.4 0.3 0.2 0.1 0

1 0.8 P=1.5

R=50 R=200

0.6

P=2.5

0.4

P=4

θ

R=400 R=1000

P=5.5 0.2 0

0

0.2

0.4

0.6

0.8

0

1

0.5

1

y

y

Fig.12: The temperature  with R for   0 . 1 P=1.5, E=0.01, M=2, D=500,  =0.01, a=0.78

Fig.13: The temperature  with P for  0 . 1 R=50, E=0.01, M=2, D=500,  =0.01, a=0.78 1

1

0.8

0.8

M=2

D=500 0.6

0.6

D=1000

θ 0.4

M=5

θ

D=3000

M=8

0.4

M=10

D=5000

0.2

0.2

0

0 0

0.2

0.4

0.6

0.8

0

1

0.2

0.4

0.6

0.8

1

y

y

Fig.14: The temperature  with D for   0 . 1 P=1.5, E=0.01, M=2, R=50,  =0.01, a=0.78

Fig.15: The temperature  with M for   0 . 1 P=1.5, E=0.01, R=50, D=500,  =0.01, a=0.78

1

1

0.8

0.8 α=0.01

E=0.01 0.6

0.6

E=0.02

θ

α=0.03

θ

E=0.03

0.4

α=0.04

0.4

α=0.05

E=0.04 0.2

0.2

0

0 0

0.2

0.4

0.6

0.8

1

0

y

Fig.16: The temperature  with E for   0 . 1 P=1.5, R=50, M=2, D=500,  =0.01, a=0.78

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0.2

0.4

0.6

0.8

1

y

Fig.17: The temperature  with  for  0 . 1 P=1.5, E=0.01, M=2, D=500, R=50, a=0.78

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Steady Mhd Couette Flow Of An Incompressible…

Table I R 50 200 400 1000

D



M

I 0.004225 0.0062458 0.008315 0.015615

II 0.005821 0.006825 0.009525 2.018347

I 500 0.01 2

II 1000 0.01 2

III 0.006534 0.007263 0.009845 0.024345 III 3000 0.01 2

IV 0.005258 0.006314 0.008531 0.016315

V 0.005445 0.006555 0.008825 0.017315

VI 0.008241 0.009811 0.015444 0.024554

VII 0.012225 0.015215 0.017225 0.034511

IV 500 0.03 2

V 500 0.05 2

VI 500 0.01 5

VII 500 0.01 8

Skin friction coefficient (  x ) at z=0 level for   0 . 1 Table II R 50 200 400 1000

I 0.002558 0.003125 0.004552 0.005252

II 0.003185 0.005045 0.006311 0.006341

III 0.003545 0.005569 0.006814 0.007845

IV 0.003634 0.003878 0.004875 0.005805

V 0.004535 0.004838 0.005266 0.006215

VI 0.005245 0.006835 0.007218 0.007404

VII 0.006504 0.007001 0.009215 0.014265

D  M

I 500 0.01 2

II 1000 0.01 2

III 3000 0.01 2

IV 500 0.03 2

V 500 0.05 2

VI 500 0.01 5

VII 500 0.01 8

VII 0.183244 0.366742 0.420022 0.530155

Skin friction coefficient (  x ) at z=0.5 level for   0 . 1 Table III R 50 200 400 1000

D  M

I 0.321854 0.566715 0.752055 0.834155 I 500 0.01 2

II 0.384058 0.583252 0.852055 0.943255

III 0.402445 0.620052 0.921455 0.967354

IV 0.380458 0.586585 0.788151 0.85215

V 0.402145 0.626285 0.847255 0.952698

VI 0.281445 0.424658 0.586045 0.690581

II 1000 0.01 2

III 3000 0.01 2

IV 500 0.03 2

V 500 0.05 2

VI 500 0.01 5

VII 500 0.01 8

Skin friction coefficient (  z ) at z=0 level for   0 . 1 Table IV R 50 200 400 1000

I 0.124458 0.310541 0.512414 0.655488

II 0.243458 0.526545 0.622555 0.712784

III 0.321458 0.621155 0.756841 0.821585

IV 0.282452 0.505151 0.798552 0.841585

V 0.483652 0.580150 0.884145 0.952144

VI 0.094265 0.183655 0.326555 0.420052

VII 0.052654 0.153544 0.281555 0.365559

D  M

I 500 0.01 2

II 1000 0.01 2

III 3000 0.01 2

IV 500 0.03 2

V 500 0.05 2

VI 500 0.01 5

VII 500 0.01 8

Skin friction coefficient (  z ) at z=0.5 level for   0 . 1

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Steady Mhd Couette Flow Of An Incompressible…

Table V

The Nusselt number ( Nu ) at z=0 level for   0 . 1 , at y=0 (stationary plate) .

Table V1

D  M P E

I 500 0.01 2 1.5 0.01

II 1000 0.01 2 1.5 0.01

III 3000 0.01 2 1.5 0.01

IV 500 0.03 2 1.5 0.01

V 500 0.05 2 1.5 0.01

VI 500 0.01 5 1.5 0.01

VII 500 0.01 8 1.5 0.01

VIII 500 0.01 2 2.5 0.01

IX 500 0.01 2 4 0.01

X 500 0.01 2 1.5 0.02

XI 500 0.01 2 1.5 0.03

X 500 0.01 2 1.5 0.02

XI 500 0.01 2 1.5 0.03

The Nusselt number ( Nu ) at z=0 level for   0 . 1 , at y=1 (moving plate) Table VI1

D  M P E

I 500 0.01 2 1.5 0.01

II 1000 0.01 2 1.5 0.01

III 3000 0.01 2 1.5 0.01

IV 500 0.03 2 1.5 0.01

V 500 0.05 2 1.5 0.01

VI 500 0.01 5 1.5 0.01

VII 500 0.01 8 1.5 0.01

VIII 500 0.01 2 2.5 0.01

IX 500 0.01 2 4 0.01

The Nusselt number ( Nu ) at z=0.5 level for   0 . 1 , at y=0 (stationary plate)

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Page 30

Steady Mhd Couette Flow Of An Incompressible… .

D  M P E

Table VIII

I 500 0.01 2 1.5 0.01

II 1000 0.01 2 1.5 0.01

III 3000 0.01 2 1.5 0.01

IV 500 0.03 2 1.5 0.01

V 500 0.05 2 1.5 0.01

VI 500 0.01 5 1.5 0.01

VII 500 0.01 8 1.5 0.01

VIII 500 0.01 2 2.5 0.01

IX 500 0.01 2 4 0.01

X 500 0.01 2 1.5 0.02

XI 500 0.01 2 1.5 0.03

The Nusselt number ( Nu ) at z=0.5 level for   0 . 1 , at y=1 (moving plate) ACKNOWLEDGEMENTS : The authors very much thankful to authorities of JNTU, Anantapur, Andhra Pradesh, India, providing necessary facilities to have done this work and IJES journal for the support to develop this document.

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Appendix: C1 

1 (e

 PR 

 EP  m1 m 2  1)  ( e  e 1)



2

1

 (e

 PR 

 EP  m1 m 2  1)  ( e  e 1)

m1 

 M 2 ( R )  4   

 e



2

2 m1

)

2 ( 2 m 1  PR  )

(m1  m 2 )



 R 

 PR 

2 m 1m 2 (e



C

m 1 (e

m 1 (e

2 m1

 PR 

 1)

2m1m

Sin

2

2

)

m 2 (e

 PR 

2(2m

 e

( m1  m 2 )

2

 e

2

2

(e



m

2

(e

)

2(2m

( m1  m 2 )

2

2m2



 1)

 PR  )

 1)

 ( PR  )( m 1  m

2m2

)

 PR  )

 ( PR  )( m 1  m 2 )

2

2 ( 2 m 1  PR  )

(m1  m



 2

)



  1 



  1 

 D 1

D

  

2  R 

m2 

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 M 2 ( R )  4   

2

Sin

2

 D 1

D

  

2

The IJES

Page 32

Steady Mhd Couette Flow Of An Incompressible…

C3 

D1 D

D2

, C4 

D

, C5 

D3 D

D4

, C6 

D

D   a 4 (  a 10   a 9 )  a 5 ( m 4 a 10   a 8 )  a 6 ( m 4 a 9   a 8 )     a 3 (  a 10   a 9 )  a 5 ( m 3 a 10   a 7 )  a 6 ( m 3 a 9   a 7 )     a 3 ( m 4 a 10   a 8 )  a 4 ( m 3 a 10   a 7 )  a 6 ( m 3 a 8  m 4 a 7 )    a 3 ( m 4 a 9   a 8 )  a 4 ( m 3 a 9   a 7 )  a 5 ( m 3 a 8  m 4 a 7 ) 

D 1     a 4 (  a 10   a 9 )  a 5 ( m 4 a 10   a 8 )  a 6 ( m 4 a 9   a 8 )  D 2    a 3 (  a 10   a 9 )  a 5 ( m 3 a 10   a 7 )  a 6 ( m 3 a 9   a 7 )  D 3     a 3 ( m 4 a 10   a 8 )  a 4 ( m 3 a 10   a 7 )  a 6 ( m 3 a 8  m 4 a 7 )  D 4   a 3 ( m 4 a 9   a 8 )  a 4 ( m 3 a 9   a 7 )  a 5 ( m 3 a 8  m 4 a 7 )  a3  e

m3

, a4  e

a 7  m 3e

m3

m4

, a8  m 4e

 R 



a5  e ,

,

m4

( R )

2

m3 

a6  e



,



, a 9   e , a 10    e

  4  

2

D  M

2



2

Sin

D  1   



D 2

 R 

( R )

m4 

2

  4  

2

D  M

2

2

Sin

  

D 2

C 7  D C 3 m 3  RD  C 3 m 3  ( D 

2

 M

2

C 8  D C 4 m 4  RD  C 4 m 4  ( D 

2

 M

2

3

2

3

C 9  D C 5

3

C 10   D C 6 

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 D 1

2

 RD  C 5 

3

2

 RD  C 6 

 ( D

2

2

 ( D

 M

2

2

Sin

Sin

Sin

 M

2

2

  1 )C 3 m 3

2

  1 )C 4 m 4

2

  1 )C 5 

Sin

The IJES

2

  1 )C 6 

Page 33

Steady Mhd Couette Flow Of An Incompressible…

C 11 

C 12 

m6

R ( B  Ae (e

m6

m5

 e

)( e

m1

(e

m5

 e

m6

)( e

 e

m5

R ( B  Ae

)

m1

m2

)

)

 e

m2

)

A  ( B1  B 2  B 3  B 4 )  ( B 5  B 6  B 7  B 8 ) B  (B1e

( m1  m 3 )

 B2e

 (B5e

 R  m5 

B3 

B4 

B5 

B6 

 B6e

 2 ( R )  4   

( R )

m6 

B2 

(m2  m3 )

 B3e

( m1   )

(m2  m4 )

D  M

2

2

 B4e

 B7e

Sin

2

( m1  )

) 

(m2  )

 B8e

(m 2  )

)

 D 1   

D

2

 R 

B1 

( m1  m 4 )

2

  4  

2

D  M

2

Sin

2

 D 1   

D

2

D C 3m1 D ( m 1  m 3 )  RD  ( m 1  m 3 )  (  D  M 2

2

2

2

Sin

 D  1)

D C 4m1 D ( m 1  m 4 )  RD  ( m 1  m 4 )  (  D  M 2

2

2

2

Sin

 D  1)

D C 3m1 D ( m 1   )  RD  ( m 1   )  (  D  M 2

2

2

Sin

2

 D  1)

2

 D  1)

D C 4m1 D ( m 1   )  RD  ( m 1   )  (  D  M 2

2

2

Sin

D C 3m 2 D ( m 2  m 3 )  RD  ( m 1  m 3 )  (  D  M Sin  D  1 ) 2

2

2

2

D C 4m 2 D ( m 2  m 4 )  RD  ( m 2  m 4 )  (  D  M Sin  D  1 )

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2

2

The IJES

2

2

Page 34

Steady Mhd Couette Flow Of An Incompressible…

B7 

B8 

a1 

a2 

D C 3m 2 D( m2   )

2

D( m2   )

2

 RD  ( m 2   )  ( 

2

D  M

2

2

D  M

2

Sin

2

 D 1)

2

 D 1)

D C4m2

 PR  

 RD  ( m 2   )  ( 

( PR  )

2

 4

2

2

 4

2

Sin

2

 PR  

( PR  ) 2

F   A 1  A 2  A 3  A 4  A 5  A 6  A 7  A 8  A 9  A 10  A 11  A 12 

 A 13  A 14  A 15  A 16  A 17  A 18  A 19  A 20 G   A1 e

m 3  PR 

 A7 e

 A2e

m 4  2 m1

 A 13 e

 A8 e

m1  m 2  m 3

m 4  PR 

 A3e

  PR 

 A4e

m4 2m2

 A9 e

  2 m1

 A 10 e

 A 14 e

m1  m 2  m 4

 A 15 e

 (   PR  )

 2m2

m1  m 2 

 A 18 e

f ( y )   A1 e

( m 3  PR  ) y

 A5e  A 10 e

(  2 m 2 ) y

 A 14 e

 A 18 e

A1 

A2 

( m 3  2 m1 ) y

 A 11 e

(m1  m 2  m 4 ) y

(m1  m 6 ) y

 A2e

2

P ( m 3  PR  )

2

www.theijes.com

4

 PR  )

2

(m3  2m2 ) y

(  2m1 ) y

 A3e

(m1  m 2   ) y

(m1  m 5 ) y

 A 20 e

(  2m 2 ) y

 A 16 e

2

m3 2m2

  2 m 2

m1  m 2 

 A 17 e

m1  m 5

m1  m 5

 A4e

( m 4  2 m1 ) y

 A6 e

 A 12 e

 A 19 e

(   PR  ) y

 A7e

 A 12 e

 A 16 e

m 3  2 m1

  2 m1

 A 11 e

m1  m 6

 A 13 e

 A 20 e

 (   PR  ) y

 A8 e

(m1  m 2   ) y



(m4 2m2 ) y

(m1  m 2  m 3 ) y

 A 17 e

m1  m 6

 A9 e

(  2 m1 ) y





(m1  m 5 ) y



(m1  m 6 ) y

R C 3 C 2 2

 PR  ( m 3  PR  )  

P (m

 A6e

 A 15 e

 A 19 e

( m 4  PR  ) y

 A5e

2

R C 4 C 2 2

 PR  ( m

4

 PR  )  

2

The IJES

Page 35

Steady Mhd Couette Flow Of An Incompressible…

P R C 5 C 2 2

A3 

(   PR  )

2

 PR  (   PR  )  

A5 

A6 

A7 

A8 

A9 

A 10 

A 11 

A 12 

A 13 

(   PR  )

EPRm m1

(e

1

 e

m2

EPRm m1

(e

m1

2

 e

m1

(e

 e

m2

(e

(e

 e

 e

(e

)

(e

2

)

2

2 m2

 e

e

)

m2

)

 PR  (   PR  )  

 PC 3 m 1 2 B1 (m 1  m   2 m 1  PR 

 PC 3 m 2 2 B 5 (m   2 m 2  PR 

 PC 4 m 1 2B  2 m  PR  1 

 PC 4 m 2 2B   2 m 2  PR 

2

2

6

(m

2

2

 )   (m

3

 m

(m 1  m

2

 PC 5 m 2 2 B7 (m   2 m 2  PR 

 m

4

2

 2m1)

3

 )   (m

4

 )   (m

 PR  ( m

2

 2m1)  

3

 2m

3

2

)

 PR  ( m

3

 2m

2

)  

2

4

 PR  ( m

2

 2m1)  

4

2

1  2m

4

2

)

2

 PR  ( m

4

 2m

2

 PR  ( 

 2m1)  

2

 2m

2

)

 PR  ( 

 2m

2

)  

2

 2m1)

 PR  (  

 2m1)  

 PC 6 m 2  1 2 B8 (m 2   )  2   2 m 2  PR    (    2 m 2 )  PR  (    2 m 2 )  

 PC 3 m 1 m 2  m 1B5 ( m 2  m 3 )  m 2 B1( m 1  m 3  )  ( m 1  m 2  PR  ) 2

 1  2   ( m 1  m 2  m 3 )  PR  ( m 1  m 2  m 3 )  

2

2

   

   

   

1 2

)  

   

1 2

2

   

   

1  2m1)

 2m1)

The IJES

2

1 2

1

   )    (

   

1

 )   (m

3

  PC 6 m 1 2 B 4 (m1   )    2 m 1  PR   (

*

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2

2

  PC 5 m 1 2 B3 (m1   )    2 m 1  PR    (

2

1 m 2

2 EPR m1

2

2 m 2

EPRm m1

)

1 m 2

EPRm m1

2

)

m2

EPRm m1

)

2

 e

EPRm m1

2

1

 e

EPRm (e

2

)

m2

EPRm (e

2

P R C 6 C 2 2

A4 

2

2

2

   

 ) * 

   

Page 36

Steady Mhd Couette Flow Of An Incompressible…

A 14 

2 EPR (e

m1

 e

m2

)

2

 PC 4 m 1 m 2  m 1B 6 (m   ( m 1  m 2  PR  )

    (m 1  m

*

A 15 

2 EPR (e

m1

 e

m2

)

2

    (m 1  m

A 16 

(e

 e

m2

)

2

*

A 17 

A 18 

A19 

A 20 

2 EPm (e

m1

2 EPm (e

m1

2 EPm (e

m1

2 EPm (e

m1

1

m 5 C 11

e

1

2

)

m2

)

m 5 C 11

e

2

m2

m 6 C 12

e

m2

)

m 6 C 12

e

m2

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2

 m4)

 PR  ( m 1  m

2

2

 m4)  

2

  )

   

   )  m 2 B 3 (m 1   ) * 

1 2

2

 PR  ( m 1  m

2

  )  

2

   

  PC 6 m 1 m 2  m1B8 (m 2   )  m 2 B 4 (m1   ) *   ( m 1  m 2  PR  ) 

2 EPR m1

  m 4 )  m 2 B 2 (m 1  m 4 ) * 

1 2

 PC 5 m 1 m 2  m 1B 7 (m  ( m  m 2  PR  ) 1 

*

2

)

 1  2   ( m 1  m 2   )  PR  ( m 1  m 2   )  

 1  2   ( m 1  m 5 )  PR  ( m 1  m 5 )  

 1  2   ( m 1  m 6 )  PR  ( m 1  m 6 )  

 1  2   ( m 1  m 5 )  PR  ( m 1  m 5 )  

 1  2   ( m 1  m 6 )  PR  ( m 1  m 6 )  

The IJES

2

2

2

2

2

   

   

   

   

   

Page 37