Hall and Ion-Slip Effects on MHD Free Convection Flow through An Oscillatory Porous Medium With Cons

K.V.B.Rajakumar, International Journal of Advanced Trends in Computer Applications (IJATCA) Special Issue 1 (1), July - 2019, pp. 159-166 ISSN: 2395-3519

International Journal of Advanced Trends in Computer Applications www.ijatca.com

Hall and Ion-Slip Effects on MHD Free Convection Flow through An Oscillatory Porous Medium With Constant Suction Velocity And Radiation 1

K.V.B.Rajakumar, 2P. V. Sivaji, 3K. S. Balamurugan 1,2 Department of Mathematics, Kallam Haranadhareddy Institute of Technology, Guntur, India 3 Department of Mathematics, R V R & J C college of Engineering, Guntur, India 1 kvbrajakumar@gmail.com, 2shivaji25@gmail.com, 3muruganbalaks@gmail.com

Abstract: Current work focused on perturbation method for the influence of Hall and Ion-slip current on unsteady Magneto Hydrodynamic free convection flow through an oscillatory porous medium with constant suction velocity and chemical reaction in sway of radiation. The numerical outcomes involved in velocity, Skin friction, temperature, Nusselt number, concentration as well as Sherwood number are demonstrated graphically for pertinent parameters entering into the problem. It is revealed that the velocity as well as Skin friction declined with the enhancement of Hall and ion-slip current. Meanwhile temperature diminished with the rise in Prandtl number, but contrast result was eventuated in case of Nusselt number. In addition Concentration and Sherwood number decremented with ascend in chemical reaction as well as Schmidt number.

Keywords: Prandtl number, Hall, Ion-slip current, Perturbation method.

I.

INTRODUCTION

Due to versatile technological as well as manufacturing applications, it is great understand to examine the MHD flow. The foremost objective of MHD principles is to interrupt the flow field in a required direction by fluctuate the formation of the boundary layer. Thus, with the intention of modify the flow kinematics; the idea to execute MHD seems to be more flexible and reliable. In pharmaceutical as well as environmental science, MHD has been playing a essential role in the application of fluid dynamics and medical sciences, owing to its implications in chemical fluids as well as metallurgical fields. Rudraiah [1] analyzed hydro magnetic free convection flow through a porous medium between two parallel plates. In this paper, it was found that velocity as well as temperature diminished with rise in Darcy dissipations and porous parameter. Abdul Hakeem [2] considered analytical solutions for two-dimensional oscillatory flow on free convective radiation through a highly porous medium bounded by an infinite perpendicular plate. Hamza et al. [3] in paper it was observed that the velocity

declined with the reduced in magnetic field, porous parameter and Grashof number but reverse effect was shown in case of slip parameter. In this investigation, analytical model (Perturbation method) was utilized for solving the governing equations. M.A. El-Hakiem et al.

[4] and Makinde [5] examined constant suction velocity on MHD free convection oscillatory flow on radiation through a porous medium.

Raptis [6] analyzed effect of non-constant 2D free convective flow throughout the motion of a viscous incompressible fluid through a very much porous medium. Gholizadeh [7] and Swati [8] analyzed the thermal and mass diffusion effects on MHD oscillatory flow past a vertical porous plate through a porous medium in the presence of heat source. In the above all investigation Hall and Ion lip current was not taken into considered.

Set et al. [9] have discussed, affect of rotational system on unsteady hydro magnetic flow which is natural convective flow over impulsively affecting erect plate thereby ramped temperature embedded in porous medium by taking thermal diffusion as well as heat absorption.

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K.V.B.Rajakumar, International Journal of Advanced Trends in Computer Applications (IJATCA) Special Issue 1 (1), July - 2019, pp. 159-166 ISSN: 2395-3519

comparison with that in y* - direction. Hall, Ion Slip current and heat source has been taken into contemplated. The magnetic Reynolds number of the flow is purloined to be small adequately, thus the induced magnetic field can be omitted. The

II. MATHEMATICAL FORMULATION AND SOLUTION OF THE PROBLEM Contemplate unsteady two-dimensional free convection fluid flow through a highly porous medium which is bounded by a perpendicular infinite plane surface in vicinity of a crosswise magnetic field as well as thermal radiation. The fluid is hypothetical to be a gray, absorbing emitting but non-scattering medium. As stated in axial system of co-ordinates the x*-axis is purloined along the plane surface by means of a way reversal to the direction tendency of gravity and y* axis is normal to it. The physical variables are functions of y* as well as the time t* only. The radiate heat flux in the x*- direction is considered negligible in

K

r is taken into homogeneous chemical reaction contemplated. Thence, the equation of the conservation of mass, momentum, energy and concentration are given by Equation of continuity:

v 0 y*

(1)

 v  v0

(2)

*

*

Equation of Momentum: 2  * B0  e eu*  e w*  u* * u* 1 p*  2u* * * * * *    v     g  ( T  T )  g  C  C  u     2 t* y*  y* k*    e2  e2  y*

  2w*   *  B02 e w* * w*   e  v1*  u*    e w*  v    w   *2  * * 2 2  y t    e  e    y  k





1 p

*

 y*



(3)

 (4)

dv  * e 2 *  v1  B v k dt  e2  e2 0 1 * 1 *





(5)

Equation of Energy: *   2T * 1 qr*  Q0 * * T * * T (T  T ) * v *   2  k *  t y 0 y  y*





(6)

C C D  K r (C *  C* ) *2 t* y *

2

*

(7)

The related boundary conditions are

 y*  0: u *  0, w*  0, T * T* , C * C* * * * i  t* * * * * *  As y : u  v1 U 0 (1 e ), w 0 T T , C C ,

at

(8)

From the equation (3), (4) & (4) then we get



u* * u* dv1*  2u* * *  g  T  T * v *  *  *2 t y dt y



B02 e  * * v  u  1  2 K*  1  i  e    e2



  2 w*   B e w w*  v* *    2   w*  * 2 k * y t  1  i  e    e2  y  *



qr*  

2 0



  v e

4 T 3k1 y*

* 1

u

*

 1     v u    w  i

* 1

e



*

*

e

(9)

  1     w  *

i

e

(10)

*4

(11)

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K.V.B.Rajakumar, International Journal of Advanced Trends in Computer Applications (IJATCA) Special Issue 1 (1), July - 2019, pp. 159-166 ISSN: 2395-3519

Now Non-dimensional quantities are defined as

   K * 2 * * * * * * * *  K , T  T   Tw T  , C  C  C Cw C  V02  *

v u* w* V y* V 2t* U* f  ,g  y  0 , v1  1 , t  0 , U  U0 U0  U0  U0

(12)

After substituting the boundary conditions and non-dimensional variables in the governing equations (3), (4), (6) & (7) then we get,

B0  e f f dv1  2 f     t y dt y 2 V 2  1    2   2  i e e 0 2



g g  g 1    t y y 2 k 2

g

2 B0  e

V0  2

1  i e 

2

 e

2

2   1   4 R     Pr   1 (  )3  2  4 R( t y  3  y







 1     v  f   e g   Gr  GmC  k1  v  f  i

e

1

1

(13)

 e v1  f   1  i e  g  (14)

     ) 2        y   2

(15)

C 1  C   Sc   Kr C t y 2 2

(16)

The boundary conditions are

At y 0 f 0, g 0,  1, C 1  As y  f 1 ei  t , g 0,  0, C 0 

(El-Hakiem et al [4] )

(17)

Take F=f+i g

(18)

2F

F F dv1     Gr  N  v1 F  t y dt y2 Gr 

 g Tw*  T*  V02U 0

3 4 Tw*  T*    B02 Q0 k1  ,M  ,  2 , K r  2 , Pr  , R  ,  V02 V0  V0  k0 k1

 T 1   * N  * k Tw  T   * 

(19)

M

 1     i

e

2

  e2



     

   1     i   i e e  , Sc   D 

So as to solve the deferential equation (15), (16) & (19) it was suppose that F  F0  y    eit F1  y  ......  0  y    eit1  y  .........C  C0  y    eit C1  y 



(20)

(21)

From equation (15), (16), (19) & (21) then we get;

F0"  F0'  NF0  Gr0  GmC0  N

(22)

F1"  F1'   N  i   F1  ( N  i  )  Gr1  GmC1

(23)

 4 3  '' 2 ' 2 ' 1  3 R(0   )  0  4 R(0   ) (0 )  Pr 0  Pr 0  0

(24)

 4 3  '' 2 ' 2 2 ' ' '  1  3 R( 0   )  1  8 R( 0   ) ( 0 ) 1  8 R( 0   )  01  Pr 1    2 ''  4 R(0   )  0 1  i Pr 1  Pr 1  0 

(25)

C  Sc K r C0  0 " 0

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K.V.B.Rajakumar, International Journal of Advanced Trends in Computer Applications (IJATCA) Special Issue 1 (1), July - 2019, pp. 159-166 ISSN: 2395-3519

C1"  Sc  K r  n  C1  0

(27)

The corresponding boundary conditions are

F0  0, F1  0, 0  1,1  0, C0  1, C1  0, at y  0   F0  1, F1  1, 0  0, 1  0, C0  0, C1  0 as y   

(28)

Initially solve equation (25) &equation (26) then we obtain,

C0  e







ScKr y

& C1  0

(29)

If we suppose that the radiation parameter R to be small, we expand (23)-(25) as

F0  F01  y   R F02  y  ........F1  F11  y   R F12  y  0  01  y   R 02  y  ..........1  11  y   R 12  y  

(30)

From the equation (22)-(25) & equation (30) then we obtain: F ''  F '  NF  Gr  G C  N

(31)

F02''  F02'  NF02  Gr02

(32)

F11''  F11'  ( N  i ) F11  ( N  i )  Gr11  GmC1

(33)

F12''  F12'  ( N  i ) F12  Gr12

(34)

11''  Pr 11'    i  Pr 11  0

(35)

01''  Pr 01'  Pr01  0

(36)

4  02''  ( 01   )3 01''  4( 01   ) 2 (  01' ) 2  Pr  021  Pr   02  0 3

(37)

4  12''  (01   )311''  8(01   )2 (01' )211  8(01   )201111  Pr 12'  3  2 ''  4(01   ) 011  Pr   i 12  0 

(38)

01

01

01

01

m

0

The corresponding boundary conditions are

F01  0, F02  0,01  1,02  0, F11  0, F12  0 11  0, 12  0 F01  1, F02  1,  01  0,  02  0, F11  1, F12  0 11  0, 12  0

at y  0   at y   

(39)

Solve (31) – (38) subject to boundary condition (39), & (29) we get velocity, temperature and concentration  A3 e  R3 y  N 6e  R2 y  N 7 e 4 R2 y   R3 y  R2 y l y F  A1 e  N1e  N10e 1  R    eit  e  R4 y  3 R2 y 2 R2 y     N8 e  N9e   (40)







  e R y  R A2 e 2

C e





 R2 y

 N 2e

4 R2 y



 N 3e

 R2 y

 N 4e

3 R2 y

 N 5e

2 R2 y





(41)

ScKr y

(42)

2.1 Skin friction: F w    R3 A1  R2 N1  l N10  1  R y y 0





  R3 A3  R2 N 6  4 R2 N 7 e 4 R2 y   R4 eit  2 R2 y    3R2 N8  2 R2 N9e  

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(43)

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K.V.B.Rajakumar, International Journal of Advanced Trends in Computer Applications (IJATCA) Special Issue 1 (1), July - 2019, pp. 159-166 ISSN: 2395-3519

2.2 Nusselt number:  Nw    R2  R  R2 A2  4 R2 N 2  R2 N3  3R2 N 4  2 R2 N5 y y 0



 (44)

2.3 Sherwood number:

Sh 

c   y y 0



ScKr

 (45)

very near to the plate. Fig 2: Reflects that the velocity diminished owing to enhancement of distinct values of

III. RESULTS AND DISCUSSION



Fig 1: Represents the behaviour of velocity for

Ion-slip parameter i owing to the fact that the reduced the resistive force inflict by magnetic field.



disparate values of Hall parameter e . From this figure it was found that the enhancement of various values of hall parameter it leads to reduced in velocity and it is 1.2

1.2

1

1

0.8

0.8

0.6

0.6

F

F

 =2, 4, 6, 8 e

0.4

0.4

0.2

0.2

0

0

-0.2 0

2

4

6

 =0.2, 0.4, 0.6, 0.8 i

-0.2 0

8

2

4

6

8

y

y Figure 1: Influence of

i

e

on Velocity

Figure 2: Influence of

 i on Velocity

1

1 Pr=0.71, 0.81, 0.91, 1.0

0.8

0.8 R=0.2, 0.4, 0.6, 0.8 0.6





0.6 0.4

0.4

0.2

0.2

0 0

1

2

3

4

5

6

7

8

y

0 0

2

4

6

8

10

y

Figure 3: Effect of Pr on Temperature

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Figure 4: Effect of R on Temperature

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K.V.B.Rajakumar, International Journal of Advanced Trends in Computer Applications (IJATCA) Special Issue 1 (1), July - 2019, pp. 159-166 ISSN: 2395-3519

1

1

0.8

Sc=0.42, 0.60, 0.81, 0.94

0.8

Kr=0.2, 0.4, 0.6, 0.8

0.6

C

C

0.6

0.4

0.4

0.2

0.2

0 0

2

4

6

8

10

0 0

12

2

4

6

y Figure 5: Effect of Kr on Concentration

10

12

Figure 6: Effect of Sc on Concentration

10

7.5  =2, 4, 6, 8

 =0.2, 0.4, 0.6, 0.8

e

i

9

w

7 w

8

y

6.5

8 7

6 5.5 0

2

4

6

8

10

12

6 0

14

2

4

6

8

10

12

14

y

y Figure 7: Effect of βe on Skin-friction

Figure 8: Effect of βi on Skin-friction

3

-4.85 Pr=0.91

2

-4.9 Pr=0.81

-4.95

0

Nw

Nw

1

Pr=0.71 -1

-5

-2

Pr=0.61

-5.05

-3 -4 0

=0.001, 0.003, 0.005, 0.007

2

4

6

8

10

12

14

-5.1 0

y

5

10

15

y

Figure 9: Effect of Pr on Nusselt number

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Figure 10: Effect of ղ on Nusselt number

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K.V.B.Rajakumar, International Journal of Advanced Trends in Computer Applications (IJATCA) Special Issue 1 (1), July - 2019, pp. 159-166 ISSN: 2395-3519 -0.28

-0.25

-0.3

-0.3

-0.32

-0.35

-0.34

-0.4

-0.36

Sh

Sh

Sc=0.42, 0.60, 0.81, 0.94

-0.45

-0.38

-0.5

-0.4

-0.55

-0.42

-0.6

-0.44 0

5

10

15

Kr=0.2, 0.4, 0.6, 0.8

-0.65 0

5

10

15

y

y Figure 11: Effect of Sc on Sherwood numbers

Figure 12: Effect of Kr on Sherwood number

Fig 3: Illustrated that for dissimilar values of radiation parameter rises that it leads to declined in temperature.

Fig 4: Established that for disparate values of Prandtl number (Pr) rises then it leads to reduced in temperature. This is owing to the reality that a higher Prandtl number fluid has comparatively low thermal conductivity, which diminishes the conduction as an outcome temperature reduces. As a result higher Prandtl number Pr show the way to more rapidly cooling of the plate. Fig 5: Demonstrate the performance of concentration for disparate estimators of chemical reaction (Kr). The results obtained from this figure it was perceived that the concentration reduced due to rise in chemical reaction parameter. For the reason that the chemical reaction reinforce momentum transfer and consequently accelerates the flow. For incongruent estimators of the Schmidt number on the fluid concentration is exposed in the Fig 6: From this figure it was found that the outcomes indicate that the enhancement of Sc leads to diminished in concentration. Due to the effect of concentration buoyancy to declined, yielding reduce in the velocity.

The depletion in the concentration is accompanied by instantaneous depletion in the concentration boundary layers, which is perceptible from the Fig 6. The influence of hall (βe) and ion slip (βi) current was illustrated in the Fig 7: and Fig 8: Form these figures it was determined that for heterogeneous incremental values of hall (βe) as well as ion slip (βi) leads to declined in the Skin-friction. For incongruent values of Prandtl number (Pr) on the Nusselt number is showing in the Fig 9: From the figure it is obvious that the outcomes illustrates rise in the Pr outcome Nw becomes rises. The impact of heat source parameter(ղ) on the nusselt number is presented on Fig 10: From this figure it is evidently that as ղ rises subsequently it leads to diminish in Nw. Fig 11: & Fig 12: Described the influence of Schmidt number(Sc) as well as Chemical reaction (Kr) on the Sherwood number(Sh). Here the results indicated that Sherwood number diminished with the enhancement of Sc as well as Kr.

IV. CONCLUSIONS  The velocity distribution declined with the enhancement of Hall and Ion-slip parameter.  The temperature profile is reduced with the enhancement of Radiation parameter and Prandtl number.  The concentration diminished with amplifies in chemical reaction parameter and Schmidt number.  As incremental values of βe & βi, Skin friction becomes declined.  The Nusselt number rises with the enhancement of Pr but contrast effect was occurred in case of ղ.  The Sherwood number reduced with the accelerated values of Sc as well as Kr. Conflict of Interest Both the authors have equal contribution in this work and it is declared that there is no conflict of interest for this publication.

Acknowledgment The authors express their sincere gratitude to the referees for their valuable suggestions towards the improvement of the paper.

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Nomenclature Coordinate axis along the plate(m), x* , y * Co-ordinate axis normal to the plate(m) Magnetic induction (A.m-1) B0 T Tw

Tw*

Temperature of the fluid (K) Fluid temperature at walls (K) Fluid temperature at the wall (K)

T

Dimensional free stream temperature (K)

u

components of velocity vector in x direction(m.S-1) Velocity component in y direction (m.S-1) acceleration due to gravity (m.S-2) Heat generation parameter Prandtl number Radiation heat flux density (W.m-2)

v g* ɳ Pr

qr*



t* M K Gr β

 

R

 i

,

Frequency of vibration of the fluid Dimensional time (S) Magnetic parameter Permeability of porous medium local temperature Grashof number Spin gradient viscosity (K-1) Fluid dynamic viscosity Density of the fluid (kg.m-3) The radiative parameter Temperature difference parameter Ion-slip parameter

e

Hall parameter

τw Nw Sh

Skin friction Nusselt number Sherwood number

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