MHD Flow of Second Grade Fluid over an Infinite Permeable

Plate Embedded In a Porous Medium

K Raghunath

Assistant professor, Department of Mathematic,

Bheema Institute of Technology and Science, Adoni, A.P, India

Email- kraghunath25@gmail.com

M Obulesu

Research Scholar, Department of Mathematics,

S K University, Anantapur, A.P, India

Email- mopuriobulesu1982@gmail.com

A Shareef

Assistant Professor, Department of Mathematics,

Santhiram Engineering College, Nandyal, A.P, India

Email- Shareef.maths@gmail.com

Abstract- In this paper, investigated heat transfer in a laminar MHD flow of heat generating /absorbing second grade

fluid on permeable surface. The dimensionless governing equations are solved using regular perturbation technique. The

impact of different taking interest parameters are shown graphically and clarified in detail. It is noticed that, far away

from the bounding surface it is seen that a wide spreading in the velocity profiles is observed. Such a phenomena could

not be observed in boundary layer region as the porosity of the bounding surface decreases, not much of change in the

velocity field is noticed. Additionally, as the pore size of the fluid bed diminishes the velocity apparently is expanding. The

Shear stress and Nusselt number are calculated and discussed with reference to governing parameters.

Keywords – MHD, Porous medium, Radiation absorption, second grade fluids

Nomenclature:

(u, v) the velocity components along the (x, y) directions

A Suction coefficient

B0 Electro-magnetic induction

pC Specific heat

g Acceleration due to gravity

K1 thermal conductivity

k permeability of the porous medium

0Q Heat absorption

T temperature of the fluid

wT fluid temperature at the surface (wall)

T fluid temperature in the free stream

0U Constant velocity

U the free stream velocity

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0w suction velocity

t time

q complex velocity

M Hartmann number

K Permeability parameter

S second grade fluid parameter

E Ekman number

Gr thermal Grashof number

Pr Prandtl number

Greek symbols

Dimension less temperature

Ω Angular velocity

Electrical conductivity of the fluid

Density of the fluid

Kinematic viscosity

the frequency of oscillation

The coefficient of volume expansion

Heat source parameter

Sub scripts

w conditions on the wall

free stream conditions

I. INTRODUCTION:

The study related to free convective flow in presence of heat source has drawn considerable attention of

many researchers during last few decades because of its wide application in astrophysical sciences, cosmical studies

etc. These types of flows play vital role in chemical engineering, aerospace technology etc. Unsteady oscillating

flows have applications in many fields such as paper industry and many other technological fields. Asghar et al. [1]

have studied the flow of a non-Newtonian fluid induced due to the oscillations of a porous plate. Choudhury and

Das [2] investigated visco-elastic magnetohydrodynamic (MHD) free convective flow through porous media in

presence of radiation and chemical reaction with heat and mass transfer. Deka et al. [3] have discussed free

convection effects on MHD flow past an infinite vertical oscillating plate with constant heat flux. Das et al. [4] have

investigated mass transfer effects on free convective MHD flow of a viscous fluid bounded by an oscillating porous

plate in the slip flow regime with heat source. A study has been done by Hayat et al. [5] on the flow of a visco-

elastic fluid on an oscillating plate. Manna et al. [6] have discussed effects of radiation on unsteady MHD free

convective flow past an oscillating vertical porous plate embedded in a porous medium with oscillatory heat flux.

Shen et al. [7] investigated RayleighStokes problem for a heated generalized second grade fluid with fractional

derivative model. Singh and Gupta [8] have studied MHD free convective flow of viscous fluid through a porous

medium bounded by an oscillating porous plate in slip flow regime with mass transfer. Jhansi Rani and Murthy [9]

discussed the radiation absorption effects on an unsteady convective flow past a semi-infinite inclined permeable

plate embedded in a porous medium with heat and mass transfer. Recently, Krishna et al. [10-13] discussed the

MHD flows of an incompressible and electrically conducting fluid in planar channel. The effect of thermal radiation

on MHD nanofluid flow between two horizontal rotating plates is studied by Sheikholeslami et al. [14]. Rashid et al.

[15] discussed a mathematical model for two-dimensional fluid flow under the influence of stream wise transverse

magnetic fields in laminar regime is simulated. Hall current effects on MHD convective flow past a porous plate

with thermal radiation, chemical reaction and heat generation /absorption was studied by Obulesu et al. [16]. Heat

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and mass transfer on MHD flow of Non-Newtonian fluid over an infinite vertical porous plate was studied by

Raghunath et al [17]. Effect of Inclined Magnetic Field and Radiation Absorption on Mixed Convection Flow of a

Chemically Reacting and Radiating Fluid Past a Semi Infinite Porous Plate was studied by Obulesu et al. [18].

Raghunath K et al. [19] Discussed Hall Effects on MHD Convective Rotating Flow of Through a Porous Medium

past Infinite Vertical Plate. Raghunath K et al. [20] have investigated Heat and Mass Transfer on Unsteady MHD

Flow of a Visco-Elastic Fluid Past an Infinite Vertical Oscillating Porous Plate.

II. FORMULATION AND SOLUTION OF THE PROBLEM

We have considered an unsteady 2D MHD convective flow of a viscous laminar heat engrossing second

grade fluid over a semi-infinite vertical porous moving plate embedded in a uniform porous medium. The physical

model is as shown in the Fig.1

Fig. 1. Physical configuration of the problem

All thermo physical properties are supposed to be invariable of the linear momentum equation estimated

according to the Boussinesq approximation. The plate is extended to infinitely then all the physical variables are

functions of z and the time t only. The governing equations with respect to the frame are given by

0u u

x y

(1)

22 3

0

2 2

12 ( )

Bu u p u uw Ωv u u g T T

t z x z z t k

(2)

22 3

0

2 2

12

Bv v p v vw Ωu v v

t z y z z t k

(3)

2

1 02

1( )

p

T T Tw k Q T T

t z C z

(4)

We assume that the porous plate budges with a invariable velocity in the direction of fluid flow. Also, the

temperature at the wall and the suction velocity is exponentially varying with time.

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The boundary conditions are

0 0, ( )e at 0i t

wq U T T T T z

(5)

, asq U T T z (6)

From Eqn.(1), the suction velocity at the plate is either a stable or a fraction of time. Hence, the suction

velocity normal to the plate is assumed in the form

0(1 e )i tw w A (7)

Where, A and are optimistic invariables, which satisfies the condition 1A and 0w is size of

suction velocity and non-zero constructive constant. The negative sign indicates the suction is towards the plate.

Combining equations (2) & (3) , set andq u iv x iy we obtain

22 3

0

2 2

12 ( )

Bq q p q qw iΩq q q g T T

t z z z t k

(8)

Outside the boundary layer, equation (8) gives

2

01 UpU

t k

(9)

We introduce the non-dimensional variables are

2 2* 20 0 0

2

0 0 0 0

, * , z* , * , , ,w

U z tU BU T Tq wq* w U t M

w U U T T U

2 2 2

0 0 0 1

2 3 2 2

1 0 0

( ),Pr , ,Gr , ,

p w

p

CkU U g T T Q UK E S

k U C U

Making use of non dimensional variables, the governing equations are reduced to

2 31 2

2 2

12 (1 e ) Gri t duq q q qiE q A S M q

t z dt z z t K

(10)

2

2

1(1 e )

Pr

i tAt z z

(11)

The boundary conditions are

0, 1 at 0i tq U e z (12)

0, 0 asq z (13)

By making use of perturbation technique, the velocity and temperature are assumed by

2

1( ) ( ) ( )i tq q z e q z O

(14)

2

0 1(z) e ( ) ( )i t z O (15)

Substituting equations (14) and (15) in equations (10) and (11) respectively, we obtain, zeroth and first

order are

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22 10

0 02

12 Gr

d q dqM iE q

dz dz K

(16)

2

0 002

Pr Pr 0d d

dz dz

(17)

22 11 1

1 12

1(1 ) 2 Gr

dqd q dqSi M iE q A i

dz dz K dz

(18)

2

01 112

Pr ( ) Pr Prdd d

i Adz dz dz

(19)

Corresponding boundary conditions are

0 1 1, 0, 1, 1 at 0pq U q z (20)

0 1 0 10, 0, 0, 0 atq q z (21)

Solving equations (17),(18),(19),(20) with respect to the boundary conditions (21),(22),we obtain

3 2 4 1 2

7 8 5 3 4 6 3 5 4 6{( ) ( ) }m z m z m z m z m zi tq a e a e e a a a a e a a e a e a

(22)

and

1 2 1

1 2e ( )m z m z m z i ta e a e e

(23)

The non-dimensional skin friction at the plate 0z in term of amplitude and phase angle is given by

ti

zzz

edz

dq

dz

dq

dz

dq 0

1

0

0

0

3 7 2 8 4 5 3 4 6 1 3 5 2 4{ ( ) ( ) }i tm a m a e m a a a a m a a m a (24)

The xz and yz components of skin friction at the plate are given by

0

1

0

0

zz

xzdz

dv

dz

du

and

0

1

0

0

zz

yzdz

du

dz

dv

The rate of heat transfer co-efficient at the plate 0z in term of amplitude and phase angle is given by

0 11 1 2 2 1

0 00

i t i t

z zz

dT dTdTNu e m a m a m e

dz dz dz

(25)

Where,

2

1

Pr Pr 4Pr

2m

,

2

2

Pr Pr 4 Pr( )

2

im

,

2 1

3

11 1 4 2

2

M iEK

m

,

2 1

4

11 1 4(1 ) 2

2

Si M iEK

m

11 2

1 1

Pr1

Pr ( ) Pr

A ma

m m i

, 1

2 2

1 1

Pr

Pr ( ) Pr

A ma

m m i

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13

2 2 1

1 1

1(1 ) 2

Ama

Si m m M iEK

14

2 2 1

2 2

Gr

1(1 ) 2

aa

Si m m M iEK

25

2 2 1

1 1

Gr

1(1 ) 2

aa

Si m m M iEK

, 6

2 1 12

ia

M iEK

72 2 1

1 1

Gr

12

pa U

m m M iEK

, 8

2 2 1

1 1

Gr

12

a

m m M iEK

III. RESULTS AND DISCUSSION

The flow governed by the non-dimensional parameters M, K, E, Gr, , S, Pr, and time t. The velocity

and temperature profiles are shown in Figures (2-10) and Figs (11) respectively. Tables (1-2) represent the skin

friction and Nusselt number for different variations in the governing parameters. Computational purpose we are

fixing the values 1 0 01A , . . The results are good agreement with the result of Jhansi Rani et al. [9].

From the Figs. 2, the velocity components u reduces and v increases with increasing the Hartmann number

M. The Lorentz force decelerates the fluid flow consequently resulted in thinning of boundary layer. The resultant

velocity retards with increasing M. The velocity component u enhances for 1.75z and then gradually reduces for

1.75z throughout the flRaRuid region and v reduces with increasing permeability parameter K. i.e., the resultant

velocity enhances with increasing K throughout the fluid region (Figs. 3). The reversal behaviour is observed with

increasing Ekman number E (Figs. 4). The velocity components u increases and v reduces with increasing S or Gr

(Figs. 5-6). The resultant velocity enhances with increasing S or Gr. From the Figures (7-9) the velocity component

u continuously retardation in its magnitude and v is boost up with increasing Pr or or . The resultant velocity

retards with Pr or or . Presence of source reduces the velocity, significantly in all the layers. The velocity

component u enhances and v reduces with increasing time t (Figs. 10). The resultant velocity also enhances with

increasing time t.

Figs. (11) exhibit the effects of Prandtl number, , and t on . The temperature reduces with

increasing Pr or . The temperature increases with increasing and t throughout the fluid region.

Table (1) represent the values of skin friction components xz and yz . The increase of M or leads to

decrease xz and yz . Since, elastic property in visco-elastic second grade fluid reduces the frictional drag. This

is in good agreement with Jhansi Rani [9]. Further, an increase in K or Gr, leads to exert greater skin friction

components xz and yz on the boundary. The component

xz retards with increasing Ekman number E or Prandtl

number Pr whereas opposite behavior observed for yz . Finally we noticed that the skin friction components xz

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boost up and the magnitude of yz reduces with increasing S, and t. This coincides with the result of Jhansi Rani

[9].

From the Table 2, accelerate in Pr or or leads to an increase in Nusselt number. Nusselt number

reduces with increasing time.

Figs. 2 velocities for u and v against M 0.5,Gr 5, 1, 0.1,Pr 0.71, 1, / 6, 0.1K S E t

Figs. 3 The velocities for u and v against K 0.5, Gr 5, 1, 0.1,Pr 0.71, 1, / 6, 0.1M S E t

Figs. 4 velocities for u and v against E 0.5, 0.5,Gr 5, 1,Pr 0.71, 1, / 6, 0.1M K S t

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Figs. 5 velocities for u and v against S 0.5, 0.5,Gr 5, 0.1,Pr 0.71, 1, / 6, 0.1M K E t

Figs. 6 velocities for u and v against Gr 0.5, 0.5, 1, 0.1,Pr 0.71, 1, / 6, 0.1M K S E t

Figs. 7 velocities for u and v against Pr 0.5, 0.5,Gr 5, 1, 0.1, 1, / 6, 0.1M K S E t

Figs. 8 velocities for u and v against 0.5, 0.5,Gr 5, 1, 0.1,Pr 0.71, / 6, 0.1M K S E t

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Figs. 9 velocities for u and v against 0.5, 0.5,Gr 5, 1, 0.1,Pr 0.71, 1, 0.1M K S E t

Figs. 10 velocities for u and v against t 0.5, 0.5,Gr 5, 1, 0.1,Pr 0.71, 1, / 6M K S E

Figs. 10 Temperature against Pr, , and t

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Table 1. Skin friction

M K E S Gr Pr t xz

yz

0.5 0.5 0.1 1 5 0.71 1 / 6 0.1 1.075580 -0.097994

1 0.784617 -0.078272

1.5 0.413659 -0.059883

1 1.622850 -0.154857

1.5 1.882170 -0.195509

0.2 1.065600 -0.184985

0.3 1.049610 -0.269974

2 1.075700 -0.095594

3 1.077030 -0.093868

6 1.497700 -0.111836

7 1.919830 -0.125678

3 -0.073138 -0.110867

7 -0.540916 -0.138832

2 0.803362 -0.085081

3 0.628892 -0.081597

/ 4 1.078780 -0.097660

/ 3 1.080860 -0.097281

0.2 1.075590 -0.097993

0.3 1.075600 -0.097990

Table 2. Nusselt number

Pr t Nu

0.71 1 / 6 0.1 1.27111

3 3.79754

7 7.90163

2 1.60044

3 1.85930

/ 4 1.27148

/ 3 1.27202

0.2 1.27109

0.3 1.27107

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IV.CONCLUSION

The velocity is reduces with M and increases with K. As the pore size of the fluid bed decreases the velocity is seen

to be increasing. The Grashoff number contributes to the velocity increasing to peak. When viscosity of a fluid

dominates over conductivity then the rate of heat transfer increases significantly.

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[13] VeeraKrishna/M and G.Subba Reddy Unsteady MHD convective flow of Second grade fluid through a porous medium in a Rotating parallel plate channel with temperature dependent source, IOP Conf. Series: Materials Science and Engineering, vol. 149, p. 012216, 2016. DOI: 10.1088/1757-899X/149/1/012216.

[14] Sheikholeslami Mohsen, Domiri Ganji Davood, Younus Javed M, Ellahi. R., Effect of thermal radiation on magnetohydrodynamics nanofluid flow and heat transfer by means of two phase model, Journal of Magnetism and Magnetic Materials, vol. 374, pp. 36-43, 2015. Doi: 10.1016/j.jmmm.2014.08.021.

[15] Rashidi.S, M. Dehghan, R.Ellahi, M.Riaz, M.T.Jamal-Abad, Study of stream wise transverse magnetic fluid flow with heat transfer around a porous obstacle, Journal of Magnetism and Magnetic Materials, vol. 378, pp. 128–137, 2015. Doi: 10.1016/j.jmmm.2014.11.020

[16] Obulesu M, Siva Prasad R; Hall Current Effects on MHD Convective Flow Past A Porous Plate with Thermal Radiation, Chemical Reaction and Heat Generation /Absorption , To Physics Journal Vol 2 (2019) ISSN: 2581-7396.

[17] Raghunath K, R Sivaprasad, GSS Raju, Heat and mass transfer on MHD flow of Non-Newtonian fluid over an infinite vertical porous plate, International Journal of Applied Engineering Research ISSN 0973-4562 Volume 13, Number 13 (2018) pp. 11156-11163

[18] Obulesu M, Siva Prasad R;,Effect of Inclined Magnetic Field and Radiation Absorption on Mixed Convection Flow of a Chemically Reacting and Radiating Fluid Past a Semi Infinite Porous Plate, .i-Manager's Journal on Mathematics; Nagercoil Vol. 7, Iss. 4, (Oct/Dec 2018): 39-49. DOI:10.26634/jmat.7.4.1556.

[19] Raghunath K, R Sivaprasad, GSS Raju, Hall Effects on MHD Convective Rotating Flow of Through a Porous Medium past Infinite Vertical Plate Annals of Pure and Applied Mathematics Vol. 16, No. 2, 2018, 353-263 ISSN: 2279-087X (P), 2279-0888(online), DOI: http://dx.doi.org/10.22457/apam.v16n2a1.

[20] Raghunath K, M.V Krishna, R Sivaprasad, GSS Raju, Heat and Mass Transfer on MHD Flow of a Visco-Elastic fluid past an infinite verticle oscillating Porous plate, British Journal of Mathematics, ISSN: 2231-0851,Vol.: 17, Issue.: 6, DOI : 10.9734/BJMCS/2016/25872

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