Natural Convection Magnetohydrodynamic Flow of a Micro ...ripublication.com/ijaes17/ijaesv12n9_05.pdfplate absorbs the polar fluid with a suction velocity varying with time and the

International Journal of Applied Environmental Sciences

ISSN 0973-6077 Volume 12, Number 9 (2017), pp. 1693-1706

© Research India Publications

http://www.ripublication.com

Natural Convection Magnetohydrodynamic Flow of a

Micro Polar Fluid past a Semi infinite Vertical

Porous Flat Moving Plate

S. Panda1, S. S. Das2 and N. C. Bera3

1Department of Physics, KISS, KIIT Campus-10, Patia, Bhubaneswar-751 024 (Odisha), India.

*2Department of Physics, KBDAV College, Nirakarpur, Khordha-752 019 (Odisha), India.

3Department of Physics, KIIT University, Patia, Bhubaneswar-751 024 (Odisha), India.

Abstract

The objective of this paper is to analyze an unsteady natural convection flow

of a viscous incompressible electrically conducting micro polar fluid past a

semi-infinite vertical porous flat moving plate in presence of transverse

magnetic field. The vertical porous plate is subjected to move with a uniform

velocity in the upward direction in its own plane and the free stream velocity

follows an exponentially increasing or decreasing small perturbation law. The

porous plate absorbs the polar fluid with a suction velocity varying with time.

The governing equations for velocity, angular velocity and temperature of

flow field have been solved using perturbation technique. The effects of the

important flow parameters such as Grashof number for heat transfer Gr, magnetic parameter M, viscosity ratio , plate velocity Up and Prandtl number Pr etc. on the velocity, angular velocity and temperature profiles of the flow field are discussed with the help of figures. This paper has some relevance in

geothermal and oceanic circulation.

Keywords: Unsteady, magnetohydrodynamic, free convection, micro polar,

moving plate

mailto:[email protected]

1694 S. Panda, S.S. Das and N.C. Bera

NOMENCLATURE

A suction velocity parameter B0 magnetic flux density Cp specific heat at constant pressure Gr Grashof number for heat transfer g acceleration due to gravity k thermal conductivity M magnetic field parameter Nu Nusselt number n dimensionless exponential index Pr Prandtl number T temperature t dimensionless time U0 scale of free stream velocity u,v components of velocities along and perpendicular to the plate, respectively V0 scale of suction velocity x, y distances along and perpendicular to the plate, respectively

Greek symbols

Fluid thermal diffusivity Dimensionless viscosity ratio f Coefficient of volumetric thermal expansion of the working fluid Spin-gradient viscosity Scalar constant (1) Electrical conductivity Fluid density Λ Coefficient of gyro-viscosity Fluid dynamic viscosity Fluid kinematic viscosity r Fluid kinematic rotational viscosity Dimensionless temperature Angular velocity vector

Superscripts

differentiation with respect to y

* dimensional properties

Subscripts

p plate w wall condition free stream condition

Natural Convection Magnetohydrodynamic Flow of a Micro Polar Fluid past… 1695

1. INTRODUCTION

The phenomenon of flow and heat transfer in an electrically conducting polar fluid

past a vertical porous plate under the influence of a magnetic field has been a subject

of interest of a good number of researchers because of its applications in several fields

of science and technology for example in the boundary layer control in the field of

aerodynamics (Aero et al. [1]), geothermal energy extractions, oil exploration and in plasma studies. Physically, micro polar fluids represent fluids consisting of randomly

oriented particles suspended in a viscous medium (Dep [2] and Lukaszewicz [3]).

Several Darcian porous MHD studies have been carried out considering the effects of

magnetic field on electrically conducting fluid with or without heat transfer in various

configurations. Chamkha [4] investigated the unsteady convective heat and mass

transfer flow of an electrically conducting fluid in presence of transverse magnetic field

past a semi-infinite vertical permeable moving plate with heat absorption. Mbeledogu

and Ogulu [5] analyzed the effect of heat and mass transfer on unsteady natural

convection flow of a rotating electrically conducting fluid past a vertical porous flat

plate in the presence of radiative heat transfer and transverse magnetic field. Das and

his co-workers [6] estimated the effect of mass transfer on magnetohydrodynamic flow

and heat transfer past a vertical porous plate through a porous medium under oscillatory

suction and heat source and observed that a growing magnetic parameter reduces the

velocity of the flow field at all points of the flow field.

The transient free convection MHD flow past an infinite vertical porous flat plate in

presence of mass transfer has been studied by Panda and his team [7]. Das et al. [8] reported the free convection effects on unsteady viscous flow of an electrically

conducting fluid past an infinite vertical porous plate with heat source/sink in presence

of magnetic field. Patil and Kulkarni [9] observed the effect of chemical reaction on

free convective flow of a polar fluid through a porous medium in the presence of

internal heat generation. Das and his group [10] analyzed unsteady mixed convective

flow of a polar fluid past a semi-infinite vertical porous moving plate in presence of

transverse magnetic field and noticed that an increase in magnetic parameter enhances

the velocity as well as angular velocity of the flow field at all points of the flow field.

Recently, Das and his associates [11] discussed the magnetohydrodynamic convective

mass transfer flow of a polar fluid past a semi infinite vertical porous flat moving plate

embedded in a porous medium and reported that their results are in good agreement

with those of the previous group [10].

The present study investigates an unsteady natural convection flow of a viscous

incompressible electrically conducting micro polar fluid past a semi-infinite vertical

porous flat moving plate which is subjected to move with a uniform velocity in

upward direction in its own plane in presence of a transverse magnetic field. The

plate absorbs the polar fluid with a suction velocity varying with time and the free


stream velocity follows an exponentially increasing or decreasing small perturbation

law. The effects of the flow parameters on the flow field across the boundary layer

have been discussed and analyzed with the aid of figures. This paper has some

relevance in geothermal and oceanic circulation.

2. MATHEMATICAL FORMULATION OF THE PROBLEM

Consider an unsteady viscous incompressible flow of an electrically conducting micro

polar fluid past a semi-infinite vertical porous moving plate embedded in a porous

medium and subjected to a transverse magnetic field in the presence of a pressure

gradient. The transversely applied magnetic field and magnetic Reynolds number are

very small and hence the induced magnetic field is negligible (Cowling [12]). Viscous

and Darcy’s resistance terms are taken into account with constant permeability of the

porous medium. The hole size of the porous plate is assumed to be significantly larger

than a characteristic microscopic length scale of the porous medium. The geometry of

the problem is shown in Figure A. Following Yamamoto and Iwamura [13], we

assume the porous medium as an assembly of small identical spherical particles fixed

in space. The flow variables are functions of y* and t* only.

FigureA. Geometry of the problem


Under the above conditions, the governing equations for mass, momentum and energy

conservation in Cartesian coordinates are given by:

Continuity:

,yv

0

(1)

Linear momentum:

*r20f2

2

r y2uBTTg

yu

xp1

yuv

tu

(2)

Angular momentum:

2

2

*

*

*

**

*

**

yyv

tj

, (3)

Energy:

2

2

***

* yT

yTv

tT

, (4)

where , , r , g , f , , B0, *j , * , , T and are respectively the density,

kinematic viscosity, kinematic rotational viscosity, acceleration due to gravity,

coefficient of volumetric thermal expansion of the fluid, electrical conductivity of the

fluid, magnetic induction, micro-inertia density, component of the angular velocity

vector normal to xy-plane, spin-gradient viscosity, temperature and effective fluid

thermal diffusivity; x and

y are the dimensional distances longitudinal and perpendicular to the plate respectively and

u , v are the components of dimensional

velocities along x and

y directions respectively.

It is assumed that the porous plate moves with a constant velocity (*pu ) in the

longitudinal direction and the free stream velocity (*U ) follows an exponentially

increasing or decreasing small perturbation law. We, further assume that the suction

velocity ( *v ) and the plate temperature (T) vary exponentially with time. For small velocities, the heat due to viscous dissipation is neglected in the energy equation (4).

With the above assumptions, the boundary conditions for velocity and temperature

fields are

*t*n

ww*p

* eTTTT,uu , 22

*

*

*

*

yu

y

at

*y 0


*t*n** eUUu 10 , 0,TT* as

*y , (5)

where 0

U is a scale of free stream velocity and *n is a scalar constant.

It is clearly seen from the continuity equation (1) that the suction velocity normal to

the plate is a function of time only and therefore, we shall take it in the form:

*t*nAeV*v 10 , (6)

where A is a real positive constant, and A are small and less than unity and 0

V is a scale of suction velocity which is a non-zero positive constant. Outside the boundary

layer, equation (2) gives

*20*

*

*

*UB

dtdU

dxdp1

(7)

We now introduce the following non-dimensional variables:

TTTT

,Vt

t,VU

,Uu

U,UU

U,yV

y,Vvv,

Uuu

w

**

*p

p

**** 20

0000

0

00

, ,Vnn

20

*

*2

20 j

Vj

,

kC

P pr is the Prandtl number, 20

20

VB

M

is the magnetic

parameter,2

00VU

)TT(gG wfr

is the Grashof number for heat transfer. (8)

Again, the spin-gradient viscosity which gives the relationship between the

coefficients of viscosity and micro-inertia is defined as

)(jj)A( ** 2

11

2, (9)

where denotes the dimensionless viscosity ratio which is defined as

(10)

Here, Λ is the coefficient of gyro-viscosity or vortex viscosity.

Using equations (6)-(10), the governing equations (2)-(4) are reduced to the following

non-dimensional form:

y

2uUMGy

u1dt

dUyuAe1

tu

r2

2nt

, (11)


2

2nt

y1

yAe1

t

, (12)

2

211

yPyAe

t rnt

, (13)

where

22j (14)

The corresponding boundary conditions now take the form

2

2

1y

uy

,e,Uu ntp

at y=0,

0,0,Uu as y (15)

3. METHOD OF SOLUTION

In order to solve the equations (11)-(13) by reducing these partial differential

equations to a

system of ordinary differential equations in non-dimensional form, we assume the

linear

velocity, angular velocity and temperature as

210 Oyueyuu nt (16)

210 Oyey nt (17)

210 Oyey nt (18)

Substituting equations (16)-(18) in equations (11)-(13) and equating the harmonic and

non-harmonic term, neglecting the coefficient of O(2 ), we get the following pairs of

equations for 000 ,,u and 111 ,,u .

00r00"0 2GMMuuu1 (19)

11r011"1 2GuAnMunMuu1 (20)

000 (21)

0111 An (22)

000 rP (23)

0111 rrr APnPP (24)


Here the primes denote differentiation with respect to y.

The corresponding boundary conditions now reduce to:

1,1,u,u,0u,Uu 1011001p0 at y=0

0,0,0,0,1u,1u 101010 as y (25)

The solutions of equations (19)-(24) satisfying boundary conditions (25) are given by

y5m4y

53yrP

210 eAeAAeAAyu (26)

y7m14

y5m12

y3m1311

y1m10

y9

yrP871 eAeAeAAeAeAeAA)y(u

(27)

y50 eAy (28)

y3m14y

61 eAeAy (29)

yrPey 0 (30)

y1m0yrP

01 eA1eAy

(31)

where,

r2

rr1 P4PP21m ,

r2

rr2 P4PP21m ,

n421m 23 ,

n421m 24 , 1M4112

1m5 ,

1M41121m6 ,

1nM41121m7 , 1nM4112

1m8 ,

r2r1

2r

0 PmPmAPA

,

651 mm

MA , r6r5

r2 PmPm

GA

,

653 mm

2A ,

p53214 UAAAAA ,

2

532

3

p2125

2r2

5 mAA

UAAmPAA ,

43

52

6 mmAA

A ,

877 mm

nMA , 8r7r

r28 mPmP

PAAA

,

87

6539 mm

A2AAAA ,

8171

0r10 mmmm

A1GA

,

83313

11 mmmmm2

A

, 8575

24

12 mmmmAA

A

,

2723113

2512

2110

29

2r86

13 mmAmmAmAAPAA

A

2723113

121098727

mmAmAAAAAm

,


1213111098714 AAAAAAAA . (32)

3.1. Skin friction

The skin friction at the wall of the plate is given by

000

y

*w

w yu

VU (33)

Using Equations (16), (26) and (27) in Equation (33), the skin friction becomes

1471251311310198rnt

45532rw AmAmAAmAmAAPeAmAAAP (34)

3.2. Rate of heat transfer

The rate of heat transfer or the heat flux at the wall in terms of Nusselt number is

given by

0

yu y

N (35)

Using Equations (18), (30) and (31) in Equation (35), the heat flux becomes

01r0nt

ru A1mPAePN (36)

4. DISCUSSIONS AND RESULTS

The study reported herein investigates an unsteady natural convection flow of a

viscous incompressible electrically conducting micro polar fluid past a semi-infinite

vertical porous flat moving plate which is subjected to move with a uniform velocity

in upward direction in its own plane in presence of a transverse magnetic field. The

governing equations for linear velocity, angular velocity and temperature of the flow

field have been solved employing perturbation technique and the effects of the flow

parameters on the linear velocity, angular velocity and temperature of flow field

across the boundary layer have been discussed and analyzed with the aid of velocity

profiles (1-4), angular velocity profiles (5-6) and temperature profile (7) as detailed

below.

4.1. Velocity field

The flow parameters affecting the velocity of the flow field are Grashof number for

heat transfer Gr, magnetic parameter M, viscosity ratio and the plate velocity Up. The velocity of the flow field is found to change more or less with the variation of the


above parameters. The effects of these parameters on the velocity field are discussed

with the help of Figures 1-4.

0

1

2

3

4

5

0 1 2 3 4 5

y

u

Gr= -5

Gr= 0

Gr=2

Gr=5

Figure1. Effect of Grashof number for heat transfer Gr on Velocity profiles against y

The effect of Grashof number for heat transfer Gr on the velocity field is presented in Figure 1. The Grashof number for heat transfer is found to accelerate the velocity of

the flow field near the plate. Figure 2 depicts the effect of magnetic parameter M on the velocity field. A growing magnetic parameter is noticed to enhance the velocity of

the flow field at all points.

0

2

4

6

8

10

12

14

0 1 2 3 4 5y

u

M=0

M=0.5

M=2

M=10

Figure2. Effect of magnetic parameter M on Velocity profiles against y


0

5

10

15

20

25

30

0 1 2 3 4 5y

u

Figure3. Effect of viscosity ratio on Velocity profiles against y

The effect of viscosity ratio on the velocity of the flow is shown in Figure 3. The

viscosity ratio is observed to decelerate the velocity of the flow field at all points.

Figure 4, elucidates the effect of the plate velocity Up on the velocity field. From the curves of the said figure, it is clearly observed that an increase in plate velocity has an

accelerating effect on the velocity of the flow field near the plate. The results shown

in velocity profiles 1, 2 and 3 are in good agreement with those of Das et al. [11] and Figure 4 closely matches with those of Das and his group [10].

0

1

2

3

4

5

6

7

0 1 2 3 4 5y

u Up=0 Up=0.2Up=2 Up=4

Figure4. Effect of plate velocity Up on Velocity profiles against y


4.2. Angular velocity field

The effect of magnetic parameter M and the viscosity ratio on the angular velocity of the flow field has been shown in Figures 5-6. The profiles of Figure 5 clearly show

that a growing magnetic parameter M leads to enhance the angular velocity of the flow field at all points. Figure 6 elucidates the effect of plate velocity Up on the angular velocity of the flow field. It is observed that an increase in of plate velocity

leads to decrease the angular velocity of the flow field. The angular velocity profiles

are in good agreement with those of Das et al. [10, 11].

0

5

10

15

20

25

30

0 1 2 3 4 5y

M=0 M=0.5

M=2 M=10

Figure5. Effect of magnetic parameter M on Angular velocity profiles against y

0

2

4

6

8

10

12

14

0 1 2 3 4 5y

Up=0 Up=0.2

Up=2 Up=4

Figure6. Effect of plate velocity Up on Angular velocity profiles against y


4.3. Temperature field

The plot of temperature of the flow field against y for different values of the Prandtl number Pr is shown in Figure 7. Comparing the curves of the said figure, it is observed that an increase in Prandtl number reduces the temperature of the flow field

at all points. The temperature profiles shown in Figure 7 closely matches with those

of Das et al. [11].

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 1 2 3 4 5y

Pr=0.71 Pr=1 Pr=5

Figure7. Effect of Prandtl number Pr on Temperature profiles against y

5. CONCLUSIONS

We summarize the conclusions drawn from the above analysis on the linear velocity,

angular velocity and temperature of the flow field.

1. The Grashof number for heat transfer Gr accelerates the velocity of the flow field near the plate.

2. A growing magnetic parameter M enhances the linear velocity as well as angular velocity of the flow field at all points.

3. The effect of increasing viscosity ratio is to retard the linear velocity of the

flow field at all points.

4. The plate velocity Up is observed to enhance the linear velocity of the flow field near the plate while it shows reverse effect in case of angular velocity.

5. An increase in Prandtl number Pr decreases the temperature of the flow field at all points.


REFERENCES

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[2] Dep, N. V., 1968, “Equations of a fluid boundary layer with couple stresses”,

J. Appl. Math. Mech., 32(4), pp. 777-783.

[3] Lukaszewicz, G., 1999, “Micropolar Fluids- Theory and Applications”,

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[4] Chamkha, A. J., 2004, “Unsteady MHD convective heat and mass transfer past

a semi-infinite vertical permeable moving plate with heat absorption”, Int. J.

Engng. Sci., 42 (2), pp. 217-230.

[5] Mbeledogu I.U., and Ogulu , A., 2007, “Heat and mass transfer of an unsteady

MHD natural convection flow of a rotating fluid past a vertical porous flat

plate in the presence of radiative heat transfer” , Int. J. Heat Mass Transfer, 50

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porous medium under oscillatory suction and heat source”, Int. J. Heat Mass

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[8] Das, S. S., Mishra, S., and Tripathy, P., 2016, “Effect of free convection on

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convective flow of a polar fluid through a porous medium in the presence of

internal heat generation”, Int. J. Thermal Sciences, 47 (8), pp.1043-1054.

[10] Das, S. S., Biswal, S. R., and Panda, J. P., 2009, “Hydromagnetic unsteady

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moving plate”, Far East J. Ocean Research, 2 (1), pp. 51-65.

[11] Das, S. S., Saran M. R., and Pradhan, B., 2017, “MHD convective mass

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Natural Convection Magnetohydrodynamic Flow of a Micro ...ripublication.com/ijaes17/ijaesv12n9_05.pdfplate absorbs the polar fluid with a suction velocity varying with time and the