Natural convection unsteady magnetohydrodynamic mass transfer flow past an infinite vertical porous plate in presence of suction and heat sink

This paper investigates the natural convection unsteady magnetohydrodynamic mass transfer flow of a viscous incompressible electrically conducting fluid past an infinite vertical porous flat plate in presence of constant suction and heat sink. Using multi parameter perturbation technique, the governing equations of the flow field are solved and approximate solutions are obtained. The effects of the flow parameters on the velocity, temperature, concentration distribution and also on the skin friction and rate of heat transfer are discussed with the help of figures and table. It is observed that a growing magnetic parameter or Schmidt number or heat sink parameter leads to retard the transient velocity of the flow field at all points, while the Grashof numbers for heat and mass transfer show the reverse effect. It is further found that a growing Prandtl number or heat sink parameter decreases the transient temperature of the flow field at all points while the heat source parameter reverses the effect. The concentration distribution of the flow field suffers a decrease in boundary layer thickness in presence of heavier diffusive species (growing Sc) at all points of the flow field. The effect of increasing Prandtl number Pr is to decrease the magnitude of skinfriction and to increase the rate of heat transfer at the wall for MHD flow, while the effect of increasing magnetic parameter M is to decrease their values at all points

E NERGY AND E NVIRONMENT

Volume 3, Issue 2, 2012 pp.209-222

Journal homepage: www.IJEE.IEEFoundation.org

Natural convection unsteady magnetohydrodynamic mass transfer flow past an infinite vertical porous plate in

presence of suction and heat sink

S S Das1, S Parija2, R K Padhy3, M Sahu4

1

Department of Physics, K B D A V College, Nirakarpur, Khurda-752 019 (Orissa), India

2

Department of Physics, Nimapara (Autonomous) College, Nimapara, Puri-752 106 (Orissa), India

3

Department of Physics, D A V Public School, Chandrasekharpur, Bhubaneswar-751 021 (Orissa),

India

4

Department of Physics, Jupiter +2 Women’s Science College, IRC Village, Bhubaneswar-751 015

(Orissa), India

Abstract

This paper investigates the natural convection unsteady magnetohydrodynamic mass transfer flow of a viscous incompressible electrically conducting fluid past an infinite vertical porous flat plate in presence

of constant suction and heat sink Using multi parameter perturbation technique, the governing equations

of the flow field are solved and approximate solutions are obtained.The effects of the flow parameters on the velocity, temperature, concentration distribution and also on the skin friction and rate of heat transfer are discussed with the help of figures and table It is observed that a growing magnetic parameter or Schmidt number or heat sink parameter leads to retard the transient velocity of the flow field at all points, while the Grashof numbers for heat and mass transfer show the reverse effect It is further found that a growing Prandtl number or heat sink parameter decreases the transient temperature of the flow field at all points while the heat source parameter reverses the effect The concentration distribution of the flow field

suffers a decrease in boundary layer thickness in presence of heavier diffusive species (growing S c) at all

points of the flow field The effect of increasing Prandtl number P r is to decrease the magnitude of skin-friction and to increase the rate of heat transfer at the wall for MHD flow, while the effect of increasing

magnetic parameter M is to decrease their values at all points

Copyright © 2012 International Energy and Environment Foundation - All rights reserved

Keywords: Natural convection; Magnetohydrodynamic; Mass transfer; Suction; Heat sink

1 Introduction

The phenomenon of natural convection flow with heat and mass transfer in presence of magnetic field has been given much importance in the recent years in view of its varied applications in science and technology The study of natural convection flow finds innumerable applications in geothermal and energy related engineering problems Such phenomena are of great theoretical as well as practical interest in view of their applications in diverse fields such as aerodynamics, extraction of plastic sheets, cooling of infinite metallic plates in a cool bath, liquid film condensation process and in major fields of glass and polymer industries

In view of the above interests, Hashimoto [1] discussed the boundary layer growth on a flat plate with

suction or injection Sparrow and Cess [2] analyzed the effect of magnetic field on a free convection heat

transfer Gebhart and Pera [3] studied the nature of vertical natural convection flows resulting from the

combined buoyancy effects of thermal and mass diffusion Soundalgekar and Wavre [4] investigated the

unsteady free convection flow past an infinite vertical plate with constant suction and mass transfer

Hossain and Begum [5] estimated the effect of mass transfer and free convection on the flow past a

vertical plate Bestman [6] analyzed the natural convection boundary layer flow with suction and mass

transfer in a porous medium Pop et al [7] reported the conjugate MHD flow past a flat plate

Singh [8] discussed the effect of mass transfer on free convection MHD flow of a viscous fluid Raptis

and Soundalgekar [9] analyzed the steady laminar free convection flow of an electrically conducting

fluid along a porous hot vertical plate in presence of heat source/sink Na and Pop [10] explained the free

convection flow past a vertical flat plate embedded in a saturated porous medium Takhar et al [11]

discussed the unsteady flow and heat transfer on a semi-infinite flat plate in presence of magnetic field

Chowdhury and Islam [12] developed the MHD free convection flow of a visco-elastic fluid past an

infinite vertical porous plate Raptis and Kafousias [13] analyzed the heat transfer in flow through a

porous medium bounded by an infinite vertical plate under the action of a magnetic field Sharma and

Pareek [14] described the steady free convection MHD flow past a vertical porous moving surface Das

and his co-workers [15] estimated numerically the effect of mass transfer on unsteady flow past an

accelerated vertical porous plate with suction Recently, Das and his associates [16] investigated the

hydromagnetic convective flow past a vertical porous plate through a porous medium in presence of

suction and heat source

In the present problem, we analyze the natural convection unsteady magnetohydrodynamic mass transfer

flow of a viscous incompressible electrically conducting fluid past an infinite vertical porous flat plate in

presence of constant suction and heat sink Approximate solutions are obtained for the velocity,

temperature, concentration distribution, skin friction and the rate of heat transfer using multi parameter

perturbation technique and the effects of the important parameters on the flow field are analyzed with the

help of figures and a table

2 Formulation of the problem

Consider the unsteady natural convection mass transfer flow of a viscous incompressible electrically

conducting fluid past an infinite vertical porous plate in presence of constant suction and heat sink and a

transverse magnetic field B 0 The x′-axis is taken in vertically upward direction along the plate and the

y′-axis is chosen normal to it Neglecting the induced magnetic field and the Joulean heat dissipation and

applying Boussinesq’s approximation the governing equations of the flow field are given by:

Continuity equation:

0

y

v

'

'

=

∂

∂

Momentum equation:

(T T ) g (C C ) B u g

y

u y

u

v

t

2

2

′

−

′

−

′ +

′

−

′ +

′

∂

′

∂

=

′

∂

′

∂

′

+

′

∂

′

∂

∞

σ β

β

Energy equation:

( − ∞)

′ +

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

′

∂

′

∂ +

′

∂

′

∂

=

′

∂

′

∂

′

+

′

∂

′

∂

' T ' T S y

u C y

T k y

T

v

t

p 2

, (3)

Concentration equation:

2 2 y

C D y

C

v

t

C

′

∂

′

∂

=

′

∂

′

∂

′

+

′

∂

′

∂

(4)

The initial and boundary conditions of the problem are:

( ) ( ) i t

w w t

i w w

v

v

,

0

u′= ′=− ′ ′= ′ +ε ′ − ∞′ ω ′ ′= ′ +ε ′ − ∞′ ω ′ at y′=0,

,

0

u′ → T′→T∞′, C′→C∞′ as y′→∞ (5)

Introducing the following non-dimensional variables and parameters,

, ,

v

u u , v

4 , 4

v t

t

,

v

y

0 2

0

2 0 0

ρ

η ν ω

ν ω ν

′

=

′

′

=

′

′

=

′

′

T T

T T T

∞

′

−

′

′

−

′

C C

C C C

w ∞

∞

′

−

′

′

−

′

=

2 0

2 0

v

B M

′

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

ρ

σ

,

( ) ( )

( ∞)

∞

∞

′

−

′

′

=

′

′

=

′

′

−

′

=

′

′

−

′

=

=

=

T T C

v E

, v

S S , v

C C g G , v

T T g G , D

S

,

k

P

w p

2 0 c

2 0 3

0 w

* c

3 0

w r

c

r

ν β

ν β

ν ν

ν

(6)

in Eqs (2)-(4) under boundary conditions (5), we get

Mu C G T G y

u y

u

t

u

4

1

c r 2

2

− + +

∂

∂

=

∂

∂

−

∂

∂

, (7)

2 c 2

2

u E ST 4

1 y

T P

1 y

T

t

T

4

1

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

∂

∂ + +

∂

∂

=

∂

∂

−

∂

∂

, (8)

2 2

c y

C S

1 y

C

t

C

4

1

∂

∂

=

∂

∂

−

∂

∂

, (9)

where g is the acceleration due to gravity, ρ is the density, σ is the electrical conductivity, ν is the

coefficient of kinematic viscosity, β is the volumetric coefficient of expansion for heat transfer, β* is the

volumetric coefficient of expansion for mass transfer, ω is the angular frequency, η0 is the coefficient of

viscosity, k is the thermal diffusivity, T is the temperature, T' w is the temperature at the plate, T'∞ is the

temperature at infinity, C is the concentration, C' w is the concentration at the plate, C'∞ is the

concentration at infinity, C p is the specific heat at constant pressure, D is the molecular mass diffusivity,

G r is the Grashof number for heat transfer, G c is the Grashof number for mass transfer, M is the magnetic

parameter, P r is the Prandtl number, , S is the heat sink parameter,Sc is the Schmidt number and E c is the

Eckert number

The corresponding boundary conditions are:

t i t

i

e 1 C , e 1

T

,

0

u = = +ε ω = +ε ω at y=0,

0

T

,

0

3 Method of solution

To solve Eqs (7)-(9), we assume ε to be very small and the velocity, temperature and concentration

distribution of the flow field in the neighbourhood of the plate as

( )y , t T ( )y e T ( )y

( )y , t C ( )y e C ( )y

Substituting Eqs (11) - (13) in Eqs (7) - (9) respectively, equating the harmonic and non-harmonic terms

and neglecting the coefficients ofε2, we get

Zeroth order:

0 c 0 r 0

0

0 u Mu G T G C

2 0 c r 0

r

0

r

0

y

u E P T 4

S P

T

P

⎠

⎞

⎜⎜

⎝

⎛

∂

∂

−

= +

′

+

0

C

S

First order:

1 c 1 r 1 1

4

i

u

⎠

⎞

⎜

⎝

−

′

+

, (17)

⎠

⎞

⎜⎜

⎝

⎛

∂

∂

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

∂

∂

−

=

− ω

−

′

+

′′

y

u y

u E P T

S i

P

T

P

r

1 0 1

1

4

, (18)

0 C 4

S

i

C

S

1

c

(19)

The corresponding boundary conditions are

1 1 0 1 1 0

0 0 = 0 = 0 = 1 = 1 = 1 =

= : u , T , C , u , T , C

0 0 0 0 0

∞

→ : u , T , C , u , T , C

Solving Eqs (16) and (19) under boundary condition (20), we get

,

e

,

e

C m1y

1

−

Using multi parameter perturbation technique and assuming Ec<<1, we assume

01 00

01 00

11 10

11

10

Now using Eqs (23)-(26) in Eqs (14), (15), (17) and (18) and equating the coefficients of like powers of

c

E and neglecting those ofE , we get the following set of differential equations: c2

Zeroth order:

0 c 00 r 00

00

00 u Mu G T G C

1 c 10 r 10 10

4

i M

u

⎠

⎞

⎜

⎝

−

′

+

, (28)

0 T 4

S P

T

P

(i S)T 0 4

P

T

P

The corresponding boundary conditions are,

1 0 1 0

0 00 = 00 = 10 = 10 =

= : u , T , u , T

0 0

0

0 00 10 10

∞

→ : u , T , u , T

First order:

01 r 01 01

11 r 11 11

4

i M

u

⎠

⎞

⎜

⎝

−

′

+

( )2 00 r 01 r 01

r

4

S P

T

P

∂

∂

⎟⎟

⎞

⎜⎜

⎛

∂

∂

−

=

−

−

′

+

′′

y

u y

u P 2 T S i 4

P

T

P

The corresponding boundary conditions are,

0 0

0 0

0 0

0

∞

Solving Eqs (27)-(30) subject to boundary condition (31) we get,

y 1 3 y S 2 y

3

m

1

y

3

m

00 e

y n 6 y m 5 y

m

4

y

m

e

10

−

Solving Eqs (32)-(35) subject to boundary condition (36) we get,

7 y 1 3 m 6 y c S 1 5 y c S 3 m 4 y 5 m 2 3 y 3 m 2 2 y

S

1

T = − + − + − + − + + − + + − + − − , (41)

6 y c 1 m 5 y c 5 m 4 y 3 3 m 3 y 3 m 1 m 2 y 5 m

3

m

1

( ) ( ) ( ) m y

10 y 3 1 9 y 1 1 m 8 y 1 5

m

( ) (1 c)y

5 y c 3 m 4 y n 3 y m 2 2 y

S

1

( ) n y

8 y m 7 y 1 3

m

6 y c 1 m 5 y c 5 m 4 y 3 3 m 3 y 3 m 1 m 2 y 5 m 3

m

1

11 y m 10 y 3 1 9 y 1 1 m 8 y 1 5

m

Substituting the values of C0 and C1 from Eqs (21) and (22) in Eq (13) the solution for concentration

distribution of the flow field is given by

y m t i y

S

e e

e

3.1 Skin friction

The skin friction at the wall is given by

0

=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

∂

∂

=

τ

y

w

y

u

=−m 3 A 1−S c A 2+n 1 A 3−E c[ b 1 S c+ b 2 m 3+ b 3 n 1+b 4(m 3+S c)+b 5(n 1+S c)

( ) ] { 5 4 1 5 3 6 c[ ( 3 5) 1

t i 1 8 3 7 1 3

+(m 1+m 3)D 2+(m 3+n 3)D 3 +(m 5 +S c)D 4 +(m 1+S c)D 5 +(n 3+S c)D 6

+(m 5 +n 1)D 7 +(m 1+n 1)D 8 +(n 1+n 3)D 9 +m 5 D 10 −n 3 D 11] } (46)

3.2 Heat flux

The heat flux at the wall in terms of Nusselt number is given by

0

y

u

y

T

N

=

⎟⎟

⎞

⎜⎜

⎛

∂

∂

=

=−m 3−E c[ a 1 S c+ a 2 m 3+ a 3 m 5−a 4(m 3+S c)+a 5(n 1+S c)+a 6(m 3+n 1)−a 7 m 3]

+εe iωt{−m 5 −E c[ (m 3+m 5)B 1+(m 1+m 3)B 2 +(m 3+n 3)B 3 +(m 5 +S c)B 4 +(m 1+S c)B 5

+(n 3+S c)B 6 +(m 5+n 1)B 7+(m 1+n 1)B 8+(n 1+n 3)B 9−m 5 B 10] }, (47)

where

⎥⎦

⎤

⎢⎣

c c

2

1

⎥⎦

⎤

⎢⎣

c c

2

1

⎥⎦

⎤

⎢⎣

r r

2

1

⎥⎦

⎤

⎢⎣

r r

2

1

( )⎥⎦⎤

⎢⎣

2

1

⎢⎣

2

1

2

1

n 1= + + , [ 1 1 4 M]

2

1

2= − + + ,

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

⎟

⎠

⎞

⎜

⎝

⎛ + +

+

=

4

i M 4

1

1

2

1

3

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

⎟

⎠

⎞

⎜

⎝

+ +

−

=

4

i M 4 1 1 2

1 4

( 1 3)( 2 3)

r 1

m n m n

G A

+

−

S n S n

G A

+

−

2

1

A = + ,

( 3 5)( 4 5)

r 4

m n m n

G A

+

−

( 3 1)( 4 1)

c 5

m n m n

G A

+

−

( 3 c)( 4 c)

2 2 2 c r 1

S m S m

A S P a

+

−

( 4 3)

2

1

3

r

2

m

2

m

A

m

P

a

+

−

2 3 2 1 r 3

n m n m

A n P a

+

−

( 3 4 c)

r 3 2 1 4

S m m

P m A A 2 a

+ +

−

S n m S n m

n A A S P 2 a

+ +

−

−

−

3 2

1

r

6

n m

m

m A

A

P

2

a

+

+

= ,a 7 =a 1+a 2+a 3+a 4 +a 5 +a 6,

6 1 r 1

m m m

A A P 2 B

+ +

−

( 5 3 r 1 1)(5 6 1 3 3 1)

2

m m m m m m

m m A A P 2 B

+ +

−

−

5 4 2

r

4

S m

m

m A

A

P

2

B

+ +

−

m m n m m n

n m A A P 2 B

+ + +

−

1 5 2 c r 5

S m m S m m

m A A S P 2 B

+ +

−

−

( 3 5 c)( 3 6 c)

3 6 2 c r

6

S m n S m

n

n A A S P

2

B

+ + +

−

5 4 3 r 7

m m n

m A A P 2 B

+ +

1 1 5 3 r 8

m m n m m n

n m A A P 2 B

+ +

− +

( 3 1 5)( 3 1 6)

3 1 6 3 r

9

m n n m n

n

n n A A P

2

B

+ + +

+

= ,B 10=B 1+B 2+B 3+B 4+B 5 +B 6 +B 7 +B 8+B 9,

( 1 c)( 2 c)

1 r 1

S n S n

a G b

+

−

( 1 3)( 2 3)

2

r

2

m 2 n

m

2

n

a

G

b

+

−

( 2 1)

1

3 r 3

n n n a G b

+

−

4 r 4

S m n S m n

a G b

+ +

−

−

5 r 5

S n n S

a G b

+ +

−

( 2 1 3)

3

6

r

6

m n

n

m

a

G

b

+

+

−

( 3 1)( 2 3)

7 r 7

m n n m

a G b

+

−

= ,b 8=b 1+b 2+b 3+b 4+b 5+b 6+b 7,

1 r 1

m m n m m n

B G D

+ +

−

−

( 3 3 1 r)(2 4 3 1)

2

m m n m

m

n

B G D

+ +

−

−

3

3 r 3

m n n m

B G D

+ +

−

4 r 4

S m n S m n

B G D

+ +

−

−

5 r 5

S m n S m

n

B G D

+ +

−

−

( 4 3 c)

c

6 r 6

S n n S B G D

+ +

−

7 r 7

m n n m n n

B G D

+ +

−

−

8 r 8

m n n m

n

n

B G D

+ +

−

−

1

9 r 9

n n n n

B G D

+ +

−

( 3 5 r)(10 4 5)

10

m n m n B G D

+

−

10 9 8 7 6 5 4 3 2

1

4 Results and discussions

The problem natural convection unsteady magnetohydrodynamic mass transfer flow of a viscous incompressible electrically conducting fluid past an infinite vertical porous flat plate in presence of constant suction and heat sink has been investigated The governing equations of the flow field are solved employing multi parameter perturbation technique and the effects of the flow parameters on the velocity, temperature, concentration distribution and also on the skin friction and rate of heat transfer in the flow field are analyzed and discussed with the help of velocity profiles 1-5, temperature profiles 6-7,

concentration distribution 8 and Table 1 respectively

4.1 Velocity field

The velocity of the flow field suffers a substantial change in magnitude with the variation of the flow parameters The important parameters affecting the velocity of the flow field are magnetic parameter M,

Grashof numbers for heat and mass transfer G r, G c; heat sink parameter S and Schmidt number S c Figures 1-5 discuss the effects of these parameters on the velocity of the flow field

0 1 2 3 4 5 6 7

y

u

M=0 M=0.5 M=5 M=10

Figure 1 Velocity profiles against y for different values of M with G r=3, G c=3, S= -0.1, S c=0.60, P r=0.71,

E c=0.002, ω=5.0, ε=0.2, ωt=π/2

The effect of magnetic parameter M on the velocity field is discussed in Figure 1 Curve with M=0

corresponds to the case of non-MHD flow Comparing the curves of Figure 1, it is observed that a growing magnetic parameter retards the velocity of the flow field at all points due to the dominant effect

of the Lorentz force acting on the flow field In Figures 2 and 3, we observe the effect of Grashof numbers for heat and mass transfer G r, G c respectively on the velocity field Curves with G r <0

correspond to heating of the plate, while those with G r >0 correspond to cooling of the plate Analyzing

the curves of Figures 2 and 3, we come to a conclusion that both the parameters G r and G c enhance the velocity of the field at all points Figure 4 elucidates the effect of heat sink/source parameter S on the

velocity of the flow field Curves with S<0 and S>0 correspond to the presence of heat sink and heat

source respectively in the flow field The heat source parameter (S>0) is found to accelerate the velocity

of the flow field at all points while the presence of heat sink (S<0) reverses effect The effect of Schmidt

number S c on the velocity field is discussed in Figure 5 The heavier diffusive species (growing S c) has a decelerating effect on the velocity of the flow field at all points

Figure 2 Velocity profiles against y for different values of G r with G c=3, M=1, S= -0.1, S c=0.60, P r=0.71,

E c=0.002, ω=5.0, ε=0.2, ωt=π/2

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

y

u

G r =5

G r =3

G r =1

G r = -1

G r= -3

G r = -5

Figure 3 Velocity profiles against y for different values of G c with G r=3, M=1, S= -0.1, S c=0.60, P r=0.71,

E c=0.002, ω=5.0, ε=0.2, ωt=π/2

0 1 2 3 4 5

y

u

S=0.5 S=0 S= -0.05 S= -0.2 S= -0.5

Figure 4 Velocity profiles against y for different values of S with G r=3, G c=3, E c=0.002, M=1, S c=0.60,

P r=0.71, ω=5.0, ε=0.2, ωt=π/2

-4 -3 -2 -1 0 1 2 3 4 5

u

G c =5

G c =3

G c =1

G c =-1

G c =-3

G c =-5

Figure 5 Velocity profiles against y for different values of S c with G r=3, G c=3, E c=0.002, M=1, S= -0.1,

P r=0.71, ω=5.0, ε=0.2, ωt=π/2

4.2 Temperature field

The temperature field is found to change appreciably with the variation of Prandtl number P r and heat sink parameter S These variations have been shown in Figures 6 and 7 respectively On close

observation of the curves of both the figures, we notice that the effect of increasing the magnitude of heat sink parameter and the Prandtl number is to decrease the temperature of the flow field at all points; while the heat source parameter reverses the effect

Figure 6 Temperature profiles against y for different values of P r with G r=3, G c=3, M=1, S= -0.1,

E c=0.002, ω=5.0, ε=0.2, ωt=π/2

0 1 2 3 4 5 6 7

y

u

S c =0.22

S c =0.3

S c =0.6

S c =0.78

S c =1.004

0 0.2 0.4 0.6 0.8 1 1.2

y

T

P r =1

P r =2

P r =7

P r =9

P r =11