EFFECT OF VARIABLE SUCTION AND CHEMICAL …JORIND 13(2) December, 2015. ISSN 1596-8303. EFFECT OF VARIABLE SUCTION AND CHEMICAL REACTION ON MHD ...transcampus.org/JORINDV13DEC2015/Jorind

JORIND 13(2) December, 2015. ISSN 1596-8303. www.transcampus.org/journal; www.ajol.info/journals/jorind

EFFECT OF VARIABLE SUCTION AND CHEMICAL REACTION ON MHD OSCILLATORY

FLOW THROUGH A VERTICAL POROUS PLATE WITH HEAT GENERATION

A.S. Idowu, A. Jimoh, K. M. Joseph and L.O. Ahmed

Department of Mathematics, University of Ilorin, Ilorin, Nigeria

E-mail:- [email protected]

Abstract

The effect of variable suction and chemical reaction on MHD oscillatory flow through a vertical porous

plate with heat generation has been investigated. The governing equations of the flow field are solved

employing perturbation technique and the expressions for the velocity, temperature and species

concentration are obtained. The effect of flow parameters on the flow field has been studied and the

results are presented graphically and discussed quantitatively.

Keywords: MHD, chemical reaction, variable suction, heat generation

Introduction

Oscillatory flows has known to result in higher

rates of heat and mass transfer, many studies have

been done to understand its characteristics in

different systems such as reciprocating engines,

pulse combustors and chemical reactors.

The applications of variable suction and chemical

reaction play important role in the design of

chemical processing equipment, formation and

dispersion of fog, distribution of temperature and

moisture over agricultural fields and groves of

fruits damage of crops due to freezing, food

processing and cooling of towers. Investigation of

periodic flow through a porous medium is

important from practical point of view because

fluid oscillations maybe expected in many magneto

hydrodynamic devices and natural phenomena,

where fluid flow is generated due to oscillating

pressure gradient or due to vibrating walls.

Consequently, Ayuba et al (2015) studied the

effect of variable suction on magneto

hydrodynamic couette flow through porous

medium in the slip flow regime.

Joseph et al (2015) examined the problem of

unsteady MHD mixed convictive oscillatory flow

of an electrically conducting optically thin fluid

through a planer channel filled with saturated

porous medium. The effect of buoyancy, heat

source, thermal radiation and chemical reaction of

the fluid were taken into considerations with slip

boundary condition, varying temperature and

concentration. The closed-form analytical solutions

are obtained for the momentum, energy and

concentration equations. Chamkha (2003) studied

the MHD flow of a numerical of uniformly

stretched vertical permeable surface in the

presence of heat generation/absorption and

chemical reaction. Idowu et al (2013) studied the

effect of chemical reaction on MHD oscillatory

flow through a vertical porous plate with heat

generation. Hady et al (2006) researched on the

problem of free convection flow along a vertical

wavy surface embedded in electrically conducting

fluid saturated porous media in the presence of

internal heat generation.

Recently, considerable attention has also been

focused on new applications of MHD and heat and

mass transfer such as metallurgical processing

Kishore et al (2013). In melt refining, the magnetic

field is used to control excessive heat and mass

transfer rate. The effect of radiative heat and mass

transfer on unsteady natural convection coquette

flow of a viscous incompressible fluid in the slip

flow regime in present of variable suction and

radiative heat source was analyzed by Das et al

(2012). Rao et al (2013), analyzed the unsteady

free convection heat and mass transfer flow

through a non-homogeneous porous medium with

variable permeability bounded by an infinite

porous vertical plate in slip flow regime taking in

to account the radiation, chemical reaction and

temperature gradient dependent heat source.

The problem of oscillatory MHD slip flow along a

porous vertical wall in a medium with variable

suction in the presence of radiation was analyzed

numerically by Ogulu and Prakash (2004).

Makinde(2005) investigated the free convection

flow with thermal radiation and mass transfer past

a moving vertical porous plate. Das and his co-

workers (2005) discussed the laminar flow of an


elastico-viscous Rivlin-Ericksen fluid through

porous parallel plates with suction and injection,

the lower plate being stretched. Ogulu and Motsa

(2005) investigated the problem ofradiative heat

transfer to magnetohydrodynamiccouette flow with

variable wall temperature. Cortell (2005) studied

the flow and heat transfer of a fluid through a

porous medium over astretching surface with

internal heat generation/absorption and suction/

blowing. Das and his co-workers (2008) analyzed

the effect of heat source and variable magnetic

field on unsteadyhydromagnetic flow of a viscous

stratified field past a porous flat moving plate in

the slip flow regime. In a separate paper Das et al.

(2008) studied the hydromagnetic three

dimensional couetteflow and heat transfer.

Recently, Das and his associates (2008)estimated

the effect of mass transfer on free convective MHD

flow of a viscous fluid bounded by an oscillating

porous plate inthe slip flow regime in presence of

heat source. Sharma and Singh (2008) investigated

the unsteady MHD free convective flow and heat

transfer along a vertical porous plate with

variablesuction and internal heat generation. S. S.

Das, J. Mohanty, S. Panda and B. K. S. Pattanaik

(2013) studied the effect of variable suction and

radiative heat transfer on MHD couette flow

through a porous medium in the slip flow regime.

Sib Sankar Manna, Sanatan Das and RabindraNath

Jana et al (2012) investigated the effects of

radiation on unsteady MHD free convective flow

past an oscillating vertical porous plate embedded

in a porous medium with oscillatory heat flux. K.

sarada and B. shanker (2013) analyzed the effect of

chemical reaction on an unsteady MHD free

convection flow past an infinite vertical porous

plate with variable suction.

This present paper studied theeffect of variable

suction and chemical reaction on MHD oscillatory

flow through a vertical porous plate with heat

generation.

Formulation of the problem

Consider unsteady two – dimensional

hydrodynamic laminar, incompressible, viscous,

electrically conducting fluid and heat source past a

semi – infinite vertical moving heated porous plate

embedded in a porous medium and subjected to a

uniform transverse magnetic field in the presence

of thermal diffusion and thermal radiation effect.

According to the coordinate system, the x – axis is

taken along the plate in upward direction and y –

axis is normal to the plate. The fluid is assumed to

be gray, absorbing – emitting but non – scattering

medium. It is assumed that there is no applied

voltage of which implies the absence of an electric

field. The transversely applied magnetic field and

magnetic Reynolds number are very small and

hence the induced magnetic field is negligible.

Viscous terms are taken into account the constant

permeability porous medium. The MHD term is

derived from an order – of – magnitude analysis of

the full Navier – stoke equation. It is assumedhere

that the hole size of the porous plate is

significantly larger than a characteristic

microscospic length scale of the porous medium.

The fluid properties are assumed to be constant

except that the influence of density variation with

temperature has been considered in the body –

force. Since the plateis semi – infinite in length,

therefore all physicalquantities are functions of y

and t only. Hence, by the usual boundary layer

approximations, the governing equations for

unsteady flow of a viscous incompressible fluid

through a porous medium are

Continuity equation

(1)

Linear momentum equation

( ) (

)

(2)

Energy equation

(

)

(3)


Concentration equation

(

) (4)

The boundary conditions are

(5)

Where ( ) is suction velocity, - the velocity components in the directions

respectively, - the kinematics viscosity, - the thermal conductivity, - the coefficient of volume

expansion due to temperature, - the coefficient of volume expansion due to concentration, - the

density, - the electrical conductivity of the fluid, - the acceleration due to gravity, - the temperature,

- wall temperature of the fluid, - the radiation heat flux, -the concentration,

- wall

concentration of the fluid and - chemical reaction parameter, − mean free path, − specific heat at a

constant pressure and - mass diffusivity.

It should be noted that by using the Roseland approximation the present analysis is limited to optically

thick fluids. If temperature differences within the flow are sufficient small

( )

(

) (6)

Where is the radiation absorption coefficient at the wall and is the Planck’s function.

Method of solution of the problem

In order to write the governing equations and the boundary conditions in dimensionless form, the

following non – dimensional quantities are introduced

( )

,

( )

( )

(7)

As consequence of equation (7), equations (1) – (4) reduce to

(8)

( )

[

( )

] (9)

( )

( ) (10)

( )

(11)

And the boundary condition in dimensionless form are given by the following

(12)


Where and are dimensionless velocities, is

the time, is the frequency of oscillation, is

the Reynolds number, is the Prandtl number,

is the permeability parameter, is the thermal

Grashof number, is the modified Grashof

number, is the heat source parameter, is the

Eckert number, are real positive

constants, is the Schmidt number.

Solution

In order to solve the partial differential equations

(9) – (11) subject to the boundary conditions in

equation (12), we used the following two term

perturbation technique

( ) ( ) ( ) ( ) (13)

( ) ( ) ( ) ( ) (14)

( ) ( ) ( ) ( ) (15)

Substituting equations (13), (14) and (15) into equations (9), (10) and (11). Equating the coefficients of

harmonic and non-harmonic term and neglecting the coefficients of higher order of we get:

( )

( ) ( ) ( ) ( ) (16)

( )

( ) ( ) ( ) ( ) ( ) (17)

Where, (

)

( )

( ) ( ) (18)

Where ( )

( )

( ) ( ) ( ) (19)

Where [

( )]

( )

( ) ( ) (20)

( )

( ) ( ) ( ) (21)

Where, [

]

And the corresponding boundary conditions are

(22)

We now solve equations (16) – (21) under the

relevant boundary conditions (22) for the mean

flow and unsteady flow separately.

The mean flows are governed by the equations

(16), (18) and (20) where and are called

the mean velocity, mean temperature and mean

concentration respectively. The unsteady flows are

governed by equations (17), (19) and (21) where

and are the unsteady components.


These equations are solved analytically under the

relevant boundary conditions (22) as follows;

Solving equations (16), (18) and (20) subject to the

corresponding relevant boundary conditions in

(22), we obtain the mean velocity, mean

temperature and mean concentration as

( )

(23)

( )

(24)

( )

(25)

Similarly, solving equations (17), (19) and (21) under the relevant boundary conditions in (22), the

unsteady velocity, unsteady temperature and unsteady concentration becomes

( )

(26)

( )

(27)

( )

(28)

Therefore, the solutions for the velocity, temperature and species concentration profiles are

( )

[

] (29)

( )

[

] 30)

( )

[

] (31)

Where,

(

), ( ), [

( )],

[

],

√

,

√

, ,

√

,

√

,

, (

), ( ),

,

, √

,

√

,

, ,

√

,

√

,

,

,

(

), ( ),

, ,


√

,

√

,

,

,

,

,

, ,

√

,

√

,

,

,

,

,

,

,

,

,

,

(

), ( ),

,

.

Discussion of results

In order to get physical insight of the effect of

variable suction and chemical reaction on MHD

oscillatory flow through a vertical porous plate

with heat generation. The governing equations of

the flow fluid are solved for velocity, temperature

and concentration. The effect of flow parameters

such as Schmidt number , heat source , thermal

Grashof number , chemical parameter ,

Hartmann number , Prandtl number ,

Reynolds number are presented graphically and

discussed numerically with the aid of MATLAB.

Figure 1 shows the effect of Schmidt number on

velocity profile . It is observed that the velocity

decreases with increase in Schmidt number. To this

effect, the velocity distribution across the boundary

layer is decreased.

Figure 2 depicts the effect of heat source parameter

on velocity profile . It is noticed that the

velocity increases with increase in heat source.

The effect of thermal Grashof number on

velocity profile is shown in figure 3. It can be

depicted that the velocity increases with increase in

thermal Grashof number. Here the thermal Grashof

number represent the effect of free convection

currents. Physically, means heat of fluid of

cooling of the boundary surface, means

cooling of the fluid of heating of the boundary

surface and corresponds to the absence of

free convection current.

Figure 4 shows the effect of chemical parameter

on velocity profile . It is seen that the velocity

increases with increase in chemical parameter.

The effect of Hartmann number on velocity

profile is presented in figure 5. It can be

observed that the velocity decreases with increase

in Hartmann number. This infers that the

turbulence at the boundary layer is suppressed.

Figure 6 shows the effect of Prandtl number on

velocity . It is seen that the velocity decreases

with increase in .

Figure 7 depicts the effect of Reynolds number

on velocity . It is observed that the velocity

increases with increase in Reynolds number.

Figure 8 depicts the effect of Prandtl number on

temperature . The numerical results show that the

effect of increasing values of Prandtl number

results in a decreasing fluid temperature. To this

effect it decreases the thermal boundary layer

thickness and in general lower average temperature

within the boundary layer. The reason is that

smaller values of Prandtl numbers are equivalent to

increase in the thermal conductivity of the fluid

and therefore heat is able o diffuse away from the

heated surface more rapidly for higher values of

Prandtl number as the thermal is thicker and rate of

heat transfer is reduced.

Figure 9 shows the effect of chemical parameter

on concentration distribution . It is observed

that the concentration distribution increases with

increase in .


Figure 10 shows the effect of Schmidt number

on concentration distribution. It can be seen that

the concentration distribution decreases with

increase in . This is because smaller values of

Schmidt number are equivalent to the chemical

molecular diffusivity.

Fig. 1: Effect of on


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

y

Velo

city u

Effect of Schmidt number Sc on Velocity u

Sc=1

Sc=2

Sc=3

Sc=4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.2

0

0.2

0.4

0.6

0.8

1

1.2Effect of Heat Source on Temperature

Tem

pera

ture

y

=1

=2

=3

=4




0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

y

Velo

city u

Effect of thermal Grashof number Gr on Velocity u

Gr=1

Gr=2

Gr=3

Gr=4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

y

Velo

city u

Effect of Chemical parameter Kr on Velocity u

Kr=1

Kr=2

Kr=3

Kr=4




0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

y

Velo

city u

Effect of Hartmann number M on Velocity u

M=1

M=2

M=3

M=4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

y

Velo

city u

Effect of Schmidt number Sc on Velocity u

Sc=1

Sc=2

Sc=3

Sc=4




0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.2

0

0.2

0.4

0.6

0.8

1

1.2Effect of Heat Source on Temperature

Tem

pera

ture

y

=1

=2

=3

=4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

y

Velo

city u

Effect of thermal Grashof number Gr on Velocity u

Gr=1

Gr=2

Gr=3

Gr=4




0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

y

Velo

city u

Effect of Chemical parameter Kr on Velocity u

Kr=1

Kr=2

Kr=3

Kr=4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

y

Velo

city u

Effect of Hartmann number M on Velocity u

M=1

M=2

M=3

M=4



Fig. 7: Effect of

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

y

Velo

city u

Effect of Prandtl number Pr on Velocity u

Pr=0.71

Pr=0.81

Pr=1.0

Pr=2.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

y

Velo

city u

Effect of Reynolds Number Re on Velocity u

Re=1

Re=2

Re=3

Re=4


Fig. 8: Effect of

Fig. 9: Effect of

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.2

0

0.2

0.4

0.6

0.8

1

1.2Effect of Prandtl number Pr on Temperature

Tem

pera

ture

y

Pr=0.71

Pr=0.81

Pr=1

Pr=2

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

y

Specie

s C

oncentr

ation C

Effect of Kp on Species Concentration C

Kp=1

Kp=2

Kp=3

Kp=4


Fig. 10: Effect of

Summary and conclusion

In the present paper, we investigated the effect of

variable suction and chemical reaction on MHD

oscillatory flow through a vertical porous plate

with heat generation. The governing equations of

the flow field are solved by using Perturbation

method and the expressions for velocity profile,

temperature profile and concentration distribution

are obtained.

The conclusions of the study are as follows:

1. Increase in Schmidt number, Hartmann

number, heat source and Prandtl number

have decelerating effect on the velocity

profile.

2. The velocity increases with decrease in

chemical parameter, thermal Grashof

number, heat source and Reynolds

number.

3. Increase in Prandtl number has

decelerating effect on temperature profile.

4. Increase in chemical parameter has

accelerating effect on concentration

distribution and increase in Schmidt

number has decelerating effect on

concentration distribution.

This paper may have potential role in the design of

chemical processing equipment, distribution of

temperature over agricultural fields, food

processing etc.

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Documents

EFFECT OF VARIABLE SUCTION AND CHEMICAL …JORIND 13(2) December, 2015. ISSN 1596-8303. EFFECT OF VARIABLE SUCTION AND CHEMICAL REACTION ON MHD ...transcampus.org/JORINDV13DEC2015/Jorind