RADIATION EFFECT ON NATURAL CONVECTIVE MHD ......Convective MHD Flow through a Porous Medium with...

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International Journal of Civil Engineering and Technology (IJCIET)

Volume 9, Issue 5, May 2018, pp. 441–449, Article ID: IJCIET_09_05_049

Available online at http://www.iaeme.com/ijciet/issues.asp?JType=IJCIET&VType=9&IType=5

ISSN Print: 0976-6308 and ISSN Online: 0976-6316

© IAEME Publication Scopus Indexed

RADIATION EFFECT ON NATURAL

CONVECTIVE MHD FLOW THROUGH A

POROUS MEDIUM WITH DOUBLE DIFFUSION

IN THE PRESENCE OF CHEMICAL REACTION

Pudhari Srilatha

Department of Mathematics, Institute of Aeronautical Engineering, Hyderabad, TS, India

M N Raja Shekar

Department of Mathematics, JNTUH College of Engineering, Nachupally, Jagitial, TS, India

ABSTRACT:

The influence of radiation on natural convective MHD flow through a porous

medium in the presence of chemical reaction and transverse magnetic field bounded

by a vertical infinite surface is studied. The basic equations governing the flow, heat

and mass transfer are reduced to a set of ordinary differential equations by suitable

transformations. The resulting set of coupled non-linear ordinary differential

equations are solved by using the MATLAB in-built numerical solver bvp4c for

velocity, temperature and concentration, and that has been presented graphically for

different values of needed parameters. It is observed that results of magnetic

parameter and radiation parameter in the flow field change the flow significantly.

Keywords: Natural Convective MHD, Chemical Reaction, Double Diffusion

Cite this Article: Pudhari Srilatha and M N Raja Shekar, Radiation Effect on Natural

Convective MHD Flow through a Porous Medium with Double Diffusion in the

Presence of Chemical Reaction, International Journal of Civil Engineering and

Technology, 9(5), 2018, pp. 441–449.

http://www.iaeme.com/IJCIET/issues.asp?JType=IJCIET&VType=9&IType=5

1. INTRODUCTION

The natural convection flow past a vertical surface was examined widely for its application in

designing and ecological process. In many transport forms in nature and modern application

in which heat and mass exchange is an outcome of buoyancy force because of diffusivity of

heat, chemical substances and concentration. In decades ago, the investigation of heat and

mass transfer in MHD natural convective flow with chemical reaction and radiation has

received a growing interest because of its significance in a many engineering, geophysical,

and astrophysical applications, for example, polymer production, cooling of atomic reactors,

underground energy transport, plasma flow, etc. Heat and mass transfer with radiation

Radiation Effect on Natural Convective MHD Flow through a Porous Medium with Double

Diffusion in the Presence of Chemical Reaction


assumes a main part in the manufacturing industries, in atomic power plants, gas turbines and

different drive gadgets for air craft’s, rockets, satellites, and space vehicles.

Rajesh and Varma [3] examined the radiation effects on MHD flow through a porous

medium with variable temperature or variable mass diffusion. The effect of viscous

dissipation on the mixed convection heat transfer from an exponentially stretching surface

was studied Partha et al. [4]. Sahoo et al. [5] analyzed the magneto hydrodynamic unsteady

free convection flow past an infinite vertical plate with constant suction and heat sink.

Ibrahim et al. [6] presented the effect of the chemical reaction and radiation absorption on the

unsteady MHD free convection flow past a semi-infinite vertical permeable moving plate with

heat source and suction. Y.J. Kim [7] studied an unsteady MHD convective heat transfer past

a semi-infinite vertical porous moving plate with variable suction. The radiation effect on

MHD mixed convection flow about a permeable vertical plate was studied by Orhan Aydn

[8]. Chen [9] discussed the problem of magnetohydrodynamic mixed convective flow and

heat transfer of an electrically conducting, power-law fluid past a stretching surface in the

presence of heat generation/absorption and thermal radiation. B.Shankar Goud., M.N. Raja

Shekar[10] studied Finite element study of Soret and Radiation effects on mass transfer flow

through a highly porous medium with heat generation and chemical reaction. Influence of

chemical reaction and radiation on unsteady MHD free convection flow and mass transfer

through viscous incompressible fluid past a heated vertical plate immersed in porous medium

in the presence of heat source was studied by Sharma et.al [11]. J. Chamkha and I. Camille

[12] discussed the effects of heat generation/absorption and thermophoresis on hydromagnetic

flow with heat and mass transfer over a flat surface. Sudheer Babu and Satyanarayan [13]

presented the effect of chemical reaction and mass transfer on MHD unsteady free convection

flow past semi-infinite vertical plate with constant/variable suction and heat sink. Hady et al.

[14] have analyzed the MHD free convection flow along a wavy surface with heat generation

or absorption effect. Sankar Reddy et al. [15] studied the heat and mass effects on MHD flow

of a continuously moving vertical surface with uniform heat and mass flux.

In view of the above discussion the objective of the present paper is radiation effect on

natural convective MHD flow through a porous medium with double diffusion in the presence

of chemical reaction and the resulting set of coupled non-linear ordinary differential equations

are solved by using the MATLAB in-built numerical solver bvp4c.

2. MATHEMATICAL ANALYSYS

Consider a steady incompressible viscous free convective MHD fluid flow with thermal

radiation embedded in a permeable medium in a semi-infinite region bounded by a vertical

endless surface, which experiences a homogeneous chemical reaction. The x¢-axis is taken the

surface an upward way and the y¢ axis is perpendicular to it. A uniform Magnetic field is also

connected toward the path opposite to the plate. The induced magnetic field and Hall effect

are ignored. Here, there is no applied voltage, which implies there is no electric current. The

equations describing the above flow are given by

Equation of continuity:

0h

¢¶=¢¶

v (1)

Momentum equation:

2

22

( ) ( )su

u u b bh rh

¥ ¥

¢ ¢¶ ¶¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢= + - + - - -

¢ ¢¶ ¢¶

oBu vg T T g C C u u

K (2)

Pudhari Srilatha and M N Raja Shekar


Energy equation:

2

22 1a uu

h r r h hh

æ ö¢ ¢ ¢¶¶ ¶ ¶ ÷ç¢ ÷= - + ç ÷ç ÷ç¢ ¢ ¢¶ ¶ ¶¢ è ø¶

r

p p p

qT T u

c c c (3)

Diffusion equation:

2

2

uh h

¢ ¢¶ ¶¢ ¢ ¢= -

¢¶ ¢¶

C CD Kr C (4)

Equation (1) gives0

constu u¢= = - (5)

Where uo is scale of suction velocity which is a nonzero positive constant. The negative

sign indicates that suction is towards the plate

With the appropriate initial and boundary conditions are given by

0, , 0

0, ,

h

h¥ ¥

ü¢ ¢ ¢ ¢ ¢ ¢= = = = ïïýï¢ ¢ ¢ ¢ ¢ ¢® ® ® ® ¥ ïþ

w wu T T C C at

u T T C C at (6)

The radiative heat flux term by using the Rosseland approximation is given [1] by 4

4

3

s

h

¢ ¢¶= -

¢¶r

s

Tq

K (7)

Where s ¢is the Stefan – Boltzmann constant and sK is the mean absorption coefficient.

For sufficiently small temperature difference within the flow we can expressed 4

¢T as a linear

function of the temperature and expanding

4 434 3¥ ¥¢¢ ¢@ -T T T T (8)

By using equation (5) and (6) 3 2

2

16

3

s

h h

¥¢ ¢¶ ¶

=¶ ¶

r

s

q T T

k (9)

Introducing the following non – dimensional quantities,

0

0

* 2

0

3 3

0 0

2 2 3

0 0

2 2 2

0 0

, , , , Pr ,

( ) ( ), ,

( )

16, , , ,

3

uru h uh q

u u a

u b u b u

u u

u s u s u

u ru a u

¥ ¥

¥ ¥

¥ ¥

¥

¥

üï¢ ¢ ¢ ¢ ¢¢ - - ï= = = = = = ïï¢ ¢ ¢ ¢- - ïïïï¢ ¢ ¢ ¢- - ïï= = = ýï¢ ¢- ïïïï¢ ¢ ¢ ¢ ï= = = = ïïïïþ

p

w w

w w

p w

s

CT T C Cuu C Sc

T T C C D

g T T g C CGr Gc Ec

C T T

K B T KrK M R Ko

K

(10)

In the view of above equation, the basic flow field equations can be expressed in the

following form:

2

2

10

u uGr GcC M u

K

(11)

22

2

Pr0

1

uEc

R

(10)




2

2

10

C CKoC

Sc y

(12)

Where , , Pr, , ,Gr Gc Sc Ko R ,and K are the thermal Grashof number, Solutal Grashof

number, Prandtl number, Schmidt number, chemical reaction parameter ,radiation parameter,

heat generation parameter, and permeability of the porous medium respectively.

0, 1, 1 0

0, 0, 0

q h

q h

ü= = = = ïïýï® ® ® ® ¥ ïþ

u C at

u C as (13)

Knowing the velocity field, the skin-friction coefficient at the plate is given by

0

u

Other important physical quantity of interest is the rate of heat transfer in terms of Nusselt

number at the plate which is non-dimensional form is given by0

Nu

An important phenomenon in this study is to study the Sherwood number at the plate is

given by0

CSh

Skin friction, Nusselt number and Sherwood number

B. R. Rout [2] Present Study

5 5 1 1 1 0.71 0.22 1 0.05 6.984 0.2736 0.5918 6.925611 0.192981 0.591831

10 5 1 1 1 0.71 0.22 1 0.05 11.1821 0.1866 0.5918 11.17526 -0.10063 0.591831

-10 5 1 1 1 0.71 0.22 1 0.05 -4.3021 0.3443 0.5918 -4.28978 0.289517 0.591831

5 10 1 1 1 0.71 0.22 1 0.05 10.3326 0.1558 0.5918 10.21914 -0.00582 0.591831

5 -10 1 1 1 0.71 0.22 1 0.05 -2.5309 0.3135 0.5918 -2.61209 0.343863 0.591831

5 5 0.1 1 1 0.71 0.22 1 0.05 3.0104 0.3158 0.5918 3.006049 0.347049 0.591831

5 5 0.1 0.5 1 0.71 0.22 1 0.05 3.0682 0.3518 0.4564 3.063835 0.346017 0.460073

5 5 0.1 0.5 3 0.71 0.22 1 0.05 2.8204 0.3522 0.4594 2.81654 0.350068 0.460073

5 5 0.1 0.5 1 7.0 0.22 1 0.05 2.4325 0.353 0.4594 2.300862 3.415401 0.460073

5 5 0.1 0.5 1 0.71 0.8 1 0.05 2.8089 1.8393 1.1483 3.063835 0.346017 0.460073

5 5 0.1 0.5 1 0.71 0.22 2 0.05 3.1274 0.2351 0.4594 3.114497 0.247633 0.460073

5 5 0.1 0.5 1 0.71 0.22 1 0.1 3.0694 0.3493 0.4594 3.066757 0.326441 0.460073

5 5 0.1 0.5 1 0.71 0.22 1 -0.05 3.0658 0.3578 0.4594 3.058039 0.384889 0.460073

The comparison of variety of skin friction, Nusselt number, and Sherwood number is

given in the table. Increasing the mass Grashof number and the thermal Grashof

number , skin friction increases, but the reverse effect is considered for Nusselt number,

though there is no impact on Sherwood number. When and are negative, the skin

friction becomes negative. Porosity parameter increases the skin friction, but decreases the

Nusselt number without influencing the Sherwood number. At the point when chemical

reacting substances increases, then there is a small decreases in skin contact, however

opposite behaviour is appeared for Nusselt number and Sherwood number. For getting higher

estimation of magnetic parameter and Prandtl number , skin friction diminishes, so far

Nusselt number increases. Skin friction and Nusselt number increase, yet Sherwood number

decreases with increment in Schmidt number. At the point when radiation factor and Eckert

number increment, skin friction increments, yet Nusselt number reduces, while Sherwood

number stays unaltered.



3. RESULTS AND DISCUSSIONS

In Fig. 1 the curves demonstrates that the velocity starts from a minimum value on the surface

and increments till achieves a peak values close to the plate, at that point starts decreasing

toward the boundary layer. The ebb and flow in Fig. 2 mirrors that with increment in there

is increment in fluid speed because of enhancement of buoyancy force. The positive

estimation of shows the cooling of the plate and it is observed that velocity increments

quickly close to the wall of the plate and after that decays to the free stream velocity. At the

point when is negative, the plate becomes hotter and there is retardation in the fluid

velocity. A comparable impact of mass Grashof number on fluid velocity is also

considered in Fig. 2. Eckert number, on the velocity profile is examined in Fig. 3 also, it

demonstrates that the fluid velocity increments because of increment in the Eckert number. It

is found out that the fluid velocity increments gradually and at a short distance from the wall

it accomplishes the maximum value also, decreases to zero for positive estimation of the

Eckert number, i.e., for . This demonstrates there is a hot layer close to the plate

with a cooler fluid far from the plate. At the point when . i.e., for negative Eckert

number, when the temperature of the fluid close to the wall is not as much as the surrounding

temperature, the fluid velocity increments and attaining a peak values decreases step by step

to zero. The curve and flow of Pr and Sc versus velocity in Fig. 4 specifies that the velocity

profile diminishes with increment in Pr and Sc. This occurs because of predominance of

thermal diffusion and molecular diffusion over the viscous diffusion. The diagram of Fig. 5

demonstrates that with increment in porosity parameter there is increment in velocity profile,

i.e., if the porosity parameter increases, at that point the resistive power for the fluid flow

diminishes, which prompts the improvement of the fluid velocity. Closer to the wall the fluid

flow achieves a peak value then slowly diminishes. The effects of the chemical reaction

parameter on the velocity profile are studied in Fig. 6. It is observed that for i.e.,

generative reaction constructs the fluid flow velocity yet i.e., destructive reaction

decreases the liquid stream speed. The velocity profiles accomplish most extreme close to the

wall also, decrease step by step. In Fig. 7, it is fascinating to take note of that in a steady flow

an increase in the radiation parameter prompts an increase in the extent of the fluid velocity

inside the boundary layer.

The variety of temperature with and is considered in Fig. 8, for positive value of

Gr, i.e., for externally cooled plate the temperature increments close to the wall yet for

negative value of the i.e., for externally heated plate, the temperature decreases. In the two

cases, with increment in there is an increase in the temperature profile. Comparable

impacts are additionally examined for mass Grashof number Figure 9 describes that with

increasing chemical reaction parameter there is an increment in the temperature profile. In

Fig. 10, it is seen that the values of the Prandtl number Pr is inversely proportional to the

amount of temperature flow. If the value of Pr is smaller, there is an increase in thermal

conductivity of the fluid, which enables to flow in more normal temperature. Along these

lines, amount of smaller Prandtl number the thermal boundary is thicker, for which the

temperature profile increments. In any case, for the radiation parameter 'R' reverse behavior

was appeared by the liquid, i.e., for larger thermal radiation parameter the thermal boundary

layer is thicker, so the temperature profile expands, which is shown in Fig. 11.

The variety of focus as for the concentration with respect to the chemical reaction

parameter and Schmidt number is examined in Figs. 12 and 13. Figure 12 shows that

destructive, i.e., for reduces the concentration because of the commitment of mass

dissemination in concentration equation, meanwhile, in the generative reaction, i.e., for

the reverse effect is observed. The effect of Schmidt number on focus profile is

defined in Fig. 13. It is observed that the concentration profile diminishes monotonically and




the boundary layer thicknesses decrease with the increase of Schmidt number. Figure 14 show

the effect of skin friction with various value of M. it is observed that increasing M decreases

skin friction.

. CONCLUSIONS:

The governing partial differential equations for the studying incompressible natural

convective MHD flow with thermal radiation introduced in a permeable medium in a infinite

region bounded by a vertical surface, which experiences homogeneous chemical reaction with

magnetic field, are changed into a set of ordinary differential equations by utilizing the

MATLAB in-built numerical solver bvp4c and under given boundary conditions. Some of the

important conclusions of the present examination are as per the following:

Radiation parameter improves the fluid velocity and the fluid temperature.

Velocity diminishes with increment in the magnetic field parameter.

The back flow is observed for the negative estimations of thermal and mass Grashof

number.

Figure 1 Velocity profiles versus Figure 2 Velocity profiles versus

variation of M variation of &


variation of variation of Pr and Sc




variation of variation of

Figure 7 Velocity profiles versus Figure 8 Temperature profiles versus

variation of R variation of and

Figure 9 Temperature profiles versus Figure 10 Temperature profiles versus





Figure 11 Temperature profiles versus Figure 12 Concentration profiles versus


Figure 13 Concentration profiles versus Figure 14 Skin friction profiles versus


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on natural convective MHD flow through a porous medium with double diffusion, Journal

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