RADIATION EFFECT ON HYDRO MAGNETIC CONVECTIVE HEAT ...€¦ · to the forced convection with heat generating source. AbdEl – Naby et al [17] studied the effects of radiation on

Open Journal of Applied & Theoretical Mathematics (OJATM)

Vol. 2, No. 4, December 2016, pp. 380~399

ISSN: 2455-7102

Journal homepage: http://ojal.us/ojatm/

380

RADIATION EFFECT ON HYDRO MAGNETIC CONVECTIVE HEAT

TRANSFER THROUGH A POROUS MEDIUM IN A VERTICAL CHANNEL

WITH HEAT SOURCES

S.T.DINESH KUMAR

DINESH. P.A

&

P.RAVEENDRA NATH

Assistant professor, Department of Mathematics, Govt.Science college, Chitradurga, Karnataka

Associate Professor, Department of Mathematics, M.S.Ramaiha Institute of Technology, Bengaluru

Lecturer in Mathematics, Department of Mathematics, Sri Krishnadevaraya University College of Engineering and Technology, S.K. University, Anantapur - 515 055, A.P., India.

E-mail: [email protected]

Article Info ABSTRACT

The present study concentrates an attempt to

study an unsteady convective heat and mass transfer

of a flow of a viscous electrically conducting fluid

through a porous medium confined in a vertical

channel bounded by flat walls in the presence of heat

generating sources, by taking into account Radiation

effect. An oscillatory pressure gradient in the x-

direction is considered in the flow region. A

perturbation method is employed to solve the

equations governing the flow heat and mass transfer

has been solved to obtain the expression for velocity,

temperature and concentration. The skin frictions, the

rate of heat and mass transfer have been evaluated for

different variations of the governing parameters.

Keyword:

Porous medium, Vertical channel, Radiation effect, Heat source

mailto:[email protected]



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Copyright © 2015 Open Journal of Applied & Theoretical Mathematics (OJATM)

All rights reserved.

1. INTRODUCTION:

The application of electromagnetic fields in controlling the heat transfer as in

aerodynamic heating leads to the study of magneto hydrodynamics heat transfer this

MHD heat transfer has gained significance owing to advancement of space technology.

The MHD heat transfer can be divided into two sections. One contains problems in

which the heating is an incidental by product of the electromagnetic fields as in the

MHD generators and pumps etc., and the second contains of problems in which the

primary use of electromagnetic fields is to control the heat transfer [1]. With the fuel

crisis depending all over the world there is great concern to utilize the enormous power

beneath the earth’s crust in the geothermal region. Hence the study of interaction of the

geomagnetic field with the fluid in the geothermal region is of great interest, thus

leading to interest in the study of Magneto hydrodynamic convection flows through

porous medium. The effects of system temperature on heat and mass transfer

investigated, by Atul kumar Singh et al [2] heat and mass transfer in MHD flow of a

viscous fluid past a vertical plate under oscillatory suction velocity. Convection fluid

flows generated by traveling thermal waves have also received attention due to

applications in physical problems.

Coupled heat and mass transfer driven by buoyancy due to temperature and

concentration variation in a saturated porous medium has several important

applications in geothermal and geophysical engineering such as the migration of

moisture through the air contained in fibrous insulation, the extraction of geothermal

energy, underground disposal of nuclear wastes and the spreading of chemical

contaminants through eater saturated soul.

A systematic derivation of the governing equations with various types of

approximations used in applications has been presented. Heat and mass transfer by



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free convection in a porous medium under boundary layer approximations has been

studied by Bejan and Khair [3], Lai and Kulacki [4], Nakayama and Hossain [5] and

Singh and Queeny [6].

Angirasa et al [7] have presented the analysis for combined heat and mass transfer

by natural convection for aiding and opposing buoyancies in fluid saturated porous

enclosures. Tanamy Basak et al[8] have analysed the natural convection flows in a

square cavity filled with a porous matrix for uniformly and non-uniformly heated

bottom wall and adiabatic top wall maintaining crust temperature of cold vertical walls

Darcy-Forchheimer model is used to simulate the momentum transfer in porous

medium.

The unsteadyness in the fluid can also be generated by imposing an oscillatory

flux in the fluid region. Several authors notably, Roslinda Nazar et al [9], Shohel

mahmud and Pop[10], Nawaf et al [11] discussed the mixed convective heat and mass

transfer flow through a porous medium in different channels and different

configurations.

In view of this several authors, Lai F.C. [12,13] ,Angirasa et al [14] Abdul [15]

have studied Natural convection in differentially heated vertical enclosures is of

fundamental interest to many practical applications. Several investigators have

presented analytical and experimental results on convection in the rectangular cavity

with vertical walls at constant temperatures, the horizontal walls being insulated

[16,13]. Reviewed the extensive work and mentioned about [13] who have contributed

to the forced convection with heat generating source. AbdEl – Naby et al [17] studied

the effects of radiation on unsteady free convective flow past a semi-infinite vertical

plate with variable surface temperature using Crank – Nicolson finite difference

method. Chamkha et al. [18] analyzed the effects of radiation on free convection flow

past a semi-infinite vertical plate with mass transfer, by taking into account the

buoyancy ratio parameter N.Ganesan and Loganadhan [19] studied the radiation and

mass transfer effects flow of incompressible viscous fluid past a moving vertical

cylinder using Rosseland approximation by the Crank – Nicolson finite difference

method. Takhar et al. [20] considered the effects of radiation on MHD free convection

flow of a radiating gas past a semi-infinite vertical plate.

2. Formulation of the problem.

We analyze the unsteady flow of an incompressible, viscous, electrically conducting

fluid through a porous medium confined in a vertical channel. A uniform transverse



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magnetic field of strength Ho is applied normal to the walls of the channel. The

unsteadiness in the flow is due to an oscillatory pressure gradient. The walls are

maintained at constant temperature. The Boussinesq approximation is used so that the

density variation will be retained only in the buoyancy force. The viscous dissipation is

neglected in comparison to the heat flow by conduction and convection. We choose a

Cartesian co-ordinate system O(x,y,z) with x-axis in the vertical direction and the walls

are taken at Ly ,2L being the mean distance between the plates .A linear density

variation is assumed with e, Te as the equilibrium density and temperature .The

equations governing the unsteady MHD flow ,heat and mass transfer in Cartesian

coordinate system in the absence of input electric field are

guHuky

u

x

p

t

uoee

22

2

2

(2.1)

y

qTTQ

y

Tk

t

TC r

epe 2

2

1 (2.2)

2

2

1y

TD

t

C (2.3)

eeee CCTT (2.4)

Invoking Rosseland approximation for radiative heat flux we get

y

Tq

R

r

)(4 4

(2.5)

and expanding 4T about Te by Taylor’s series and neglecting higher order terms we

obtain

434 34 ee TTTT (2.6)

Where is the Stefan-Boltzman constant and R is the mean absorption coefficient.

In the equilibrium state

0

g

x

pe

e (2.7)

We choose the imposed oscillatory pressure gradient as



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)( tiCospx

pe

(2.8)

The boundary conditions relevant to the problem are

u=0,T=T1exp(i t) on Ly (2.9)

u=0,T=T2 exp(i t) on Ly (2.10)

Introducing the non-dimensional variables (u*,t*,y*,*,C*,p*) as

e

e

e

e

CC

CCC

TT

TT

L

pp

L

yytt

L

uu

11

2,,

/

,,)/(

(2.11)

the equations (2.1)-(2.3) and (2.4) reduce to (on dropping the asterisk)

NCGuDMx

p

y

u

t

u

12

2

22 (2.12)

12

22

1

ytP (2.13)

2

22

y

C

t

CScP (2.14)

With the boundary conditions

1)exp()exp(,0 yonitcitu (2.15)

)exp(,)exp(,0 itncitmu on y= -1 (2.16)

Where

2

3

1 )(

LTTgG e

(Grashof number)

222

2 LHM oe

(Hartman

number)

22 L

(Wormsely number) k

LD

21



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(Porous parameter) 1k

CP

p

(Prandtl number)

1

2

k

QL (Heat source parameter)

1DSc

(Schmidt

Number) )(

)(

2

2

e

e

TT

CCN

(Buoyancy ratio)

e

e

TT

TTm

1

2 (Non-

dimensional temperature ratio) e

e

CC

CCn

1

2 (Non-dimensional concentration ratio)

314 e

R

TN

(Radiation parameter)

43

3

43

3

1

11

1

11

N

N

N

PNP

In view of the imposed oscillatory pressure gradient we take

)exp(0 ituu (2.17)

)exp(0 it (2.18)

C=Co exp (it) (2.19)

)exp(0 itpp (2.20)

Substituting (2.17)-(2.20) in (2.12)-(2.14) we obtain

00

2

12

0

2

Guy

u

(2.21)

00

2

22

0

2

y (2.22)

00

2

32

0

2

C

y

C (2.23)

Where

2122

1 iDM

2

11

2

2 Pi

22

3 iSc



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The relevant boundary conditions are

11,0 00 yonu (2.24)

1,0 00 yonmu (2.25)

Solving (2.21)-(2.23) subject to the boundary conditions (2.24) and (2.25)

)(

)(

)(

)()(

)(

)(

)(

)()(1

)(

)(

2

2

1

128

1

1

2

227

1

1

2

1

0

Ch

yCh

Ch

yChCha

Sh

ySh

Sh

yShSha

Ch

yChu

(2.26)

)(

)(

)(

)((

2)

)(

)(

)(

)(5.0

2

2

2

2

2

2

2

20

Ch

yCh

Sh

yShm

Ch

yCh

Sh

ySh (2.27)

)(

)(

)(

)((

2)

)(

)(

)(

)(5.0

3

3

3

3

3

3

3

30

Sh

ySh

Ch

yChn

Ch

yCh

Sh

yShC (2.28)

3. Shear Stress and Nusselt Number.

The shear stress on the plates is given by

1

y

dy

du (3.1)

which in the non-dimensional form is

12

2

y

dy

du

L

(3.2)

and the corresponding expressions are

)1()()()( 11211111 ShaCha (3.3)

)1()()()( 11211111 ShaCha

(3.4)

The rate of heat transfer (Nusselt Number) at the plates in the non-dimensional form is

given by



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1

ydy

dNu

(3.5)


)()()( 222121 ShaChaNu y

(3.6)

)()()( 222121 ShaChaNu y

(3.7)

The rate of mass transfer (Sherwood Number) at the plates in the non-dimensional

form is given by

1

ydy

dCSh

(3.8)


)()()( 363531 ShaChaSh y

(3.9)

))()(()( 363131 ShaChaSh y

(3.10)

where

)()()()()1( 3310339228227 ShaChaShaCha

(3.11)

)()()()()1( 3310339228227 ShaChaShaCha

(3.12)

4. Discussion of the Numerical results

The velocity, temperature and concentration have been discussed for variations

in different parameters governing the flow. The applied pressure gradient is chosen to

be negative so that the actual flow is yield along x-direction. In this analysis we choose

the non-dimensional temperature and convection ratios have been chosen to be -1. The

Grashof number ‘G’ is positive or negative according as the boundary temperature T, is



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greater or lesser than the equilibrium temperature Te. The velocity u is shown in Figs 1

– 6 for different values of G, D-1, M, R,N, and .

Fig.1 Variation of u with G Fig.2 u with D-1&M

D-1=102,M=2,=2,Sc=1.3 M=2,=2,G=103

I II III IV V VI I II III IV V

G 103 3x103 5x103 -103 -3x103 -5x103 D-1 102 3x102 5x102 102 102

M 2 2 2 4 6

Fig.3 u with N , M=2,=2,Sc=1.3 Fig.4 u with R,M=2,N=1, =2

I II III IV I II III

N 1 2 -0.5 -0.8 R 10 20 30

-6

-4

-2

0

2

4

6

8

-1 0 1

y

u

-1.6

-1.1

-0.6

-0.1

0.4

0.9

1.4

1.9

-1 -0.5 0 0.5 1

u

y

-2

-1

0

1

2

3

4

5

-1 -0.5 0 0.5 1

y u -4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

-1 0 1

u

y



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Fig.5 u with , M=2, N=1, =2 Fig.6 u with α,N=1,M=2

I II III I II III

2 4 6 α 2 4 6

Fig.7 with R,M=2,N=1, =2 Fig.8 with ,M=2,N=1, =2

I II III I II III

R 10 20 30 2 4 6

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-1 0 1

y

u

-6

-4

-2

0

2

4

6

-1 0 1

y

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-1 -0.5 0 0.5 1

y

-1.7

-1.2

-0.7

-0.2

0.3

0.8

1.3

1.8

-1 -0.5 0 0.5 1

I

II

III

y

u



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Fig.9 with ,N=1,M=2 Fig.10 with N,N=1, M=2

I II III I II III IV

2 4 6 N 0.5 1.5 5 10

Fig.11 C with Sc,M=2,N=1, =2 Fig.12 C with ,N=1,M=2

I II III IV I II III

Sc 1.3 2.01 0.24 0.6 2 4 6

Table.1 Shear Stress (

G/ I II III IV V VI VII VIII

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-1 -0.5 0 0.5 1

y

-7

-5

-3

-1

1

3

5

7

-1 0 1

c

y -7

-5

-3

-1

1

3

5

7

-1 0 1

c

y

-1.5

-1

-0.5

0

0.5

1

1.5

-1 -0.5 0 0.5 1

i

ii

iii

iv



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103 -

30.672

-

77.047

107.605 138.722 287.636 105.585 106.602 107.993

2x103 -

61.570

-

156.11

162.714 179.182 315.775 158.673 160.708 163.491

-103 31.123 81.097 -2.612 57.801 231.356 -0.5921 -1.609 -3.0008

-2x103 62.021 160.17 -67.721 17.341 203.217 -53.680 -55.715 -58.498

Table.2 Shear Stress ( ) at y = -

I II III IV V VI VII VIII

103 -

45.176

-

86.963

102.815 134.399 283.919 104.835 103.818 102.427

2x103 -

90.578

-

105.12

153.134 170.536 308.342 157.175 155.141 152.357

-103 45.626 90.014 2.1770 62.124 235.073 2.4571 1.6742 1.5658

-2x103 91.028 178.03 -48.141 25.987 210.650 -52.182 -50.148 -47.364

I II III IV V VI VII VII

D-1 3x102 5x103 103 103 103 103 103 103

R 10 10 10 10 10 10 10 10

M 2 2 2 4 6 2 2 2

Sc 1.3 1.3 1.3 1.3 1.3 0.24 0.6 2.01

Table.3 Shear Stress ( ) at y =+1 P=0.71,N1=0.5

I II III IV V VI

103 107.993 110.008 104.009 103.280 70.541 12.7551

2x103 163.491 167.520 155.505 154.063 100.121 20.970

-103 -3.008 -5.0156 2.992 1.7132 11.382 -3.673

-2x103 -58.498 -62.521 -50.512 -49.070 -18.197 -11.888

Table.4 Shear Stress ( ) at y = -

I II III IV V VI

103 102.427 100.412 106.420 107.141 63.467 6.2655

2x103 152.357 148.328 160.343 161.785 85.973 7.9873

-103 2.5658 4.5806 -1.4271 -2.1481 18.456 2.8169



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-2x103 -47.364 -93.335 -55.350 -56.792 -4.049 1.0932

I II III IV V VI

N 1 2 -0.5 -0.8 1 1

2 2 2 2 4 6

Table.5 Shear Stress ( ) at y =+1P=0.71,N1=0.5,N=1.0

I II III IV V

103 12.755 -16.080 -7.0051 106.195 107.079

2x103 20.970 -32.172 -14.011 165.635 170.452

-103 -3.673 16.104 7.005 -12.684 -22.668

-2x103 -11.888 32.197 14.011 -72.124 -87.041

Table.6 Shear Stress ( ) at y = -1P=0.71,N=1.0,N1=0.5

I II III IV V

103 6.265 -20.886 -11.812 101.404 101.687

2x103 7.989 -41.785 -23.622 156.054 160.869

-103 2.816 20.911 11.812 -7.893 -17.876

-2x103 1.092 41.809 23.622 -62.542 -77.458

I II III IV V

R 10 20 30 10 10

2 2 2 4 6

Table.7 Shear Stress (

I II III IV

103 12.755 13.867 13.998 14.2342

2x103 20.970 21.1234 23.4563 24.6754

-103 -3.673 -3.9886 -4.1243 -4.7865

-2x103 -11.888 -12.2345 -13.245 -13.9876

Table.8 Shear Stress ( ) at y= -



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I II III IV

103 6.265 5.7657 5.3452 4.1234

2x103 7.9896. 7.01234 6.9886 6.3245

-103 2.816 2.4567 2.1234 1.9886

-2x103 1.092 1.01234 0.9876 0.8976

I II III IV

N1 0.5 1.5 5 10

Table.9 Nusselt Number (Nu) at y =+1P=0.71,N1=0.5

I II III IV V

2 87.047 222.64 623.36 618.06 595.25

4 100.75 241.11 649.17 643.94 621.43

6 115.09 260.04 675.33 670.16 647.94

Table.10 Nusselt Number (Nu) at y = -1P=0.71.N1=0.5

I II III IV V

2 -43.556 -111.32 -311.67 -309.03 -297.62

4 -50.401 -120.55 -324.58 -321.97 -310.72

6 -57.567 -130.02 -337.66 -335.08 -323.97

I II III IV V

R 10 20 30 10 10

2 2 2 4 6

I II III IV

2 87.047 88.2341 89.4563 90.6574

4 100.75 102.5643 103.5643 103.9886

6 115.09 116.3456 117.4563 118.5643



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Table.12 Nusselt Number (Nu) at y = -

I II III IV

2 -43.556 -

45.7865

-

55.7865

-

66.8976

4 -50.405 -

55.7688

-59.897 -

69.6754

6 -57.567 -

63.6789

-

69.4657

-

74.7996

I II III IV

N1 0.5 1.5 5 10

Table.13 Nusselt Number (Nu) at y = - 1P=0.71,M=2,

I II III IV V

2 -43.556 -

42.6754

-

41.4568

-

40.6874

-

39.5678

4 -50.401 -

49.5674

-

48.6785

-

48.1234

-

47.6754

6 -57.567 -

56.7865

-

55.4563

-

54.3456

-

52.3545

Table.14 Sherwood Number (Sh) at y =+1P=0.71

I II III IV

2 1.8957 2.2115 -0.109 0.9393

4 3.2014 4.0056 1.5834 2.3294

6 4.8388 6.0149 2.2586 3.2632

Table.15 Sherwood Number (Sh) at y = -1P=0.71

I II III IV

2 -3.375 -4.114 -0.9112 -2.061

4 -6.438 -8.017 -2.8713 -4.451

6 -9.675 -12.02 -4.2472 -6.561



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I II III IV

Sc 1.3 2.01 0.24 0.6

From Fig.1 we notice that for G>103 the velocity changes from positive to

negative as we move from the right boundary to the left boundary in the neighborhood

of y = -1, there by indicating the reversal flow in the region -0.8 y -0.4. This region of

reversal flow shrinks in its size in the horizontal direction and enhances in the

transverse direction. In case of G < 0 the reversal flow appear in the region -0.6 y

-0.8 for |G| = 103 and for higher |G| 2 x 103 it confines to the region 0.4 y 0.8.

Also the magnitude of u enhances with G> 0 and depreciates with G< 0 with maximum

at y = 0.8 and this maximum value enhances with increase in |G|.

The variation of u with D-1 and M reveals that the reversal flow which appears in

the vicinity of y = -1 shrinks in its size with increase in D-1 or M. Also lesser the

permeability of porous medium or higher the Lorentz force smaller the magnitude of u

in the entire flow region Fig .2

Fig.3 represents the behavior of u with respect to buoyancy ratio ‘N’. It is found

that when the molecular buoyancy force dominates over the thermal buoyancy force,

the magnitude of u depreciates when the buoyancy forces act in the same direction and

for the forces acting in opposite direction |u| enhances in the flow region.

From Fig.4 we find that the reversal flow which appears in the region -0.8 y -

0.2 for R=10 reduces in its size with increase in R. Also |u| depreciates with increase in

R.

It is noticed that Fig.5 shows the An increase in the wormsely number results in a

depreciation of |u|.

It is found that the magnitude of u experiences an depreciation in the left half and

enhances in the right half of the channel. An increase in the strength of heat generating

source reduces the magnitude of the axial velocity u Fig.6.

The temperature distribution is shown in Figs 7 – 9. For different values of R,

,N and .It is found that actual temperature is greater than the equilibrium temperature

in the left half and T is less than Te in the right half. An increase in the Reynolds number

‘R’ Fig.7

Fig.8 shows the leads to depreciation in the actual temperature in the left half and

enhances in the right half. The behavior of ‘’ with respect to shows that the actual



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temperature experiences the depreciation in the left half and an enhancement in the

right half with increase in .

Also, the actual temperature reduces in the left half and enhances in the right half of

the channel with increase in the strength of the heat generating sources Fig.9.

An increase in the radiation parameter N reduces the actual temperature in the left

half and enhances in the right half Fig.10.

The Non dimensional concentration distribution C is shown in Figures. 11 & 12 For

different values of Sc and . It is to be noted that the concentration C is positive or

negative according as the actual concentration C is greater or lesser than the

equilibrium concentration. It is found from Fig.11.

Fig.12 show the higher the molecular diffusivity lesser the actual concentration in

the left half and higher the concentration in the right half of the channel. The variation

of C with shows that for smaller values of the concentration is positive in the left half

and negative in the right half. And for higher it is positive in the vicinity of y = -1 and

negative in the region -0.6 y - 0.2 and positive in the region 0.2 y 0.6. It is found

that higher the values of lesser the actual concentration in the vicinity of y = -1 and

higher the concentration in the remaining portion of the channel region.

The shear stress at the boundaries y = 1 have been calculated for different

values of G, D-1, M, R, Sc, and are shown in Tables. 1 – 8. It is found that the shear

stress at both the boundaries enhances with increasing |G|. The variation of ‘’ with D-1

& M reveals that lesser the permeability of the porous medium or higher the Lorentz

force larger the magnitude of ‘’.The behavior of ‘’ with Schmidt number ‘Sc’. Shows

that higher the molecular diffusivity larger at y = +1 and smaller at y = -1 .Tables 1 &

2.

The molecular buoyancy force dominates over thermal buoyancy force, the

stress experiences an enhancement at y = +1and depreciates at y= -1, when the

buoyancy forces act in the same direction and for the forces acting in opposite

directions decreases at y = +1 and enhances at y = -1. Also | depreciates with

increase in the Wormsely number .Tables 3&4.

From tables. 5 and 6 we find that for values of R 20, at y = 1 experiences

an enhancement and for higher R30 the stress depreciates at both the wall.

The stress experiences a enhancement in its magnitude at y=1and a

depreciation at y=-1 with increase in the radiation parameter N .Tables.7&8.



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The rate of heat transfer at y = 1 is shown in Tables. 9 & 10 for different, R &

. It is found that the rate of heat transfer at y = + 1 is positive and is negative at y = -

1.An increase in or R leads to an enhancement in |Nu| at both the walls. At y = 1 the

“Nusselt number” enhances with increase in 4 and depreciates with 6,while at y =

-1 it depreciates with . An increase in the radiation parameter N enhances the

magnitude of Nu at both the walls. Tables.11 & 12.

The Sherwood number which measures the rate of mass transfer across the

boundaries has been depicted in Tables.13&14. for different Sc & .

Table.15 shows the Higher the molecular diffusivity larger the rate of heat

transfer at y = 1 Also |Sh| experiences an enhancement with increase in the

‘Wormsely number . In general we observe that the rate of heat and mass transfer at y

= -1is greater than that at y =1

5. References

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Nomenclature

u Velocity (m/s)

p Pressure(N/m2)

T Temperature in the flow region(K)

Te Temperature in equilibrium state (K)

C Concentration (Kg/m3)

Ce Concentration in equilibrium state (Kg/m3)

k Porous permeability(H/m)

Q Strength of the heat source (W)

Cp Specific heat at constant pressure(J/kg K)

qr Radiative heat flux (W/m2)

D1 Molecular diffusivity

Greek Symbols

e Density of the fluid in the equilibrium state (Kg/m3)

Density (Kg/m3)

Dynamic viscosity (Ns/m2)

β Coefficient of thermal expansion (K-1)

* Volumetric coefficient of expansion with mass fraction (m3)

Electrical conductivity (ms3A2/kg)

e Magnetic permeability(H/m)

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RADIATION EFFECT ON HYDRO MAGNETIC CONVECTIVE HEAT ...€¦ · to the forced convection with heat generating source. AbdEl – Naby et al [17] studied the effects of radiation on