International Journal of Scientific & Engineering Research, Volume 4, Issue 4, April-2013 65

ISSN 2229-5518

Effect of Hall Currents and Thermo-diffussion on convective Heat and Mass Transfer flow of a Viscous, Chemically reacting rotating fluid through a porous medium past a Vertical porous plate

D.Venkatesulu U. Rajeswara Rao

Abstract: The combined effects of Hall currents and radiation on the convective heat and mass transfer of a viscous electrically conducting rotating fluid through a porous medium past a vertical porous plate. in the presence of heat generating sources. The equations governing the flow, heat and mass transfer are solved exactly to obtain the velocity, temperature and concentration and are analyzed for different variations of the governing parameters G,D-1,M,m,N,N1 ,Sc, So and

.The shear stress and the rate of heat and mass transfer are numerically evaluated for different sets of the parameters. The influence of the Hall currents, thermo- diffusion and radiation effects on the flow, heat and mass transfer characteristics are discussed in detail.

Key words: Hall Currents and Thermo-diffussion; Heat and Mass Transfer; Viscous, Chemically reacting rotating fluid; Vertical porous plate.

.

1. INTRODUCTION:

he flow of Free convective heat and mass transfer flow in

T porous medium has received considerable attention due to

its numerous applications in geophysics and energy related
problems. Such types of applications include natural circulation in isothermal reservoirs, acquifires, porous insulation in heat storage bed, grain storage, extraction of thermal energy and thermal insulation design. Studies associated with flows through porous medium in rotating environment have some relevance in geophysical and geothermal applications. Many aspects of the motion in a rotating frame of reference of terrestrial and planetary atmosphere are influenced by the effects of rotation of the medium. Also buoyancy and rotational effects are often comparable in geophysical processes. Convective transport in a rotating atmosphere over a locally heated surface gives rise to dust storms (typhoons) and other atmosphere circulations.
Magneto-hydrodynamics of rotating fluids is highly important due to its varied and wide applications in the areas of Geophysics, Astrophysics and fluid engineering. An order of magnitude analysis shows that in the basic field equations, the effects of corialis force are more significant as compared to that of inertial forces. Furthermore, it may be noted that the corialis and Magneto-hydrodynamic forces are comparable in magnitude and corialis force induces secondary flow in the fluid. The unsteady flow of a rotating viscous fluid in a channel maintained by non- torsional oscillations of one or both the boundaries has been studied by several authors to analyse the growth and development of boundary layers associated with geothermal flows for possible applications in geophysical fluid dynamics [2, 3,
4, 6-8, 9, 11].

Research Scholar, Dept. of Mathematics, Sri Krishnadevaraya University, Anantapur -515003, India.

Mail: dvenkatesulu1@gmail.com

Professor, Department of Mathematics, Sri Krishnadevaraya University, Anantapur- 515003, India. Mail: urajeswararao.professor@gmail.com

Later Singh et al [12] studied free convection in MHD flow of a rotating viscous liquid in porous media. Singh et al [13] have also studied free convective MHD flow of a rotating viscous fluid in a porous medium past an infinite vertical porous plate.
In all these investigations, the effects of Hall currents are not considered. However, in a partially ionized gas, there occur Hall currents when the strength of the impressed magnetic field is very strong. These Hall effects play a significant role in determining the flow features. Yamanishi [15] have discussed the Hall effects on the steady hydromagnetic flow between two parallel plates. Debnath [3] has studied the effects of Hall currents on unsteady hydromagnetic flow past a porous plate in a rotating fluid system and the structure of the steady and unsteady flow is investigated. Alam et al [1] have studied unsteady free convective heat and mass transfer flow in a rotating system with Hall currents, viscous dissipation and Joule heating. Taking Hall effects into account Krishna et al [5] have investigated Hall effects on the unsteady hydromagnetic boundary layer flow. Rao et al [9] have analyzed Hall effects on unsteady hydromagnetic flow. Siva prasad et al [14] has studied Hall effects on unsteady MHD free and forced convection flow in a porous rotating channel. Seth et al (11a) have discussed the effect of Hall currents on the unsteady hydromagnetic couette flow within a porous channel.

2. FORMULATION AND SOLUTION OF THE PROBLEM

We consider the steady flow of an incompressible, viscous, electrically conducting, and rotating fluid through a porous medium bounded by a vertical plate. In the undisturbed state, both the plates and fluid rotate with same angular velocity Ω and are maintained at constant temperature and concentration. Further the plates are cooled or heated by constant temperature gradient in some direction parallel to the plane of the plates. We
choose a Cartesian co-ordinate system O (x,y,z) such that the plate

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International Journal of Scientific & Engineering Research, Volume 4, Issue 4, April-2013 66

ISSN 2229-5518

is at z=0 and the z-axis coincide with the axis of rotation of the plates .The steady hydro magnetic boundary layer equations of
motion with respect to a rotating frame moving with angular

Where m= ee is the Hall parameter.

On solving equations (2.7) & (2.8) we obtain

 H

velocity Ω under Boussinesq’s approximations are

2

j e 0 (v mu)

x 1  m 2

(2.9)

W u  2v   1 p  u  ( H J u)   )u   g


0 z

x

z 2

e 0 y (

k

(2.1)

j  e H 0 (mv u)

y 1  m2

(2.10)

By using Rosseland approximation the radiative heat flux is

W

v  2u

2 v

 ( H

J v)  )v

qr  

4

(T 4 )

(2.11)

0 z

z 2

e 0 x k

3R y

The energy equation is

2

(2.2)

2 2

It should be observed that by using Rosseland approximation the present analysis is limited to optically thick fluids. We assume that the temperature differences

C (W T) k

T Q(T

T)  (u2 v2 )  e Ho (u2 v2 )

0 p 0 z

f z2 

z z

0

(2.3)

within the flow are sufficiently small so that T may be
expressed as a linear function of the temperature. This is

4

The diffusion equation is
accompanied by expanding

T in a Taylor series about T

C 2 C

2 T

and neglecting higher order terms as

p (W0


)  D1

z z 2

k1C k11

z 2

(2.4)

T 4  4T 3  3T 4

(2.12)

 

Equation of State is

   (T T) 

(C C )

(2.5)

Where is Stefan-Boltzmann constant and mean absorption coefficient.
Far away from the plate we have

R is the

Where u, v are the velocity components along x and y directions respectively is the pressure including the centrifugal force,is the

p

0   g

x

(2.13)

density is the permeability constant, is the dynamic viscosity, kf
is the thermal diffusivity,D1 is the chemical molecular diffusivity,
is the coefficient of thermal expansion, is the volumetric
Using the equations (2.5), (2.12) & (2.13) the equations of motions governing the flow through a porous medium with respect to a rotating frame are given by

2 2 2

coefficient of expansion with mass fraction, Q is the strength of

Wo



u  2v u  (

e H0 )

(mvu)  (

)u

the heat source , k11is the cross diffusivity and qr is the radiative z
heat flux.

z 2

(1 m2 )

g(T T)

k

g(C C)

When the strength of the magnetic field is very large
we include the Hall current so that the generalized Ohm’s law is modified to

2 2 2

(2.14)

v v

(e H0 )






J ee JxH ( E e q xH )

(2.6)



Wo  2u

z z 2

(v mu)  ( )v

(1  m2 ) k

Where q is the velocity vector. H is the magnetic field intensity

(2.15)

vector. E is the electric field; J is the current density vector,

e is

Combining (2.14) & (2.15) we get
the cyclotron frequency, e is the electron collision time,is the q

( 2 H 2 )

2 q

W0

 2iq  


e 0 (1  im)q  (

2

)q

2

fluid conductivity and

e is the magnetic permeability. z

o (1  m )

k z

Neglecting the electron pressure gradient, ion-slip and thermo- electric effects and assuming the electric field E=0, equation (2.6)
reduces
Where

q = u + i v

(2.16)

j x mj y  e H 0 v

(2.7)

The non-dimensional variables are

j y mj x   e H 0u

(2.8)

z  

z

(/ Wo )

, , q  q

Wo

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 (T T1 )

Tw T

(C C )

C 1

C w C

C  exp(m1 z)

 (1  a3 ) exp(m2 z)  a3 exp(m1 z)

The equations governing the flow, heat and mass transfer are
(dropping the dashes)

2

Where

2


q  q 2 q  G( 2 NC)

(2.17)

1 1 4 2

( ) 4

z 2 z

2 M2 D1 2iE1

m   

2

mSc

Sc

2

2

z 2

P

z

   PEc

( q

z

. q

z

)  PEcM 2 (q.q )

P

m2

P2  4

2

a2  1  a3

(2.18)

a3

Q1

m2 Pm 

a4 a5

a6

a7

2 C C

ScSo 2

1 1 1 1

z 2

Sc

z

kC  

N

z 2

(2.19)

a  

G(1  a3 )

a Ga3

5 m2 m

2

6 m2 m

2

Where

2  ( D 1  2iE 1

M 2 (1  im)


,

2 2

a7

GN

m 2 m

1 1

 2

1  m 2 1 1

E (Ekman number)

L2

2

3. SHEAR STRESS, NUSSELT NUMBER & SHERWOOD NUMBER

D 1 L

k

(Darcy parameter),
Shear stress on the wall z = 0 is given by

dq

gL4 (T

T )

 (

) z0

G w

2

(Grashof number)

dz

 m3a4 m2 a5 m1 (a6 a7 )

(C w C )

N (Buoyancy ratio),

Local rate of heat transfer across the walls (Nusselt

P

k

(Tw T)

(Prandtl number),
Number) is given by

( Nu) z  0

 ( d)

dz

z 01

QL2

k1

(Heat source parameter)

 m2 (1  a3 )  a3 m1

Rate of mass transfer (Sherwood Number) is given by

Sc  

D1

(Schmidt number),

(Sh)

z 0

 ( dC )

dz

z 0

  m1

So k11

(Soret Number),

4. DISCUSSION OF THE NUMERICAL RESULTS



The effect of Hall currents and thermo-diffusion

2

k1 L

/ D1 (Chemical reaction parameter)

on convective Heat and Mass transfer flow of a viscous fluid
The boundary conditions are
past a vertical plate is investigated. We take P = 0.71 in this

q  0

 1 ,

C  1

on z = 0
on z = 0
analysis.
The velocity component (u) is shown in figs 1-8 for

q 0,

0

, C 0

as z 

(2.20)

different G, D-1, M, m, E-1, Sc, S0, N, and. Fig.1 represents
the velocity with Grashof number G. It is found that u > 0 for G > 0
Solving equations (2.17)-(2.19) subject to the boundary
conditions (2.20) we obtain

q  (a6 ) exp(m1z)  a5 exp(m2 z)  a4 exp(m3z)

and u < 0 for G < 0. |u| experiences an enhancement with increase in |G| ( 0). The variation of u with D-1 or M shows that lesser the
permeability of the porous medium / higher the Lorentz force

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smaller |u| in the flow region. [An increase in the Hall current parameter (m) enhances |u| in the region]. With respect to variation of u with m shows that the velocity changes from negative to positive as we move away from the plate. The region of transition from negative to positive enhances with increase in m (fig. 2 & 3). From fig.4 we find that u reduces with E-1 in the entire flow region. The variation of u with Schmidt number (Sc) shows that lesser the molecular diffusivity smaller |u| in the flow region and for further lowering of the molecular diffusivity the velocity changes from positive to negative as we move away from the wall (fig.5). With respect to Soret parameter S0 we observe that u > 0 for S0 > 0 and u < 0 for S0 < 0. |u| enhances with increase in |S0| (<0) and reduces with S0 (<0) (fig.6). When the molecular buoyancy force dominates over the thermal buoyancy force the velocity depreciates in the flow region irrespective of the directions of the buoyancy forces (fig.7).
The non-dimensional temperature () is shown in figs 8
for different the temperature gradually reduces from its prescribed value 1 on y = 0 and attains the value ‘0’ for away from the wall. An increase in the heat source parameter reduces the temperature.
The concentration distribution (C) is exhibited in fig 9 -
11 for different Sc, S0, and N From fig. 9 we find that lesser the molecular diffusivity smaller the concentration in the flow region. An increase in S0 > 0 enhances the actual concentration while an increase in S0 < 0 leads to depreciation in the actual concentration (fig.10). When the molecular buoyancy force dominates over the thermal buoyancy force the actual concentration depreciates when the buoyancy forces act in the same direction while for the forces acting in opposite directions it experiences an enhancement in the flow region. An increase in the radiation parameter enhances u in the neighbourhood of the wall and it depreciates for away from the wall (fig.11).
The shear stress at y = 0 is shown in tables 1 & 2 for different G, D-1, M, m, and N. It is found that the shear stress enhances with increase in G. Lesser the permeability of the porous medium / higher the Lorentz force larger |Sh| at y = 0. An increase in the Hall parameter m reduces || at the wall. Also it experiences an enhancement with increase in D-1
(table 1). From table 2, we find that lesser the molecular diffusivity smaller || at y = 0 and for further lowering of the molecular diffusivity larger ||. The variation of with Soret Parameter S0 shows that || enhances with |S0| |( 0). Also it enhances with N > 0 and reduces with increase in |N| (<0) an increase in leads to a depreciation in || (table 3).
The Nusselt number (Nu) at the wall y = 0 is
shown in table 4. It is found that the rate of heat transfer
enhances with increase in the radiation parameter N1 and heat source parameter .
The Sherwood number (Sh) at y = 0 is shown in table 5 for different parametric values. It is found that the rate of mass transfer reduces in magnitude with 5 and enhances with higher 10. Lesser the molecular diffusivity larger |Sh| at the wall. Also it enhances with increase in the Soret parameter |S0| ( 0). When the molecular buoyancy force dominates over the thermal buoyancy force the rate of mass transfer experiences an enhancement in magnitude irrespective of the directions of the buoyancy forces.

5. REFERENCES

[1] Alam, M. M and Sattar, M. A: Unsteady free convection and mass transfer flow in a rotating system with Hall currents, viscous dissipation and Joule heating., Journal of Energy heat and mass transfer,V.22,pp.31-39(2000)
[2] Claire Jacob: Transient motions produced by disc oscillatory
torsionally about the state of rigid rotation, ZAMM, V, 24, p.221 (1971)
[3] Debnath, L: Exact solutions of unsteady hydrodynamic and
hydromagnetic boundary layer equations in a rotating fluid system, ZAMM, V.55, p.431 (1975)
[4] Krishna, D.V. Prasada Rao, D. R.V, Ramachandra Murty, A.
S: Hydromagnetic convection flow through a porous medium in a rotating channel. J. Engg. Phy. and Thermo.Phy,V.75(2),pp.281-291(2002)
[5] Krishna, D.V and Prasada Rao, D. R.V: Hall effects on the unsteady hydromagnetic boundary layer flow. Acta Mechanica, V.30, pp.303-309(1981)
[6] Murthy, K.N.V. S: Oscillatory MHD flow past a flat plate.
Ind, J, Pure and Appl, Mthd. V.9, p.501 (1979)
[7] Mohan. M: Combined effects of free and forced convection on MHD flow in a rotating channel. V, Ind. Acad .Sci. v, 85, pp.393-401(1977).
[8] Mohan and Srivastava, K.K: Combined convection flows through a porous channel rotating with angular velocity, Proc.Ind.Acad.Sci, V.87, p.14 (1978)
[9] Rao D.R.V, Krishna, D.V and Debnath, L: Combined effect of free and forced convection on MHD flow in a rotating porous channel, Int.J.Maths and Sci, V.5, pp.165-182(1982)
[10] Rao, D.R.V and Krishna, D.V: Hall effects on unsteady hydromagnetic flow. Ind.J. Pure and Appl. Maths, V.12 (2), pp.270-276(1981)
[11] Sarojamma and Krishna, D.v: Transient hydromagnetic convective flow in a rotating channel with porous boundaries, Acta Mrchanica, V.39, p.277 (1981)

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International Journal of Scientific & Engineering Research, Volume 4, Issue 4, April-2013 69

ISSN 2229-5518

[11a] Seth, G.S, Singh, J.K and Mahato, G.K: Unsteady hydromagnetic coquette flow within a porous channel with hall effects. Int. J. Engg, Sci and Tech, V,3, No.6, pp.172-183 (2011)
[12] Singh, N.P, Gupta, S.K and Singh, Atul Kumar: Free convection in MHD flow of a rotating viscous liquid in porous medium past a vertical porous plate, Proc. Nat. Acad.Sci, India, V.71A. pp.149-157(2001)
[13] Singh,N.P,Singh,Ajay Kumar,Yadav,M.K and Singh Atul Kumar : Hydromagnetic oscillatory flow of a viscous liquid past a vertical porous plate in a rotating
system.,Ind.Theor.Phy.,V.50,pp.37-43(2002)
[14] Siva prasad, Prasada Rao. D.R.V and Krishna. D.V: Hall effects on unsteady MHD free and forced convection flow in a porous rotating channel. Ind. J. Pure and Appl.Maths, V.19 (2) pp.688-696(1988)
[15] Yamanishi: Hall effects on hydromagnetic flow between two parallel plates. Phy.Soc., Japan, Osaka, V.5, p.29 (1962)

GRAPHS:

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International Journal of Scientific & Engineering Research, Volume 4, Issue 4, April-2013 70

ISSN 2229-5518

o_s ,------------------------------------------,

0_4

0_3

0_ 2

0_ 1 I

;-- r,;-- ;: rv

-0_2

-0_3

-0_ 4 J _J

y y

I n

05 LO

m

-05

IV

- 1_0

0_ 4 ,- ----------------------------------------,

0_ 35

0_ 3

025

0_ 2

0_ 15

0_ 1

o_ ---r---,----r---r--=::::::==-!

0_3 .- ----------------------------------------

025

0_ 2

0_ 15

I u 0_ 1

rv o_ -- ---,----r-=====:

-O _ QS

-0_1

-0_ 05 -- -- J

y

-0_ 15

------------------------------------ J

y

Fig_ 7 Variation ofu wtth N

Fi.g _S Variation ofu '\.vith ot.

I

n

m

IV

I

n

m

IV

N

2

-0_ 5

-0_8

2

4

6

8

I I IV

y y

Fig9 V.a::ri.ati..oo. o£"9 --i'th. ex.&. N

F.ig_ lO Va.ria.ti.on ofC ·i:tb. Sc

I

n

m

IV

I n

m

C1

2

4

2

2

0_24- 0_ 6

L3

Nl

2

2

4

6

1_2

o_s

0_6

c 0_4

0_ 2

0

-0_ 2

-0_4

y

1

o_s

0_6

c 0_4

0_2

0

-0_2

-0_4 J J

y

Fig_ l 2 Varia:ti.OD. ofC '"--ith. N

I

n

m

IV

I

n

m

IV

0_5

LO

-0 _5

- 1 _ 0

N

1

2

-0_.5

-0_ 8

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Tabll' -1

Shl'l'l' Sb'i'SS ('t) )at z = 0

G

I

II

m

IV

v

VI

VII

vm

10,

27.2077

44.8 8

58.2840

36.6831

495986

25.0449

24_

2602

27.2297

3x10,

81.6230

1345755

174.8520

110.0494

148.

7960

75.1349

72.

7806

81.6892

-10'

-27.2077

-44.

8 85

-58.2840

-36.6831

-495

986

-2.5.0449

-24

_2602

-27.2297

-3x10'

-81.6230

-1345755

-174.8520

- ll0.0494

-148

.7960

-75.1349

-72

.7806

-81.6892

M

2

2

2

4

6

2

2

2

m

0.5

0.

5

0.5

0.5

0

.5

1.5

25

0.5

F'

0.5

0.

5

0.5

0.5

05

0 .5

0.5

1.5

Tabll'- ::

Shi'er SU'i'Ss ('t) )at z = 0

G I II m IV v VI VII vm IX X

to> 27.2077 27.2298 -15.7463 -8.742 121.3533 -66.9379 - ll4.0108 29.4449 23.8518 21.0525

3x10' 81.6230 86.1296 -47.2388 -26.2276 364.0599 -200.8139 -342.0324 88.3346 71.555 65.9846

-10"' -27_2077 -27.2298 1_7463 8.7425 -121.3533 66.9379 ll4.0108 -29.4449 -23.8518 -21.0 25

-3xlo> -81.6230 -86.1296 47_2388 26.2276 -364.0599 200.8139 342.0324 -88.3346 -71.5555 -65.9846

Sc L3 2.01 0_24 0.6 l.3 L3 1.3 13 1.3 l.3

So 0. 0 .5 0. 0 .5 l.S 0. -1.0 05 0 .5 0.5

1 1 1 1 1 1 1 2 -0.5 -0.8

Tabll' -3

Shl't'r SU'i'Ss ('t) )at z = 0

Tablt> --'

1\ussdr 1\umbt>r (Nu) at z = 0

Table - 5

Sherwood Kumber (Sh) )at z = 0

Table- 6

Sherwood Kumber (Sh) )at z = 0

a.

I

IT

ill

N

2

-0.43618

-0.8681

-3.02765

-3.37978

5

-0.02724

-0.6636

-3.84553

-4.46986

10

0.43663

-0.4417

-4.77325

-5.36636

N

1

2

-0.5

-0.8

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