Finite Element Analysis of Convective Heat And Mass Transfer Flow of a Viscous Fluid through a Porous Medium with Heat Generating Source and Quadratic Density Temperature Variation

Finite Element Analysis of Convective Heat And Mass Transfer Flow of a Viscous Fluid through a Porous

Medium with Heat Generating Source and Quadratic Density Temperature Variation

Y. MADHUSUDHANA REDDY ¹ and D. R. V. PRASADA RAO ²

1 Associate Professor, Department of Mathematics, Sri Venkateswra Institute of Technology, Anantapur, INDIA.

2 Professor, Department of Mathematics, S. K. University, Anantapur, INDIA.

(Received on: April 15, 2012) ABSTRACT

In this analysis we investigate the effect of quadratic density temperature variation on convective heat and mass transfer flow of a viscous fluid through a porous medium in vertical channel with heat generating source. By using galarkin finite element analysis the equations governing flow heat and mass transfer are solved with three noded line segments. The effect of various fluid sources on velocity, temperature concentration is analyzed. The rate of heat and mass transfer on the boundaries are evaluated numerically.

Keywords: Vertical Channel, Nondarcy , Finite element analysis, Porous Medium, Heat and Mass Transfer.

1. INTRODUCTION

Flow and heat transfer in porous medium has been attracting the attention of an increasingly large number of investigators in recent years. The need for fundamental studies in porous media heat transfer stems from the fact that a better under standing of a host of thermal engineering applications in which porous

materials present is required. The

accumulated impact of these studies is two

fold, first to improve the performance of

existing porous media-related thermal

systems, second is to generate new ideas and

explore new avenues with respect to the use

of porous media in heat transfer

applications. Some examples of thermal

engineering disciplines which stand to

benefit from a better understanding of heat

and fluid flow process through porous materials are geothermal systems, thermal insulations, grain storage, solid matrix heat exchangers, oil extractions and the manufacturing of numerous products in the chemical industry. Darcy’s equation give satisfactory results for closely packed porous medium but does not explain the flow through sparsely distributed porous medium in later situation, Brinkman ⁵ proposed an alternate model by adding a term which accounts for the viscous shear in addition to the Darcy’s equation. The first theoretical investigation of natural convection in porous enclosure using Brinkman model was made by Chan et al ⁶ . Later on a series of investigations were carried out using the Brinkman model by a few authors notably Poulikakos and Bejan ¹⁴ .

Convective flows driven by temperature and concentration differences have been studied extensively in the fast and various extensions of the problem have been reported in the literature with both temperature and concentration interacting simultaneously, the convection has become quite complex. Combined heat and mass transfer along a vertical plate in natural convection flows have been studied in great detail. Bejan and Khair ⁴ have investigated the vertical free convection boundary layer flow embedded in a porous medium resulting from the combined heat and mass transfer. Similarity solutions for buoyancy induced heat and mass transfer for vertical surfaces in porous media were presented.

Lai and Kulacki ¹¹ have used an integral method to solve the problem of Bejan and Khair ⁴ . Nelson and Wood ¹³ has numerically and analytically investigated the combined heat and mass transfer for both boundary

conditions. Mamou ¹² et al have analyzed the problem of thermosolutal convection in a rectan cell filled with a brinkman porous medium saturated by a binary fluid.

Sugunama ¹⁸ and Ravindranath reddy ¹⁵ have analyzed the free convective heat transfer in vertical channel taking dissipative terms.

Sulochana ¹⁹ has analyzed the convective heat and mass transfer through a porous medium confined in a vertical channel with viscous and Darcy dissipations.

In all the above investigations, the variation of density is taken in the linear form

( ) ^{∆ T}

−

=

∆ ρ ρβ (1.1)

Where β is the co-efficient of thermal expansion and is 2,07 x 10 ⁴ (OC) ^-1+ . This is valid for temperature variation near 20 ⁰ c.

But this analysis is not applicable to the study of the flow of water at 4 ⁰ c, the density of water is a maximum at atmospheric pressure and the above relations (1.1) does not hold good. The modified form of (1.1) applicable to water at 4 ⁰ c is given by

( ) ^{∆ T 2}

−

=

∆ ρ ργ (1.2)

Where γ = 8 x 10 ^-6 (OC) ^-2 . Taking this fact

into account, Goren showed in this case,

similarity solutions for the fee convection

flow of water at 4 ⁰ c past a semi-infinite

vertical plate exist. An approximate solution

for velocity and temperature has been

obtained by using Karman – Pohlhausen

method together with the method of finding

similarity solution. Following a quadratic

density temperature variation applebaum ³

have discussed the laminar free convection

flow through coaxial circular cylinders with and without heat sources. Taking non-linear density temperature variation Sarojamma ^16a has analyzed the hydro-magnetic free convection flow in a cylindrical geometry.

Vajravelu ^19a have solved the problem of free convection between vertical walls by taking the non-linear density temperature variation, viz.

( ) 1 ( ) ²

0 g T − T e − T − T e

−

=

∆ ρ ρβ ρβ _(1.3)

Where β 0 and β 1 are the constants.

Bhargawa and Agarwal ^2a have investigated the fully developed laminar free convection flow in the presence of constant heat sources in a circular pipe taking the same density temperature relationship (1.3). It is found that the flow and heat transfer both depend upo a new parameter

∆ T

 



 

= 

0 1

β β γ

in addition to the heat source parameter and free convection parameter k.

Keeping the applications in view Jaffer ⁹ and Venkataramana ²⁰ have studied the effect of Quadratic density temperature variation on convective heat transfer in vertical channel. Recently Alivene ¹ and Rao ¹⁶ have studied the effect of quadratic density temperature variation on convective heat and mass transfer through a porous medium in vertical channel under varied conditions.

In this paper we investigate the convective heat and mass transfer through a porous medium confined in a vertical channel in the presence of heat generating sources with quadratic density temperature

variations. The equations governing the velocity, energy, and diffusion are non- linear coupled. By employing Galerkin finite element analysis with quadratic interpolation functions the equations are solved. The velocity, temperature, and concentration distributions are analyzed for different variations of the parameters , viz. Gm, Gc, D ^-1 , S, Sc, and α. The rate of heat transfer and mass transfer are evaluated numerically for different variations of the parameters.

2. FORMULATION OF THE PROBLEM We consider the convective heat and mass transfer flow of a viscous, incompressible, non-conducting, fluid through a Darcy, isotropic, homogenous porous medium. The x-axis is directed along the vertical surface and the y-axis is transverse to this. Both the vertical surface and the fluid are maintained initially at the same temperature and concentration.

Instantaneously they are raised to a temperature Tw and concentration Cw which remain unchanged. The walls are situated at y = ± L. Assuming the Concentration at a low level the soret and Doufour effects are neglected. Incorporating viscous heating effects, wall mass flux and buoyancy, under the Boussinesque approximation, the boundary layer equations may be presented as follows:

v 0

∂ = y

∂ (2.1)

( ) ( )

2 2

2 _e * _e

p

u u

v g T T g C C u

y y K

υ β β υ

∂ = ∂ + − + − −

∂ ∂

(2.2)

2

2 ( _e )

T T

v Q T T

y λ y

∂ = ∂ − −

∂ ∂ (2.3)

2

m 2

C C

v D

y y

∂ = ∂

∂ ∂ (2.4)

where u is the axial velocity in the porous region, T & C are the temperature and concentrations of the fluid, k is the permeability of porous medium, g is the acceleration due to gravity, K p is the permeability of porous medium , Q is the hear source, υ is the kinematic viscosity , ρ is the dencity of the fluid, α is the thermal diffusivity , D m is the coefficient of mass diffusitivity, C s is the concentration susceptibility, C p is the specific heat, β is the coefficient of thermal expansion, β ^* is the coefficient of volume expansion.

The boundary conditions are

y = 0; u = 0, T = T 1 , C = C 1 , (2.5) y = 1, u = 0 ,

T = T 2, C = C 2.

We introduce the non-dimentional variables as

(x ¹ , y ¹ ) = (x, y)/L

2

1 2

T T T T θ = ⁻

− ₁ ² ₂

C C C C φ = ⁻

−

The conservation equations are now transformed into the following system of coupled, non-linear ordinary differential equations of f, θ, φ.

1 2

yy y m c

u + Su − D u ⁻ = G θ + G φ

(2.6)

Pr 0

yy S y

θ + θ αθ − =

(2.7)

= 0

+ _y

yy SSc φ

φ (2.8)

The corresponding boundary conditions are u = 0, θ = 1, φ = 1 at y = L (2.9) u = 0, θ = 0, φ = 0 at y = -L.

where

Gm = 2 ³ ( 1 2 )

g β T T

υ −

(Local temperature Grashof number) Gc = 2 ³ ( 1 2 )

*

g β C C

υ ⁻ (Local mass

Grashof number) D ^-1 = K p /L ² (Darcy’s number) α = QL ² /λ (Heat source parameter)

v L 0

S ⁼ υ (Suction Reynolds number)

P r γ

= λ

(Prandtl number) Sc Dm

= ν

(Schmidt number)

3. FINITE ELEMENT ANALYSIS OF THE PROBLEM

To solve these differential equations with the corresponding boundary conditions, we assume

If u ⁱ , θ ⁱ , and φ ⁱ are the approximations of u,

θ, and φ we define the errors (residual)

i i i

i E E E

E ₁ , ₂ , ₃ , ₄ as

2 1

( )

i i

i i i i

u

d du du

E S Gm Gc D u

dy  dy  dy θ φ ⁻

=   + − − −

 

(3.1) Pr

i i

i d d d i

E S

dy dy dy

θ = ^  θ ^  + θ αθ −

  (3.2)

i i

i d d d

E SSc

dy dy dy

φ ₌ ^ φ ^ ₊ φ

 

  (3.3)

where

u ⁱ =

∑

= 3

1 k

k

f k ψ

θ ⁱ =

∑

= 3

1 k

k k ψ θ

(3.4)

φ ⁱ =

∑

= 3

1 k

k k ψ φ

these errors are orthogonal to the weight function over the domain of e ⁱ . under Galerkin finite element technique we choose the approximation functions as the weight function. Multiply both sides of the equations (3.1) – (3.3) by the weight function i.e., each of the approximation

function

i

ψ j

and integrate over the typical two nodded linear element ( ^{η e , η e + 1} ) we obtain

1

0

e

e

i i

u j

E dy

η η

+ ψ

∫ =

(j = 1, 2, 3, 4) (3.5)

1

0

e

e

i i

E j dy

η θ η

+ ψ

∫ =

(j = 1, 2, 3, 4) (3.6)

1

0

e

e

i i

E j dy

η φ η

+ ψ

∫ =

(j = 1, 2, 3, 4) (3.7)

1

2 1

( ) 0

e

e

i i

i i i i

j

d du du

S Gm Gc D u dy

dy dy dy

η η

θ φ ψ

+

   − 

+ − − − =

   

 

 

∫

(3.8)

1

Pr 0

e

e

i i

i i

j

d d d

S dy

dy dy dy

η η

θ θ αθ ψ

+    

+ − =

   

 

 

∫

(3.9)

1

0

e

e

i i

i j

d d d

SSc dy

dy dy dy

η η

φ φ ψ

+    

+ =

   

 

 

∫

(3.10) following the Galerkin weighted residual method and integration by parts method to the equations (3.8 – 3.10), we obtain

1 1 1 1 1

1

1, 2,

e e e e e

e e e e e

i

j k k i i i i

j k j k j k j j j

d du du

dy S dy Gm dy Gc dy D u dy Q Q

dy dy dy

η η η η η

η η η η η

ψ ψ θ ψ φ ψ ψ

+ + + + +

+ − − − − = +

∫ ∫ ∫ ∫ ∫

(3.11)

Where

( ) ( )

1,

k

j j e e

Q du

ψ η dy η

− =

,

( ) ( )

2, 1 1

k

j j e e

Q du

ψ η ₊ dy η ₊

=

1 1 1

1, 2,

Pr

e e e

e e e

i

j k k i i

j k j j j

d d d

dy S dy dy R R

dy dy dy

η η η

η η η

ψ θ θ ψ αθ ψ

+ + +

+ − = +

∫ ∫ ∫

(3.12)

where

( ) ( ) ( ) ( )

1,

k k

j j e e j e e

d d

R dy dy

θ φ

ψ η η ψ η η

− = +

,

( ) ( ) ( ) ( )

2, 1 1 1 1

k k

j j e e j e e

d d

R dy dy

θ φ

ψ η ₊ η ₊ ψ η ₊ η ₊

= +

1 1

1, 2,

e e

e e

i

j k k i

j j j

d d d

dy ScS dy S S

dy dy dy

η η

η η

ψ φ φ ψ

+ +

+ = +

∫ ∫

(3.13)

where

( ) ( ) ( ) ( )

1,

k k

j j e e j e e

d d

S dy dy

φ θ

ψ η η ψ η η

− = +

,

( ) ( ) ( ) ( )

2, 1 1 1 1

k k

j j e e j e e

d d

S dy dy

φ θ

ψ η ₊ η ₊ ψ η ₊ η ₊

= +

making use of equations (3.4), we can write above equations as

1 1 1 1

1 3 3 3 3

3

1

1 , 2 ,

1 1 1 1

1

e e e e

e

e e e e

e i

j k k i i i i

k k j k k j k k j k k j

k j j

k k k k

k

d d d

d y S u d y G m d y G c d y D u d y

u Q Q

d y d y d y

η η η η

η

η η η η

η

ψ ψ

⁺

ψ ψ ψ θ

⁺

ψ ψ φ

⁺

ψ ψ

⁺

ψ ψ

+ −

= = = =

=

+ ∑ − ∑ − ∑ − ∑ = +

∑ ∫ ∫ ∫ ∫ ∫

(3.14)

1 1

1 3 3

3

1, 2,

1 1

1

Pr

e e

e

e e

e

i

j k k i i

k k j k k j

k j j

k k

k

d d d

dy S dy dy R R

dy dy dy

η η

η

η η

η

ψ ψ θ ψ ψ ψ α θ ψ ψ

θ ^{+ + +}

= =

=

+ ∑ − ∑ = +

∑ ∫ ∫ ∫

(3.15)

1 3 1

3

1, 2 ,

1 1

e e

e e i

j k k i

k k j

k j j

k k

d d d

dy Sc S dy S S

dy dy dy

η η

η η

ψ ψ φ ψ ψ ψ

φ ^{+ +}

= =

+ ∑ = +

∑ ∫ ∫

(3.16)

choosing different ψ ⁱ _j ’s corresponding to each element η e in the equation (3.14) yields a

local stiffness matrix of order 3× 3 in the form

1 2

2 1

( f _i ^k _j )( u _i ^k ) − δ G g ( _{i j} ^k )( θ _i ^k + NC _i ^k ) + δ D ⁻ ( m _{i j} ^k )( u _i ^k ) + Λ δ ( n _{i j} ^k )( u _i ^k ) = ( Q ^k _j ) ( + Q ^k _j ) (3.17) Likewise the equation (3.15) &(3.16)gives

rise to a Stiffness matrices

=

− Pr ( )( ) )

)(

( e _i ^k _j θ _i ^k Du t _i ^k _j u _i ^k R ₂ ^k _j + R ₁ ^k _j ^(3.18)

=

− ( )( ) )

)(

( l _i ^k _j φ _i ^k ScSr t _i ^k _j u _i ^k S ₂ ^k _j + S ₁ ^k _j

(3.19) where ( f _i ^k _j ) , ( g _i ^k _j ) , ( m _i ^k _j ) , ( n _i ^k _j ) , ( e _i ^k _j ) ,

)

( l _i ^k _j and ( t _i ^k _j ) are 3 × 3 matrices and )

( Q _{2 j} ^k , ( Q _{1 j} ^k ) , ( R _{2 j} ^k ) & ( R _{1 j} ^k ) , ( S _{2 j} ^k )

& ( S _{1 j} ^k ) are 3 × 1 column matrices and

such stiffness matrices (3.17) - (3.19) in terms of local nodes in each element are assembled using inter element continuity and equilibrium conditions to obtain the coupled global matrices in terms of the global nodal values of f, h, θ, and φ. In case we choose n quadratic elements then the global matrices are of order 2n+1. The ultimate coupled global matrices are solved to determine the unknown global nodal values of the velocity , temperature and concentration in fluid region. In solving these global matrices an iteration procedure has been adopted to include the boundary and effects in the porous region.

The shape functions corresponding to

32 ) 8 )(

4

1 (

1

−

= y − y

ψ

32 ) 16 )(

12

1 (

2

−

= y − y

ψ 32

) 24 )(

20

1 (

3

−

= y − y

ψ

8 ) 4 )(

2

2 (

1

−

= y − y ψ

8 ) 8 )(

6

2 (

2

−

= y − y

ψ 8

) 12 )(

10

3 (

2

−

= y − y

ψ 32

) 8 3 )(

4 3

3 (

1

−

= y − y ψ

32

) 16 3 )(

12 3

3 (

2

−

= y − y ψ

32 ) 24 3 )(

20 3

3 (

3

−

= y − y

ψ 2

) 2 )(

1

4 (

1

−

= y − y

ψ 2

) 4 )(

3

4 (

2

−

= y − y

ψ 2

) 6 )(

5

4 (

3

−

= y − y ψ

32 ) 8 5 )(

4 5

5 (

1

−

= y − y

ψ

32

) 16 5 )(

12 5

5 (

2

−

= y − y

ψ 32

) 24 5 )(

20 5

5 (

3

−

= y − y ψ

4. STIFFNESS MATRICES

The global matrix for θ _is

3 3

3 X B

A = (3.20)

The global matrix for C is

4 4

4 X B

A = (3.21) The global matrix for u is

5 5

5 X B

A = (3.22)

The equilibrium conditions are

2 0

1 1

3 + R =

R , R ₃ ² + R ₁ ³ = 0 ,

4 0

1 3

3 + R =

R , R ₃ ⁴ + R ₁ ⁵ = 0 ,

2 0

1 1

3 + Q =

Q , Q ₃ ² + Q ₁ ³ = 0

,

4 0

1 3

3 + Q =

Q , Q ₃ ⁴ + Q ₁ ⁵ = 0 ,

2 0

1 1

3 + S =

S , S ₃ ² + S ₁ ³ = 0 ,

4 0

1 3

3 + S =

S , S ₃ ⁴ + S ₁ ⁵ = 0

,(3.23) Solving these coupled global matrices for temperature, Concentration and velocity (3.20-3.22) respectively and using the iteration procedure we determine the unknown global nodes through which the temperature, concentration and velocity at different radial intervals at any arbitrary axial cross sections are obtained.

The above results coincide with those of Reddy ¹⁶ for S 0 =0 and Du=0.

5. DISCUSSION OF THE NUMERICAL RESULTS

In this analysis, we discuss the effect of Non linear density temperature variation on the convective Heat and Mass transfer flow of a viscous fluid through a porous medium confined in vertical channel in presence of heat generating sources.

Taking quadratic polynomials as shape functions the analysis has been carried out by finite element technique. The velocity, temperature and concentration distributions are analyzed for different values of G m , G c , R, α, Sc. The velocity distribution is exhibited in fig 1-4. From fig 1, it is found that the velocity enhances with increase in

G m ≤ 10, and depreciates for higher G m ≥ 20.

The variation of u with G c shows that the velocity changes from negative to positive as we move from the left boundary to right boundary. The region of transition enlarges with increase in G c. Also u experiences an enhancement with increase in G c with maximum attained in the mid-half of the left region. Fig-3, represents the variation of u with Reynolds number R. It is found that for smaller values of R, u changes from negative to positive in the neighborhood of the right boundary y=1. For higher R≥ 200, the velocity is completely positive everywhere in the fluid region. u experiences an enhancement in the region -0.8 ≤ y≤ -0.2, while it depreciates in the remaining region.

The variation of u with S c shows that the velocity is positive for Sc≤ 0.8 and it changes from negative to positive in the region adjacent to y=+1. Also |u|

depreciates with increase in S c ≤ 0.8 and enhances for higher S c =1.3 and for further higher values of Sc, it depreciates in the entire fluid region. The influence of the heat generating sources on u is shown in fig-5. It is observed that for α=0 and α=4, the velocity is negative except in a narrow region adjacent to y=1. For lower and higher values of α it is completely positive.

Also the magnitude of u enhances with α≤4 and depreciates with higher α ≥ 6.

The Non – dimensional temperature

distribution (θ) is shown in figures 6-10. We

follow the convention that, the temperature

is positive or negative according as the

actual temperature is greater or lesser than

the ambient temperature T 2 . Fig 6 represents

the variation of θ with G m . It is noticed that

for G m ≥30, θ is negative and is positive for

lower values of G m <30. Also the actual

temperature experiences a marginal increment in the flow region. From fig 6, we notice that for all values of G c , the actual temperature is greater than T 2 . From fig 8, we observe that the actual temperature reduces in the left half and enhances in the right half with increase in Reynolds number R. With reference to Schmidt Number S c , we notice that lesser an molecular diffusivity, smaller the actual temperature in the left half, while in the right half it depreciates with Sc≤ 0.8 and enhances with higher Sc≥1. The influence of the heat generating sources on θ is exhibited in fig.10. For α =0, θ is negative in the region - 0.6<y<0 and positive in the remaining region. The actual temperature depreciates in the left half for all α, while in the right half, it enhances with α ≤ 4 and depreciates with α ≥ 6.

The concentration distribution(C) is shown in fig.11-15. Fig-11 shows that the concentration is positive for all Gm except for Gm=30. The actual concentration enhances with increase in Gm, the variation of C with Gc exhibits that the actual concentration enhances in the left region and depreciates in the right region. With respect to the variation, with R, we find that the concentration is positive for all R except for R = 300 where it changes form positive to negative in the region -0.8≤y≤0.8. The actual concentration depreciates except in the narrow region adjacent to y=1 for R≤200, while for higher values of R, it enhances in the entire flow region. From Fig.14, we conclude that lesser the molecular diffusivity smaller the actual concentration and for further lowering of the diffusivity, it enhances in the region - 0.8≤y≤-0.4 and depreciates in the remaining

region. The variation of C with heat source parameter α shows that in the absence of the heat sources the concentration is negative in the left half and positive in the right half.

For α=2, it is negative except in the narrow region adjacent to left boundary y = -1. For higher values of α it enhances in the entire flow region and for further increase in α≥6 it depreciate in the flow region.

The Nusselt Number (Nu) which measures the rate of heat transfer is exhibited in table 1-4 for different parameters Gm, Gc, R, Sc and α . It is observed that the rate of heat transfer at y=1 depreciates with R≥200 and enhances with higher R≥300 and at y= -1, a reversed effect is observed. A variation of Nu with Gm shows that |Nu| reduces with Gm≤10 and enhances with Gm ≥20 and at y= -1, |Nu|

enhances with Gm≤10 and reduces with Gm≥20 for all R. With respect to the behavior of Nu with Gc, we find that for R≤100, |Nu| reduces with Gc at y=1 and enhances with Gc at y= -1 while for higher R≥200, it enhances at y=1 and reduces at y=

-1. The variation of Nu with Schmidt Number Sc shows that lesser the molecular diffusivity higher the rate of heat transfer and further lowering of diffusivity smaller

|Nu| at R≤ 200 while for R≥300 a reversed effect is observed. At y = -1, smaller the diffusivity, lesser the rate of heat transfer and for further the lowering of diffusivity, higher |Nu|. An increase in the ion strength of heat generating source enhances |Nu| at y=1 and reduces at y= -1.

The sherwood number(Sh) which

measures the rate of mass transfer is shown

in table 5-8 for different values of

parameters Gm, Gc, R, Sc and α. It is

observed that Sh enhances with R≤200 and

reduces with higher R≥ 300 and at y= -1 it reduces with R≤ 200 and enhances with R≥300. An increase in Gm, enhances |Sh| at y=1 and reduces |Sh| at y= -1. With respect to the behavior of Sh with Gc, we notice an enhancement in |Sh| at y=1 and at y= -1 it reduces with Gc≤20 and enhances with Gc≥20(tables 5 and 7). From tables 6 and 8 we find that for R≤200, lesser the molecular

diffusivity higher |Sh| and for further

lowering of diffusivity lesser |Sh|. At y= -1

it depreciates with Sc≤0.8 and enhances with

Sc≥1.3 for all values of R. Also an increase

in the strength of heat generating source

results in an increment in |Sh| at both the

boundaries. In general, we notice that the

rate of heat transfer at y=1 is greater than

that at y=-1.

Table 3

6. REFERENCES

1. Alivene: Effect of quadratic temperature variation on MHD convective heat and mass transfer of a viscous electrically conducting fluid in a vertical channel

with heat sources,M.Phil Dissertation, S.

K. University, Anantapur,(2010).

2. Angirasa. D, Peterson.G.P, Pop.I:

Combined heat and mass transfer by

natural convection with buoyancy

effects in a fluid saturated porous

medium, Int. J. of Heat and Mass Transfer. V.40, No.12. Pp 2755-2773, (1997).

2a. Bhargava. R and Agarwal. R. S: Int. J.

Pure and Appl. Maths, 10 (3), p.357, (1979).

3. Applebaum, M.A: investigation of buoyancy induced convective heat transfer in an inclined channel, M.S Thesis, Department of mechanicaland aero space engineering, Arizona State University. (1984).

4. Bejan, A and Khair,K.R: Heat and mass transfer by natural convection in a pourous medium., Int.J. Heat and Mass Transfer, V.28. PP. 908-818 (1985).

5. Brinkman H.C: A Calculation of the viscous force eternal by a flowing fluid on a dense swarm of particles. Appl.

Science Research, Ala, p81 (1948).

6. Chen.T. S, Yuh. C. F, and Moutsoglo.

H): Combined heat and mass transfer in a mixed convection along a vertical and inclined planes, , Int. J. of Heat and Mass Transfer, V.23, Pp 527-537 (1980).

7. Costa, V.A.F: Double diffusive natural convection in a square enclosure with heat and mass diffusive walls. Int. J. of Heat and Mass Transfer, V.40, Pp 4061- 4071 (1997).

8. Jang J.Y and Chang W.J: The flow and vertex instability of horizontal natural convection in a porous medium resulting from combined heat and mass buoyancy effects. Int. Heat and Mass Transfer, V.

31, pp.769 - 777 (1987).

9. Jaffer: Effect of Quadratic density temperature variation on convective heat transfer of a viscous fluid in a vertical channel with walls at asymmetric

temperature, M. Phil dissertation, S.K.

University, Anantapur, (2009).

10. Kuan-Tzong Lee: natural convection heat and mass transfer in partially heated vertical parallel plates. Int. Heat and Mass Transfer, V. 42, pp.4417 - 4425 (1999).

11. Lai, F.C and Kulachi, F.A: Coupled heat and mass transfer by natural convection from vertical surface in porous medium.

Int. Heat and Mass Transfer, V. 34, p.1189 - 1194 (1991).

12. Mamou,m : Stability analysis of double diffusive convection in the presence of vertical Brinkman porous enclosure. Int.

Comm, Heat and Mass Transfer, V. 25, pp.491 - 500 (1997).

13. Nelson. D. J, and Wood. B. D. Filly developed Combined heat and mass transfer in a natural convection between vertical parallel plates with asymmetric boundary conditions. Int. J. of Heat and Mass Transfer, V. 32, Pp 1789-1792 (1989).

14. Poulikakos, D and Bejan, A; ”A departure from Darcy flow in natural convection in a vertical porous layer”, Physics Fluids 28. pp. 3477-3484 (1985).

15. Ravindranath Reddy, A: Computational techniques in hydromagnetic convective flows through a porous medium, Ph.D thesis, S.K University, Anantapur. India (1997).

16. Rao: Convective heat and mass transfer of a viscous fluid in a vertical channel with Quadratic density temperature radiation effect, M.Phil dissertation, S.

K. University, Anantapur ( 2010).

16a.Sarojamma.G: Ph.D Thesis, S. K.

University, Anantapur, (1981).

17. Sudarsan reddy: Combined influence of thermal diffusion and diffusion thermo effects on convective heat and mass transfer through a porous medium in channels/ cylindrical ducts with heat sources, Ph.D thesis, S.K. University, Anantapur, (2010).

18. Sugunamma,V: Finite element method for convection flow and heat transfer in channels through porous medium. Ph.D thesis, S.K University, Anantapur. India (1997).

19. Sulochana,C: Convective heat and mass transfer through a porous medium in channels – A finite element study, Ph.D thesis, S.K University, Anantapur. India (2002).

19a.Vajravelu. K and Debnath. L : Acta Mech.V. 59, p. 233-249, (1986).

20. Venkataramana: Free convective heat

transfer of a viscous fluid through a

porous medium in vertical channel with

quadratic density temperature variation,

M. Phil dissertation, S.V. University,

Tirupati, (2009).