Cmjv04i04p0300 by Journal of Computer and Mathematical Sciences

J. Comp. & Math. Sci. Vol.4 (4), 300-314 (2013)

Combined Effect of Chemical Reaction and Soret Effect of Convective Heat and Mass Transfer Through Porous Medium in an Annular Region Between Two Vertical Co-axial Circular Pipes T. KOTESWARA RAO1, D. CHITTI BABU2 and D.R.V. PRASADA RAO3 1

Department of Mathematics, VRS & YRN College of Engg. & Tech., Chirala, INDIA. 2 Department of Mathematics, Government College(A), Rajahmundry, A.P., INDIA. 3 Department of Mathematics, S.K.University, Anantapur, A.P., INDIA. (Received on: July 13, 2013) ABSTRACT In this paper we investigate the effects of chemical reaction and Soret effect on convective heat and mass transfer through a porous medium confined in a circular annulus in the presence of heat generating heat source with non-linear density-temperature variation. 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. G, N, M, Sc, So, Îł and Îą. The Shear stress, the rate of heat transfer and mass transfer are evaluated numerically for different variations of the parameters. Keywords: Convective heat transfer, mass transfer, porous medium , radiation. Mathematics Subject Classification Nos. 76R10,76S05.

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 accumulated impact of these studies is two fold, first to improve the performance of

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T. Koteswara Rao, et al., J. Comp. & Math. Sci. Vol.4 (4), 300-314 (2013)

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. heat exchangers, oil extractions and the manufacturing of numerous products in the chemical industry. The majority of the studies pertain to fluid flow and heat transfer in porous medium based on the Darcy’s flow model3. Brinkman1 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.2. Vafai and Tien12, Vafai13, Kim and Vafai4 have worked on the problem of convective heat transfer in porous media relaxing some or all the limitations of Darcy’s model. Later on a series of investigations were carried out using the Brinkman model by a few authors notably Poulikakos and Bejan6, Tong and Subramanian11, Prasad and Tuntomo7. Yan et al.14,15,16 have investigated numerically the laminar mixed convection flow in the vertical channel and simultaneous influence of the combined buoyancy effects of the thermal and mass diffusion for an air – water system. Both boundary conditions of uniform wall temperature/uniform wall concentration and uniform heat flux/uniform mass flux are considered. Mamou et al.5 have analyzed the problem of thermosolutal convection in a rectan cell filled with a brinkman porous medium saturated by a binary fluid. Sugunama9 and Ravindranath Reddy8 have analyzed the free convective heat transfer in vertical channel taking dissipative terms.

Sulochana10 has analyzed the convective heat and mass transfer through a porous medium confined in a vertical channel with viscous and Darcy dissipations. In this paper we investigate the effect of chemical reaction and Soret effect on convective heat and mass transfer through a porous medium confined in a circular annulus in the presence of heat generating heat source with non-linear density-temperature variation. The equations governing the velocity, energy, and diffusion are non-linear coupled. The velocity, temperature, and concentration distributions are analyzed for different variations of the parameters, viz. G, N, M, Sc, So, γ and α. The Shear stress, the rate of heat transfer and mass transfer are evaluated numerically for different variations of the parameters. 2. FORUMLATION We analyse the fully developed, steady laminar free convective flow of a viscous, fluid through a porous medium confined in an annular region between two vertical co-axial circular pipes in the presence of heat generating sources. We choose the cylindrical polar coordinates system O(r,θ,z) with the inner and outer cylinders at r =a and r = b respectively. Pipes being sufficiently long all the physical quantities are independent of the axial coordinate z. Also the motion being rotationally symmetric the azimuthal velocity V is zero. The equation of motion governing the MHD flow through the porous medium are ur + u / r = 0 ρe u ur = – pr + µ (u r r +ur / r- u / r2 )

Journal of Computer and Mathematical Sciences Vol. 4, Issue 4, 31 August, 2013 Pages (202-321)

(2.1) (2.2)

T. Koteswara Rao, et al., J. Comp. & Math. Sci. Vol.4 (4), 300-314 (2013)

ρe u wr = – pz + µ (w r r +wr / r )– ρ g-

302

ρC p (uTr ) = k f (Trr + Tr / r ) + Q(T − T0 ) (2.4)

- ρe g - p e , z = 0 (2.9) where ρe and pe are the density and pressure in the static case and hence - ρg - pz = - ( ρ - ρe ) g – pd ,z (2.10) where pd is the dynamic pressure Also substituting (2.10) in (2.2) we find

(uCr ) = D1 (C rr + C r / r ) − k1C

∂ pd = f (r ) ∂r

(

σµ H 2 e 2

2 o

+ k11 (Trr + Tr / r )

(2.3)

(2.5)

ρ - ρe= – β0 ( T – T0 )- β 1 ( T – T0 )2- β • ( C – C0 ) (2.6) where (u,w) are the velocity components along 0(r,z) directions respectively , ρ is the density of the fluid , p is the pressure , T,C are the temperature and concentration, µ is the coefficient of viscosity , Cp is the specific heat at constant pressure , σ is the electrical conductivity ,µe is the magnetic permeability , Ho is the strength of the magnetic field and ρe , Te ,Ce are density, temperature and concentration in the equilibrium state. where kf is the coefficient of thermal conductivity ,D1 is the molecular diffusivity, β0, β1 are the β • coefficients of thermal expedition is the volumetric expansion with mass fraction, k1 is chemical reaction coefficient, k11 is the cross diffusivity and Q is the strength of the heat generating source (suffices r and z indicate differentiation w.r.t .the variables ). The boundary conditions are w(a ) = w(b) = 0 (2.7a)

dT T(a) = Ti and ( ) r =b = Q1′ dr C(a) = Ci and (C ) r =b = C0

(2.7b) (2.7c)

In the hydrostatic state equation (2.3) gives

(2.11)

Using the relation (2.8) – (2.11) in (2.1) – (2.4) the equations governing free convective heat transfer flow under no pressure gradient are w r r + wr / r + ( ( β0 g / υ ) (T – T0 )+ β1 g/υ) (T – Te)2 + (β• g / υ ) (C - Ce ) - (

σµe2 H o2 r2

)w = 0

(2.12)

Trr + Tr / r + Q(T − T0 ) / k f = 0

(2.13)

C rr + C r / r − k1 , C + k11 (Trr + Tr / r ) = 0

(2.14)

Introducing the non-dimensional variables (r ′ , w ′ , θ ′ ) as r ′ = r /a , w ′ = w (a /ν ) ,

θ =

C − Ce T − Te , C′ = Ci − Ce Ti − Te

(2.15)

the equations (2.13) and (2.14) reduce to w r r + (1 /r ) wr – (M2/r2) w = -G (θ + γθ2 + N C) (2.16) θ rr + θ r / r − αθ = 0 (2.17) C rr + C r / r − γ 1 C + (

ScS o )(θ rr + θ r / r ) = 0 N

(2.18)

The corresponding boundary conditions are w = 0 , θ =1 , C=1 on r =1 w(s) = 0,

(

dθ ) r = s = Q1 , C ( s ) = 0 dr

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3. FINITE ELEMENT ANALYSIS The finite element analysis with quadratic polynomial approximation functions is carried out along the radial distance across the circular duct. The behavior of the velocity, temperature and concentration profiles has been discussed computationally for different variations in governing parameters. The Gelarkin method has been adopted in the variational formulation in each element to obtain the global coupled matrices for the velocity, temperature and concentration in course of the finite element analysis. Choose an arbitrary element ek and k k let u , θ and Ck be the values of u, θ and C in the element ek We define the error residuals as E pk =

d  dw r dr  dr

  + Gr (θ k + γ (θ k ) 2 NC k ) − ( M 2 / r ) w k 

(3.1)

E kθ =

d  dθ r dr  dr

Ekc =

d  dC k r dr  dr

  − αθ 

(3.2)

  − rSc γ 1 C k  Sc S 0 d  dC k   +( ) r N dr  dr 

θ k = θ 1kψ 1k + θ 2kψ 2k + θ 3kψ 3k C k = C1kψ 1k + C 2kψ 2k + C 3kψ 3k

Following the Gelarkin weighted residual method and integrating by parts equations (3. 1) - (3. 3) we obtain rB1

∫r

rA1

k dw k dψ j dr − dr dr

rB1

G ∫ r (θ k + γ (θ k ) 2 NC k )ψ kj dr + rA1 rB1

( M 1 ) ∫ w kψ kj (dr / r ) 2

rA1

rB1

= Q 2k j + Q1k j − P ∫ rψ kj dr

(3.3)

(3.4)

rA1

 dw k − Q1k j =   dr  dw k − Q2k j =   dr

  (rψ kj )  rA1 ,     (rψ kj )  rB1  

dθ k dψ j ∫ r dr dr dr = rA1 k

rB1

where wk, θk & Ck are values of w, θ& C in the arbitrary element ek. These are expressed as linear combinations in terms of respective local nodal values.

w k = w1kψ 1k + w3kψ 1k + w3kψ 3k

where ψ 1k , ψ 2k --------- etc are Lagrange’s quadratic polynomials.

rB1

− α ∫ rθ kψ j dr = + R2k j + R1k j k

rA1

 dθ k   (rψ kj ) rA1 , − R1k j =   dr   k  dθ   ( rψ kj ) rB1 R2k j =   dr  

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(3.5)

304

T. Koteswara Rao, et al., J. Comp. & Math. Sci. Vol.4 (4), 300-314 (2013) rB1 dC k dψ j dr − γ Sc ∫ rC kψ kj dr dr dr rA1 k

rB1

∫r

rA1

rB Sc S 0 1 dθ k dψ j ) ∫r dr N rA1 dr dr k

(3.6)

+ S 2k j + S1k j

 dC k − S1k j =   dr

  (rψ kj ) rA1  

  dC k  (rψ kj ) rB1 S 2k j =   dr   Expressing uk , θk , Ck in terms of local nodal values in (3. 4) - (3. 6) we obtain dψ dψ dr − dr dr

rB1

k i

∑w ∫r k

i =1

rA1

k j

rB1

i =1

rA1

G ∑ (θ ik + γ (θ k ) 2 NCik ) ∫ rψ ikψ kj dr

(3.7)

i =1 rA1

rB1

=Q +Q

− P ∫ rψ ikψ kj dr rA1

rB1

∑ θ ik ∫ r i =1

rA1

k d ψ dψ j dr − dr dr k i

(3.8)

rB1

α ∑ θ ik ∫ rψ ikψ kj dr = R2k j + R1k j i =1

∑C ∫ r i =1

k i

rA1

dψ ik dψ dr − dr dr k j

γSc∑ ∫ rC kψ kj dr + i =1 rA1

rB k Sc S o 3 k 1 dψ ik dψ j )∑ θ i ∫ r dr N i =1 rA1 dr dr

= S 2k j + S1k j

(3.14)

(3.9)

(3.15)

4. SOLUTION OF THE PROBLEM Solving these coupled global matrices for temperature, concentration and velocity (3.13)-(3.15) 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 respective expressions are given by

1 ≤ r ≤ 1 + S * 0. 2

3 rB1

A2 X 2 = B2

1 θ (r ) = ψ 11 θ 11 + ψ 21 θ 12 + ψ 31 θ 13

rA1

rB1

(3.13)

The global matrix for C is

A3 X 3 = B3

3 rB1

k 1 j

A1 X 1 = B1

The global matrix for u is

+ M 12 ∑ ∫ψ ikψ kj ( dr / r ) j 2 j

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 medium. The global matrix for θ is

= ψ 12 θ13 +ψ 22 θ14 +ψ 32 θ15 1 + S * 0 .2 ≤ r ≤ 1 + S * 0 .4 = ψ 13 θ15 + ψ 23 θ16 +ψ 33 θ17 1 + S * 0 .4 ≤ r ≤ 1 + S * 0 .6 = ψ 14 θ 17 + ψ 24 θ 18 + ψ 34 θ 19 1 + S * 0 .6 ≤ r ≤ 1 + S * 0 .8

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= ψ 115 θ 19 + ψ 25 θ 20 + ψ 35 θ 21 1 + S * 0.8 ≤ r ≤ 1 + S 1 C ( r ) = ψ 11 C11 + ψ 21 C12 + ψ 31 C13 1 ≤ r ≤ 1 + S * 0. 2 = ψ 12 C13 + ψ 22 C14 + ψ 32 C15 1 + S * 0 .2 ≤ r ≤ 1 + S * 0 .4 = ψ 13 C15 + ψ 23 C16 + ψ 33 C17 1 + S * 0 .4 ≤ r ≤ 1 + S * 0 .6 = ψ 14 C17 + ψ 24 C18 + ψ 34 C19 1 + S * 0 .6 ≤ r ≤ 1 + S * 0 .8 = ψ 115 C19 + ψ 25 C 20 + ψ 35 C 21 1 + S * 0.8 ≤ r ≤ 1 + S 1 u ( r ) = ψ 11 u11 + ψ 21 u12 + ψ 31 u13 1 ≤ r ≤ 1 + S * 0. 2 = ψ 12 u13 + ψ 22 u14 + ψ 32 u15 1 + S * 0 .2 ≤ r ≤ 1 + S * 0 .4 = ψ 13 c15 + ψ 23 c16 + ψ 33 c17 1 + S * 0 .4 ≤ r ≤ 1 + S * 0 .6 = ψ 14 u17 + ψ 24 u18 + ψ 34 u19 1 + S * 0 .6 ≤ r ≤ 1 + S * 0 .8 = ψ 115 u19 + ψ 25 u 20 + ψ 35 u 21 1 + S * 0.8 ≤ r ≤ 1 + S

5. SHEAR STRESS, NUSSELT NUMBER AND SHERWOOD NUMBER The shear stress ( τ ) is evaluated using the formula τ = (

du ) r =1,1+ s dr

The rate of heat transfer (Nusselt number) and the rate of mass transfer (Sherwood

number) is evaluated using the formula

dθ ) r =1,1+ s dr dC Sh = −( ) r =1,1+ s dr

Nu = −(

6. DISCUSSION OF THE RESULTS In this analysis we discuss the effect of chemical reaction and thermo-diffusion on non-darcy convective heat and mass transfer flow of a viscous, electrically conducting fluid in a circular annulus between the cylinders r = a and r = b with non-linear density temperature variation. The equations governing the flow, heat and mass transfer are solved by employing Galerkin Finite Element Analysis with three noded line segments. The axial velocity (w) is shown in figs. 1 to 6 for different variations of pramaters. Fig.1 represents w with Grashof number G, it is found that w exhibits a reversal flow for G<0 and the region of reversal flow shrinks in its size with increase in G<0. w Experiences an enhancement with increase in G>0 and reduces with G (<0). Higher the Lorentz force larger the magnitude of w (Fig.2). An increase in the strength of the heat source (α) leads to depreciation in w (fig.3). The molecular buoyancy force dominates over the thermal buoyancy force w enhances when the buoyancy forces act in the same direction and for the forces acting in opposite directions w depreciates (Fig.4). The variation of w with Soret parameter So

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T. Koteswara Rao, et al., J. Comp. & Math. Sci. Vol.4 (4), 300-314 (2013)

shows that w exhibits a reversal flow for higher So=1. Higher So larger w in the flow region(Fig.5). Fig.6 represents w with chemical reaction parameter γ, it is observed that in the degenerating chemical reaction

case w reduces with γ≤1.5 and enhances with higher γ≥2.5 while in the generating chemical reaction case w experiences a reduction in the entire flow region.

0.3

1 0.5 w

0 1

1.2

1.4

1.6

1.8

0.2

w 0.1

III

-0.1 1

III 1.2

-0.5 Fig 1: Variation of velocity w with G I II III IV G 103 3x103 -103 -3x103

0.3 0.25 0.2 w 0.15 0.1 0.05 0

I II III

1.2 1.4 r 1.6 1.8

0 -0.2 1 1.2 1.4 1.6 1.8 2 r

1.2 1 0.8 w 0.6 0.4 0.2 0

I II …

1.2

III

1.4 r 1.6

1.8

Fig 4: Variation of velocity w with N I II III N 1 2 -0.5

0.6

w 0.2

1.8

Fig 2: Variation of velocity w with M I II III M 2 4 6

Fig 3: Variation of velocity w with α I II III α 2 4 6

0.4

1.6 r

1.4

I II III IV

Fig 5: Variation of velocity w with So I II III IV So 0.5 1 -0.5 -1.0

0.6 0.5 0.4

I II III

w 0.3 0.2 0.1 0

1.2

1.4 1.6 r

1.8

Fig 6: Variation of velocity w with γ I II III IV γ 0.5 1.5 -0.5 -1.5

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T. Koteswara Rao, et al., J. Comp. & Math. Sci. Vol.4 (4), 300-314 (2013) 1.2 1 0.8 θ 0.6 0.4 0.2 0

1.5

II III IV

0.5

III

1.2

1.4 r 1.6

1.8

Fig 7: Variation of temperature θ with G I II III IV G 103 3x103 -103 -3x103 1.2 1 θ0.8 0.6 0.4 0.2 0

I II

1.2 1.4 r 1.6 1.8

I II III IV 1

1.2 1.4 1.6 1.8 r

1.8

I II III IV

Fig 10: Variation of temperature θ with N I II III IV N 1 2 -0.5 -0.8

1.5

1.6

1 1.2 1.4 1.6 1.8 2 r

Fig 9: Variation of temperature θ with α I II III α 2 4 6

0.5

1.2 1 0.8 0.6 0.4 0.2 0

θ 1

1.4

Fig 8: Variation of temperature θ with M I II III M 2 4 6

III 1

1.2

1.2 1 0.8 0.6 0.4 0.2 0

I II III IV 1

Fig 11: Variation of temperature θ with So I II III IV So 0.5 1 -0.5 -1.0

1.2 1.4 r 1.6 1.8

Fig 12: Variation of θ with γ I II III γ 0.5 1.5 -0.5

IV -1.5

1.2

1.5

I 1

c 0.5

III IV

0 1

1.2 1.4 r 1.6 1.8

c 0.6 0.4

III

0.2 0 1

Fig 13: Variation of C with G I II III G 103 3x103 -103

0.8

IV -3x103

1.2

1.4

1.6

1.8

Fig 14: Variation of C with M I II M 2 4

III 6

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T. Koteswara Rao, et al., J. Comp. & Math. Sci. Vol.4 (4), 300-314 (2013) 1.2

1.2 1 0.8 c 0.6 0.4 0.2 0

0.8

III

c 0.6

I II III IV

0.4 0.2

1.2

1.4

1.6

1.8

1 1.21.41.61.8 2 r

Fig 15: Variation of C with α I II III α 2 4 6

1.2 1 0.8 c 0.6 0.4 0.2 0

1.2 1 0.8

I II IV 1.2

1.4 r 1.6

1.8

Fig 17: Variation of C with Sc I II III Sc 0.24 0.6 1.3

I II III

c 0.6 0.4 0.2 0

III

Fig 16: Variation of C with N I II III IV 1 2 -0.5 -0.8

IV 2.01

1.4 r 1.6

1.8

Fig 18: Variation of C with So I II III IV 0.5 1 -0.5 -1.0

1.2 1 0.8 c 0.6 0.4 0.2 0

I II III IV 1

1.2

1.2 1.4 r 1.6 1.8

Fig 19: Variation of C with γ I II III IV 0.5 1.5 -0.5 -1.5

The non-dimensional temperature(θ) is shown in Figs.7 to 12 for different parametric values. Fig.7 represents θ with G. It is found that θ enhances with increase in G . Higher the Lorentz force larger the temperature in the entire flow region (Fig.8).

An increase in the strength of the heat source α≤4 leads to a marginal depreciation in θ and for higher α≥6 we notice an enhancement in θ (Fig.9). The variation of θ with buoyancy ratio N shows that the temperature enhances with increase in N>0

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T. Koteswara Rao, et al., J. Comp. & Math. Sci. Vol.4 (4), 300-314 (2013)

when the buoyancy forces act in the same direction and for the forces acting in the opposite directions θ depreciates with N (<0) (Fig.10). An in the Soret parameter So ( <> 0 ) results in an enhancement in θ (Fig.11). The variation of θ with chemical reaction parameter γ shows that θ depreciates with γ ≤1.5 and enhances with higher γ ≥2.5 while it reduces in the generating chemical reaction case γ (<0) (Fig.12). The Concentration distribution (C) is shown in Figs.13-19 for different parametric values. Fig.13 represents C with G. It is found that the actual concentration reduces with increase in G>0 and enhances with G<0. Higher the Lorentz force or strength of the heat source larger the actual concentration in the flow region (Figs.14&15).The actual concentration enhances with increase in N>0 and reduces with N (<0) in the entire flow region (Fig.16). The variation of C with Sc shows that lesser the molecular diffusivity larger the actual concentration (Fig.17). An increase in the Soret parameter So ( <> 0 ) results in an enhancement in the actual concentration (Fig.18). The actual concentration enhances with increase in γ>0 when the buoyancy forces act in the same direction and for the forces acting in opposite directions it reduces in the flow region (Fig.19). The Shear Stress(τ) at r = 1 & 2 is evaluated for different values of G, M, α, N, Sc, So, γ, Q and K and is shown tables 1to 6. An increase in G enhances τ at both the

cylinders. The variation of τ with M shows that higher the Lorentz force larger τ at r =1&2 in the heating case and in the cooling case τ enhances at r = 1 and reduces at r = 2. An increase in the strength of the heat source results in a depreciation in the stress at both the cylinders. With reference to N we find that when the molecular buoyancy force dominates over the thermal buoyancy force τ enhances at r = 1&2. When the buoyancy forces act in the same direction and for the forces acting in opposite directions the stress reduces at r = 2 and enhances at r = 1(tables1&4) with reference to Sc we find that lesser the molecular diffusivity smaller τ at r =1&2 and for further lowering the molecular diffusivity, τ

depreciates for

G>0 and enhances for G<0 at the both cylinders. An increase in the Soret parameter So ( <> 0 ) leads to an enhancement in τ at r=1&2. An increase in the radiation absorption parameter Q reduces at r = 1 and enhances at r = 2 for G>0 and for G<0, τ enhances at both the cylinders (tables 2&5). The variation of τ with chemical reaction parameter K we find that τ reduces at r = 1&2, τ

reduces with K for G>0 and

enhances for G<0 in the degenerating chemical reaction case and in the generating case, τ enhances at r = 1 and reduces at r = 2 in the heating case while in the cooling case, it enhances at both the cylinders. An increase in the density ratio K enhances τ at r = 1&2 in the heating case while in the cooling case it reduces at r = 1 and enhances at r = 2(tables 3&6).

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G = 10 3 and at G = 3 × 10 3 , it enhances

The rate of mass transfer (Sherwood Number) at y = ±1 is shown in the tables 7 to 12 for different parametric values. It is found that the rate of mass transfer enhances at y = ±1 with increase in G ( <> 0 ). Higher

with Sc≤0.6 and reduces with Sc≥1.3. For the cooling of the channel walls, Sh reduces at both the walls with increase in Sc. An increase in So>0 enhances at G = 10 3 and reduces at G = 3 × 10 3 and enhances with So (<0) at y = ±1. At y = -1, Sh

the Lorentz force larger Sh at y = ±1. With reference to heat source parameter α, we find that an increase in α enhances Sh at y

reduces for G>0 and enhances for G<0 with increase in So>0 and enhances with So

= +1 and reduces at y = -1. When the molecular buoyancy force dominates over the thermal buoyancy force the rate of mass transfer enhances at y = +1 and reduces at y = -1. When the buoyancy forces act in the same direction and for the forces acting in opposite directions, Sh at y = +1reduces

(<0). An increase in the radiation parameter Q, results in an enhancement in Sh at both the walls (tables 11 &14). The variation of Sh with chemical reaction parameter K shows that the rate of mass transfer at y = +1, reduces for G>0 and enhances for G<0 in the degenerating chemical reaction case and reduces in the generating case. At y = -1, Sh reduces in both degenerating

for all G while at y = -1, it enhances for G>0 and reduces at y = -1 (table 7&10). The variation of Sh with Sc shows that at y = +1, the rate of mass transfer reduces with Sc at G = 10 3 and at G = 3 × 10 3 , Sh enhances

and generating cases. An increase in the density ratio K results in an enhancement in Sh at both the walls (tables 11&12).

with Sc≤0.6 and enhances with higher Sc≥1.3. At y = -1, Sh enhances with Sc at

TABLE-1 G 10

1 3

Shear Stress (τ) at r = 1 4

2.2135

2.216

2.25

0.4919

0.4191

3.1761

4.663

23.792

3 x 103

29.629

29.137

30.14

1.5918

1.3162

51.629

6.559

29.915

-0.878

7.528

7.9

-0.460

-0.4031

-1.7397

-3.459

-5.053

-3 x 03

15.23

15.39

16.4

-1.297

-1.1625

23.685

-35.361

-5.342

-0.5

-0.8

-10

Journal of Computer and Mathematical Sciences Vol. 4, Issue 4, 31 August, 2013 Pages (202-321)

311

T. Koteswara Rao, et al., J. Comp. & Math. Sci. Vol.4 (4), 300-314 (2013) TABLE-2 Shear Stress (τ) at r = 1 G 103 3x103 -103 -3x103 Sc So Q

1 23.792 29.915 -5.053 -5.342 1.3 0.5 1

2 3.9228 21.473 -1.790 21.374 0.24 0.5 1

3 3.1313 9.293 -1.567 13.888 0.6 0.5 1

4 1.6092 4.5354 4.1975 28.103 2.01 0.5 1

TABLE-3 G 103 3x103 -103 -3x103 γ K

1 2.213 29.09 -0.87 15.26 0.5 0.001

2 2.174 26.71 -0.54 12.50 1.5 0.001

3 2.138 24.38 -0.40 10.28 2.5 0.001

5 -0.1219 12.0074 23.5026 29.696 1.3 1.0 1

6 20.577 26.078 -4.786 -5.666 1.3 -0.5 1

7 24.463 60.351 -10.76 -24.73 1.3 -1.0 1

8 2.4959 25.381 6.686 9.6306 1.3 0.5 1.5

9 2.2423 24.096 26.066 34.823 1.3 0.5 2.5

Shear Stress (τ) at r = 1

4 2.156 31.23 -0.993 11.592 -0.5 0.001

5 2.201 33.42 -1.03 19.091 -1.5 0.001

6 2.356 35.448 -1.036 21.314 -2.5 0.001

7 2.239 36.027 -0.865 10.899 0.5 0.003

8 2.264 48.28 -0.843 5.431 0.5 0.005

9 2.283 73.041 -0.844 4.821 0.5 0.007

TABLE-4 Shear Stress (τ) at r = 2 G 103 3 x 103 -103 -3 x 103 M α N

1 -2.6971 -36.204 1.037 -18.063 2 2 1

2 -2.6991 -36.323 1.0367 -17.64 4 2 1

3 -2.7023 -36.523 1.0358 -16.956 6 2 1

4 0.6018 -1.991 0.5508 1.5161 2 4 1

5 -0.5137 -1.6516 0.4816 1.3547 2 6 1

6 -3.8899 -65.375 2.0694 -29.6272 2 2 2

7 -5.3075 -73.236 4.2125 -43.029 2 2 -0.5

8 -2.90 -36.60 3.1132 6.434 2 2 -0.8

TABLE-5 Shear Stress (τ) at r = 2

G 103 3 x 103 -103 -3 x 103 Sc So Q

1 -2.9067 -36.606 3.1132 6.4354 1.3 0.5 1

2 -4.6271 -26.621 2.0505 -25.846 0.24 0.5 1

3 -3.732 -11.47 1.818 -22.79 0.6 0.5 1

4 -2.0095 14.5354 -5.2695 -68.413 2.01 0.5 1

5 0.1874 -15.272 -29.824 -37.584 1.3 1.0 1

6 -23.58 -30.00 5.153 5.4368 1.3 -0.5 1

7 -16.164 -28.697 11.8462 26.8592 1.3 -1.0 1

8 -3.041 -31.64 -8.138 -11.70 1.3 0.5 1.5

Journal of Computer and Mathematical Sciences Vol. 4, Issue 4, 31 August, 2013 Pages (202-321)

9 -2.739 -36.03 -31.25 -41.60 1.3 0.5 2.5

312

T. Koteswara Rao, et al., J. Comp. & Math. Sci. Vol.4 (4), 300-314 (2013) TABLE-6 Shear Stress (τ) at r = 2 G 103 3 x 103 -103 -3x 103 γ K

1 -2.691 -36.24 1.037 -18.07 0.5 0.001

2 -2.73 -38.81 0.8792 -16.86 1.5 0.001

3 -2.761 -39.410 0.6172 -13.97 2.5 0.001

4 -2.6329 -38.913 1.1917 -13.851 -0.5 0.001

5 -2.5685 -34.496 1.2428 -16.889 -1.5 0.001

6 -2.513 -29.92 1.250 -18.97 -2.5 0.001

7 -2.729 -45.47 1.0233 -11.14 0.5 0.003

8 -2.7628 -61.917 1.01083 -27.008 0.5 0.005

9 -2.7965 -96.333 1.29272 45.490 0.5 0.007

TABLE-7 Sherwood Number (Sh) at r = 1 G 103 3 x 103 -103 -3 x 103 M α N

1 -0.3141 -4.9118 0.5525 28.9815 2 2 1

2 -0.3145 -4.9274 0.5531 29.0802 4 2 1

3 -0.3146 -4.9535 0.5541 29.245 6 2 1

TABLE-8

4 1.4592 1.6228 1.3153 1.186 2 4 1

5 1.5452 1.7264 1.3771 1.2199 2 6 1

6 6.078 -6.4632 0.8744 31.9478 2 2 2

7 43.1307 48.638 8.8734 8.4499 2 2 -0.5

8 2.4964 2.7199 1.8455 1.6567 2 2 -0.8

Sherwood Number (Sh) at r = 1

G 103

1 2.4964

2 0.9938

3 0.3965

4 2.893

5 3.9437

6 13.1483

7 18.496

8 -0.618

9 -1.2412

3 x 103

2.7199

-7.805

-7.885

-3.6662

-2.574

14.3479

23.449

-4.043

-5.7488

1.8455

1.2159

0.8431

7.4767

5.7318

6.4971

47.3403

6.8467

25.0967

1.6567

1.7471

3.4715

22.1158

6.5351

6.4179

35.752

29.432

30.0102

1.3 0.5 1

0.24 0.5 1

0.6 0.5 1

2.01 0.5 1

1.3 1.0 1

1.3 -0.5 1

1.3 -1.0 1

1.3 0.5 1.5

1.3 0.5 2.5

-10

-3x 10

Sc So Q

TABLE-9 G 103 3 x 103 -103 -3 x 103 γ K

1 -0.314 -4.918 0.552 28.985 0.5 0.001

2 -0.307 -4.901 1.094 52.07 1.5 0.001

3 -0.285 -4.864 2.468 66.43 2.5 0.001

Sherwood Number (Sh) at r = 1 4 -0.326 -4.896 6.364 17.66 -0.5 0.001

5 -0.317 -4.857 5.764 11.59 -1.5 0.001

6 -0.3156 -4.7968 5.3852 7.9084 -2.5 0.001

7 -0.3188 -6.1652 0.5621 35.3582 0.5 0.003

8 -0.3236 -8.3247 0.5717 46.738 0.5 0.005

Journal of Computer and Mathematical Sciences Vol. 4, Issue 4, 31 August, 2013 Pages (202-321)

9 -0.3286 -12.6004 0.5817 43.334 0.5 0.007

313

T. Koteswara Rao, et al., J. Comp. & Math. Sci. Vol.4 (4), 300-314 (2013) TABLE-10 G 103 3 x 103 -103 -3 x 103 M α N

1 2.7550 7.9949 1.6688 -4.258 2 2 1

2 2.7556 8.0159 1.6678 -42.68 4 2 1

Sherwood Number (Sh) at r = 2 3 2.7564 8.0512 1.6661 -42.97 6 2 1

TABLE-11 G 103 3 x 103 -103 -3 x 103 Sc So Q

1 -30.76 -33.99 -1.811 -1.549 1.3 0.5 1

2 1.2435 11.4932 0.9604 -0.023 0.24 0.5 1

3 1.9551 11.6414 1.3976 -3.2091 0.6 0.5 1

4 0.6673 0.4999 0.8031 0.9139 2 4 1

5 0.5782 0.4239 0.7112 0.8249 2 6 1

6 2.1145 6.0251 1.3919 -1.179 2 2 2

7 -5.694 -10.47 -2.083 -2.023 2 2 -0.5

8 -30.76 -33.99 -1.811 -1.546 2 2 -0.8

Sherwood Number (Sh) at r = 2 4 2.9959 6.3448 -7.6724 -10.407 2.01 0.5 1

5 -2.7963 5.2835 -72.402 -82.905 1.3 1.0 1

6 -15.724 -17.647 -5.1694 -5.0372 1.3 -0.5 1

7 -23.602 -30.408 -18.681 -15.059 1.3 -1.0 1

8 3.0601 6.8873 -6.9385 -9.1298 1.3 0.5 1.5

9 2.5573 6.499 -31.46 -38.22 1.3 0.5 2.5

TABLE-12 Sherwood Number (Sh) at r = 2 G 103 3 x 103 -103 -3 x103 γ K

1 2.750 7.999 1.668 -4.258 0.5 0.001

2 2.623 7.614 1.051 -7.43 1.5 0.001

3 2.472 7.201 1.007 -13.4 2.5 0.001

4 2.8802 8.3392 -2.0447 -27.226 -0.5 0.001

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3. Darcy, H : Les Fontaines published de la ville de Dijon victor dalmont, Paris (1956). 4. Kim, S. J and Vefai, K: Analysis of natural convection about a vertical plate embedded in a porous medium. 5. Mamou, M. et al.: 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).

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T. Koteswara Rao, et al., J. Comp. & Math. Sci. Vol.4 (4), 300-314 (2013)

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Journal of Computer and Mathematical Sciences Vol. 4, Issue 4, 31 August, 2013 Pages (202-321)