Cmjv02i02p0358 by Journal of Computer and Mathematical Sciences

J. Comp. & Math. Sci. Vol.2 (2), 358-371 (2011)

Free Convectivre Heat and Mass Transfer Past A Porous Vetical Plate with Periodic Temperature and Concentration M. SREEVANI* and D. R. V. PRASADA RAO** *Department of Mathematics, S.K.U. College of Engineering & Technology, S. K. University, India **Department of Mathematics, S. K. University, Anantapur, Andra Pradesh. India ABSTRACT In this chapter we analyse the Convective heat and mass transfer flow of a viscous fluid through a porous medium bounded by vertical porous plate. We assume that the wall temperature and concentration are taken to be spanwise co-sinosoidal. We neglect the soret and duffer effect. The viscous and darcy dissapation are taken into account. By applying a regular perturbation method the expressions for velocity, temperature and concentration distributions are obtained. The skin friction, the Nusselt number and the sherwood number have been evaluated for different variation of the governiing parameters. Keywords: Viscous and Darcy dissipation, porous medium, Vertical plate.

1. INTRODUCTION The phenomenon of heat and mass transfer has been the object of extensive research due to its applications in Science and technology. Such phenomenon are observed in buoyancy induced motions in the atmosphere ,in bodies of water, quasisolid bodies such as earth and so on. The flow of a viscous incompressible fluid bounded by one or two infinite planes with porous walls has gained considerable importance in view of its applications to reduce boundary layer growth. One of the ways in which a transition from the laminar

boundary layer to a turbulent may be suppressed is to reduce mass from the boundary layer through pores or slits on the boundary. The development on this subject has been compiled by Lachmann4 .An important design consideration in such a profile is the geometry and configuration of the outlets through which the suction is effected. One possible suction distribution is a transverse sinusoidal one. Gerstein and Grosh3 have studied the effects of transverse sinusoidal suction velocity on flow and heat transfer along an infinite vertical porous wall. It is seen that three regions exits for the Prandtl number

Journal of Computer and Mathematical Sciences Vol. 2, Issue 2, 30 April, 2011 Pages (170-398)

359 o < Pr <

M. Sreevani, et al., J. Comp. & Math. Sci. Vol.2 (2), 358-371 (2011)

π (π − 1)1/ 2

1.211 < Pr <

= 1.211

π = 1.467 π −1

1.467 < Pr < ∞

F3 ( R, Pr ) > 0 F3 ( R, Pr ) >< 0 F3 ( R, Pr ) > 0

In the case when Pr <1.211, a situation similar to that occurring for Cfx obtaining i.e.,the quasi two dimensional values are too large when Pr=2,the Reynolds anology holds. Nu = Cfx ,Pr=1.In the case for Pr=0,the Nusselt number is dependent of the Reynolds number and the periodic suction velocity; Nu = 1 , Pr=0. In other words, the periodicity of the suction velocity has no effect on the heat transfer. When Pr>1.467,the quasi two dimensional values are too small. In the intermediate region,1.211<Pr<1.467,the heat transfer decreases for small Reynolds numbers and increases for larger Reynolds number compared to be quasi two dimensional values. To obtain any desired reduction in the drag increasing suction alone is uneconomical as the energy consumption of the suction pump will be more. Therefore the method of “ cooling of the wall” in controlling the laminar flow together with applications of suction has become more usefull. Singh et al.5 have investigated the effect of wall shear and heat transfer of the flow caused by the periodic suction velocity perpendicular to the flow direction when the difference between the wall temperature and the free stream temperature gives rise to a buoyancy force in the direction of free stream. Later Singh et al 6 have investigated the effect of periodic variation of suction

velocity on free convection flow and heat transfer through a porous medium. The problem becomes three dimensional due to variation of suction velocity in transverse direction on the wall .A series expansion method is used to get the solution of the governing equations and the expressions for velocity and temperature fields are obtained. Choudary and Sharma2 have analysed the free convective flow of a second –order fluid along a porous vertical plate subjected to a transverse sinusoidal velocity. Another possible wall temperature in such type of problems can be spanwise co-sinusoidal. Taking the wall temperature to be spanwise co-sinusoidal Acharya and Padhy1 have analysed the free convective flow of a viscous fluid past a hot vertical porous wall under the assumption that the suction velocity is constant and normal o the plate .Approximate solutions of the equation of motion and energy have been obtained by the method of regular perturbation. Prasad7 has discussed the free convective flow of a viscous incompressible fluid past a vertical wall assuming the wall temperature to be spanwise co-sinusoidal. 2. FORMULATION OF THE PROBLEM We consider the steady flow of an incompressible viscous fluid past a vertical porous plate. We choose a Cartesian coordinate system O(x , y , z) with the wall lying in the (x , z) plane ,y-axis normal to it and the positive direction of x-axis is vertically upwards .Let (u , v , w) be the components of velocity of the fluid at any point(x , y , z ) .Since V = -Vo through out, w is independent of z and we assume w = 0 through outlet the fluid velocity parallel to the x-axis at infinity tends to U∞ and the

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θw

spanwise wall temperature concentration Cw be θ w = θ 0 (1 + ε c o s ( (1 + ε c o s (

πz L

and

)), Cw = Co

−Vo (

))

where ε is a small positive number, L is the

∂u ∂2u ∂2u =ν ( 2 + 2 ) ∂y ∂y ∂z

(1)

1 ∂p − ρ g − ( )u − ( ) k ρ ∂x ∂T ∂ 2 T ∂ 2T ) = k1 ( 2 + 2 ) ∂y ∂y ∂z ∂u ∂u + µ (( ) 2 + ( )2 ) ∂y ∂z

− ρ C pVo (

(2)

+ β • g (C − C∞ ) − ( )(u − U ∞ ) k

∂C ∂ 2C ∂ 2C ) = D( 2 + 2 ) ∂y ∂y ∂z

(3)

the equation of State

ρ − ρ ∞ = − β (T − T∞ ) − β • (C − C∞ ) (4)

πz L

( y ′, z ′, u ′, θ ′, C ′) as

( y ′, z ′) = ( y, z ) / L , u ′ = u / U ∞ , θ ′ = (T − T∞ ) / θo , C ′ = (C − C∞ ) / Co reduce

∂θ 1 ∂ 2θ ∂ 2θ =− ( + )− ∂y PR ∂y 2 ∂z 2 Ec ∂u 2 ∂u 2 (( ) + ( ) ) R ∂y ∂z

)),

(5)

)) on y = 0

u = U ∞ , T = 0, C = 0 as

y → ∞ (6)

∂u 1 ∂2u ∂2u = − ( 2 + 2 )− ∂y R ∂y ∂z

∂C 1 ∂ 2C ∂ 2C =− ( + ) ∂y RSc ∂y 2 ∂z 2

The boundary conditions are u = 0 , T = θ o (1 + ε cos(

We define the non-dimensional variables

G D −1 (θ + NC ) + (u − 1) R R

the dissipation equation

C = C o (1 + ε cos(

(7)

The governing equations (dropping the dashes)

the energy equation

− Vo (

∂u ∂2u ∂2 u ) = ν ( 2 + 2 ) + β g (T − T∞ ) ∂y ∂y ∂z

wave length, 0 , o are constants. The equations governing the steady flow of a viscous fluid are The momentum equation −Vo

By applying the condition at infinity(6) and the equation of state (5),the momentum equation reduces to

The boundary conditions are

corresponding

(8)

(9)

(10) boundary

u = 0 , θ = 1+ ε cos(π z), C = 1+ ε cos(π z) on y = 0

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(11)

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M. Sreevani, et al., J. Comp. & Math. Sci. Vol.2 (2), 358-371 (2011)

u = 1 , θ = 0 , C = 0 as y → ∞

(12)

where

Vo L

is theSuction Re ynolds number

ν β gθ o L2 G= is the Grashof number U ∞ν

(14)

L2 is the Darcy parameter k µC p P= is the Pr andtl number k1

(15)

ψ o'' + RSc ψ o' = 0

(16)

2 o

y = 0 :

u o = 0 , φo = 1 , ψ o = 1

y→∞:

uo = 1 , φo = 0 , ψ o = 0 (17)

and

∂ 2u1 ∂ 2u1 ∂u + 2 + R 1 = −G (θ + Nψ 1 ) + D −1u1 2 ∂y ∂z ∂y (18)

U ∞2 is the Ec ker t number C pθ 0

φ + PR φ = − − PE c (u ) ' o

with the boundary conditions

β • Co N= is the buoyancy ratio βθ o

Sc =

u o'' + Ru o' = − G (θ o + N ψ o ) + D −1 (u 0 − 1) '' o

D −1 =

Ec =

functions of both y and z . Substituting (13) in (9)-(11) we obtain the zeroth and first order equations as

is the Schmidt number

For solving (9)-(11) we use the following perturbed forms for the velocity, temperature and concentration distributions

∂u ∂u ∂ 2θ1 ∂ 2θ1 ∂θ + 2 + PR 1 = −2 PEc ( o . 1 ) 2 ∂y ∂z ∂y ∂y ∂y (19)

u = u o ( y ) + ε u1 ( y , z ) + O (ε 2 )

∂ 2C1 ∂ 2C1 ∂C + 2 + RSc 1 = 0 2 ∂y ∂z ∂y

θ = φo ( y ) + εθ 1 ( y , z ) + O (ε )

with the boundary conditions

(20)

C = ψ 0 ( y ) + εC1 ( y , z ) + O (ε ) 2

(13)

where u0 , φo ,ψ 0 are functions of y only and represent the velocity, temperature and concentration fields respectively when the wall temperature and concentration are constant and equal to φo andψ 0 respectively (i. e., the case ε=0) where as u1 , φ1 ,ψ 1 are

y =0 :

u1 = 0 , θ1 = cos(πz ) , C1 = cos(πz )

y → ∞:

u1 = 0 , θ1 = 0 ,

C1 = 0 (21)

Now we take

u 1 = V ( y ) c o s (π z )

θ 1 = φ1 ( y ) c o s (π z ) C 1 = ψ 1 ( y ) c o s (π z )

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M. Sreevani, et al., J. Comp. & Math. Sci. Vol.2 (2), 190-197 (2011)

in (19)-(22).The equations for V , φ1 and ψ 1 are given by

V '' + RV ' − ( π 2 + D −1 )V = (22)

−G (φ1 + Nψ 1 ) + D V −1

362

Using (26) in(14)-(25) we obtain theO(Ec0) and O(Ec1) equations and the boundary conditions as '' uoo + R uoo' − D−1uoo

= −G(φoo + Nψ oo )

(27)

φoo'' + PR φoo' = 0

(28)

(24) and the corresponding boundary conditions are

ψ oo'' + RScψ oo' = 0

(29)

V = 0 , φ1 = 1,ψ 1 = 1

on y = 0

V o'' + R V o' − ( π

V = 0 , φ1 = 0 , ψ 1 = 0

as y → ∞ (25)

= − G (φ 1 0 + N ψ 1 0 )

φ1'' + PR φ1' − π 2φ1 = 2 PEc (u o' V ′)

(23)

ψ 1'' + RScψ 1' − π 2ψ 1 = 0

Now the solutions of the equations(14)-(16) and (22)-(24) satisfying the boundary conditions(18)and(25) respectively yield the functions characterizing the velocity , temperature and concentration fields. We solve (14)-(16) and (22)-(24) by perturbation technique choosing the Eckert number Ec to be the perturbation parameter(Ec<<1).We assume

u 0 = u00 + Ecu01 + O ( Ec 2 )

φ0 = φ00 + Ec φ01 + O ( Ec 2 ) ψ 0 = ψ 00 + Ec ψ 01 + O ( Ec )

ψ1 =ψ 01 + Ecψ11 + O(Ec )

(26)

(30)

φ10'' + PR φ10' − π 2φ10 = 0

(31)

ψ 10'' + RSc ψ 10' − π 2ψ 10 = 0

(32)

with uoo = 0 , φoo = 1,ψ oo = 1, V0 = 0, φ10 = 1,ψ 10 = 1 on y = 0 uoo = 1, φoo = 0 ,ψ oo = 0, V0 = 0, φ10 = 0,ψ 10 = 0 as y → ∞ and

= −G (φo1 + Nψ o1 )

V = V0 + EcV1 + O( Ec2 ) 2

+ D − 1 )V 0

uo'' 1 + R uo' 1 − D −1uo1

φ1 = φ01 + Ec φ11 + O(Ec )

(33)

(34)

φo''1 + PRφo'1 =0

(35)

ψ o''1 + RScψ01' = 0

(36)

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M. Sreevani, et al., J. Comp. & Math. Sci. Vol.2 (2), 358-371 (2011)

V1'' + RV1' − (π 2 + D −1 )V1 = − G (φ11 + N ψ 11 ) (37)

φ11'' + PR φ10' − π 2φ11 = 0

(38)

ψ 11'' + RScψ 11' − π 2ψ 11 = 0

(39)

uo1 = 0 , φo1 = 0 ,ψ o1 = 0, V1 = 0, φ11 = 1,ψ 11 = 0 on y = 0 uo1 = 0 , φo1 = 0 ,ψ o1 = 0, V1 = 0, φ11 =

0,ψ 1 = 0 as y → ∞ On solving (27)-(31) and (34)-(39) subject to the conditions (33) and (40) we get

with

u 00 = a3e − m1 y − a1e − h1 y − a 2 e − h2 y + 1

φ00 = e − h y 1

ψ 00 = e − h y 2

u 01 = a12 e − m1 y − a13 e − h1 y + a14 e − 2 m1 y + a15 e − 2 h1 y + a16 e − 2 h2 y − − a17 e − ( m1 + h1 ) y − a18 e − ( m1 + h2 ) y + a19 e − ( h1 + h2 ) y

φ01 = a5 e − h y − a6 e −2 m y − a7 e −2 h y − a8 e −2 h y + a9 e − ( m + h ) y + 1

+ a10 e − ( m1 + h2 ) y − a11e − ( h1 + h2 ) y

ψ 01 = 0 V0 = a 20 e − m 4 y − a 21e − m 2 y − a 22 e − m3 y

φ10 = e − m y ψ 10 = e − m y 2

V1 = a33 e − m4 y − a34 e − m32 y + a35 e − ( m1 + m4 ) y − a36 e − ( m1 + m2 ) y − a37 e − ( m1 + m3 ) y − a38 e − ( h1 + m4 ) y + a39 e − ( h1 + m2 ) y + a40 e − ( h1 + m3 ) y − a41 e − ( h2 + m4 ) y + a42 e − ( h2 + m2 ) y + a43 e − ( h2 + m3 ) y

θ11 = a 23e − m y − a 24 e − ( m + m 2

2)y

(40)

+ a 25 e − ( m1 + m2 ) y + a 26 e − ( m1 + m3 ) y

+ a 27 e − ( h1 + m4 ) y − a 28 e − ( h1 + m2 ) y − a 29 e − ( h1 + m3 ) y + a30 e − ( h2 + m4 ) y − a31e − ( h1 + m2 ) y − a32 e − ( h2 + m3 ) y

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364

ψ 11 = 0 where the constantsa1,a 2,……….,a43,h1,h2,h3,h4,m1,……,m4 are given in the appendix 3. SKIN FRICTION, NUSSELT NUMBER AND SHERWOOD ∂C Sh = ( ) y = 0 NUMBER

∂y

The skin friction at the wall y = 0 is given by

τ = µ(

∂u ) y =0 ∂y

which in the non-dimensional form reduces to

τ′ = (

( µC L / L )

)=(

∂u ) y =0 ∂y

' ' = ( u 00 (0) + Ec u 01 (0) + ε cos(π z )

(V a' (0) + EcV1' (0)) y = 0 = a44 + Ec a45 + ε cos(πz )(a46 + Ec a47 ) The rate of heat transfer(Nu) and the rate of mass transfer (Sh) in the non-dimensional form is given by

Nu = (

∂θ ) y =0 ∂y

= φ00' (0) + Ec φ01' (0) + ε cos(π z ) (φ10' (0) + Ecφ11' (0)) = − h1 + Eca48 + ε cos(π z ) ( − m2 + Eca49 )

' = ψ 00 (0) + Ecψ 01' (0) + ε cos(π z )

(ψ 10' (0) + Ecψ 11' (0)) = − h2 − ε cos(π z ) m3 4. DISCUSSION 0F THE RESULTS The velocity, temperature and concentration distributions are analysed for different sets of the governing parameters G, D, N, R, Sc. We choose the Prandtl number P to be 0.71 . The corresponding profiles are plotted in figs.(1)-(11). It is to be noted that the axial variation of the suction contributes perturbed velocity u1 over the mean flow u0 generated in the state of uniform suction(ε=0). Figs.(1)-(4) show the variation of the axial velocity u for different values of the parameters .It is found that the velocity u is always positive for all variations .The velocity steeply rises in a layer near the boundary and falls to its prescribed value 1 far away from the boundary .We find that the velocity u enhances with increase in the thermal buoyancy(G) with maximum attained at y = 0.1(fig.1). An increase in the permeability of the porous medium reduces u for D-1 ≤ 3x103 and enhances with D-1≥ 5x103. Also u enhances with R(fig.2) .When the concentration buoyancy dominates over the thermal buoyancy force the velocity u experiences an enhancement in the entire

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fluid region .The variation of u with Sc shows that an increase in the molecular diffusivity enhances u in the fluid region(fig.3). Also we find that the inclusion of the viscous dissipation depreciates the velocity u (fig.4). The temperature(θ) in the fluid region is exhibited in figs.5-9 for different values of G,D,N,R Sc & Ec .We find that the temperature is positive for all variations .The temperature gradually reduces from its prescribed value 1 on the boundary to attain its value zero far away from the boundary. An increase in the thermal buoyancy (G) enhances θ in the entire fluid region. For D-1 ≤2x102, the temperature reduces and for D1 ≥ 4x102, it enhances with decrease in the permeability of the porous medium(fig.7). When the concentration buoyancy dominates over the thermal buoyancy force the temperature experiences an enhancement irrespective of the directions of the buoyancy forces . Also it decreases with R(fig.6).The variation of θ with Sc shows that an increase in the molecular diffusivity decreases the temperature in the entire fluid region. Also the variation of θ with Ec implies that the inclusion of the viscous dissipation enhances θ in the fluid region.(fig.9). The concentration distribution (C) is exhibited in figs. (10) & (11) for different variations of the governing parameters. For all variations in the parameters we find that the concentration is positive .An increase in the Reynolds number R reduces C in the

fluid region(fig.10).Also the effect of the molecular diffusivity is to enhance the concentration in the fluid region(fig.11). The shear stress(τ ),the rate of heat transfer (Nusselt number) and the rate of mass flux (Sherwood number) on the boundary y = 0 have been evaluated for different variations of the parameters G ,D, N, R, Sc and Ec and are presented in tables.1-5. It is found that the stress increases with increase in the thermal buoyancy (G) or the Reynolds number R. A decrease in the permeability of the porous medium reduces the shear stress .When the concentration buoyancy force dominates over the thermal buoyancy force the shear stress enhances when the two forces act in the same direction but when they act in opposing directions, τ reduces with |N| for G≤4x102 and for higher G ≥ 6x102,it enhances with |N|(table.1).Also we find that an increase in the molecular diffusivity reduces τ at the boundary .The variation of τ with Ec shows that the viscous dissipation enhances the shear stress(table.2). The rate of heat transfer( Nusselt number) at the boundary is presented in tables.3&4.It is found that the rate of heat transfer decreases with increase in the thermal buoyancy G ≤ 2x102 but for higher G ≥4x102 it enhances with it .Also for G ≤ 2x102 a decrease in the permeability of the porous medium enhances Nu while forG ≥4x102 it reduces with D-1. With the concentration force dominating over the thermal force the rate heat transfer reduces

Journal of Computer and Mathematical Sciences Vol. 2, Issue 2, 30 April, 2011 Pages (170-398)

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for G ≤ 2x102 but for higher G ≥4x102it enhances with it irrespective of the directions of the two buoyancy forces .Also the Nusselt number enhances with R and decreases with increase in either Sc or Ec.

Fig.1

The rate of mass flux (Sherwood number) exhibited in table 5 shows that it depends only on R and Sc. We find that the rate of mass transfer increases with increase in R. Also an increase in the molecular diffusivity reduces Sh on the boundary (table.5).

Fig.2 u with D-1 and R

Variation of velocity (u) with G

D-1=103,Sc=1.3,N=1,R=35,P=0.71 πz = π/4 , Ec = 0.01 I II III IV G 103 3×103 5×103 104

366

D-1 R

G = 103 ,P=0.71 ,N=1 I II III IV V VI VII 103 3x103 5x103 104 2x104 103 103 35 35 35 35 35 70 140

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Fig.3 u with N and Sc D-1 =103 , G = 103 , N =1 I II III IV V VI VII VIII N 1 0 2 -0.5 -0.8 1 1 1 Sc 1.3 1.3 1.3 1.3 1.3 2.01 0.24 0.6

Fig .5

Variation of temperature(θ θ) with G I II III IV G 103 3x103 5x103 104

D-1

Fig.4 u with Ec I II III IV 0.01 0.05 0.1 0

Fig.6 θ with D-1 and R I II III IV V 103 2x103 3x103 5x104 104

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Fig.7

θ with N

Fig.8 θ with Sc

D-1 = 103 , G =103 , Sc=1.3 I II III IV V N 1 0 2 -0.5 -0.5

Fig.9 θ with Ec Ec

I 0.01

II 0.05

III 0.1

I II III 1.3 2.01 0.24

IV 0.6

Fig.10 Profiles of concentration( C ) with R IV 0

SC=1.3, P=0.71 I II R 5 10

III 15

IV 20

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Fig.11 C with Sc R=15,P =0.71 I II III IV Sc 2.01 1.3 0.6 0.24 Table.1 Shear Stress(τx10-1) at y = 0 P= 0.71,Sc=1.3,R=5,Ec=0.01,πz = π/4 G

I 20.7568

II 20.6429

III 20.5385

IV 27.3409

V 13.7803

VI 10.1448

VII 7.9165

3X103

41.2392

41.0226

40.8545

58.0664

21.2726

6.4567

3.0392

68.3537 105.478

68.0893 105.237

67.9739 105.035

102.461 167.283

23.6507 28.6009

-2.6738 -5.3456

-19.739 -69.856

103 5X10 104

D-1 N

103 1

3x103 1

5x103 1

103 2

103 0

103 -0.5

103 -0.8

Table.2 Shear Stress(τx10-1) at y = 0 D-1=103,N=1,P=0.71 G 103 3x103 5x103

I 41.7584 83.0628 137.768

II 83.6484 166.467 276.221

III 17.8332 30.7076 41.0721

IV 43.5184 75.6342 92.6698

V 24.0672 28.4835 1.3896

VI 22.7662 58.2518 126.909

VII 25.2532 79.5177 200.103

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VIII 20.2572 36.8859 53.7148

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140

1.3

2.01

0.24

0.6

1.3

0.01

0.05

0.1

Table.3 Nusselt Number(Nu) at y = 0 R=5,Sc=1.3,P= 0.71,πz = π/4 G

I -16.2957

II -17.0338

III -17.8263

IV -11.0212

V -17.9869

VI -17.4887

VII -16.7598

3X103

13.5673

10.6941

07.6002

35.1852

06.2825

08.0149

10.7744

62.7228

56.3183

49.4149

111.753

45.9419

49.6469

55.7388

131.172

119.839

107.618

218.682

100.991

107.401

118.387

103 5X10 104

D-1 N

103 1

3x103 1

5x103 1

103 2

103 0

103 -0.5

103 -0.8

Table.4 Nusselt Number(Nu) at y = 0 D-1=103,N=1,P=0.71 I

III

VII

103

-31.341

-61.887

-17.368

06.6765

06.6173

26.4803

79.9503

-26.97

3x103

31.8152

66.6308

08.7481

95.9748

53.8055

175.795

378.581

-26..97

5x103

135.792

278.232

51.4722

244.328

151.102

431.574

870.135

-26.98

104

280.588

572.918

110.807

450.637

286.874

763.813

997.615

-26.99

140

1.3

2.01

0.24

0.6

1.3

0.01

0.05

0.1

Table.5 Sherwood Number(Sh) at y = 0 R

III

-81.0123

-77.3991

-23.1461

-50.0716

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-100.1108

-87.5678

-43.5674

-87.3456

-200.2054

-267.235

-67.6578

-98.7865

1.3

2.01

0.24

0.6

CONCLUSION

5. RFERENCES

The axial velocity G enhances with increase in G & N and reduces with D-1. An increase in molecular diffusivity enhances U in the entire flow region. Also u depreciates with increase in Eckert number Ec. An increase in thermal buoyancy G & porous parameter D-1 and N enhances the temperature in the fluid region. An increase in the molecular diffusivity decreases temperature in the fluid region. Also the temperature enhances with increase in Ec. The concentration reduces with Reynolds number R and enhances with Sc. An increase in G and R increase the stress Ď&#x201E; and reduces with increase in D-1 Ď&#x201E; enhances with increase in buoyancy ratio N and Eckert number Ec.The Sherwood number increases with increase in R and reduces with Sc.

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Journal of Computer and Mathematical Sciences Vol. 2, Issue 2, 30 April, 2011 Pages (170-398)