Cmjv02i03p0493

J. Comp. & Math. Sci. Vol.2 (3), 493-504 (2011)

Finite Element Analysis of Viscous Heat Generating Fluid in a Vertical Channel with Quadratic Density Temperature Variation and Radiation G. VENU GOPALA KRISHNA1 and D. CHITTI BABU2 1

Department of Mathematics, VRS & YRN College of Engg. & Tech., Chirala, A.P. India 2 Department of Mathematics, Government College (A), Rajahmundry, A.P., India.

ABSTRACT 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 temperature variation and radiation. The equations governing the velocity, energy and diffusion are non-linear coupled. By applying Garlekin Finite element analysis with quadratic interpolation functions the equations are solved. The velocity, temperature and concentration distributions are evaluated for different parameters. The rate of heat transfer and mass transfer are evaluated for different variations of parameters.

Keywords: Convective heat transfer, mass transfer, porous medium.

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 understanding of a host of thermal engineering applications in which porous materials present is required. The accumulated impact of the studies in two fold, first to improve the performance of existing porous media – related thermal system, 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 understand to benefit from a better understanding of heat and fluid flow process through porous materials and geo thermal system, thermal insulation, grain storage, solid matrix heat exchangers, 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 flow6. Darcy’s equation gives satisfactory results for closely packed porous medium but does not explain

Journal of Computer and Mathematical Sciences Vol. 2, Issue 3, 30 June, 2011 Pages (399-580)

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G. Venu Gopala Krishna, et al., J. Comp. & Math. Sci. Vol.2 (3), 493-504 (2011)

the flow through sparsely distributed porous medium in later situation. Brinkman2 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 from Brinkman model was made by Chen et al.5.

Vafai&Tien16, Vafai17, Kim & Vafai11 have worked on the problem of convective heat transfer in porous media relaxing some of all the limitations of Darcy’s model. Later on a series of investigation were carried out using the Brinkman model by a few authors notably Poulikakos. D & Bejan12, Tong & Subramanian15, Prasad & Tuntomo13.

Fotcheiner7 extended Darcy’s model to the natural convection flow through process medium.

Keeping the applications in view Jaffer10, Venkata Ramana18 has studied the fact of quadratic density temperature variation on convective heat transfer in vertical channel. Recently Alivene1 and Rao14 have studied the effect of quadratic density temperature variation convective heat and mass transfer through a porous medium in vertical channel under conditions.

The role of thermal radiation is of major importance in some industrial applications such as glass production and furnace design and in space technology applications, such as cosmical flight aerodynamics rocket, propulsion systems, plasma physics and space craft reentry aero thermo dynamics which operate at high temperatures. When radiation is taken into account, the governing equations become quite complicated and hence many difficulties arise while solving such

equations. Grief et al.8 shows that in the optically thin limit the physical situation can be simplified, and then they derived exact solution to fully developed vertical channel for a radiative fluid. Hossain & Takhar9 studied the radiation effects on mixed convection along a vertical plate with uniform surface temperature using Keller Box finite difference method. Chamkha3 considered the thermal radiation effects on MHD forced convection flow of an electrically conducting and heat generating or absorbing fluid over an isothermal wedge.

Chamkha4 investigated thermal radiation and buoyancy effects on hydro – magnetic flow over an accelerating permeable surface with heat source of sink.

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 and the effect of radiation. The equations governing of the velocity, energy and diffusion are non – linear coupled. By employing Garlekin finite element analysis with quadratic interpolating functions and equations are solved. The velocity temperature and concentration distributions are analyzed for different variations of the parameters viz… Gm, Gc, D-1, S, Sc, N and α.

The rate of heat transfer and mass transfer are evaluated numerically for different variations of the governing parameters.

FORMULATION OF THE PROBLEM We consider the convective heat and mass transfer flow of a viscous, incompressible, non conducting fluid through a Darcy isotropic, homogeneous porous medium with radiation. The X-axis is

Journal of Computer and Mathematical Sciences Vol. 2, Issue 3, 30 June, 2011 Pages (399-580)

G. Venu Gopala Krishna, et al., J. Comp. & Math. Sci. Vol.2 (3), 493-504 (2011)

directed to the vertical surface and 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 the temperature Tw and concentration Cw which remain unchanged. The walls are situated at y = 1. 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)

∂u ∂ 2u = v 2 + β g (T − Te ) 2 + β * g (C − Ce ) ∂y ∂y (2.2) v − u kp

v

v

∂T ∂ 2T ∂q =λ 2 +Q− r ∂y ∂y ∂y

v

∂C ∂ 2C = Dm 2 ∂y ∂y

(2.3) (2.4)

where u is the axial velocity in the porous region T & C are the temperature and concentration of the fluid g is the acceleration due to gravity is the permeability of porous kp medium Q is the heat source is the kinematics viscosity ν ρ is the density of the fluid α is the thermal diffusivity Dm is the coefficient of diffusitivity β is the coefficient of the thermal expansion

β*

495

is the coefficient of volume expansion

The boundary conditions are y=0 u=0 T=T1 C = C1 y=1 u= ± L T=T2 C = C2 Introducing Rosseland’s approximation for − 4σ ∂T 4 radiative heat flux, qr = 3 β r ∂u r Expanding T4 about Te in Taylor series T4 ≅ 4T Te4 Introducing the non-dimensional variables (x1, y1) = (x, y)/L θ=

T −T 2 , T 1−T2

φ=

C − C2 C1 − C2

The conservation equations are now transformed in to the following system of coupled, non linear ordinary differential equations of u, θ, φ . u yy + su y − D −1u = Gm (θ + N1φ ) 2 + Gcφ

(2.5)

θ yy + SP1θ y−α1θ = 0

(2.6)

φ yy + SScφ y = 0

(2.7)

The corresponding boundary conditions are u = 0 θ = 1 φ = 1 at y = L u = 0 θ = 0 φ = 0 at y = -L (2.8) Where Gm =

gβ * (T1 −T 2) v2

(Local

Temperature

Grashof Number) Gc =

gβ * (C1 − C2 ) v2

(Local Mass Grashof

Number)

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G. Venu Gopala Krishna, et al., J. Comp. & Math. Sci. Vol.2 (3), 493-504 (2011)

α=

3

kp

D −1 =

(Darcy’s Number)

L2 QL2

(Heat Source Parameter)

λ β T3 N1 = R3 e 3 4σ Te

α= S=

(Radiation Parameter)

QL2

(Heat Source Parameter)

λ v0 L v

(Suction Reynolds Number)

γ λ v Sc = Dm p =

α1 =

(Prandtl Number) (Schmidt Number)

3 N1α 3N + 4

P1 =

3 N1 P 3N + 4

i   + S du − Gm (θ i + N1φ i ) 2  dy (3.1) 

− Geφ i − D −1u i Eθ i =

d  dθ i dy  dy

i   + P1S dθ − α1θ i  dy 

(3.2)

Eφ i =

d  dφ i dy  dy

i   + SS c dφ  dy 

(3.3)

u = i

3

∑ f kψ k

k =1

3

θ = ∑ θ kψ k i

k =1

The errors are orthogonal to the weight functions over the domain of ei under Galerkin finite element technique we choose the approximation functions as the weight functions. Multiply both sides of the equations (3.1)–(3.3) by the weight functions i.e. each of the approximation function and integrate over the typical two nodded linear element (ηe,ηe+1) we obtain ηe +1

i i ∫ Eu ψ j dy = 0 (j = 1, 2, 3, 4)

ηe +1

To solve these differential equations with the corresponding boundary conditions we assume ui, θi, and Øi are the approximations of u, θ, and Ø. We define the errors (residual) Eu i , Eθ i , Eφ i , as d  du i dy  dy

(3.4)

k =1

(3.5)

ηe

FINITE ELEMENT ANALYSIS OF THE PROBLEM

Eu i =

φ i = ∑ φkψ k

i i ∫ Eθ ψ j dy = 0 (j = 1, 2, 3, 4)

(3.6)

ηe

ηe +1

i i ∫ Eφ ψ j dy = 0 (j = 1, 2, 3, 4)

(3.7)

i   d  du i     + S du   dy  dy  dy  η e+1   i i i 2 ∫  − Gm (θ + N1φ ) ψ j dy = 0  ηe  i −1 i  − Geφ − D u       

(3.8)

ηe

η e+1 

d  dθ ∫  dy  dy ηe  

η e +1 

i

d  dφ i  ∫   η e  dy  dy

i    + P1S dθ − α1θ i ψ j i dy = 0 (3.9)   dy  

i   + SSc dφ  dy 

 i ψ j dy = 0  

Journal of Computer and Mathematical Sciences Vol. 2, Issue 3, 30 June, 2011 Pages (399-580)

(3.10)

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G. Venu Gopala Krishna, et al., J. Comp. & Math. Sci. Vol.2 (3), 493-504 (2011)

Following the Garlekin weighted residual method and integration by parts method to the equations (3.8 – 3.10) we obtain η e +1 d ψ i η e +1 du k j du k dy + ∫ S ψ j i dy ∫ dy

ηe

−

dy

dy

ηe

η e +1

η e +1

i i ∫ G mθ kψ j dy − ∫ G eφ kψ j dy

ηe

−D

ηe

−1

η e +1

η e +1

dψ

∫

duk (η e ) dy du = ψ j (η e +1 ) k (η e +1 ) dy

i j

dy

ηe

η e +1 dθ k dθ k ψ j i dy dy + ∫ P1S dy dy ηe

η e +1

i ∫ α 1θ kψ j dy = R1, j + R2 , j (3 .12 )

ηe

Where dθ k dθ (η e ) + ψ j (η e ) k (η e +1) dy dy dθ k (η e +1 ) R2, j = ψ j (η e +1 ) dy dθ + ψ j (η e +1 ) k (η e +1 ) dy R1, j = ψ j (η e )

i

ηe+1

dθ

dθ

i k k ∫ dy dy dy + ∫ S c S dy ψ j dy = S1, j + S 2, j ηe ηe (3.13) Where

S1, j

η e +1 3 du k dψ k dy + ∑ Su k ∫ ψ k ψ j i dy dy dy dy K =1 ηe

ηe

i

j

3

η e+1

3

ηe +1

ηe

k =1

ηe

− Gm ∑ θ k ∫ ψ kψ j i dy − Gc ∑ φ k ∫ ψ kψ j i dy k =1 3

ηe +1

k =1

ηe

(3.14)

− D −2 ∑ u k ∫ ψ kψ j i dy = Q1, j + Q2, j

ηe+1

Q2, j

dψ

∫

dψ

∫ u kψ j dy = Q1, j + Q 2 , j ( 3 .11 )

Where Q1, j = ψ jη e

ηe+1

η e+1

i

ηe

−

Making use of equations (3.4) we can write above equations

j

dθ dθ = ψ j (η e ) k (η e ) + ψ j (η e ) k (η e +1 ) dy dy

S 2, j = ψ j (η e +1 )

dθ k dθ (η e +1 ) + ψ j (η e +1 ) k (η e +1 ) dy dy

∫

ηe

ηe+1 3 dψ j i dθ k dψ k i dy + P1 ∑ Sθ k ∫ ψ k ψ j dy dy dy dy k =1 ηe 3

η e+1

k =1

ηe

− α ∑θ k

ηe +1

dψ

i

(3.15)

∫ ψ kψ j dy = R1, j + R2, j i

dθ

3

ηe +1

dψ

j i k k ∫ dy dy dy + Sc ∑ Sφk ∫ ψ k dy ψ j dy (3.16) k =1 ηe ηe

= S1, j + S 2, j

Choosing different ψji’s corresponding to each element ηe in the equation (3.14) yields a local stiffness matrix of order 3x3 in the form

(u )− δG( g k

i

k i, j

)(θ i k + NCi k ) + δ (mi , j k )(ui k )

+ δ 2 Λ (ηi , j k )(ui k ) = (Q2, j k ) + (Q1, j k )

(3.17) The equations (3.15) & (3.16) give rise to stiffness matrices

(θ )− P D (t )(u ) = R k

i

k

r

u i, j

k

i

(θ )− S S (t )(u ) = S k

i

k

c r i, j

k

i

k

2, j

k 2, j

+ R1, j k

(3.18)

+ S1, j k

(3.19)

( f i , j k ), ( g i , j k ), (mi , j k ), (ni , j k ), (ei , j k ), (li , j k ) and (ti , j k )

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G. Venu Gopala Krishna, et al., J. Comp. & Math. Sci. Vol.2 (3), 493-504 (2011)

are 3x3 matrices and k

k

k

0.6

k

k

k

(Q1, j ), (Q2, j ), ( R1, j ), ( R2, j ), ( S 2, j ), ( S1, j )

are 3x1 column matrices and such stiffness matrices (3.17)–(3.19) in terms of local nodes in each element are assembled using inter element continuity equilibrium conditions to obtain the coupled global matrices in terms of the global nodal of f, g, h, θ and φ . In case we choose n quadratic elements then the global matrices are of 2n+1. The ultimate coupled global matrices are solved to determine the unknown global values of the velocity, temperature and concentration in fluid region, in solving these matrices an iteration procedure has been adopted to include the boundary and effects in the region.

Solving the coupled global matrices for temperature, concentration and velocity 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.

DISCUSSION OF THE NUMERICAL RESULTS In this analysis we will discuss the effects of non-linear density temperature variations on convective heat and mass transfer flow of a viscous fluid through a porous medium confined vertical channel in the presence of heat generating sources and with the effect of radiation. Taking quadratic polynomials as shape functions, the analysis has been carried out by finite element technique. The velocity, temperature and concentrations are analyzed for different values of Gm, Gc, S, α, Sc, and N1. The velocity profiles are exhibited in figs. 1-5.

0.5 I

0.4

II

u 0.3

III IV

0.2 0.1 0 -1

-0.75

-0.5

-0.25

0 r

0.25

0.5

0.75

1

Fig 1. Variation of u with D-1

Gm = 10 Gc= 4 Sc = 1.3 S = 102 α = 2 I II III IV D-1 102 3X102 5X102 103 0.4 0.3 0.2 I

0.1 u 0 -0.1

II III

-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

IV

-0.2 -0.3 -0.4

r

Fig 2. Variation of u with S

S

Gm = 10 Gc= 4 Sc = 1.3 α = 2 I II III IV 1X102 2X102 3X102 5X102

0.35 0.3 0.25

I II

u 0.2

III

0.15

IV

0.1 0.05 0 -1

-0.75

-0.5

-0.25

0 r

0.25

0.5

0.75

Fig 3. Variation of u with Sc

S = 102 Gc= 4 Gm = 10 α = 2 I II III IV Sc 0.4 0.8 1.3 2

Journal of Computer and Mathematical Sciences Vol. 2, Issue 3, 30 June, 2011 Pages (399-580)

1

G. Venu Gopala Krishna, et al., J. Comp. & Math. Sci. Vol.2 (3), 493-504 (2011) 1 0.9 0.8 0.7 I

0.6

II

u 0.5 0.4

III

0.3 0.2 0.1 0 -1

-0.75

-0.5

-0.25

0 r

0.25

0.5

0.75

1

Fig 7. Variation of temperature θ with S

Fig 4. Variation of u with α

α

I 2

II 4

III 6

S

Gm = 10 Gc = 4 Sc = 1.3 I II III IV 102 2X102 3X102 5X102

0.3 0.25 0.2 0.15 I

0.1 u

II

0.05 0 -0.05 -1

III

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

IV

-0.1 -0.15 -0.2 -0.25

r

Fig 5. Variation of u with N1

Fig 8. Variation of temperature θ with Sc

Gm = 10 Gc= 4 Sc = 1.3 I II III IV N1 0.24 0.5 1 1.5

S = 102 Gc = 4 Gm = 10 α = 2 I II III IV Sc 0.4 0.8 1.3 2

Fig 6. Variation of temperature θ with D-1

Fig 9. Variation of temperature θ with N1

Gm = 10 Gc = 4 Sc = 1.3 S = 102 I II III IV D-1 102 3X102 5X102 103

Gm = 10 Gc = 4 Sc = 1.3 S = 102 I II III IV N1 0.24 0.5 1 1.5

Journal of Computer and Mathematical Sciences Vol. 2, Issue 3, 30 June, 2011 Pages (399-580)

499

500

G. Venu Gopala Krishna, et al., J. Comp. & Math. Sci. Vol.2 (3), 493-504 (2011)

1.5 1 0.5 0 -0.5 -1 c

I

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

II

-1

III

-1.5

IV

-2 -2.5 -3 -3.5

r

Fig 10.Variation of concentration C with D-1

Fig 13. Variation of concentration C with N1

Gm = 10 Gc = 4 Sc = 1.3 S = 102 I II III IV N1 0.24 0.5 1 1.5

Gm = 10 Gc = 4 Sc = 1.3 S = 102 I II III IV D-1 102 3X102 5X102 103 1.2 1 0.8

I II

c 0.6

III IV

0.4 0.2 0 -1

-0.75

-0.5

-0.25

0 r

0.25

0.5

0.75

1

Fig 11. Variation of concentration C with S

S 1.5 1 0.5 0 -0.5 -1 -1 c -1.5 -2 -2.5 -3 -3.5 -4

Gm = 10 Gc = 4 Sc = 1.3 I II III IV 102 2X102 3X102 5X102

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

I II III IV

r

Fig 12. Variation of concentration C with Sc

S = 102 Gc = 4 Gm = 10 α = 2 I II III IV Sc 0.4 0.8 1.3 2

From figure (1) it shows that velocity u enhances as the value of D-1 increases. u increases in the region (-0.75, 0.5) while it depreciate in the remaining region. Figure (2) represents the variations of u with Reynolds number. It is observed that small values of R, enhances when R ≤ 2X102. For higher ≥ 102, the velocity u depreciates in the fluid region. The variations of u with Sc show that the velocity is positive and depreciates for higher values of Sc. The influence of heat generating sources on u is shown in figure (4). It is obtained that velocity enhances with an increased in α. Figure(5) represents the variation of u with radiation parameter N1. It is found that u changes from negative to positive in the neighborhood of right boundary y = 1. The variation of u with N1 shows that the velocity is negative for N1 ≤ 0.5 and it changes from negative to positive in the region adjacent to y = +1. Also u depreciates with increase in N1 ≤ 0.5 and enhances for higher N1 ≥ 1.

The non-dimensional temperature distribution is shown in figures 6–9. We follow the convection that the temperature is

Journal of Computer and Mathematical Sciences Vol. 2, Issue 3, 30 June, 2011 Pages (399-580)

G. Venu Gopala Krishna, et al., J. Comp. & Math. Sci. Vol.2 (3), 493-504 (2011)

positive of negative according as the actual temperature is greater or lesser than the ambient temperature T2. From figure(6) we observe that the temperature enhances with an increase in the value of D-1. Figure(7) shows that the temperature reduces in the left half and enhances in the right half with an increasing in the suction Reynolds number S. with reference to Schmidt number Sc, we notice that lesser the 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 ≥ 1.3. Figure(9) represents the variation of temperature θ with radiation parameter N1. The temperature enhances for different values of N1 and it reaches maximum at 0.75.

The concentration distribution C is shown in figures 10–13. From figure(10) we observe that concentration enhances with an increase in D-1. From figure(11) we noticed that concentration C reduces with an increase in the suction Reynolds number S. From figure(12) we conclude that lesser the molecular diffusivity smaller the actual concentration and depreciates in the fluid region. From figure(13) we noticed that the concentration increases for an increase in the radiation parameter N1.

The Shear Stress (τ) at boundaries y = ±1 is shown in the tables 1–4 for different values of R, Sc, α and N1. The variation of τ with Sc reduces that τ depreciates for Sc ≤ 0.6 and enhances with higher Sc ≥ 0.8 at y = ±1. The influence of heat generating source α, Gc is shown in tables 1 and 2. It is observed that with an increase in the strength of heat source, we notice that depreciation in τ for all values of R

501

(Tables 1 and 2). At y = -1 the stress τ enhances with an increase in the radiation parameter N1 where as at y = 1 we observed that τ increases for N1 ≤ 0.5 and decreases for N1 ≤ 1.5.

The Nusselt Number (Nu) which measures the rate of heat transfer is exhibited in tables 5–6 for different parameters α and N1. It is observed that the rate of heat transfer at y = -1 depreciates with R ≥ 80 and enhances with higher R ≥ 100 and at a reversed effect is observed at y = 1. A variation of Nu with Gm shows that Nu reduces with Gm ≤ 20 at y = -1 and enhances for Gm ≥ 20 at y = 1 with respect to the behavior of Nu with Gc, we find that Nu enhances for Gc ≤ 10 and depreciates for Gc ≥ 20 at y = -1, where as for y = 1 we observe that Nu depreciates for Gc ≤ 20.

The variation of Nu depreciates for R < 80 and enhances with an increase in R ≥ 100.

We notice that Nu enhances with an increase in Sc ≥ 1.3 at y = -1 and Nu enhances for R ≥ 100 at y = -1. We find that at y = 1 higher the diffusivity Sc, higher the rate of heat transfer, an increase in the ion strength of heat generating source enhances Nu at y = 1 and reduces at y = -1. The rate of heat transfer Nu depreciates for R ≤ 50 and enhances for R ≥ 100. We notice that Nu enhances with an increase in the radiation parameter N1 at y = -1 and y = 1 (Table 5–6).

The Sherwood number (Sh) which measures the rate of mass transfer is shown in tables (7–8) for different values of

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G. Venu Gopala Krishna, et al., J. Comp. & Math. Sci. Vol.2 (3), 493-504 (2011)

parameters. It is observed that Sh enhances

Sh reduces with an increase in the Sc at y

for R ≤ 100 at y = -1 and y = +1, with respect to behavior of Sh with Gm we observe that an increase in Gm depreciates Sh at y = -1 and y = +1. We notice that an

= ±1. Also an increase in the strength of heat generating source results in an increment in Sh at both the boundaries. In general we

increase of Gc, Sh enhances at y = -1 and y = +1. The rate of mass transfer Sh with molecular diffusivity Sc, Sheer wood number

R 50 80 100 Sc α

R 50 80 100 Sc α

observe that the rate of mass transfer at y = +1 is greater than y = -1. The rate of mass transfer reduces for N1 ≤ 0.5 and enhances for N1 ≥ 1 at both the boundaries.

I 0.03154 0.00101 0.00043 0.4 2

Table 1 Shear Stress (τ) at r =1, Gm = 2, Gc = 2.

II III IV V VI 0.01724 0.03252 0.55672 8.77456 4.00291 0.00093 0.00085 0.00063 4.51013 2.00417 0.00051 0.00038 0.00030 3.7842 1.2114 0.6 0.8 1.3 1.3 1.3 2 2 2 0 4

VII 2.00210 1.00339 0.01813 1.3 6

I 1.95468 0.28752 0.13369 0.4 2

Table 2 Shear Stress (τ) at r =2, Gm = 2, Gc = 2.

II III IV V VI 1.84439 1.75515 3.93810 2.80952 0.65503 0.27555 0.26478 0.23421 7.71017 0.12148 0.1285 0.12386 0.11154 8.74217 0.05889 0.6 0.8 1.3 1.3 1.3 2 2 2 0 4

VII 0.47974 0.09282 0.04539 1.3 6

R 50 80 100 N1

Table 3 Shear Stress (τ) at r =1, Sc = 1.3, α = 2.

I II III -2.03581 3.38913 4.63219 1.49438 2.9517 5.84600 2.3764 3.80841 6.49303 0.24 0.5 1

R 50 80 100 N1

Table 4 Shear Stress (τ) at r =2, Sc = 1.3, α = 2.

I II III -1.44625 2.71367 -8.40751 1.06499 -2.766 -3.56746 -0.80 -2.9373 1.14087 0.24 0.5 1

IV 5.80416 6.93352 7.00187 1.5

IV -2.36226 -4.25467 2.48257 1.5

Journal of Computer and Mathematical Sciences Vol. 2, Issue 3, 30 June, 2011 Pages (399-580)

G. Venu Gopala Krishna, et al., J. Comp. & Math. Sci. Vol.2 (3), 493-504 (2011)

503

Table 5 Nusselt Number (Nu) at r =1, Sc = 1.3, α = 2.

R 50 80 100 N1

I 2.86123 -7.6562 -8.6130 0.24

II -8.02013 -4.7766 -3.13976 0.5

III 1.24909 -3.53424 -1.59715 1

IV 0.64879 -2.65571 -1.25775 1.5

Table 6 Nusselt Number (Nu) at r =2, Sc = 1.3, α = 2.

R 50 80 100 N1

I -4.84076 -3.79605 -2.611 0.24

II 1.40978 2.8622 1.45616 0.5

III 4.51556 3.84084 2.78630 1

IV 5.19834 4.21250 3.61693 1.5

Table 7 Sheer wood Number (Sh) at r =1, Gm = 2, Gc = 2.

R 50 80 100 N1

I 1.06021 -3.6827 -4.8351 0.24

II -1.87033 -4.98143 -4.99682 0.5

III 1.49662 -8.132404 -3.68101 1

IV 2.40851 1.65830 0.25826 1.5

Table 8 Sheer wood Number (Sh) at r =2, Gm = 2, Gc = 2.

R 50 80 100 N1

I 1.38465 -3.1769 -2.715 0.24

II 2.79921 -2.8096 -3.3873 0.5

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