thermal radiation effects on mhd viscoelastic fluid flow ...

THERMAL RADIATION EFFECTS ON MHD VISCOELASTIC FLUID FLOW OVER A STRETCHING SHEET WITH VARIABLE VISCOSITY G. C. Shit and R. Haldar Department of Mathematics, Jadavpur University, Kolkata - 700032, India Email: [email protected] Received 26 August 2011; accepted 4 April 2012

ABSTRACT The effect of thermal radiation and temperature dependent viscosity on free convective flow and mass transfer of an electrically conducting fluid over an isothermal stretching sheet is investigated. The sheet is being stretched linearly in the presence of a uniform transverse magnetic field and the flow is governed by the second-order viscoelastic fluid. The nonlinear boundarylayer equations together with the boundary conditions are reduced to a system of non-linear ordinary differential equations by using a similarity transformation. The transformed equations are solved by developing a suitable numerical technique such as finite difference scheme along with the Newton’s linearization method. The results of this study concerned with the velocity, temperature and concentration profiles as well as with the local skin-friction coefficient Cf , local Nusselt number N u and the local Sherwood number Sh for different values of the physical parameters of interest. The effect of thermal radiation and variable viscosity can lead to the decrease of heat transfer, whereas the presence of viscoelasticity of the fluid and magnetic field causes increase of the heat transfer. Keywords: Thermal radiation; Variable viscosity. 1

INTRODUCTION

The study of MHD free-convective flow and mass transfer over a stretching sheet has gained a considerable attention to the researchers, scientists and technologists due to its many engineering and industrial applications such as in polymer processing, electro-chemistry, MHD power generators, flight magnetohydrodynamics as well as in the field of planetary magnetosphere, aeronautics and chemical engineering. In the extrusion of a polymer sheet from a die, during glass-fibre and paper production, hot rolling, and the drawing of plastic films, the sheet is some times stretched. By drawing such a sheet, the rate of cooling can be controlled and the quality of the final products of the desired characteristics can be achieved. Sakiadis (Sakiadis 1961) first attempted to solve the problem of boundary-layer flow over a continuously moving surface and Crane (Crane 1970) extended this problem to a stretching sheet whose surface velocity varies linearly with the distance from a fixed point in the sheet. Int. J. of Appl. Math. and Mech. 8(14): 14-36, 2012.

Thermal Radiation Effects on MHD Viscoelastic Fluid Flow

15

Later on several investigators (Rajagopal et al. 1984; Dandapat and Gupta 1989; Vajravelu and Rollins 1991; Mahapatra and Gupta 2002; Shit 2009 and Reddaiah and Rao 2012) extensively studied on various aspects of boundary-layer flow problems over a stretching sheet. Gupta and Gupta (Gupta and Gupta 1977) examined the heat and mass transfer effects using a similarity transformation for the boundary-layer flow over a stretching sheet subject to suction or blowing. Since many materials such as polymer solutions or melts, drilling mud, blood, shampoo, paint, certain oils and greases are classified as non-Newtonian fluids, the most of the recent studies are concerned with the viscoelastic fluids. Therefore a large number of studies on viscoelastic fluid with or without considering magnetic field/heat transfer are available in the scientific literatures (Vajravelu et al. 1991, 1999; Garg and Rajagopal 1991; Andersson 1992; Char 1994; Sharma and Rao 1998; Datti et al. 2004; Cortell 2005; Misra and Shit 2009). The study of magnetohydrodynamic (MHD) flow of an electrically conducting fluid is of considerable interest in modern metallurgical and metal-working processes lies in the purification of molten metals. Another important aspect, that is, heat transfer process plays a vital role in industrial applications. This is due to the fact that the rate of cooling can be controlled by the application of magnetic field. In view of this many authors have investigated the problems on MHD flow and heat transfer in an electrically conducting viscoelastic boundary layer flow over a linearly stretching sheet (Rollins and Vajravelu 1991; Abel 2004; Zakaria 2004; cortell 2006; Abel et al. 2008). The non-uniqueness of MHD flow of a second order fluid past a stretching sheet was studied by Lawrence and Rao (Lawrence and Rao 1995). Ezzat et al. (Ezzat et al. 1996) formulated the state space approach for the one-dimensional problem of viscoelastic magnetohydrodynamic unsteady free convection boundary-layer flow with one relaxation time. Misra et al. (Misra et al. 2008) studied on flow and heat transfer of a MHD viscoelastic fluid in a channel with stretching walls. They concluded that due to the stretching of the channel walls a reversal flow takes place near the central line of the channel and this flow reversal can be eliminated by applying a strong magnetic field. However, Liu (Liu 2004) presented an analytical solution for flow and heat transfer of a laminar boundary layer flow of an electrically conducting second-grade fluid subject to a transverse magnetic field over a stretching sheet with an aim to motivate the possible industrial applications. Recently, a new idea is added to the study of viscoelastic boundary-layer fluid flow and heat transfer is the consideration of the effect of thermal radiation and temperature dependent viscosity. Thermal radiation effect might play an important role in controlling heat transfer process in polymer processing industry. In view of this, Raptis and Perdikis (Raptis and Perdikis 1998) investigated the viscoelastic fluid flow and heat transfer past a semi-infinite porous plate having constant suction in the presence of thermal radiation. Heat transfer in a viscoelastic fluid flow over a non-isothermal porous sheet influenced by a continuous suction/blowing of the fluid has been carried out by Khan (Khan 2006). Mukhopadhaya et al. (Mukhopadhaya et al. 2005) employed the study of MHD boundary-layer flow over a heated stretching sheet with variable viscosity. The radiation effect on boundary layer flow with or without applying a magnetic field under different situations were studied by Mahmoud (Mahmoud 2009) and Shateyi (Shateyi 2008), but their studies are restricted in Newtonian fluid model only. However, Salem (Salem 2007) investigated the effect of variable viscosity on MHD viscoelastic fluid flow and heat transfer over a stretching sheet, wherein he did not considered thermal radiation effect. Prasad et al. (Prasad et al. 2010) carried out the study of variable viscosity on MHD viscoelastic fluid flow and heat transfer over a stretching sheet in the absence of viscous dissipation and buoyancy Int. J. of Appl. Math. and Mech. 8(14): 14-36, 2012.

16

G. C. Shit and R. Haldar

effect. Motivated by the above mentioned investigations, we present a theoretical and numerical analysis of the effect of thermal radiation on MHD free-convective flow and mass transfer of a viscoelastic fluid over a stretching sheet with variable viscosity. In the present study viscous dissipation and buoyancy effects have been taken into account in a situation when there would be a temperature dependent viscosity. Newton’s linearization method have been employed to solve the non-linear ordinary differential equations followed by a perturbation technique. The numerical results to the flow quantities are presented graphically as well as in tabular form.

2

MATHEMATICAL FORMULATION

Let us consider the steady two dimensional boundary-layer flow of an incompressible homogeneous, electrically conducting viscoelastic fluid past over an isothermal stretching sheet under the action of an externally applied magnetic field. The flow is assumed to be in the x- direction, which is taken along the sheet and the y-axis is normal to the sheet. Two equal and opposite forces are introduced along the x-axis, so that the sheet is stretched with a velocity proportional to the distance from the fixed origin. The fluid is permeated by an externally applied uniform magnetic field of strength B0 , which acts in the direction perpendicular to the sheet. The effect of induced magnetic field has been neglected under the assumption of negligible induced electric field. The sheet is kept at a constant temperature Tw and the species concentration is maintained at a constant value Cw , where the viscoelastic fluid far away from the sheet is rest at a temperature T∞ and concentration C∞ . Since the concentration of diffusing species is very small in comparison to the other chemical species, the thermal diffusion and diffusing thermal energy effects are neglected. But the viscous dissipation and the effect of thermal radiation have been included in the energy equation. Following Lai and Kulacki (Lai and Kulacki 1990), the fluid viscosity is assumed to vary as a reciprocal of a linear function of temperature given by

1 1 = [1 + γ (T − T∞ )] µ µ∞

(1)

which yields 1 = a(T − Tr ), µ with a =

γ µ∞

(2)

and Tr = T∞ − γ1 .

In equation (2), both a and Tr are constants and their values depend on the thermal property of the fluid, i.e., γ. In general a > 0 represent for liquids, whereas a < 0 for gases. By assuming Rosseland approximation for radiation, the radiative heat flux qr is given by qr = −

4σ ∗ ∂T 4 3K ∗ ∂y

Int. J. of Appl. Math. and Mech. 8(14): 14-36, 2012.

(3)


17

where σ ∗ is the Stefan-Boltzman constant and K ∗ the mean absorption coefficient. We assume that the temperature differences within the flow are sufficiently small such that T 4 may be expressed as a linear function of the temperature as shown in Chamakha (Chamakha 1997). Expanding T 4 in a Taylor series about T∞ and neglecting higher order terms, we obtain, 3 4 T4 ∼ T − 3T∞ = 4T∞

(4)

Using (4) in (3) we obtain as 3 ∂qr 16σ ∗ T∞ ∂ 2T =− ∂y 3K ∗ ∂y 2

(5)

Owing to the above mentioned assumptions, the governing equations in the boundary-layer flow can be written as ∂u ∂v + =0 , ∂x ∂y

(6)

) ( ) } ( { ∂u ∂ ∂u ∂ 3 u ∂u ∂ 2 u ∂u ∂ 2 u ∂u ∂ 3u ρ∞ u +v = µ − k0 u +v 3 − + ∂x ∂y ∂y ∂y ∂x∂y 2 ∂y ∂y ∂x∂y ∂x ∂y 2 + ρ∞ gβt (T − T∞ ) + ρ∞ gβc (C − C∞ ) − σB02 u , (7)

( ) ∂T ∂T ∂ 2T ρ∞ C p u +v =k 2 ∂x ∂y ∂y

u

( + µ

∂u ∂y

)2

[ ( )] ∂qr ∂u ∂ ∂u ∂u − − k0 u +v , ∂y ∂y ∂y ∂x ∂y

∂C ∂C ∂2C +v =D 2 , ∂x ∂y ∂y

(8)

(9)

The boundary conditions for the present problem are assumed as u = Uw = bx, v = 0, T = Tw , C = Cw at y = 0,

(10)

u → 0, T → T∞ , C → C∞ as y → ∞,

(11)

where (u, v) are the velocity components along the (x, y) directions respectively, Uw is the velocity at the surface of the sheet, µ is the coefficient of variable viscosity, µ∞ the constant viscosity at the free stream, g the acceleration due to gravity, βt the coefficient of thermal expansion, βc the coefficient of expansion with concentration, T and C are the temperature and concentration respectively, D the thermal molecular diffusivity, k0 is the coefficient of viscoelasticity, σ is the electrical conductivity, Cp is the specific heat at constant pressure, k is the thermal conductivity, b the constant stretching rate, T∞ and ρ∞ are the free stream temperature and density.


18

3


METHOD OF SOLUTION

To examine the flow regime adjacent to the sheet, the following transformations are invoked

√

√

′

u = bxf (η); v = − bνf (η); η =

b T − T∞ C − C∞ y; θ(η) = ; ϕ(η) = ν Tw − T∞ Cw − C∞

(12)

where f is a dimensionless stream function, η is a similarity space variable, θ and ϕ are the dimensionless temperature and concentration respectively. Clearly, the continuity equation (6) is satisfied by u and v defined in equation (12). Substituting equation (12) in equations (7) - (9) gives

(

θ − θr θr

)

(

′2

f − ff

′′

)

(

+f

′′′

) ( ) ] θ′ θ − θr [ ′ ′′′ ′′ − f + K1 2f f − f f iv − f ′′2 θ − θr θr ( ) ( ) θ − θr θ − θr − (13) (Grθ + Gcϕ) + M f ′ = 0, θr θr

(

) θr (3N r + 4)θ + 3N rP rf θ − 3N rP rEc f ′′2 θ − θr ( ) −3K1 EcN rP r f ′ f ′′2 − f f ′′ f ′′′ = 0,

(14)

ϕ′′ + Scf ϕ′ = 0

(15)

′′

′

and the transformed boundary conditions are reduce to f ′ (η) = 1, f (η) = 0, θ(η) = 1, ϕ(η) = 1 at η = 0,

(16)

f ′ (η) → 0, f ′′ (η) → 0, θ(η) → 0, ϕ(η) → 0 as η → ∞,

(17)

where prime denotes differentiation with respect to η only. The dimensionless parameter θr = Tr −T∞ = −( γ(Tw1−T∞ ) ) is defined as the viscosity parameter, K1 = ρK∞0 νb the viscoelastic paTw −T∞ rameter, M =

σB02 ρ∞ b

the magnetic parameter, Pr =

µCp k

the Prandtl number, Ec =

2 Uw Cp (Tw −T∞ )

the

Eckert number, Gr = gβt (Tbw2 x−T∞ ) the local Grashof number, Gc = gβc (Cbw2 x−C∞ ) the local modi∗ µ fied Grashof number, N r = 4TkK 3 ∗ the thermal radiation parameter and Sc = ρ D the Schmidt ∞ ∞σ Int. J. of Appl. Math. and Mech. 8(14): 14-36, 2012.


19

number. Since the viscoelastic parameter K1 is very small, it is therefore, reasonable to seek a solution of equation (13) in the form

f = f0 (η) + K1 f1 (η) + O(K12 )

(18)

Substituting (18) in (13) and equating the like powers of K1 , we get ( f0′′′

+

( f1′′′

+

θ − θr θr

θ − θr θr

)

(

f0′2

−

f0 f0′′

)

( −

θ′ θ − θr

)

(

) f0′′

) θ − θr + M f0′ θr ( ) θ − θr − (Grθ + Gcϕ) = 0, θr

(

(2f0′ f1′

−

f0 f1′′

−

f0′′ f1 )

) θ′ − f1′′ θ − θr ( ) ) θ − θr ( ′ ′′′ + 2f0 f0 − f0 f0iv − f0′′2 θ (r ) θ − θr + M f1′ = 0, θr

(19)

(20)

Using (18), the boundary conditions for f0 and f1 from (16) and (17) can be written as f0 = 0, f0′ = 1 at η = 0; f0′ −→ 0, f0′′ −→ 0 as n −→ ∞

(21)

f1 = f1′ = 0 at η = 0; f1′ −→ 0, f1′′ −→ 0 as n −→ ∞.

(22)

The important characteristics of the present investigation are the local skin-friction coefficient Cf , the local Nusselt number N u and the local Sherwood number Sh defined by ] θr √ =− + 2K1 f ′′ (0), Cf = θ − θr µ∞ (bx) νb ( ) [ ] ∂u ∂ 2 u ∂u ∂u ∂ 2u τw = µ +v 2 + − k0 u , ∂y y=0 ∂x∂y ∂y ∂x ∂y y=0 τw

where

where

[

qw Nu = √ = −θ′ (0), k νb (Tw − T∞ ) √ ( ) ∂T b (Tw − T∞ )θ′ (0), qw = −k = −k ∂y y=0 ν


(23)

(24)

20


√

Sh = D ( where

4

mw = −D

∂C ∂y

)

mw b (Cw ν

= −D y=0

√

− C∞ )

= −ϕ′ (0),

b (Cw − C∞ )ϕ′ (0), ν

(25)

NUMERICAL METHODS

Several investigators (Abo-Eldahab 2001; Salem 2007; Prasad et al. 2010) and many others have used numerical technique for the solution of such two point boundary value problem is the fourth order Runge-Kutta scheme along with the shooting method. Although, this method provides satisfactory results, it may fail when applied to problems in which the differential equations are very sensitive to the choice of its missing initial conditions. Moreover, difficulty arises in the case in which one end of the range of integration is at infinity. In this case, the end point of integration is usually approximated by a large but finite number and it is obtained by estimating a value at which the solution will reach its asymptotic state. Owing to the above mentioned difficulties, we attempted to present a numerical method, which has better stability, simple, accurate and more efficient. The essential features of this technique is that it is based on a finite difference scheme. Finite difference technique leads to a system which is tridiagonal and therefore speedy convergence as well as economic in memory space to store the coefficients. We used Newton’s linearization method (Cebeci and Cousteix 1999) to linearize the discretized equations followed by a perturbation technique. In this method, we assumed that the values of the dependent variables at the k-th iteration are known. Then the values of the variables at the next iteration are obtained from the following equation

wik+1 = wik + δwik ,

(26)

where w stands for f0 and f1 and δwik represents the error at the k-th iteration. Using (26) in equations (19) and dropping quadratic terms in δwik , we get a tri-diagonal system. The resulting system of equations is then solved by Thomas algorithm. Since the equation (14) and (15) are second order linear differential equations, a simple finite difference method is used to solve them. We employed an iterative procedure with 10−6 as the maximum absolute error between two successive iterations. With an aim to test the accuracy of our numerical method we have compared the values of f ′ (η) (when M = N r = Gr = Gc = K1 = Sc = 0.0, θr → −∞) with the analytical results of Crane (Crane 1970) and have found excellent agreement. Int. J. of Appl. Math. and Mech. 8(14): 14-36, 2012.


21

1 0.9

Results of the present study

0.8 0.7

+++++

Results of Crane (1970)

0.6

f (η)

0.5 0.4 0.3 0.2 0.1 0 0

Fig. 1

1

η

2

3

4

5

6

7

Comparison of axial velocity profile with the analytical solution of Crane (1970) (M =Gr = Gc = Sc =P r = N r = K1 = 0.0 )

5 RESULTS AND DISCUSSION The system of ordinary non-linear and linear equations (13) - (15) subject to the boundary conditions (16) and (17) are solved numerically described in the previous section. In order to test the accuracy of our numerical solution, Fig. 1 shows the comparison of f ′ (η) with the analytical solution obtained by Crane (Crane 1970). For numerical solution it is necessary to assign some numerical values to the parameters involved in the problem under consideration. The present study has been carried out with an aim to examine the variations of different quantities of parameters in which M = 0, 1, 2, 4, 6 ; K1 = 0.01, 0.1, 0.2; P r = 1, 5, 10, 25; N r = 1, 3, 5, 10; θr = −1, −2, −5, −10; Gr = Gc = 1.0; Sc = 0.5, 1.0, 2.0, 2.5 and Ec = 0.02. Since the present study concerned with the cooling problem that is generally encountered in nuclear engineering, the value of P r is taken as 5.0. For the sake of comparison, we have also examined the cases where P r=1, 10 and 25. Figs. 2 - 7 give the distribution of the axial velocity f ′ (η) for different values of the magnetic parameter M , the thermal radiation parameter N r, the Prandatl number P r, the viscoelastic parameter K1 , the viscosity parameter θr and the Schmidt number Sc. The effect of the magnetic parameter on the axial velocity component is shown in Fig 2. It is observed that an increase in the magnetic parameter M gives rise to decrease in the velocity. This is due to the fact that by the application of transverse magnetic field in an electrically conducting fluid produce to a resistive force known as Lorentz force. This force has the tendency to slow down the motion of the fluid in the boundary layer. Fig. 3 shows that the axial velocity decreases as the thermal radiation parameter N r increases. Increasing the thermal radiation parameter N r produces a decrease in the fluid velocity. This may attribute to the fact that the increase of the values of N r Int. J. of Appl. Math. and Mech. 8(14): 14-36, 2012.

22


causes less interaction of radiation with the momentum boundary layer. The axial velocity also decreases with the increase of the Prandatl number P r, the viscoelastic parameter K1 as well as the Schmidt number Sc as shown in Figs. 4, 5 and 7. It is interesting to note from Fig. 5 that, the axial velocity decreases with the viscoelastic parameter K1 up to a certain height above the sheet, beyond which it increases. It seems that there would be a decrease of the boundary layer thickness in the presence of thermal radiation and the viscoelasticity of the fluid. These results may have greater importance in polymer processing, as the choice of the higher order viscoelastic fluid, which will reduce the power consumption for the stretching sheet. It reveals form Fig. 6 that the axial velocity f ′ (η) increases with the decrease of viscosity parameter θr .

1 0.9

Nr = Pr = 5.0, Sc = 2.0 Gr = Gc = 1.0, K 1= 0.01 θ r = −5.0

0.8 0.7 0.6 0.5 0.4

M = 0, 1, 2, 4, 6

0.3

f ( η)

0.2 0.1 0 0

1

η

2

3

4

Fig. 2 Variation of f ′ (η) with η for different values of M Int. J. of Appl. Math. and Mech. 8(14): 14-36, 2012.

5


23

1 0.9

M = 2, Pr = 5.0, Gr = Gc = 1.0 Sc = 2.0, K 1= 0.01, θ r = −5.0

0.8 0.7 0.6 0.5 0.4

Nr = 1, 3, 5, 10

0.3 0.2

1

f ( η)

0.1 0 0

1

η

2

3

4

5

Variation of f ′ (η) with η for different values of N r

Fig. 3

1 0.9

M = 2.0, Nr = 5.0, Gr = Gc = 1.0 Sc = 2.0, K 1= 0.01, θ = −5.0 r

0.8 0.7 0.6 0.5 0.4

Pr = 1, 5, 10, 25

0.3 0.2

1

f ( η)

0.1 0 0

1

η

2

3

4

Fig. 4 Variation of f ′ (η) with η for different values of P r


5

24

G. C. Shit and R. Haldar 1

M = 2, Pr = Nr = 5.0, Sc = 2.0 Gr = Gc = 1.0, θ r = −5.0

0.8

0.6

0.4

K1 = 0.01, 0.1, 0.2

0.2

1

f ( η) 0 0

1

η

2

3

4

5

Variation of f ′ (η) with η for different values of K1

Fig. 5

1 0.9

M = 2, Nr = Pr = 5, Sc = 2

0.8

Gr = Gc = 1.0, K 1= 0.01

0.7 0.6 0.5 0.4

θ r = −1, −2, −5, −10

0.3

1

f ( η)

0.2

θr

0.1 0 0

Fig. 6

1

η

2

3

4

Variation of f ′ (η) with η for different values of θr


5


25

1 0.9

M = 2, Nr = Pr = 5.0, K 1= 0.01 Gr = Gc = 1.0, θ r = −5.0

0.8 0.7 0.6 0.5 0.4 0.3

Sc = 0.5, 1, 2, 2.5

0.2

1

f ( η)

0.1 0 0

1

η

2

3

4

5

Fig. 7 Variation of f ′ (η) with η for different values of Sc

Figs. 8 - 13 illustrate the variation of dimensionless temperature θ(η) for different values of the physical parameters involved in the present study. Fig. 8 shows the variation of temperature with magnetic parameter M . It has been observed that under the action of an external magnetic field, the temperature of the fluid increases, while from Fig. 9 that the temperature decreases with the increase of the thermal radiation parameter N r. This may attribute to the fact that the increase of the values of N r causes decrease of the interaction with the radiation effect on the thermal boundary layer. Figs 10 and 11 noticed that the dimensionless temperature θ(η) decreases with the increase of the Prandatl number P r and increases with the increasing values of viscoelastic parameter K1 . It is interesting to note from Fig. 10 that the increase of Prandatl number P r means decrease of thermal conductivity. It may also observed from Fig. 12 that the effect of thermal radiation is to enhance the temperature with increase in the fluid viscosity parameter θr . This is lies in the fact that the increase of Prandatl number P r gives rise to a decrease of the thermal boundary layer thickness. It is interesting to note that in the presence of thermal radiation, the effect of viscoelastic parameter on temperature θ(η) causes marginal significance. The effect of the chemical species on temperature distribution shown in Fig. 13. From this figure we conclude that the temperature increases with the increase of the Schmidt numbers Sc. So the temperature can be enhanced with the introduction of heavier species diffusion.


26

G. C. Shit and R. Haldar 1 0.9

Nr = Pr = 5.0, Gr = Gc = 1.0 Sc = 2.0, K 1= 0.01, θ r = −5.0

0.8 0.7 0.6 0.5 0.4 0.3

θ(η)

M = 0, 1, 2, 4, 6

0.2 0.1

M 0 0

Fig. 8

1

2

η

3

4

5

Distribution of dimensionless temperature θ(η) for different values of M 1 0.9

M = 2, Pr = 5.0, Gr = Gc = 1.0 Sc = 2.0, K 1 = 0.01, θ r = −5.0

0.8 0.7 0.6 0.5 0.4

Nr = 1, 3, 5, 10 0.3 0.2

θ(η)

0.1 0 0

1

η

2

3

4

5

Fig. 9 Distribution of dimensionless temperature θ(η) for different values of N r



27

1 0.9

M = 2, Nr = 5, Gr = Gc = 1.0 Sc = 2.0, K 1= 0.01, θ r = −5.0

0.8 0.7 0.6 0.5 0.4

Pr = 1

0.3

Pr = 5, 10, 25

0.2

θ(η)

0.1 0 0

1

2

η

3

4

5

Fig. 10 Distribution of dimensionless temperature θ(η) with η for different values of P r 1 0.9

M = 2, Nr = Pr = 5.0, Sc = 2.0 Gr = Gc = 1.0, θ r = −5.0

0.8 0.7 0.6 0.5 0.4 0.3

K1 = 0.01, 0.1, 0.2

0.2

θ(η)

0.1

K1 0 0

1

2

3

4

5

η

Fig. 11 Distribution of dimensionless temperature θ(η) with η for different values of K1


28


M = 2, Gr = Gc = 1.0, Sc = 2.0 Nr = Pr = 5.0, K 1 = 0.01

0.8 0.7 0.6 0.5 0.4 0.3

θ r = −1, −2, −5, −10

0.2

θ(η)

0.1 0 0

1

2

η

3

4

5

Fig. 12 Distribution of dimensionless temperature θ(η) with η for different values of θr

1 0.9

M = 2, Nr = Pr = 5, Gr = Gc = 1 K1 = 0.01, θ r = −5.0

0.8 0.7 0.6 0.5 0.4 0.3 0.2

θ(η)

Sc = 0.5, 1, 2, 2.5

0.1

Sc 0 0

1

η

2

3

4

5

Fig. 13 Distribution of dimensionless temperature θ(η) with η for different values of Sc The effect of the imposition of various parameters on the concentration species are shown in Figs 14 - 19. It has been observed from Fig. 14 that under the action of a magnetic field, the concentration species has an enhancing effect. It reveals from Fig. 15 that the species Int. J. of Appl. Math. and Mech. 8(14): 14-36, 2012.


29

concentration increases when the thermal radiation parameters N r increases. It is interesting to note from Figs. 16 and 17 that the concentration increases with the increase of the Prandatl

1

Nr = Pr = 5.0, Gr = Gc = 1.0 Sc = 2.0, K 1= 0.01, θ r = −5.0

0.9 0.8 0.7 0.6 0.5 0.4

M = 0, 1, 2, 4, 6 0.3

φ(η)

0.2 0.1

M

0 0

1

η

2

3

4

5

Fig. 14 Distribution of species concentration ϕ(η) for different values of M 1 0.9

M = 2, Pr = 5, Gr = Gc = 1.0 Sc = 2.0, K 1 = 0.01, θ r = −5.0

0.8 0.7 0.6 0.5 0.4 0.3

φ(η)

0.2 0.1

Nr = 1, 3, 5, 10 0 0

1

η

2

3

4

5

Fig. 15 Distribution of species concentration ϕ(η) for different values of N r Int. J. of Appl. Math. and Mech. 8(14): 14-36, 2012.

30


M = 2, Nr = 5, Gr = Gc = 1.0 Sc = 2.0, K 1= 0.01, θ = −5.0 r

0.8 0.7 0.6 0.5 0.4 0.3 0.2

φ(η)

Pr = 1, 5, 10, 25

0.1 0 0

η

1

2

3

4

5

Variation of ϕ(η) with η for different values of P r

Fig. 16 1 0.9

M = 2, Nr = Pr = 5, Sc = 2 Gr = Gc = 1.0, θ r = −5.0

0.8 0.7 0.6 0.5 0.4 0.3

K1 = 0.01, 0.1, 0.2

0.2

φ(η)

0.1

K1

0 0

η

1

2

3

4

Fig. 17 Variation of ϕ(η) with η for different values of K1


5


31

1 0.9

M = 2, Nr = Pr = 5, Sc = 2 Gr = Gc = 1.0, K 1 = 0.01

0.8 0.7 0.6 0.5 0.4 0.3

θ r = −1. −2, −5, −10

0.2

φ(η)

0.1 0 0

1

2

η

3

4

5

Fig. 18 Variation of ϕ(η) with η for different values of θr

1 0.9

M = 2, Nr = Pr = 5 Gr = Gc = 1.0 K1 = 0.01, θ r = −5.0

0.8 0.7 0.6 0.5

Sc = 0.5, 1, 2, 2.5 0.4 0.3 0.2

φ(η)

0.1 0 0

η

1

2

3

4

5

Fig. 19 Variation of ϕ(η) with η for different values of Sc number P r and the viscoelastic parameter K1 . However, the variation of viscoelasticity on the concentration is insignificant. Again Figs. 18 and 19 illustrate that the species concentration decreases when the viscosity parameter θr as well as the Schmidt number Sc decreases. The Int. J. of Appl. Math. and Mech. 8(14): 14-36, 2012.

32


Table-1. Numerical values of the local skin-friction Cf = − Pr 5.0 10.0 25.0 5.0

Nr 5.0

K1 0.1

θr -2.0

Ec 0.02

Sc 2.0

0.1

-2.0

0.02

2.0

5.0

1.0 3.0 10.0 5.0

-2.0

0.02

2.0

5.0

5.0

0.01 0.1 0.2 0.1

0.02

2.0

5.0

5.0

0.1

-2.0 -5.0 -10.0 -2.0

0.02

1.0 2.0 5.0

[

θr θ−θr

] + 2K1 f ′′ (0)

M = 0.0 M = 2.0 1.086818 0.137371 1.073959 0.117509 1.027765 0.064592 1.057054 0.141902 1.085077 0.139329 1.086954 0.135329 -0.067367 -0.770393 1.086818 0.137371 1.340476 0.51277 1.086818 0.137371 1.415251 0.197166 1.564831 0.225918 0.840175 0.119884 1.086818 0.137371 1.233495 0.141796

M = 4.0 -0.365433 -0.399346 -0.460730 -0.343073 -0.360007 -0.370254 -1.239042 -0.365433 0.125414 -0.365433 -0.407288 -0.420902 -0.358801 -0.365433 -0.385960

explanation for such behaviour of concentration field lies in the fact that the decay of concentration from the stretching sheet to the surrounding fluid takes place with decreasing rate due to decreasing values of molecular diffusivity means increasing of Sc. The important characteristics in the present study are the local skin-friction coefficient Cf , the local rate of heat transfer at the sheet (Nusselt number N u) and the rate of mass transfer (the local Sherwood number Sh ) defined in equations (23) -(25). Tables 1-3 exhibit the numerical values to the local skin-friction, local Nusselt number and the local Sherwood number respectively. It has been observed empirically that for any particular values of P r, N r, K1 , θr , Ec and Sc the local skin-friction and the rate of heat and mass transfer decreases with the increase in the magnetic parameter M . It is interesting to note note that the skin-friction coefficient increase of the thermal radiation parameter N r and the Schmidt number Sc, when no magnetic field is applied. The skin-friction is also decreases with the increase of the Prandatl number P r and the viscosity parameter θr . But the reversal trend is observed in the presence of fluid viscoelasticity and the thermal radiation. It is worthwhile to mention here that the rate of heat transfer increases with the increasing values of P r, N r, θr . However, the heat transfer rate decreases with the increase of the viscoelasticity of the fluid in the presence of magnetic field, where as it is increases in the absence of magnetic field. The rate of mass transfer decreases significantly when the thermal radiation parameter N r and Prandatl number P r increases. The rate of heat transfer decreases with the increase of Schmidt number Sc, whereas the rate of concentration increases with the increase of Sc.

6 CONCLUSIONS In the present investigation we dealt with a study concerned with the thermal radiation effect on MHD viscoelastic fluid flow over a stretching sheet. The study also concerned with the free convective flow and mass transfer with temperature dependent viscosity and viscous dissipation effects. The main finding of this paper can be summerized as follows : Int. J. of Appl. Math. and Mech. 8(14): 14-36, 2012.


Table-2. Pr 5.0 10.0 25.0 5.0

Numerical Nr K1 5.0 0.1

5.0

1.0 3.0 10.0 5.0

5.0

5.0

0.01 0.1 0.2 0.1

5.0

5.0

0.1

values θr -2.0

0.1

-2.0

-2.0

-2.0 -5.0 -10.0 -2.0

33

of local Nusselt number N u = −θ′ (0) Ec Sc M = 0.0 M = 2.0 M = 4.0 0.02 2.0 0.605187 0.492944 0.429809 1.453229 1.258542 1.125252 2.094247 1.861715 1.647112 0.02 2.0 1.045914 0.880375 0.772498 1.354924 1.166545 1.038717 1.541859 1.341522 1.203540 0.02 2.0 1.446612 1.270911 1.146811 1.453329 1.258542 1.125252 1.460785 1.245265 1.102035 0.02 2.0 1.453329 1.258542 1.125252 1.459934 1.278581 1.152058 1.462591 1.286254 1.162331 0.02 1.0 1.485394 1.287787 1.149660 2.0 1.453329 1.258542 1.125252 5.0 1.415911 1.216807 1.085377

Table-3. Numerical values of local Sherwood number Sh = −ϕ′ (0) Pr 5.0 10.0 25.0 5.0

Nr 5.0

K1 0.1

θr -2.0

Ec 0.02

Sc 2.0

0.1

-2.0

0.02

2.0

5.0

1.0 3.0 10.0 5.0

-2.0

0.02

2.0

5.0

5.0

0.01 0.1 0.2 0.1

0.02

2.0

5.0

5.0

0.1

-2.0 -5.0 -10.0 -2.0

0.02

1.0 2.0 5.0


M = 0.0 1.053553 0.992389 0.977560 1.012186 0.996101 0.989479 0.990389 0.992339 0.994666 0.992389 0.999440 1.002169 0.677496 0.992389 1.637781

M = 2.0 0.891959 0.834297 0.817520 0.855581 0.838440 0.831022 0.845123 0.834299 0.822298 0.834299 0.851679 0.858333 0.553056 0.834299 1.447654

M = 4.0 0.787294 0.737661 0.720853 0.757527 0.741662 0.734450 0.753019 0.737661 0.720650 0.737661 0.759318 0.767703 0.484297 0.737661 1.322714

34


The velocity decreases with the increase of the magnetic parameter M , Prandatl number P r, thermal radiation parameter N r, viscoelastic parameter K1 and the species diffusion parameter Sc, while it increases with the increase of viscosity parameter θr . The combined effect of increasing the thermal radiation parameter N r and Prandatl number P r lead to the decrease in the thermal boundary layer thickness. However, the concentration boundary layer thickness can be reduced as a result of the increase in the thermal radiation parameter N r. It has also observed that increasing values of the Schmidt number Sc causes reduction in the concentration distribution in the boundary layer. Thus the present study will serve as a scientific tool for understanding more complex flow problems concerning with the various physical parameters. ACKNOWLEDGEMENT The authors are thankful to DST, New Delhi for supporting this investigation through DSTPURSE program of Jadavpur University.

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