Appl. Math. Mech. -Engl. Ed., 33(11), 1403–1418 (2012) DOI 10.1007/s10483-012-1632-6 c Shanghai University and Springer-Verlag Berlin Heidelberg 2012

Applied Mathematics and Mechanics (English Edition)

Dufour and Soret effects on MHD flow of viscous fluid between radially stretching sheets in porous medium∗ M. NAWAZ1 , T. HAYAT2,3 ,

A. ALSAEDI3

(1. Department of Humanities and Sciences, Institute of Space Technology, Islamabad 44000, Pakistan; 2. Department of Mathematics, Quaid-I-Azam University, Islamabad 44000, Pakistan; 3. Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia)

Abstract The aim of this paper is to examine the Dufour and Soret effects on the two-dimensional magnetohydrodynamic (MHD) steady flow of an electrically conducting viscous fluid bounded by infinite sheets. An incompressible viscous fluid fills the porous space. The mathematical analysis is performed in the presence of viscous dissipation, Joule heating, and a first-order chemical reaction. With suitable transformations, the governing partial differential equations through momentum, energy, and concentration laws are transformed into ordinary differential equations. The resulting equations are solved by the homotopy analysis method (HAM). The convergence of the series solutions is ensured. The effects of the emerging parameters, the skin friction coefficient, the Nusselt number, and the Sherwood number are analyzed on the dimensionless velocities, temperature, and concentration fields. Key words magnetohydrodynamic (MHD) flow, radial stretching, Soret and Dufour effects, porous medium, skin friction coefficient Chinese Library Classification O343.6 2010 Mathematics Subject Classification

1

74F05

Introduction

In practice, there are a number of situations where the boundary may not be stationary rather than moving with constant or variable velocities. Sakiadis[1–2] initiated the idea of the boundary layer flow caused by a surface moving with a constant velocity. Following Sakiadis’ work[1–2] , several investigators considered the boundary layer flow over a surface moving with a constant velocity. However, there may encounter physical situations where the surface velocity does no longer remain constant, especially in flows over stretchable and flexible materials. In such flows, the assumption of a constant surface velocity is not adequate, and the consideration of a constant surface velocity leads to erroneous results. Moving surfaces with variable velocities have been encountered in industry and engineering. Flows over stretching surfaces are the ∗ Received Dec. 23, 2011 / Revised Jul. 23, 2012 Project supported by the Deanship of Scientific Research (DSR) of King Abdulaziz University of Saudi Arabia (No. HiCi/40-3/1432H) Corresponding author M. NAWAZ, Ph. D., E-mail: nawaz [email protected]

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M. NAWAZ, T. HAYAT, and A. ALSAEDI

most prominent in the studies regarding variable surface velocities. Such considerations have a key role in aerodynamic extrusion of plastic sheets, cooling of an infinite plate in a cooling bath, liquid film in condensation process, continuous filament extrusion from a dye, and fluid dynamics of a long thread traveling between a feed roll and a wind-up roll. In view of these reasons, an extensive research regarding the flows induced by a stretching surface is available. Wang[3] studied the simultaneous effects of slip and suction in the flow of a viscous fluid over a stretching surface. Ishak et al.[4] analyzed the heat transfer effects in unsteady flows induced by stretching surfaces with prescribed wall temperature. Wang[5] discussed the natural convective flow by a vertical stretching surface. Hayat and Awais[6] considered the three-dimensional flow of a Maxwell fluid. They solved the resulting nonlinear problem by the homotopy analysis method (HAM). Ariel[7] studied the axisymmetric flow of a second-grade fluid over a radial stretching surface. Salem and Abd El-Aziz[8] investigated the Hall effects on heat and mass transfer in hydromagnetic flows of viscous fluids over a vertical stretching surface. Ishak et al.[9] discussed the axisymmetric flow of a viscous fluid over a cylinder stretching with a linear velocity. Cortell[10] analyzed the effect of viscous dissipation on the stretching flow of a viscoelastic fluid. Mukhopadhyay[11] investigated the effects of thermal radiation on mixed convection flows of viscoelastic fluids over a stretching surface immersed in a porous medium. Bataller[12] investigated the effects of source/sink, thermal radiation, and work done by deformation. Tasi et al.[13] considered the heat transfer characteristics in unsteady flows over a stretching surface. It has been experimentally verified that the diffusion of energy is caused by a composition gradient. This fact is known as the Dufour effect or the diffusion-thermo effect. The diffusion of species by a temperature gradient is termed as the Soret effect or the thermal diffusion effect. In most of the studies dealing with heat and mass transfer, these effects are neglected under the assumption that these are of smaller orders of magnitude described by Fourier’s and Fick’s laws. However, recent developments show that Dufour and Soret effects are significant when the transfer of heat and mass occurs in the flow of the mixture of gases with a very light molecular weight (e.g., H2 and He) and with a medium molecular weight (e.g., N2 and air). The thermal diffusion and diffusion thermo effects are explained on the separation of isotopes and gases from their mixture, and the formulae are derived for the thermal diffusion coefficient and the thermal diffusion factor for monoatomic or polyatomic gas mixtures. The effects cannot be ignored when the density difference exists. Osalusi et al.[14] considered the simultaneous effects of thermal diffussion, diffusion-thermo, viscous dissipation, and Joule heating in the magnetohydrodynamic (MHD) flow over a rotating disk. B´eg et al.[15] obtained the numerical solutions for the free convective flow induced by a stretching surface in the presence of Dufour and Soret effects. Tsai and Huang[16] analyzed Dufour and Soret effects on the Hiemenz flow over a stretching surface immersed in a porous medium. Afify[17] studied Dufour and Soret effects in heat and mass transfer in the flow induced by a stretching surface. It is also found that chemical reactions may affect (retard/enhance) the transport of chemical reacting species and must be included in the mass transfer analysis. El-Arabawy[18] derived the exact solution for the flow problem dealing with mass transfer in the presence of a chemical reaction. Hayat et al.[19] analyzed the effect of a chemical reaction in the flow of a Maxwell fluid past a shrinking surface. Mohamed et al.[20] investigated the effects of the heat generation and chemical reaction in the flow of a Maxwell fluid saturating in a porous medium. The flows induced by a stretching surface are a subject of great interest of several recent investigators. This is due to their wide applications in polymer processing, metallurgy, drawing of plastic sheets, cable coating, continuous casting, glass blowing, spinning synthetic fibers, etc. However, literature survey reveals that no attention has been given to the Dufour-Soret effects on the MHD steady laminar flow between two radially stretching sheets. The objective of the present work is to investigate the effects of Diffusion-thermo and thermal-diffusion on the steady laminar MHD flow driven by two radially stretching sheets in a porous medium. The first-order chemical reaction is also included. The governing problems are solved analytically

Dufour and Soret effects on MHD flow of viscous fluid between radially stretching sheets

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by the HAM[21–32] . Graphical results are reported and discussed.

2

Formulation of problem

In this section, we discuss the steady and axisymmetric flow of an electrically conducting fluid between two infinite parallel stretching sheets placed at z = ±L. The fluid fills the porous space. The flow is induced by the linear radial stretching of two sheets, and is considered to be symmetric about z = 0. A uniform magnetic field B0 perpendicular to the planes of sheets is applied in the z-direction. It is assumed that the magnetic Reynolds number is very small and the induced magnetic field is neglected. There is no external electric field. Both the sheets have a constant temperature Tw and a constant concentration field Cw . The physical model and coordinate system are given in Fig. 1.

Fig. 1

Schematic representation of physical model

The velocity field is defined as follows: V = (u(r, z), 0, w(r, z)).

(1)

By virtue of the above definition of velocities, the governing equations are ∂u u ∂w + + = 0, ∂r r ∂z

(2)

u

 ∂ 2 u 1 ∂u ∂ 2 u ∂u 1 ∂p νϕm  ∂u u   σB02 +w =− +ν + + u, − + − ∂r ∂z ρ ∂r ∂r2 r ∂r ∂z 2 r2 ρ k

(3)

u

 ∂ 2 w 1 ∂w ∂ 2 w  νϕ ∂w 1 ∂p ∂w m +w =− +ν + w, − + ∂r ∂z ρ ∂z ∂r2 r ∂r ∂z 2 k

(4)

u

∂T K  ∂2T ∂ 2 T  DKT  ∂ 2 C ∂2C  1 ∂T 1 ∂C ∂T +w = + + + + + ∂r ∂z ρcp ∂r2 r ∂r ∂z 2 ρCs cp ∂r2 r ∂r ∂z 2 +

u

 ∂w 2  ∂u 2 μ   ∂u 2 u2  σB02 2 2 u + +2 + +2 2 , ρcp ρcp ∂r ∂z ∂z r

 ∂ 2C ∂C ∂ 2 C  DKT  ∂ 2 T ∂2T  1 ∂C 1 ∂T ∂C +w =D + + + − K1 C, + + ∂r ∂z ∂r2 r ∂r ∂z 2 Tm ∂r2 r ∂r ∂z 2

(5)

(6)

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M. NAWAZ, T. HAYAT, and A. ALSAEDI

where T designates the temperature field, C is the concentration field, ρ is the density, ν is the kinematic viscosity, cp is the specific heat, σ is the electrical conductivity of the fluid, p is the pressure, ϕm is the porosity, K is the thermal conductivity, D is the coefficient of mass diffusivity, Cs is the concentration susceptibility, Tm is the mean fluid temperature, KT is the thermal-diffusion ratio, K1 is the chemical reaction constant, and k is the permeability of porous medium. The boundary conditions corresponding to the flow under consideration are ⎧ ∂u ∂T ∂C ⎪ ⎨ = 0, w = 0, = 0, = 0 at z = 0, ∂z ∂z ∂z (7) ⎪ ⎩ u = ar, w = 0, T = Tw , C = Cw at z = L, a > 0. To facilitate the computational solutions, we define ⎧  ⎪ ⎨ u = raF (η), w = −2aLF (η), T ⎪ ⎩θ = , Tw

φ=

C , Cw

η=

(8)

z . L

Now, Eqs. (2)–(7) can be reduced to 

F  (η) − Re(Ha + ϕ)F  (η) + 2ReF (η)F  (η) = 0, F (0) = 0,

F (1) = 0,

F  (1) = 1,

F  (0) = 0,

⎧    2 ⎪ ⎪ θ (η) + 2ReP rF (η)θ (η) + ReHaP rEc(F (η)) ⎪ ⎪ ⎨   1 + P rEc (F  (η))2 + (F  (η))2 + DuP rφ (η) = 0, ⎪ δ ⎪ ⎪ ⎪ ⎩  θ (0) = 0, θ(1) = 1, 

(9)

(10)

φ (η) + 2ScReF (η)φ (η) + ScSrθ (η) − ScReγφ(η) = 0, φ (0) = 0,

(11)

φ(1) = 1,

where ⎧ aL2 σB02 ⎪ ⎪ Re = , Ha = , ⎪ ⎪ ν ρa ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ DKT Cw , Sc = Du = 2 ⎪ L aCs cp Tw ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ νϕm a2 r2 ⎪ ⎪ ⎩ Ec = , , ϕ= cp T w ka

Pr =

μcp , K

aL2 , D

Sr =

γ=

K1 , a

DKT Tw , L2 aTm Cw

δ=

r2 , L2

respectively, denote the Reynolds number, the Hartman number, the Prandtl number, the Dufour number, the Schmidt number, the Soret number, the local Eckert number, the porosity parameter, the first-order chemical reaction parameter, and the dimensionless radial distance. The dimensionless parameters Du and Sr correspond to the Dufour effect and the Soret effect, respectively. It is evident from the expressions of the Dufour number and the Soret number that they are arbitrary constants, provided that their product remains to be constant. This fact is stated in the attempts[14–17] .

Dufour and Soret effects on MHD flow of viscous fluid between radially stretching sheets

1407

The skin friction coefficient CF , the Nusselt number Nu, and the Sherwood number Sh are defined by the definitions given as follows: ⎧

 ∂w  ⎪ μ ∂u τrz |z=L 1  ⎪ ∂z + ∂r z=L ⎪ CF = = = F (1), ⎪ 2 2 ⎪ ρ(ar) ρ(ar) Re ⎪ r ⎪ ⎪ ⎪ ⎪  ⎨  LK ∂T Lqw ∂z z=L (12) Nu = − = − = −θ (1), ⎪ KT KT ⎪ w w ⎪ ⎪ ⎪  ⎪ ⎪  ⎪ LD ∂C LMw ⎪ ∂z z=L ⎪ ⎩ Sh = − =− = −φ (1), DCw DCw in which Rer = arL/ν is the local Reynolds number.

3

Solutions by HAM F (η), θ(η), and φ(η) in the forms of the base functions 2n+1

2n

η ,n0 , η ,n0

(13)

can be written as F (η) =

∞ 

an,n η

2n+1

,

θ(η) =

n=0

∞ 

2n

bn,n η ,

φ(η) =

n=0

∞ 

cn,n η 2n ,

(14)

n=0

in which an,n , bn,n , and cn,n are coefficients to be determined. The initial guesses F0 (η), θ0 (η), and φ0 (η) and the linear operators L1 , L2 , and L3 are chosen as follows: F0 (η) =

1 η(η 2 − 1), 2

L1 (F (η)) =

d4 F , dη 4

θ0 (η) = η 2 , L2 (θ(η)) =

φ0 (η) = η 2 ,

d2 θ , dη 2

L3 (φ(η)) =

(15) d2 φ , dη 2

(16)

where 

L1 (C1 + C2 η + C3 η 2 + C4 η 3 ) = 0, L2 (C4 + C5 η) = 0,

L3 (C6 + C7 η) = 0,

and Ci (i = 1, 2, · · · , 7) are constants. 3.1 Zeroth-order formation problems The zeroth-order deformation problems are constructed to be ⎧ (1 − q)L1 (Φ(η; q) − F0 (η)) = q1 N1 (Φ(η; q)), ⎪ ⎪ ⎨ (1 − q)L2 (Ψ(η; q) − θ0 (η)) = q2 N2 (Ψ(η; q)), ⎪ ⎪ ⎩ (1 − q)L3 (Λ(η; q) − θ0 (η)) = q3 N3 (Λ(η; q)), ⎧   ∂Φ(η; q)  ∂ 2 Φ(η; q)  ⎪ ⎪ Φ(0; q) = 0, Φ(1; q) = 0, = 1, = 0, ⎪ ⎪ ⎨ ∂η η=1 ∂η 2 η=0   ⎪ ⎪ ∂Λ(η; q)  ∂Ψ(η; q)  ⎪ ⎪ = 0, Λ(1; q) = 1, = 0. ⎩ Ψ(1; q) = 1, ∂η η=0 ∂η η=0

(17)

(18)

(19)

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M. NAWAZ, T. HAYAT, and A. ALSAEDI

In the above expressions, q ∈ [0, 1] and i = 0 (i = 1, 2, 3) are, respectively, the embedding parameter and the auxiliary parameter, and  Φ(η; 0) = F0 (η), Ψ(η; 0) = θ0 (η), Λ(η; 0) = φ0 (η), Φ(η; 1) = F (η),

Ψ(η; 1) = θ(η),

Λ(η; 1) = φ(η).

When q varies from 0 to 1, Φ(η; q) varies from the initial guess F0 (η) to F (η), Ψ(η; q) varies from the initial guess θ0 (η) to θ(η), and Λ(η; q) varies from the initial guess φ0 (η) to φ(η). The nonlinear operators N1 , N2 , and N3 are given as follows: N1 (Φ(η; q)) =

∂ 4 Φ(η; q) ∂ 2 Φ(η; q) ∂ 3 Φ(η; q) − Re(Ha + ϕ) + 2ReΦ(η; q) , ∂η 4 ∂η 2 ∂η 3

N2 (Ψ(η; q), Φ(η; q)) =

N3 (Λ(η; q)) =

∂2Λ ∂2Ψ ∂Ψ(η; q) + DuP r + 2ReP rΦ(η; q) ∂η 2 ∂η ∂η 2  1  ∂Φ(η; q) 2  ∂ 2 Φ(η; q) 2  + P rEc + δ ∂η ∂η 2  ∂Φ(η; q) 2 , + P rReEcHa ∂η

∂2Ψ ∂2Λ ∂Λ(η; q) Φ(η; q) + ScSr 2 − ScReγΛ(η). + 2ScRe 2 ∂η ∂η ∂η

In view of Taylor’s power series, one can write ⎧ ∞  ⎪ ⎪ ⎪ Φ(η; q) = F (η) + Fm (η)q m , 0 ⎪ ⎪ ⎪ ⎪ m=1 ⎪ ⎪ ⎪ ⎪ ∞ ⎨  Ψ(η; q) = θ0 (η) + θm (η)q m , ⎪ ⎪ m=1 ⎪ ⎪ ⎪ ⎪ ∞ ⎪  ⎪ ⎪ ⎪ ⎪ Λ(η; q) = φ (η) + φm (η)q m , 0 ⎩

(20)

(21) (22)

(23)

m=1

in which

 ⎧ 1 ∂ m Φ(η; q)  ⎪ ⎪ Fm (η) = , ⎪ ⎪ m! ∂η m q=0 ⎪ ⎪ ⎪ ⎪  ⎪ ⎨ 1 ∂ m Ψ(η; q)  θm (η) = , m! ∂η m q=0 ⎪ ⎪ ⎪ ⎪  ⎪ ⎪ ⎪ 1 ∂ m Λ(η; q)  ⎪ ⎪ φ (η) = . ⎩ m m! ∂η m q=0

3.2 Higher order deformation problems We write ⎧ Fn (η) = {F0 (η), F1 (η), F2 (η), F3 (η), · · · , Fm (η)}, ⎪ ⎪ ⎪ ⎨ θn (η) = {θ0 (η), θ1 (η), θ2 (η), θ3 (η), · · · , θm (η)}, ⎪ ⎪ ⎪ ⎩ φn (η) = {φ0 (η), φ1 (η), φ2 (η), φ3 (η), · · · , φm (η)}.

(24)

(25)

Dufour and Soret effects on MHD flow of viscous fluid between radially stretching sheets

Then, the so called mth-order deformation problems are  L1 (Fm (η) − χm Fm−1 (η)) = 1 R1m (Fm−1 (η)), 



Fm (0) = 0,

 Fm (1) = 0,

Fm (1) = 0,

(26)

 Fm (0) = 0,

L2 (θm (η) − χm θm−1 (η)) = 2 R2m (θm−1 (η)),  θm (0) = 0,

(27)

θm (1) = 0,

L3 (φm (η) − χm φm−1 (η)) = 3 R2m (φm−1 (η)), φm (0) = 0,

1409

(28)

φm (1) = 0,

  R1m (Fm−1 (η)) = Fm−1 (η) − Re(Ha + ϕ)Fm−1 (η)

+ 2Re

m−1 

 Fn (η)Fm−1−n (η),

(29)

n=0  R2m (θm−1 (η)) = θm−1 (η) + DuP rφm−1 (η) + 2ReP r

m−1 

 Fn (η)θm−1−n (η)

n=0

+ P rEc

m−1 

 Fn (η)Fm−1−n (η) +

n=0

+ HaP rReEc

m−1 

 12   (η) Fn (η)Fm−1−n δ

 Fn (η)Fm−1−n (η),

(30)

n=0  (η) − SReφcγm−1 (η) R3m (φm−1 (η)) = φm−1 (η) + ScSrθm−1

+ 2ScRe

m−1 

Fn (η)φm−1−n (η),

(31)

n=0

where

 χm =

0,

m  1,

1,

m > 1.

It is found that (26)–(28) have the following general solutions: F (η) = F ∗ (η) + C1m + C2m η + C3m η 2 + C4m η 3 ,

(32)

θ(η) = θ∗ (η) + C5m + C6m η,

(33)

φ(η) = φ∗ (η) + C6m + C7m η,

(34)

where F ∗ (η), θ∗ (η), and φ∗ (η) are the corresponding particular solutions. The convergence and the rate of approximations of the series solutions (32)–(34) strongly depend upon the values of the auxiliary parameters. For this purpose, the i -curves (i = 1, 2, 3) are plotted through Fig. 2. This figure shows that the admissible ranges for i are −1.2  1  −0.5,

−1.1  2  −0.7,

−1  3  −0.3.

However, the whole forthcoming calculations are done when 1 = 2 = 3 = −0.8.

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M. NAWAZ, T. HAYAT, and A. ALSAEDI

Fig. 2

i -curves (i = 1, 2, 3) of velocity field for 27th-order approximations with different parameters: (a) Ha, ϕ = 3 and Re = 1; (b) Ha, ϕ = 3, Re = 1, and P r, Ec, Sc, Sr, Du, γ = 0.5; (c) Ha, ϕ = 3 and Re, P r, Ec, Sc, Sr, Du, γ = 0.5

To ensure the convergence of the solutions, Table 1 is constructed. From this table, it is evident that the convergence is achieved at the 18th-order of approximations up to the 9th decimal places. Table 1

Convergence of HAM solutions for different orders of approximations when Ha = ϕ = 3, Re, Sc = 1, Sr, Du, P r, Ec, γ = 0.2, and δ = 12

Order of approximation

F  (1)

θ  (1)

φ (1)

1

4.051 428 571

−0.257 066 666 7

0.346 666 666 7

4

5

4.147 342 744

−0.159 328 803 5

0.238 522 896 4

10

4.147 439 247

−0.160 754 270 4

0.241 296 901 9

15

4.147 439 270

−0.160 756 306 3

0.241 300 937 7

20

4.147 439 270

−0.160 756 309 1

0.241 300 943 4

25

4.147 439 270

−0.160 756 309 1

0.241 300 943 4

30

4.147 439 270

−0.160 756 309 1

0.241 300 943 4

Results and discussion

The governing nonlinear problems given in Eqs. (9)–(11) are solved by the HAM. To get clear insight into the problem, the velocity and the temperature and concentration fields are displayed for various values of dimensionless parameters. The skin friction coefficient, the Nusselt number, and the Sherwood number are tabulated and analyzed. Here, Figs. 3–8 are sketched to see the effects of the Hartman number Ha, the Reynolds number Re, and the porosity parameter ϕ on the dimensionless velocities F  (η) and F (η). The effects of the Reynolds number Re, the porosity parameter ϕ, the Prandtl number P r, the Schmidt number Sc, the Dufour number Du,

Dufour and Soret effects on MHD flow of viscous fluid between radially stretching sheets

1411

the Soret number Sr, the local Eckert number Ec, the Hartman number Ha, and the first-order chemical reaction parameter γ on the dimensionless temperature θ(η) are shown in Figs. 9–17. Figures 18–25 depict the variation of dimensionless concentration field φ(η) for different values of the Reynolds number Re, the porosity parameter ϕ, the Prandtl number P r, the Schmidt number Sc, the Dufour number Du, the Soret number Sr, the local Eckert number Ec, the first-order chemical reaction parameter γ, and the Hartman number Ha.

Fig. 3

Effects of Ha on F  (η) with Re, ϕ = 2.0

Fig. 4

Effects of Re on F  (η) with Ha, ϕ = 2.0

Fig. 5

Effects of ϕ on F  (η) with Ha, Re = 2.0

Fig. 6

Effects of Ha on F (η) with Re, ϕ = 2.0

Fig. 7

Effects of Re on F (η) with Ha, ϕ = 2.0

Fig. 8

Effects of ϕ on F (η) with Ha, Re = 2.0

It is clear from Fig. 3 that the magnitude of the radial velocity F  (η) decreases when the Hartman number Ha increases. The term −ReHaF  (η) in Eq. (9) is due to the Lorentz force. The negative sign shows that the r-component of the Lorentz force is a drag like force. Since an increase in Ha corresponds to an increase in the magnetic field strength. This increase in the magnetic field increases the magnitude of the Lorentz force (opposing force), which results in a decrease in the magnitude of the velocity (see Fig. 3). The Reynolds number is the ratio of the inertial force to the viscous force. An increase in the Reynolds number Re indicates a decrease in the viscous force (a decrease in the viscosity). Since a less viscous fluid is not significantly influenced by the stretching of sheets, the velocity decreases with an increase in Re (see Fig. 4).

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M. NAWAZ, T. HAYAT, and A. ALSAEDI

The term −ReϕF  (η) in Eq. (9) is due to the porous medium. An increase in the porosity parameter ϕ corresponds to an increase in the resistance by the porous medium to the flow. Consequently, the velocity decreases. The magnitude of the radial velocity F  (η) decreases by increasing the porosity parameter ϕ (see Fig. 5). Figures 3–5 depict that the boundary layer thickness decreases when Ha, Re, and ϕ increase. Figure 6 elucidates that the magnitude of the axial velocity F (η) is a decreasing function of the Hartman number Ha. This is due to the fact that the Lorentz force is dragged like force and opposes to the flow. It is found from Fig. 7 that the magnitude of the axial velocity F (η) decreases by increasing the Reynold number Re. The porosity parameter ϕ is the coefficient of the term −ReHaF  (η) due to the porous medium. An increase in the porosity parameter ϕ leads to an increase in the magnitude of the resistance. Consequently, the axial velocity F (η) decreases (see Fig. 8).

Fig. 9

Effects of Re on θ(η) with Ha, ϕ = 4.0, P r, Ec, Sc, Sr, Du, γ = 0.2, and δ = 12.0

Fig. 11

Fig. 12

Fig. 10

Effects of ϕ on θ(η) with Ha = 3.0, Re = 2.0, P r, Ec, Sc, Sr, Du, γ = 0.2, and δ = 12.0

Effects of P r on θ(η) with Ha, ϕ = 3.0, Re = 2.0, Ec, Sc, Sr, Du, γ = 0.2, and δ = 12.0

Effects of Sc on θ(η) with Ha, ϕ = 3.0, Re = 2.0, P r, Ec, Sr, Du, γ = 0.2, and δ = 12.0

Fig. 13

Effects of Du and Sr on θ(η) with P r, Sc, Ec, γ = 0.2, Ha, ϕ = 3.0, Re = 2.0, and δ = 12.0

Dufour and Soret effects on MHD flow of viscous fluid between radially stretching sheets

1413

Fig. 14

Effects of Ec on θ(η) with Ha, ϕ = 3.0, Re = 2.0, P r, Sc, Sr, Du, γ = 0.2, and δ = 12.0

Fig. 15

Effects of Ha on θ(η) with Re, ϕ = 3.0, P r, Ec, Sc, Sr, Du, γ = 0.2, and δ = 12.0

Fig. 16

Effects of γ ( 0) on θ(η) with δ = 12.0, Ha, ϕ = 3.0, Re = 2.0, and P r, Sc, Sr, Ec, Du = 0.2

Fig. 17

Effects of γ ( 0) on θ(η) with δ = 12.0, Ha, ϕ = 3.0, Re = 2.0, and P r, Sc, Sr, Ec, Du = 0.2

Figure 9 represents that the dimensionless temperature θ(η) decreases by increasing the Reynolds number Re. Since the porous medium in the flow regime absorbs some heat from the fluid, θ(η) decreases (see Fig. 10). The effects of the Prandtl number P r and the Schamidt number Sc are given in Figs. 11 and 12. These figures show that θ(η) is an increase function of P r and Sc. The available studies[14–17] on the Dufour and Soret effects show that Du and Sr are arbitrary constants, which provides that their product is constant. Therefore, in Fig. 13, Du and Sr vary in such a way that their product is constant. Figure 13 shows that θ(η) increases when Du increases and Sr decreases. An increase in Du causes a concentration gradient, and this concentration gradient plays an important role in the transportation of the heat energy from the solid boundary into the fluid, which results in an increase in the temperature. Since the Eckert number is the ratio of the kinetic energy to the enthalpy, an increase in the Eckert number indicates an increase in the kinetic energy. Since temperature is the average kinetic energy of fluid particles, θ(η) increases with an increase in the Eckert number (see Fig. 14). When the electrical current passes through a medium, some of the electrical energy is converted into heat energy which is called Joule heating. The expression ReHaP rEc(F  (η))2 in the energy equation is a Joule heating term. Thus, an increase in Ha corresponds to an increase in the intensity of the magnetic field. Thus, more heat dissipates and adds to the system. Hence, θ(η) increases. This fact is displayed in Fig. 15. The effect of the chemical reaction parameter γ is shown in Figs. 16 and 17. Here, γ > 0 corresponds to the destructive chemical reaction, whereas γ < 0 is the case when the generative chemical reaction occurs in the flow regime. Here, γ = 0 represents the situation when no chemical reaction occures in the flow regime. Figure 16 reveals that θ(η) increases when γ > 0 (the destructive chemical reaction). However, the dimensionless temperature θ(η) decreases for the case of the generative

1414

M. NAWAZ, T. HAYAT, and A. ALSAEDI

Fig. 18

Effects of Re on φ(η) with Ha, ϕ = 3.0, P r, Ec, Sc, Sr, Du, γ = 0.2, and δ = 12.0

Fig. 19

Effects of ϕ ( 0) on φ(η) with Ha = 3.0, Re = 2.0, P r, Ec, Sc, Sr, Du, γ = 0.2, and δ = 12.0

Fig. 20

Effects of P r on φ(η) with Ha = 3.0, Re = 2.0, Ec, Sc, Sr, Du, γ = 0.2, and δ = 12.0

Fig. 21

Effects of Sc on φ(η) with Ha = 3.0, Re = 2.0, P r, Ec, Sr, Du, γ = 0.2, and δ = 12.0

Fig. 22

Effects of Du and Sr on φ(η) with P r, Ec, γ = 0.2, Ha = 3.0, Re = 2.0, and δ = 12.0

Fig. 23

Effects of Ec on φ(η) with Ha = 3.0, Re = 2.0, P r, Sc, Sr, Du, γ = 0.2, and δ = 12.0

Fig. 24

Effects of γ ( 0) on φ(η) with Ha = 3.0, Re = 2.0, P r, Sc, Sr, Ec, Du = 0.2, and δ = 12.0

Fig. 25

Effects of γ ( 0) on φ(η) with Ha = 3.0, Re = 2.0, P r, Sc, Sr, Ec, Du = 0.2, and δ = 12.0

Dufour and Soret effects on MHD flow of viscous fluid between radially stretching sheets

1415

chemical reaction (γ < 0) (see Fig. 17). Hence, one can conclude that generative and destructive chemical reactions have opposite effects on θ(η). The dimensionless concentration field decreases with an increase in Re (see Fig. 18), whereas it increases when the porosity parameter increases (see Fig. 19). From Figs. 18–25, we can see that the dimensionless concentration field φ(η) increases with the increases in P r, Sc, Du, Sr, Re, and Ec while decreases with the increases in γ (> 0, destructive case; < 0, generative case) and ϕ. The comparison of Figs. 9–17 with Figs. 18–25 indicates that the effects of Re, ϕ, P r, Sc, Du, Sr, and Ec on the dimensionless temperature θ(η) are quite opposite to those of Re, ϕ, P r, Sc, Du, Sr, and Ec on the dimensionless concentration field φ(η). Moreover, Figs. 17, 18, 24, and 25 indicate that the effect of γ on θ(η) is quite opposite to that of γ on φ(η). Table 2 shows the variation of the skin friction coefficient Rer CF . From this table, it is obvious that the skin friction coefficient Rer CF is an increasing function of Re, Ha, and ϕ. It means that the stresses on the surface of sheet increase by increasing the magnetic field strength and the resistance from the porous medium. Table 2

Variation of skin friction coefficient CF for different values of physical parameters

Re

Ha

ϕ

Rer CF

0.0

3.0

3.0

3.000 000 000

1.0

3.0

3.0

4.147 439 270

2.0

3.0

3.0

5.061 978 173

3.0

3.0

3.0

5.834 166 961

1.0

0.0

3.0

3.669 001 162

1.0

2.0

3.0

3.994 635 796

1.0

4.0

3.0

4.294 439 034

1.0

6.0

3.0

4.573 114 144

2.0

6.0

0.0

3.669 001 162

2.0

6.0

2.0

3.994 635 796

2.0

6.0

4.0

4.294 439 034

2.0

6.0

6.0

4.573 114 144

Table 3 is prepared for the effects of physical parameters on the Nusselt number Nu and the Sherwood number Sh. This table indicates that the Nusselt number Nu and the Sherwood number Sh are increasing functions of Re, Ha, Sc, Sr, P r, Ec, and γ. Hence, heat and mass fluxes increase by increasing Re, Ha, Sc, Sr, P r, Ec, and γ. Table 3

Variation of Nu and Sr for different values of Re, Ha, ϕ, Du, Sc, P r, Ec, and γ with δ = 12

Re

Ha

ϕ

Du

Sc

Sr

Pr

Ec

γ

−Nu

Sh

0.5

3.0

3.0

0.2

1.0

0.2

0.2

0.2

0.2

0.144 792 951 500

0.131 848 038 80

1.0

3.0

3.0

0.2

1.0

0.2

0.2

0.2

0.2

0.160 756 309 100

0.241 300 943 40

1.5

3.0

3.0

0.2

1.0

0.2

0.2

0.2

0.2

0.176 459 847 500

0.351 919 346 50

2.0

3.0

3.0

0.2

1.0

0.2

0.2

0.2

0.2

0.191 715 854 200

0.462 229 141 00

1.0

0.0

3.0

0.2

1.0

0.2

0.2

0.2

0.2

0.139 839 530 900

0.238 731 701 70

1.0

2.0

3.0

0.2

1.0

0.2

0.2

0.2

0.2

0.154 104 476 000

0.240 471 433 20

1.0

4.0

3.0

0.2

1.0

0.2

0.2

0.2

0.2

0.167 131 850 800

0.242 106 744 30

1.0

6.0

3.0

0.2

1.0

0.2

0.2

0.2

0.2

0.179 158 683 400

0.243 656 850 30

1416

M. NAWAZ, T. HAYAT, and A. ALSAEDI (Continuation of Table 3)

Re

Ha

ϕ

Du

Sc

Sr

Pr

Ec

γ

1.0

3.0

0.0

0.2

0.2

0.2

0.2

0.2

0.2

0.152 184 945 900

0.046 533 549 85

1.0

3.0

2.0

0.2

0.2

0.2

0.2

0.2

0.2

0.152 369 121 700

0.046 492 900 40

1.0

3.0

4.0

0.2

0.2

0.2

0.2

0.2

0.2

0.153 425 715 300

0.046 494 134 74

1.0

3.0

6.0

0.2

0.2

0.2

0.2

0.2

0.2

0.155 055 331 100

0.046 523 705 09

1.0

3.0

3.0

0.5

0.2

1.0

0.2

0.2

0.2

0.191 762 628 400

0.402 298 308 10

1.0

3.0

3.0

1.0

0.2

0.5

0.2

0.2

0.2

0.215 650 381 700

0.318 054 138 10

1.0

3.0

3.0

1.5

0.2

0.333

0.2

0.2

0.2

0.239 510 769 400

0.289 882 512 40

1.0

3.0

3.0

0.2

0.1

0.2

0.2

0.2

0.2

0.160 102 028 100

0.225 066 859 50

1.0

3.0

3.0

0.2

1.1

0.2

0.2

0.2

0.2

0.166 894 146 500

0.393 596 616 80

1.0

3.0

3.0

0.2

2.1

0.2

0.2

0.2

0.2

0.174 286 712 100

0.577 032 706 90

1.0

3.0

3.0

0.2

3.1

0.2

0.2

0.2

0.2

0.182 362 818 300

0.777 439 101 30

1.0

3.0

3.0

0.2

1.0

0.0

0.2

0.2

0.2

0.159 453 041 200

0.208 964 196 10

1.0

3.0

3.0

0.2

1.0

1.0

0.2

0.2

0.2

0.166 189 164 700

0.376 103 908 90

1.0

3.0

3.0

0.2

1.0

2.0

0.2

0.2

0.2

0.173 518 181 400

0.557 962 325 30

1.0

3.0

3.0

0.2

1.0

3.0

0.2

0.2

0.2

0.181 521 765 700

0.756 568 157 80

1.0

3.0

3.0

0.2

1.0

0.2

0.12

0.2

0.2

0.095 829 051 950

0.228 240 394 60

1.0

3.0

3.0

0.2

1.0

0.2

0.42

0.2

0.2

0.343 765 846 900

0.278 115 775 50

1.0

3.0

3.0

0.2

1.0

0.2

0.72

0.2

0.2

0.604 458 576 600

0.330 560 510 90

1.0

3.0

3.0

0.2

1.0

0.2

1.07

0.2

0.2

0.879 016 085 400

0.385 797 624 50

1.0

3.0

3.0

0.2

0.2

0.2

0.2

0.0

0.2

0.008 614 151 716

0.210 756 857 80

1.0

3.0

3.0

0.2

0.2

0.2

0.2

0.4

0.2

0.312 898 466 400

0.271 845 028 90

1.0

3.0

3.0

0.2

0.2

0.2

0.2

0.8

0.2

0.617 182 781 200

0.332 933 200 00

1.0

3.0

3.0

0.2

0.2

0.2

0.2

1.2

0.2

0.921 467 095 900

0.394 021 371 10

1.0

3.0

3.0

0.2

1.0

0.2

0.2

0.2

−1.0

0.073 919 134 840

−1.878 323 928 00

1.0

3.0

3.0

0.2

1.0

0.2

0.2

0.2

−0.5

0.123 435 597 100

−0.670 800 126 30

1.0

3.0

3.0

0.2

1.0

0.2

0.2

0.2

0.0

0.152 185 445 700

0.031 599 395 81

1.0

3.0

3.0

0.2

1.0

0.2

0.2

0.2

0.5

0.171 623 896 800

0.507 463 373 20

1.0

3.0

3.0

0.2

1.0

0.2

0.2

0.2

1.0

0.186 066 459 600

0.861 747 119 90

5

−Nu

Sh

Conclusions

In this investigation, we have discussed the thermal-diffusion (Soret effect) and diffusionthermo (Dufour effect) in the steady flow of a viscous fluid between two radially stretching sheets. The viscous dissipation, Joule heating, and first-order chemical reactions are considered. The main points of the presented analysis are given as follows: (i) Variations of Re and ϕ on F  (η) and F (η) are similar to that of Ha. (ii) There are opposite effects of Re, P r, Sc, Du, Sr, Ec, γ, and ϕ on θ(η) and φ(η). (iii) The effects of Ha on F  (η) and F (η) are different from those on θ(η). (iv) There is no significant effect of Ha on φ(η). (v) Qualitatively, the effects of Re, Ha, and ϕ on the skin friction coefficient CF are similar while the shear stress on the surface of stretching sheet increase when Re, Ha, and ϕ increase. (vi) Variations of Re, Ha, ϕ, Du, Sc, Sr, P r, Ec, and γ on the Nusselt number Nu and the Sherwood number Sh are similar. Moreover, the Nusselt number Nu and the Sherwood

Dufour and Soret effects on MHD flow of viscous fluid between radially stretching sheets

1417

number Sh are increasing functions of Re, Ha, Sc, Sr, P r, Ec, and γ. Hence, heat and mass fluxes increase by increasing Re, Ha , Sc, Sr, P r, Ec, and γ. Acknowledgments

The authors are grateful to the reviewers for their useful suggestions.

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