Numerical study of MHD heat and mass transfer of a Jeffrey fluid over a stretching sheet with chemical reaction and thermal radiation

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Numerical study of MHD heat and mass transfer of a Jeffrey fluid over a stretching sheet with chemical reaction and thermal radiation P.V. Satya Narayana∗, D. Harish Babu Fluid Dynamics Division, SAS, VIT University, Vellore 632014, Tamil Nadu, India

a r t i c l e

i n f o

Article history: Received 26 March 2015 Revised 8 July 2015 Accepted 14 July 2015 Available online xxx Keywords: Jeffrey fluid Stretching sheet Chemical reaction Thermal radiation Power law form of temperature and concentration

a b s t r a c t A numerical model is developed to study the effects of chemical reaction and heat source on MHD heat and mass transfer of an electrically conducting Jeffrey fluid over a stretching sheet in the presence of power law form of temperature and concentration. Similarity transformations are used to convert the governing partial differential equations to a set of coupled non-linear ordinary differential equations. The resulting equations are then solved numerically by shooting method with Runge–Kutta fourth order scheme. The influence of various dimensionless parameters on the velocity, temperature and concentration distributions are analyzed and discussed through graphs and tables. It is observed that the Deborah number (β ) and ratio of relaxation and retardation times parameter (λ) have opposite effects on the skin friction coefficient. However, the effects of β and Pr on the Nusselt number profiles are similar. Subsequently the present results are in very good agreement with the results obtained for a viscous fluid. © 2015 Taiwan Institute of Chemical Engineers. Published by Elsevier B.V. All rights reserved.

1. Introduction Boundary layer flow heat and mass transfer over a stretching sheet has gained considerable attention because of its applications in industry and manufacturing processes. Such applications include polymer extrusion, drawing of copper wires, continuous stretching of plastic films, hot rolling and cooling of an infinite metallic plate in a cooling bath etc. Heat transfer is important because the rate of cooling has a vast influence on the quality of the final product. Sakiadis [1] was the first to introduce the concept of boundary layer flow over a continuous solid surface. Crane [2] provided the classical solution for the boundary layer flow of a viscous fluid over a sheet moving with velocity varying linearly with distance from a fixed point. Later, the stretching sheet flow has been considered by the several researchers in different configurations. Ariel [3] studied the problem of boundary layer flow of a viscous fluid by a stretching sheet via homotopy perturbation method. Merkin and Kumaran [4] studied the problem of unsteady magnetohydrodynamic boundary layer flow over a shrinking sheet. Ashorynejad et al. [5] investigated the flow and heat transfer of a nanofluid due to a stretching cylinder in the presence of magnetic field. Sheikholeslami et al. [6] presented the computational study on the natural convection of nanofluids in a concentric annulus between a cold outer square and heated inner circular cylinder in the presence of a static radial magnetic field. Mukhopadhyay [7]

∗

Corresponding author. Tel.: +91 9789574488. E-mail address: [email protected] (P.V. Satya Narayana).

investigated the unsteady mixed convection flow of an incompressible viscous fluid toward a vertical permeable stretching sheet in a porous medium. Akbar et al. [8] investigated the two-dimensional stagnation point flow of an incompressible nanofluid towards a stretching surface with convective boundary condition. Fluid flow and radiative nonlinear heat transfer over a stretching sheet has been investigated by Cortell [9]. Akbar et al. [10] studied theoretically, the influence of magnetic field on two dimensional stagnation-point flow of a Prandtl fluid model in the presence of shrinking sheet. Sheikholeslami et al. [11] analyzed the MHD viscous and heat transfer flow between two horizontal plates in a rotating system by using HAM. All the above researchers however limited their investigation to the Newtonian fluid flows. The analysis of non-Newtonian fluids is significant because of several industrial and engineering applications due to the flexibility of fluid characteristics in nature [12–17]. There are many practical applications such as drilling muds, electronic chips, crystal growing, cosmetic products, dilute polymer solutions, food processing and movement of biological fluids. In view of different rheological properties of non-Newtonian fluids in nature, there are no single constitutive relationships between stress and rate of strain by which all the non-Newtonian fluids can be examined. Amongst non-Newtonian fluids the Jeffrey model is one of the simplest types of model to account for rheological effects of viscoelastic fluids. The Jeffrey model is a relatively simple linear model using the time derivatives instead of converted derivatives. Jeffrey’s model contains three constants: the zero shear rate viscosity and two time constants. We can add terms containing second, third and higher derivatives of the stress and rate of strain tensors with

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Please cite this article as: P.V. Satya Narayana, D. Harish Babu, Numerical study of MHD heat and mass transfer of a Jeffrey fluid over a stretching sheet with chemical reaction and thermal radiation, Journal of the Taiwan Institute of Chemical Engineers (2015), http://dx.doi.org/10.1016/j.jtice.2015.07.014

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Nomenclature A1 , A2 B0 C Cf cp Cw C∞ D F f K Ks Kr∗ l m M Nu Pr qr R R1 S Sc Sh T Tw T∞ Uw u, v x y

constants magnetic induction [T] concentration [kmol/m3 ] skin-friction co-efficient specific heat at constant pressure [J/kg/K] the species concentration at wall [kmol/m3 ] species concentration far from the wall [kmol/m3 ] diffusion coefficient [m2 /s] non-dimensional stream function dimensionless velocity fluid thermal conductivity [W/m/K] Rosseland mean absorption coefficient chemical reaction parameter characteristic length surface temperature parameter magnetic parameter Nusselt number Prandtl number radiative heat flux[W/m] radiation parameter Rivlin–Ericksen tensor extra stress tensor Schmidt number Sherwood number fluid temperature (K) temperature at the wall (K) temperature far away from the wall (K) shrinking velocity [m/s] velocity components in the x-, y-directions, respectively [m/s] distance along the wall [m] distance normal to the wall [m]

Greek symbols β Deborah number γ heat source/sink parameter η similarity variable λ ratio of relaxation and retardation times λ1 relaxation time [s] μ dynamic viscosity [Pa/s] υ kinematic viscosity [m2 /s] φ non-dimensional concentration ρ fluid density [kg/m] σ electric conductivity σ∗ Stefan–Boltzmann constant θ non-dimensional temperature τ Cauchy stress tensor Subscripts w sheet surface ∞ infinity Superscript ʹ differentiation with respect to η

appropriate multiplicative constants, to get a still more general linear among the stress and the rate of strain tensors. Khan et al. [18] analyzed the steady laminar flow of an incompressible non-Newtonian Powell–Eyring fluid over a rotating disk. Bhaskar Reddy et al. [19] have examined the flow of a Jeffrey fluid between torsionally oscillating disks. MHD three-dimensional boundary layer flow of an incompressible Jeffrey nanofluid by a bidirectional stretching surface is conferred by Hayat et al [20]. Abd-Alla et al. [21] investigated the in-

fluence of magnetic field and rotation on peristaltic transport of a Jeffrey fluid in an asymmetric channel. Kothandapani and Srinivas [22] studied the peristaltic transport of a Jeffrey fluid under the effect of magnetic field in an asymmetric channel. Qasim [23] investigated the heat and mass transfer in a Jeffrey fluid over a stretching sheet with heat source. Cortell [24] studied the MHD flow and heat transfer with radiation effects of a viscoelastic fluid over a stretching sheet with heat generation using non-linear approximation. Recently, Nadeem et al. [25] examined the two dimensional MHD boundarylayer flow and the heat transfer of a Maxwell fluid past a stretching sheet. Akbar et al. [26] studied the characteristics of Jeffrey fluid model for peristaltic flow of chime in small intestine with magnetic field. Few studies concerning Jeffrey fluid model over various geometries are stated in the references [27–30]. The study of heat and mass transfer with chemical reaction is of great practical importance in many branches of science and engineering such as, in damage of crops due to freezing, evaporation at the surface of a water body, energy transfer in a wet cooling tower and chemical engineering processes [31]. Anjalidevi and Kandasamy [32] analyzed the effects of chemical reaction, heat and mass transfer on laminar flow along a semi-infinite horizontal plate. Hayat et al. [33] presented the chemical reaction and radiation effects on threedimensional boundary layer flow of a second grade nanofluid past a stretching surface with heat source/ sink. Abel et al. [34] studied the effects of thermal buoyancy and variable thermal conductivity on the MHD flow and heat transfer in a power-law fluid past a vertical stretching sheet in the presence of a non-uniform heat source. Recently, Swapna et al. [35] have investigated the heat and mass transfer flow of a chemically reacting mixed convection MHD micropolar fluid with a convective surface boundary condition. Shehzad et al. [36] analyzed the influence of nonlinear thermal radiation in three-dimensional flow of Jeffrey nanofluid. Shehzad et al. [37] studied the thermophoresis particle deposition in mixed convection three-dimensional radiative flow of an Oldroyd-B fluid. More recently, Venkateswarlu and Satya Narayana [38] presented the analytical study for the flow and heat transfer of a nanofluid in a rotating system with chemical reaction. Sheikholeslami et al. [39] studied the effect of variable magnetic field on free convection heat transfer in an enclosure filled with Fe3 O4 –water nanofluid by using control volume based finite element method. They found that the rate of heat transfer is a cumulative function of Rayleigh number, nanoparticle volume fraction and magnetic number. Many recent investigators [40–51] have addressed the problem by considering features such as micropolar fluid, nanofluid, chemical reaction, thermal radiation, viscous dissipation etc. in different flow fields. For many practical claims the continuous moving surface undergoes stretching and cooling or heating that causes surface velocity, temperature and concentration variations. This motivates the present study to explore the effects of chemical reaction and heat source on MHD electrically conducting Jeffrey fluid over a continuous stretching sheet with the power law form of temperature and concentram m tion distributions (T = Tw = T∞ + A1 ( xl ) , C = Cw = C∞ + A2 ( xl ) ). The problem addressed here is a fundamental one that arises in many practical situations such as polymer extrusion process. The research work done in this direction is scant. So the objective of this study is to investigate the combined effects of heat and mass transfer in Jeffrey fluid over a stretching sheet in the presence of thermal radiation and chemical reaction. The coupled non-linear partial differential equations governing the flow field are converted to a system of non-linear ordinary differential equations by using suitable transformations. The resulting equations are then solved numerically by using fourth-order Runge–Kutta integration scheme. The organization of the remnants of the paper is as follows. In Section 2, we describe the model with its governing equations and boundary conditions. Here, we also describe the numerical method briefly. In Section 3, we present results and discussion. Finally, in Section 4, we

Please cite this article as: P.V. Satya Narayana, D. Harish Babu, Numerical study of MHD heat and mass transfer of a Jeffrey fluid over a stretching sheet with chemical reaction and thermal radiation, Journal of the Taiwan Institute of Chemical Engineers (2015), http://dx.doi.org/10.1016/j.jtice.2015.07.014

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3

The following boundary conditions on velocity are appropriate in order to employ the effect of stretching of the boundary surface causing flow in x-direction as

y

u = Uw (x) = cx,

v = 0 at y = 0

(3)

u → 0, u → 0 as y → ∞

T ,C

Uw

Tw , Cw

where c is the proportionality constant of the stretching sheet velocity, Uw (x) is the velocity of stretching surface. To solve the governing boundary layer Eq. (2), the following similarity transformations are introduced:

cx

o

x



η=

Fig. 1. Schematic diagram of the physical model and coordinate system.

summarize our results and present our conclusions. A comparison is made with the available results in the literature, and salient features of the new results are analyzed and discussed.

2.1. Flow analysis We consider a steady two-dimensional incompressible, electrically conducting Jeffrey fluid over a linear stretching sheet in the presence of chemical reaction, thermal radiation and heat source. The origin fixed as revealed in Fig. 1. The x-axis is taken in the direction along the sheet and y-axis is taken normal to it. The flow is generated, due to linear stretching of the sheet, caused by simultaneous application of two equal and opposite forces along the x-axis. Further, a uniform strength of magnetic field B0 is applied transversally to the direction of the flow. The imposition of this magnetic field stabilizes the boundary layer flow. It is also assumed that the induced magnetic field due to the motion of electrically conducting fluid is negligible. The temperature and the species concentration have power index m variations with the distance from the origin. At t = 0, the sheet is impetuously stretched with the variable velocity Uw (x). The essential equations for Jeffrey fluid can be written as [52]

v = − cυ f (η)

(4)

here η is the similarity variable and f (η) is the dimensionless stream function. Substitution of Eq. (4) in Eq. (2) results in a fourth-order non-linear ordinary differential equation of the following form:

f  + (1 + λ)( f f  − f 2 ) + β( f 2 − f f  ) − (1 + λ)M f  = 0

(5)

Here β = λ1 c is the Deborah number and M = ρ c is the Hartmann number. In view of the transformations, Eq. (3) takes the following nondimensional form:

f (η) = 0, f  (η) = 1 at η = 0 f  (η) = 0, f  (η) = 0 as η → ∞

Re1/2 x Cf =

 1   f (0) + β f  (0) . 1+λ

R1 = (∇ V ) + (∇ V ) Under these assumptions, the governing equations of continuity and momentum [17,23,53] take the following form: (1)

(2)

where u, v are the velocity components in the x and y directions, respectively. υ is the kinematic viscosity, ρ is the density of the fluid,B0 is the transverse magnetic field strength, σ is the electrically conductivity of the fluid, λ is the ratio of relaxation and retardation times and λ1 is the relaxation time.

(7)

2.2. Heat transfer By using usual boundary layer approximations, the equation of the energy in the presence of internal heat generation/absorption and thermal radiation is given by

ρ cp

where S is the extra stress tensor, τ is the Cauchy stress tensor, μ is the dynamic viscosity, λ and λ1 are the material parameters of Jeffrey fluid and R1 is the Rivlin–Ericksen tensor defined by

(6)

The physical quantities of interest are the skin-friction coefficient C f , which is defined as



τ = −pI + S,     μ ∂ R1 + V.∇ R1 R1 + λ1 S= 1+λ ∂t

  2 ∂u υ ∂ 3u ∂u ∂ 3u ∂ u +v = u + λ1 u +v 3 ∂x ∂ y 1 + λ ∂ y2 ∂ x∂ y 2 ∂y  ∂ u ∂ 2u ∂ u ∂ 2u σ B0 2 − + − u 2 ∂x ∂y ∂ y ∂ x∂ y ρ

υ

√

y, u = cx f  (η),

σ B20

2. Mathematical formulations

∂ u ∂v + =0 ∂x ∂y

c

∂T ∂T +υ u ∂t ∂y



=k

∂ 2 T ∂ qr − Q (T − T∞ ) − ∂y ∂ y2

(8)

where c p is the specific heat and k is the thermal conductivity, T is the temperature of the fluid, T∞ is the constant temperature of the fluid far away from the sheet. To solve the thermal boundary layer Eq. (8),we consider nonisothermal temperature boundary condition as

T = Tw = T∞ + A1

x m

T → T∞ as y → ∞

l

at y = 0 (9)

where A1 is the constant depends on the properties of fluid, Tw is the stretching sheet temperature m is the surface temperature parameter and l is the characteristic length. By using the Rosseland diffusion approximation the radiative heat flux, qr is given by (also see Refs. [54,55])

qr = −

4σ ∗ ∂ T 4 3Ks ∂ y

(10)

where Ks and σ ∗ are the Rosseland mean absorption coefficient and the Stefan–Boltzmann constant, respectively. We assume that the temperature differences within the flow are sufficiently small such that T4 may be expressed as a linear function of temperature. 3 4 T 4 ≈ 4T∞ T − 3T∞

(11)

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Using Eqs. (10) and (11) in the last term of Eq. (8) we get 3 ∂ qr ∂ 2T 16σ ∗ T∞ =− ∂y 3Ks ∂ y2

2.4. Numerical procedure (12)

We introduce a dimensionless temperature variable θ (η) of the form

θ (η) =

T − T∞ Tw − T∞

(13)

Substituting Eqs. (12) and (13) in Eq. (8) we obtain nondimensional thermal boundary layer equation as



[m5G;July 30, 2015;19:35]

1+

4 3

R

θ  + Pr( f θ  − m f  θ + γ θ ) = 0

where R =

3 4σ ∗ T∞ Ks k

number and γ =

A = f, B = f  , C = f  , D = f  , D = f  ; E = θ , F = θ  , F  = θ  ;

(14)

is the radiation parameter, Pr = Qυ ρcp

The system of non-linear ordinary differential equations (5), (14) and (20) under the boundary conditions (6), (15) and (21) is solved numerically by fourth order Runge–Kutta integration scheme with shooting technique. Before we apply this technique to the present problem, the coupled ODEs (5), (14) and (20), which is fourth order in f and second order in θ and φ are reduce to the system of eight simultaneous equations of initial order eight unknowns are as follows:

ρcp k

G = φ , H = φ  , H =

is the Prandtl

is the heat source/sink parameter.

Substituting Eq. (23) in Eqs. (5), (14) and (20), we get

D + (1 + λ)(AC − B2 − MB) + β(C 2 − AD ) = 0

The boundary conditions (9) become



θ (η) = 1 at η = 0 θ (η) = 0 as η → ∞

1+

(15)

4 Nux Re−1/2 = −θ  (0) 1 + R . x

(16)

3

A = B

The concentration species diffusion of the laminar boundarylayer flow with homogeneous first-order chemical reaction can be written as

B = C

(17)

where D is diffusion coefficient, Kr∗ is the chemical reaction parameter Cw and C∞ are species concentrations at the wall and far away from the wall, respectively. We consider non-isothermal concentration boundary condition as

x m

C → C∞ as y → ∞

l

(18)

where A2 is the constant that depends on the properties of fluid. We introduce a dimensionless concentration variable φ(η) of the form

C − C∞ φ(η) = Cw − C∞

(19)

Using Eqs. (4) and (19) in Eq. (17), we obtain ordinary differential equation as follows:

φ  + Sc( f φ  − m f  φ − Krφ) = 0

(20) Kr∗ δ 2

ν

is the chemical

(21)

The physical quantities of Sherwood number which is defined as

Sh Re−1/2 = −φ  (0) x

C = D D =





 1 2 1 + λ C − MB + B2 − AC ; A β

(27)

E = F F =

3 Pr

3 + 4R

(MBE − γ E − AF );

(28)

H  = Sc(mBG + KrG − AH );

(29)

where prime denotes differentiation with respective to η. The boundary conditions become

A = 0, B = 1, C = S1 , D = S2 , E = 1, F = S3 , G = 1, H = S4 at η = 0 B → ∞, F → 0, G → 0 as η → ∞

(30)

To solve Eqs. (27)–(29) with (30) as an IVP the values for D(0) i.e. f  (0), F(0) i.e. θ  (0) and H (0) i.e. φ  (0) are required; however no such values are given. Once all the eight initial conditions are determined then we solve this system of coincidental equations using fourth order Runge–Kutta integration. 3. Results and discussion

φ(η) = 1 at η = 0 φ(η) = 0 as η → ∞

(26)

G = H

at y = 0,

where Sc = υD is the Schmidt number and Kr = reaction parameter. The boundary conditions (18) become

(25)

The coupled nonlinear boundary value problem has been reduced to a system of eight simultaneous equations of first-order for eight unknowns as follows:

2.3. Mass transfer

∂C ∂ 2C ∂C +v = D 2 − Kr∗ (C − C∞ ) u ∂x ∂y ∂y

(24)



4 R F  + Pr(AF − mBE + γ E ) = 0 3

H  + Sc(AH − mBG − KrG) = 0

The local Nusselt number which is defined as

C = Cw = C∞ + A2

(23)

φ  ;

(22)

Particular case: It is worth mentioning that the present problem reduces to a regular viscous (see Ref. [53]) fluid if we choose β = λ = 0 in Eq. (5). We noticed that in the absence of magnetic fluid parameter, radiation parameter and chemical reaction parameter Eqs. (2)–(4) reduce to those of Qasim [23].

The distributions of the velocity f  (η), temperature θ (η), concentration φ(η), skin friction at the surface, Nusselt and Sherwood numbers are shown in Figs. 2–16. In order to verify the accuracy of the present model, we have compared our results with Chen [53] for the rate heat transfer with different values of m and Pr. The comparisons are found to be in good agreement, as shown in Table 1. It is also observed that increasing Pr and m is to increase the local Nusselt number. The effect of magnetic flied parameter M on the velocity profile f  (η) with η is shown in Fig. 2 for both (λ = 0.0) and (λ = 1.0) cases,

Please cite this article as: P.V. Satya Narayana, D. Harish Babu, Numerical study of MHD heat and mass transfer of a Jeffrey fluid over a stretching sheet with chemical reaction and thermal radiation, Journal of the Taiwan Institute of Chemical Engineers (2015), http://dx.doi.org/10.1016/j.jtice.2015.07.014

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Pr=.75;Sc=0.6;R=0.1;m=2.0; =0.1;Kr =0.2; =1.0

Pr=.70;Sc=0.7; =1.0;R=0.1;m=2.0; =0.1;Kr =0.2;M=1

f' ( )

f' ( )

M=5.0 M=3.0

5

M=6.0

=1.0 1.5 2.0 2.5

M=4.0

(a)

Fig. 2. Effect of M on f  . Table 1 Comparison of local Nusselt number −θ  (0) for various values of m, Pr when R = λ = 0,β = 0,M = Kr = 0.

Pr = 1

Pr = 10

−1.00003 0.58199 – 1.33334

−10.0047 2.30796 – 4.79686

−1.0000 0.5820 1.0000 1.3333

−10.0000 2.3080 3.7208 4.7969

respectively. It is observed that the velocity profiles decrease with an increase in the magnetic parameter M. This is due to the fact that, the application of transverse magnetic field has a tendency to give rise to a resistive type force called the Lorentz force and hence results in retarding the velocity profile. Further we observed that the increase of λ causes the reduction of boundary layer velocity of fluid. Physically, λ is inversely proportional to the retardation time of the non-Newtonian fluid. Hence, a rise in λ means a decrease in fluid retardation time which in result stops the rushing of fluid motion. The influence of Deborah number β on the fluid velocity, temperature, and concentration is depicted in Fig. 3(a)–(c) respectively. It is noticed from Fig. 3(a) that the fluid velocity profile increases with increasing values of β . Physically, Deborah number β is proportional to the rate of stretching sheet (β = λ1 c), the increase of β results in a higher fluid motion in the boundary layer especially adjacent to the surface of sheet. This higher fluid motion increases the thickness of the hydrodynamic boundary layer and consequently raises the fluid velocity. From Fig. 3(b) and (c) it is apparent that both temperature and concentration profiles decrease with an increasing β . Physically β is a proportional to retardation time, hence retardation time is increased when β increases. This increase in retardation time corresponds to the lower temperature and weaker thermal boundary layer thickness. Fig. 4 illustrates the effect of R,λ and β on temperature profile. The results show that temperature distributions increase with increase in the value of R. This is due to fact that thermal boundary layer thickness increases with an increase in thermal radiation. This implies that the radiation should be minimized to have the cooling process proceed at a faster rate. It is also observed that the temperature slightly rises with the increase of β in the vicinity of the sheet surface. This is due to enormous motions of the fluid particles which in effect thicken the thermal boundary layer. Nevertheless, it can be noticed that both λ and β do not have much effect on the temperature in the neighbourhood of the sheet surface (i.e. η = 0). Fig. 5 demonstrates the influence of magnetic flied parameter on temperature profiles. It is observed that the field temperature increases with increasing M values. Physically, the magnetic flied tends to retard the velocity flied which in turn induces the temperature field resulting in an increase of the temperature profiles. The magnetic field can therefore be used to control the flow characteristics.

( )

Present Pr = 10

= 0.5 1.0 1.5 2.0

(b)

( )

−2 0 1 2

Chen[55] Pr = 1

= 0.5 1.0 1.5 2.0

(c) Fig. 3. (a) Effect of Deborah number β on f  . (b) Effect of Deborah number β on θ . (c) Effect of Deborah number β on φ .

Pr=1.0;Sc=0.6;M=0.2; m=2.0; =0.1;Kr =0.2

( )

m

Pr=1.0;Sc=0.3; =1.0;M=0.6;R=0.1; m=2.0; =0.1;Kr =0.2

R=2.0 R=1.5 R=1.0 R=0.5

Fig. 4. Effects of R, λ and β on θ .

Fig. 6 shows the temperature profiles for different values of Prandtl number Pr. It is obvious from the figure that the temperature profile decreases with an increasing values of Pr. Physically, the reduction in temperature is due to thermal diffusivity because Pr depends on thermal diffusivity. Larger Prandtl number corresponds to weaker thermal diffusivity which tends to the lower temperature and thinner thermal boundary layer thickness. These results are similar to those of viscous boundary layer (see Ref. [53]).

Please cite this article as: P.V. Satya Narayana, D. Harish Babu, Numerical study of MHD heat and mass transfer of a Jeffrey fluid over a stretching sheet with chemical reaction and thermal radiation, Journal of the Taiwan Institute of Chemical Engineers (2015), http://dx.doi.org/10.1016/j.jtice.2015.07.014

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Pr =1.0;Sc=0.7; =0.6; =2;R=0.3; m=1.0; =0.4;Kr =0.2

Pr =0.70;Sc=0.7;M=2;m=2.0; =0.1;R =0.1; =1.0

( )

( )

Kr =1.0 M=0.0 0.1 0.2 0.3

Kr =0.5 Kr =0.0

Kr =1.5

Fig. 8. Effect of Kr on φ .

( )

( )

Fig. 5. Effect of magnetic field M on θ .

Pr =1.0 1.5 2.0 2.5

Fig. 6. Effect of Prandtl number Pr on θ .

Sc = 0.5 1.0 1.5 2.0

Fig. 9. Effect of Sc on φ .

1

0.6

( )

( )

=1.0;Sc=0.7; =1.0;M=0.6;R=0.2; =0.1; Pr =1.0;Kr =0.2

=0.2;Sc=0.3; =1.0;M=0.4;R=0.2; m=2.0; Pr =0.90;Kr =0.2

0.8

= 0.1 0.2 0.4 0.5

m = 0.5 1.0 1.5 2.0

0.4 0.2 = -0.1 -0.5 -1.0 -1.5 0

0.5

1

1.5

2

2.5

3

3.5

4

Fig. 7. Effect of heat source/sink parameter γ on θ .

Fig. 7 illustrates the effects of heat source parameter (γ > 0) and heat sink parameter (γ < 0) on temperature profiles. It is clear that heat source gives an increase in the temperature of the fluid. Physically, the increase of heat source in the boundary layer generates energy which causes the temperature of the fluid to increase. Whereas, heat sink provides a decrease in the temperature of the fluid. It is interesting to note that, the presence of heat sink in the boundary layer absorbs energy which results the temperature of the fluid to decrease. It is also important to note that heat sink is better suited for effective cooling of stretching sheet. Fig. 8 shows the influence of chemical reaction parameter (Kr) and the ratio of relation and retardation times (λ = 0.0, 1.0) on the concentration profiles. From the figure we see that φ(η) decreases as Kr as well as λ increases. Physically, increasing the chemical reaction parameter produces a decrease in the species concentration and its boundary layer. Fig. 9 displays the effect of Schmidt number Sc on concentration profiles. Physically, the increase of Sc means decreases of molecular diffusion. Hence, the concentration of the species is higher for small values of Sc and lower for larger values of Sc. Therefore, as the Schmidt number increases the concentration decreases. This causes the concentration buoyancy effects to decrease; consequently there is a reduction in the fluid velocity.

Fig. 10. Effect of surface temperature parameter m on θ .

( )

0

m = 1.0 2.0 3.0 4.0

Fig. 11. Effect of m on φ .

Fig. 10 is a plot of temperature distribution with η for various values of surface temperature parameter m. It can be seen that the temperature θ (η)decreases with an increase in the values of m. This is because, the fluid flow is caused by stretching of the sheet and stretching sheet temperature is greater than the free stream temperature (i.e. Tw > T∞ ), hence the temperature decreases with increasing η and the boundary layer thickness decreases with increase in m. It is also noted from Fig. 11 that the concentration profiles decrease with increase in wall concentration parameter m.

Please cite this article as: P.V. Satya Narayana, D. Harish Babu, Numerical study of MHD heat and mass transfer of a Jeffrey fluid over a stretching sheet with chemical reaction and thermal radiation, Journal of the Taiwan Institute of Chemical Engineers (2015), http://dx.doi.org/10.1016/j.jtice.2015.07.014

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R=0.70;M=0.5;m=2.0; =3; Sc=0.710;Pr =0.6;Kr =0.1

7

'

f''(0)

(0)

=0.2 0.5 0.7 0.9

R=0.70;M=0.5;m=2.0; =3; Sc=0.710;Pr =0.6; =0.1

=0.2 0.5 0.7 0.9

Pr Fig. 12. Variation of skin friction coefficient against λ for different values of β .

Fig. 15. Variation of Sherwood number against Pr for different values of β .

R=0.70; =0.5;m=2.0; =3; Sc=0.710;Pr =0.6;Kr =0.1

'

(0)

f''(0)

=0.2 0.5 0.7 0.9

=0.2 0.3 0.4 0.5 R=0.70;M=0.5;m=2.0; =3; Sc=0.710;Pr =0.6;Kr =0.1

M Fig. 16. Variation of skin friction coefficent against M for different values of β . Fig. 13. Variation of Nusselt number against λ for different values of β .

ically, increase in β implies a decrease in yield stress of the Jeffrey fluid and increase in the value of plastic dynamic viscosity, this effect creates resistance in the flow of fluid.

R=2.0 2.5 3.0 3.5

'

(0)

4. Conclusions

=0.1;M=0.5;m=2.0; =3; Sc=0.710; =0.6;Kr =0.1

Pr Fig. 14. Variation of Nusselt number against Pr for different values of R.

Figs. 12 and 13 illustrate the variation of local skin friction coefficient and the Nusselt number with λ for various values of β . It is observed that the skin friction coefficient decreases with increase of β . This is due to the fact that the increase of β leads to higher motion of fluid particles inside the boundary layer, in particular, the vicinity of the sheet surface. Hence the velocity boundary layer thickness reduces which results in lower values of skin friction coefficient. On the other hand the skin friction coefficient increases with increase of λ. Physically, increase of λ increases the non-Newtonian fluid nature and hence decreases the velocity of the fluid near the sheet surface. Therefore the thickness of hydrodynamic boundary layer increases which causes the local skin friction coefficient to increases. It is also observed that the variation of Nusselt number with β and λ is exactly opposite that of skin friction (see Ref. [16]). This is because the increase of λ adds to the thickness of thermal boundary layer and as a result, the Nusselt number reduces. Figs. 14 and 15 present the variations of the local Nusselt number and Sherwood number, with Prandtl number Pr for various values of R and β . It is observed that the Nusselt number (Nu) and Sherwood number (Sh) increases for large values of R, β and Pr. It is noticed from Fig. 16 that the local skin friction co-efficient f  (0) decreases with increase the magnitude of parameter β . Phys-

Two dimensional boundary layer MHD flow of Jeffrey fluid with chemical reaction and thermal radiation in the presence of power law form of temperature and concentration are numerically discussed in this article. The main points of present analysis are: 1. The skin friction coefficient decreases and rate of heat transfer increase with an increase inβ . 2. The fluid velocity inside the boundary layer increases with the rise ofβ , while with the increase of λ the fluid velocity reduces. 3. The thermal and concentration boundary layer thickness are decreasing function of m. 4. The present finding for the case of Newtonian fluid can be recovered by choosing β = 0. The present work may be extended for other important subclasses of non-Newtonian fluids such as Maxwell fluid, Oldroyd-B fluid, Eyring-Powell fluid and many others. 5. Increasing chemical reaction parameter is to decrease concentration profile. 6. The effect of thermal radiation is to increase temperature in the thermal boundary layer and reverse effect on the temperature is seen by increasing Prandtl number. 7. The effect of magnetic field is to decelerate the velocity of the flow field to an appreciable amount throughout the boundary layer Acknowledgments The authors are grateful to the reviewers for their suggestions that extensively enhanced our paper. References [1] Sakiadis BC. Boundary-layer behavior on continuous solid surface: I. Boundary layer equations for two-dimensional and ax symmetric flow. J Am Inst Chem. Eng 1961;7:26–8. [2] Crane LJ. Flow past a stretching plate. Z Angew Math Phys. 1970;21:645–7.

Please cite this article as: P.V. Satya Narayana, D. Harish Babu, Numerical study of MHD heat and mass transfer of a Jeffrey fluid over a stretching sheet with chemical reaction and thermal radiation, Journal of the Taiwan Institute of Chemical Engineers (2015), http://dx.doi.org/10.1016/j.jtice.2015.07.014

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Please cite this article as: P.V. Satya Narayana, D. Harish Babu, Numerical study of MHD heat and mass transfer of a Jeffrey fluid over a stretching sheet with chemical reaction and thermal radiation, Journal of the Taiwan Institute of Chemical Engineers (2015), http://dx.doi.org/10.1016/j.jtice.2015.07.014