Acta Mechanica Sinica (2013) 29(5):667–675DOI 10.1007/s10409-013-0066-6

RESEARCH PAPER

Unsteady heat and mass transfer in MHD flow over an oscillatorystretching surface with Soret and Dufour effects

Lian-Cun Zheng ··· Xin Jin ··· Xin-Xin Zhang ··· Jun-Hong Zhang

Received: 19 December 2012 / Revised: 12 April 2013 / Accepted: 2 May 2013©The Chinese Society of Theoretical and Applied Mechanics and Springer-Verlag Berlin Heidelberg 2013

Abstract In this paper, we study the unsteady coupledheat and mass transfer of two-dimensional MHD fluid overa moving oscillatory stretching surface with Soret and Du-four effects. Viscous dissipation effects are adopted in theenergy equation. A uniform magnetic field is applied ver-tically to the flow direction. The governing equations arereduced to non-linear coupled partial differential equationsand solved by means of homotopy analysis method (HAM).The effects of some physical parameters such as magneticparameter, Dufour number, Soret number, the Prandtl num-ber and the ratio of the oscillation frequency of the sheet toits stretching rate on the flow and heat transfer characteristicsare illustrated and analyzed.

Keywords Viscous conducted fluid · Magnetic field · Os-cillatory stretching surface · Heat and mass transfer · HAMsolution

1 Introduction

Boundary layer flow and heat transfer are important dueto their applications in industries and many manufacturingprocesses. In order to enhance the properties of fluids, many

The project was supported by the National Natural Science Foun-dations of China (51076012 and 51276014).

L.-C. Zheng (�) · X. JinSchool of Mathematics and Physics,University of Science and Technology Beijing,100083 Beijing, Chinae-mail: liancunzheng@ustb.edu.cnX.-X. ZhangSchool of Mechanical Engineering,University of Science and Technology Beijing,100083 Beijing, ChinaJ.-H. ZhangNaval University of Engineering,430033 Wuhan, China

researchers have analyzed the problems by including somephysical features such as magnetic field, suction/injectionand combined heat and mass transfer. Crane [1] has re-searched in the flow through a stretching sheet with linearsurface velocity and found a similarity solution to the prob-lem. Subhas Abel et al. [2–9] has studied the effects of non-uniform heat source on MHD heat transfer in a liquid filmover an unsteady stretching sheet on condition that the sur-face velocity is a function of coordinate x and time t. Kabeiret al. [10, 11] studied unsteady MHD combined convec-tion over a moving vertical sheet in a fluid saturated porousmedium with uniform surface heat flux on condition that thesurface velocity is a function of coordinate x and y.

The well-known energy flux caused by the species gra-dient (the Dufour effect or the diffusion-thermo effect) wasdefined in 1873 by Dufour. After that, the mass flux whichcaused by the temperature gradient (the Soret effect or theso-called thermal-diffusion effect), was proposed by Soretthen. Recently, some researchers have been studying theinter-diffusion. Hayat et al. [12–16] has studying heat andmass transfer with Soret and Dufour effects on mixed con-vection boundary layer flow over a stretching vertical surfacein a porous medium filled with a viscoelastic fluid. RajeshSharma et al. [17, 18] studied the cross-diffusion by usingthe element free Galerkin method.

Homotopy analysis method (HAM) was first proposedby Liao et al. [19, 20]. A lot of researchers have made useof it to solve some non-linear problems successfully. Abbaset al. [21, 22] studied mixed convection in the stagnation-point flow of a Maxwell fluid over a vertical stretching sur-face. Sweet et al. [23] used it to solve the analytical solutionfor the unsteady MHD flow of a viscous fluid between mov-ing parallel plates. In this paper, we adopted the homotopyanalysis method (HAM) to study the unsteady boundary flowover a moving oscillatory stretching surface under Soret andDufour effects with a uniform magnetic field applied verti-cally to the platform.

668 L.-C. Zheng, et al.

2 Mathematical formulations

We consider the unsteady two-dimensional magneto hydro-dynamic (MHD) laminar boundary layer flow of an incom-pressible fluid over a permeable flat surface embedded inporous medium in the presence of Soret and Dufour effects.It is assumed that a uniform magnetic field B0 is imposedin the direction of y. The sheet is stretching with a veloc-ity Uw = ax sin wt along the x-axis. The sheet temperatureTw = T∞ + Ax2 varies with the coordinate x linearly andalso varies with the concentration. In the presence of heatgeneration, mass flux and viscous dissipation, the unsteadyboundary layer equations are given by

∂u∂x+∂v∂y= 0, (1)

∂u∂t+ u

∂u∂x+ v

∂u∂y= ν

∂2u∂y2− B2

0δ

ρu, (2)

∂T∂t+ u

∂T∂x+ v

∂T∂y= α

∂2T∂y2+

DmkT

cscp

∂2c∂y2

, (3)

∂c∂t+ u

∂c∂x+ v

∂c∂y= Dm

∂2c∂y2+

DmkT

Tm

∂2T∂y2

. (4)

Subject to the boundary conditions

u = ax sinωt, v = vw, T = Tw = T∞ + Ax2,

c = cw = c∞ + Bx2, at y = 0,(5)

u = 0,∂u∂y= 0, T = T∞, c = c∞, at y→ ∞, (6)

where x and y are the Cartesian coordinates along and nor-mal to the plate, u and v are the velocity components alongx- and y-axes, respectively, T is the fluid temperature, c isthe species concentration. α is the effective thermal diffusiv-ity, Dm is the coefficient of mass diffusivity, kT is the thermaldiffusion ratio, cp is the specific heat at constant pressure, cs

is the concentration susceptibility and Tm is the mean fluidtemperature. νμ/ρ is the kinematic viscosity of fluid, ρ is thefluid density, δ is the electrical conductivity of fluid, α is thethermal diffusivity. Uw is the velocity of the wall at t � 0; vw

is the velocity of suction (vw < 0) or injection (vw > 0).We denote the ratio of the oscillation frequency to its

stretching rate of the sheet by S = ω/a. For ease of analysis,we recommend the similarity transformations

η =

√aν

y, τ = ωt, ψ =√

aνx f (η, τ), (7)

u =∂ψ

∂y= ax f ′η , v = −∂ψ

∂x= −√aν f , (8)

θ(η, τ) =T − T∞Tw − T∞

, φ(η, τ) =c − c∞

cw − c∞. (9)

Substituting Eqs. (7)–(9) into Eqs. (1)–(6), the continuityequation is satisfied automatically and we get the followingpartial differential equations

fηηη + f fηη − S fητ − ( fη)2 − M2 fη = 0, (10)

1Prθηη + Duφηη − S θτ − θ fη + f θη = 0, (11)

1S cφηη + S rθηη − Sφτ − φ fη + fφη = 0. (12)

Subjected to the boundary conditions

f (0, τ) = fw, fη(0, τ) = sin τ,

fη(∞, τ)→ 0, fηη(∞, τ)→ 0,(13)

θ(0, τ) = 1, φ(0, τ) = 1,

θ(∞, τ)→ 0, φ(∞, τ)→ 0.(14)

Here Pr = ν/αm is the Prandtl number, S c = ν/Dm is theSchmidt number, which is a ratio of kinematic viscosity overthe coefficient diffusion species. M2 = B2

0δ/(ρa) is the di-mensionless magnetic parameter, Du is the Dufour numberand S r is the Soret number, which are defined as Du =DmkT(Cw − C∞)CsCp(Tw − T∞)ν

, S r =DmkT(Tw − T∞)Tmν(Cw −C∞)

. fw = −vw/√

aν

is the dimensionless mass transfer parameter with fw > 0 forsuction and fw < 0 for injection.

3 Homotopy analysis method

Liao et al. [19, 20] proposed a new approximate analyticalsolution technique, called the Homotopy analysis method(HAM), for solving non-linear problems, which can over-come the foregoing restriction of the perturbation methods.

According to the boundary conditions (13) and (14), thevelocity f (η, τ) can be expressed by the set of base functions,{ηk sin(nτ) exp(− jη)|k � 0, n � 0, j � 0}, the temperatureθ(η, τ) and concentration φ(η, τ) can be expressed by the setbase functions {ηk exp(−nη)|k � 0, n�0} in the form

f (η, τ) = a00,0 +

∞∑n=0

∞∑k=0

∞∑j=0

a jn,kη

k sin( jτ) exp(−nη), (15)

θ(η, τ) =∞∑

n=0

∞∑k=0

bkm,nη

k exp(−nη), (16)

φ(η, τ) =∞∑

n=0

∞∑k=0

ckm,nη

k exp(−nη), (17)

in which a jn,k, bk

m,n, and ckm,n are the coefficients. The initial

guesses f0, θ0, and φ0 of f , θ, and φ are

f0(η, τ) = fw + sin τ[1 − exp(−η)], (18)

θ0(η, τ) = exp(−η), (18)

φ0(η, τ) = exp(−η). (20)

The linear operators are selected as

L f =∂3 f∂η3− ∂ f∂η, (21)

Lθ =∂2θ

∂η2− θ, (22)

Unsteady heat and mass transfer in MHD flow over an oscillatory stretching surface with Soret and Dufour effects 669

Lφ =∂2φ

∂η2− φ, (23)

which have the following properties

L f [C1 +C2 exp(η) +C3 exp(−η)] = 0, (24)

Lθ[C4 exp(η) +C5 exp(−η)] = 0, (25)

Lφ[C6 exp(η) +C7 exp(−η)] = 0, (26)

respectively, and Ci (i = 1, 2, · · · , 7) are the arbitrary con-stants.

If p ∈ [0, 1] is the embedding parameter and h f , hθ andhφ indicate the non-zero auxiliary parameter, respectively,then we construct the following zero-th order deformation

(1 − p)L f [ f̂ (η, τ; p) − f0(η, τ)] = ph f Nf [ f̂ , θ̂, φ̂], (27)

(1 − p)Lθ[θ̂(η, τ; p) − θ0(η, τ)] = phθNθ[ f̂ , θ̂, φ̂], (28)

(1 − p)Lφ[φ̂(η, τ; p) − φ0(η, τ)] = phφNφ[ f̂ , θ̂, φ̂], (29)

f̂ (0, τ; p) = fw, f̂η(0, τ; p) = 0,

f̂η(∞, τ; p)→ 0, f̂ηη(∞, τ; p)→ 0,(30)

θ̂(0, τ; p) = 1, φ̂(0, τ; p) = 1,

θ̂(∞, τ; p)→ 0, φ̂(∞, τ; p)→ 0.(31)

Based on Eqs. (9), (10), and (11), we define the non-linearoperators as

N f [ f̂ , θ̂, φ̂] = f̂ηηη + f̂ f̂ηη − S f̂ητ − ( f̂η)2 − M2 f̂η, (32)

Nθ[ f̂ , θ̂, φ̂] =1Prθ̂ηη + Duφ̂ηη − S θ̂τ − θ̂ f̂η + f̂ θ̂η, (33)

Nφ[ f̂ , θ̂, φ̂] =1

S cφ̂ηη + S rθ̂ηη − S φ̂τ − φ̂ f̂η + f̂ φ̂η. (34)

Obviously, when p = 0 and p = 1, we have

f̂ (η, τ; 0) = f0(η, τ), f̂ (η, τ; 1) = f (η, τ), (35)

θ̂(η, τ; 0) = θ0(η, τ), θ̂(η, τ; 1) = θ(η, τ), (36)

φ̂(η, τ; 0) = φ0(η, τ), φ̂(η, τ; 1) = φ(η, τ). (36)

Expanding f̂ (η, τ; p), θ̂(η, τ; p), and φ̂(η, τ; p) in Taylor’s the-orem with respect to the embedding parameter p, obtain

f̂ (η, τ; p) = f0(η, τ) +∞∑

m=1

fm(η, τ)pm, (38)

θ̂(η, τ; p) = θ0(η, τ) +∞∑

m=1

θm(η, τ)pm, (39)

φ̂(η, τ; p) = φ0(η, τ) +∞∑

m=1

φm(η, τ)pm. (40)

The convergence strongly depends on h f , hθ, and hφ.The values of h f , hθ, and hφ are chosen in a such way thatthe series (27)–(29) converge at p = 1, then we have

f (η, τ) = f0(η, τ) +∞∑

m=1

fm(η, τ), (41)

θ(η, τ) = θ0(η, τ) +∞∑

m=1

θm(η, τ), (42)

φ(η, τ) = φ0(η, τ) +∞∑

m=1

φm(η, τ), (43)

fm(η, τ) =1

m!∂m f̂ (η, τ)∂pm

∣∣∣∣∣p=0, (44)

θm(η, τ) =1

m!∂mθ̂(η, τ)∂pm

∣∣∣∣∣p=0, (45)

φm(η, τ) =1

m!∂mφ̂(η, τ)∂pm

∣∣∣∣∣p=0. (46)

The MTH order deformation problems are in the form of

L f [ fm(η, τ) − χm fm−1(η, τ)] = h f Rfm(η, τ), (47)

Lθ[θm(η, τ) − χmθm−1(η, τ)] = hθRθm(η, τ), (48)

Lφ[φm(η, τ) − χmφm−1(η, τ)] = hφRφm(η, τ), (49)

fm(0, τ) = 0,∂ fm(0, τ)

∂η= 0,

∂ fm(∞, τ)∂η

= 0,∂ f 2

m(∞, τ)

∂η2= 0,

(50)

θm(0, τ) = 0, φm(0, τ) = 0,

θm(∞, τ) = 0, φm(∞, τ) = 0,(51)

R fm = f ′′′m−1 − S

∂ f ′m−1

∂τ− M2 f ′m−1

+

m−1∑i=0

( fm−1−i f ′′i − f ′m−1−i f ′i ), (52)

Rθm =

1Prθ′′m−1 + Duφ′′m−1 − S

∂θm−1

∂τ

+

m−1∑i=0

( fm−1−iθ′i − θm−1−i f ′i ), (53)

Rφm =

1S cφ′′m−1 + S rθ′′m−1 − S

∂φm−1

∂τ

+

m−1∑i=0

( fm−1−iφ′i − φm−1−i f ′i ), (54)

χm =

⎧⎪⎪⎨⎪⎪⎩0, m � 1,

1, m > 1.(55)

The general solutions of Eqs. (47)–(51) are

fm(η, τ) = f ∗m(η, τ) +C1 +C2 exp(η) + C3 exp(−η), (56)

θ(η, τ) = θ∗(η, τ) + C4 exp(η) +C5 exp(−η), (57)

φ(η, τ) = φ∗(η, τ) +C6 exp(η) +C7 exp(−η), (58)

where f ∗(η, τ), θ∗(η, τ) and φ∗(η, τ) denote the special solu-tions and

670 L.-C. Zheng, et al.

C2 = C4 = C6 = 0, C1 = − f ∗m(0, τ) −C3,

C3 =∂ f ∗m(η, τ)

∂η

∣∣∣∣∣η=0, C5 = −θ∗m(0, τ),

C7 = −φ∗m(0, τ).

Using the above analytic approaches, we find solutionsfor boundary layer flow, heat and mass transfer of a two-dimensional laminar flow over a moving oscillatory stretch-ing surface by Mathematical analytic solution device withfw = M = 1, S = 0.25, Du = 0.6, S r = 0.1, S c = 0.6,Pr = 0.7 at τ = π/2.

f1 =18

e−η(−1 + eη − η)(2 + cos τ − 2 cos 2τ + 4 sin τ)h1,

f2 =18

e−ηh21(−1 + eη − η){2 + cos τ − 2 cos 2τ

+4 sin τ + {8[−4eη + (2 + η)2] cos τ

+ sin τ[101 − 101eη + 101 + 23η2

+12(3 − 3eη + 3η + η2) cos τ

+8(−3 + 3eη − 3η − η2) cos 2τ + 128 sin τ

−128eη sin τ + 128η sin τ + 32η2 sin τ]}},θ1 =

170

e−ηη(−36 + 35 sin τ)h1,

θ2 = − 1117 600

e−2ηh21{−1 680eηη(−36 + 35 sin τ)

+1 225[4 + eη(−4 + 9η + 3η2)] cos τ

+7 350eηη(1 + η) cos 2τ

+8[−3eηη(−2 609 + 858η)

+70(−71 + eη(71 − 213η + 54η2)) sin τ]},ϕ1 =

160

e−ηη(−23 + 30 sin τ)h1,

φ2 = − 150 400

e−2ηh21{−840eηη(−23 + 30 sin τ)

+[−525(4 + 9η + 3η2)] cos τ

+2{eη(9 057 − 3 509η)η + 1 575eηη(1 + η) cos 2τ

+70[−106+ eη(106 − 318η + 69η2)] sin τ}}.In the same way, we got the f3, θ3, φ3 and so on, which

we omitted here (the results are too long to write here).

4 Convergence of the Homotopy Solutions

For an analytic solution obtained by the homotopy analysismethod, its convergence depends on the auxiliary parameterh f , hθ, and hφ. If these parameters are properly chosen, thegiven solution is effective. Since the interval for the admis-sible values of h f , hθ, and hφ corresponding to the line seg-ments nearly parallel to the horizontal axis, we know that theadmissible value for the parameter h f is −0.3 � h f � −0.1from Fig. 1; the admissible value for the parameter hθ is−0.4�hθ�−0.1 from Fig. 2; the admissible value for the pa-rameter hφ is −0.4�hφ�−0.2 from Fig. 3. To assure the con-

vergence of the HAM solution, the values of h f , hθ, and hφshould be chosen from these ranges. From the computationwe have found that the series given by Eqs. (41)–(43) con-verge in the whole range of η when h f = hθ = hφ = −0.25.

Fig. 1 The h-curves of f ′′(0, τ) obtained by the fifth-order approx-imation of the HAM with fw = M = 1, S = 0.25, Du = 0.6,S r = 0.1, S c = 0.6, Pr = 0.7 at τ = π/2

Fig. 2 The h-curves of θ′(0, τ) obtained by the fifth-order approx-imation of the HAM with fw = M = 1, S = 0.25, Du = 0.6,S r = 0.1, S c = 0.6, Pr = 0.7 at τ = π/2

Fig. 3 The h-curves of φ′(0, τ) obtained by the fifth-order approx-imation of the HAM with fw = 1, M = 1, S = 0.25, Du = 0.6,S r = 0.1, S c = 0.6, Pr = 0.7 at τ = π/2

5 Results and discussion

Table 1 is prepared for the comparison of present results withthe former in Ref. [22]. It is found that the two results are ingreat uniformity.

Some physical characteristics are shown in Tables 2–4.

Unsteady heat and mass transfer in MHD flow over an oscillatory stretching surface with Soret and Dufour effects 671

Table 1 Comparison of values of f ′′(0, τ) for vw = 0, S = 1and M = 12 with Ref. [22] when K = 0

fw S M τ Ref. [22] Present results

0 1.0 12.0 1.5π 11.678 656 11.678 565

5.5π 11.678 707 11.678 706

9.5π 11.678 656 11.678 656

Table 2 Values of f ′′(0, τ) for different values of fw, S and Mat τ = π/2

fw S M f ′′(0,π/2)

1 0 1 −1.999 68

0.5 −2.013 71

1 −2.139 01

2 −3.889 07

−1 0.25 1 −1.001 96

−0.5 −1.187 87

0 −1.416 01

0.5 −1.687 83

1 −2.001 88

0 −1.620 56

1 −2.001 88

2 −2.271 18

Table 3 Values of −θ′(0, τ) for different values of fw, S , M, Pr,Du, S r and S c at τ = π/2

S fw M Pr Du S r S c −θ′(0, π/2)

0 1 1 0.7 0.6 0.1 0.6 0.806 89

1 0.756 25

2 0.577 51

0.25 −1 1 0.358 92

−0.5 0.436 45

0 0.534 22

0.5 0.654 81

1 0.812 12

0 0.845 97

1 0.812 12

2 0.781 62

1 0.5 0.596 62

0.7 0.812 12

1 1.088 69

0.7 0.3 0.2 0.992 11

0.6 0.1 0.812 12

0.75 0.08 0.710 32

0.6 0.1 0.6 0.812 12

1.0 0.589 71

1.5 0.366 93

Table 4 Values of −φ′(0, τ) for different values of fw, S , M, Pr,Du, S r and S c at τ = π/2

S fw M Pr Du S r S c −φ′(0,π/2)

0 1 1 0.7 0.6 0.1 0.6 0.967 66

1 0.905 95

2 0.703 38

0.25 −1 1 0.397 78

−0.5 0.490 74

0 0.614 24

0.5 0.770 88

1 0.969 26

0 1.007 76

1 0.969 26

2 0.934 95

1 0.5 0.972 05

0.7 0.969 26

1 0.946 28

0.7 0.3 0.2 0.926 97

0.6 0.1 0.969 26

0.75 0.08 0.974 19

0.6 0.1 0.6 0.969 26

1.0 1.489 69

1.5 2.113 80

Figures 4–6 show us the effects of S , M and fw on thedimensionless velocity f ′(η, τ). From Fig. 4, we can findthat with the increase of S , the velocity f ′(η, τ) decreaseswhenM = fw = 1. Figure 5 illustrates the effect of M onthe velocity. It is observed that the velocity are decreasedby increasing the values of M. The boundary layer thicknessis also decreased for large magnetic field. Figure 6 tells usthe effect of fw on f ′(η, τ). The velocity decreases when fw(> 0), whereas in the case of fw < 0, the velocity increaseswith the increasing values of | fw|.

Fig. 4 Velocity profiles for various values of S when fw = M = 1,Du = 0.6, S r = 0.1, S c = 0.6, Pr = 0.7 at τ = π/2

672 L.-C. Zheng, et al.

Fig. 5 Velocity profiles for various values of M when fw = 1,S = 0.25, Du = 0.6, S r = 0.1, S c = 0.6, Pr = 0.7 at τ = π/2

Fig. 6 Velocity profiles for various values of fw when M = 1,S = 0.25, Du = 0.6, S r = 0.1, S c = 0.6, Pr = 0.7 at τ = π/2

Figures 7 and 8 show us the temperature and concentra-tion distributions with collective variation in Soret numberand Dufour number. In Fig. 7, Du from 0.3 to 0.75 causesthe influence of mass gradients on the temperature field, sovalues of temperature are increased. In Fig. 8, concentrationincreases with the increase of S r from 0.08 to 0.2. Speciestransfer is boosted as a result of the temperature flux.

Fig. 7 Temperature profiles for various values of Du, S r (Du ·S r =0.06) when Pr = 0.7, S c = 0.6, fw = 1, S = 0.25, M = 1 at τ = π/2

The effects of S , M, and fw on the temperature andconcentration profiles are shown in Figs. 9–14. We can findthat the temperature and the concentration increase with thevalues of S and M increase. However, we find that both ofthem decrease when the injection/suction number increases.The effect of Pr on the two fields is shown in Figs. 15 and

16. Increasing Prandtl number has a weaker thermal diffu-sivity and mass flux. Due to this fact the temperature, theconcentration and the thermal boundary layer thickness aredecreasing functions of Pr. The effect of S c on the concen-tration profiles is shown in Fig. 17. We have observed thatan increase of S c causes a considerable reduction in the con-centration.

Fig. 8 Concentration profiles for various values of Du, S r (Du ·S r = 0.06) when Pr = 0.7, S c = 0.6, fw = 1, S = 0.25, M = 1 atτ = π/2

Fig. 9 Temperature profiles for various values of S with Du = 0.6,S r = 0.1, S c = 0.6, Pr = 0.7, fw = 1, M = 1 at τ = π/2

Fig. 10 Temperature profiles for various values of M whenDu = 0.6, S r = 0.1, S c = 0.6, Pr = 0.7, S = 0.25, fw = 1 atτ = π/2

Figure 18 illustrates the velocity at the four differentdistances from the surface for the five periods τ ∈ [0, 10π]when fw = M = 1, S = 0.25. We can find the amplitude nearto the sheet is larger than that far from the surface. Figures19–21 tells us the effects of M, S , and fw on the velocity at

Unsteady heat and mass transfer in MHD flow over an oscillatory stretching surface with Soret and Dufour effects 673

Fig. 11 Temperature profiles for various values of fw whenDu = 0.6, S r = 0.1, S c = 0.6, Pr = 0.7, S = 0.25, M = 1 atτ = π/2

Fig. 12 Concentration profiles for various values of S whenDu = 0.6, S r = 0.1, S c = 0.6, Pr = 0.7, fw = 1, M = 1 atτ = π/2

Fig. 13 Concentration profiles for various values of M whenDu = 0.6, S r = 0.1, S c = 0.6, Pr = 0.7, S = 0.25, fw = 1 atτ = π/2

Fig. 14 Concentration profiles for various values of fw whenDu = 0.6, S r = 0.1, S c = 0.6, Pr = 0.7, S = 0.25, M = 1 atτ = π/2

Fig. 15 Temperature profiles for various values of Pr whenDu = 0.6, S r = 0.1, S c = 0.6, fw = 1, S = 0.25, M = 1, τ = π/2

Fig. 16 Concentration profiles for various values of Pr whenDu = 0.6, S r = 0.1, S c = 0.6, fw = 1, S = 0.25, M = 1 atτ = π/2

Fig. 17 Concentration profiles for various values of S c whenDu = 0.6, S r = 0.1, Pr = 0.7, fw = 1, S = 0.25, M = 1 atτ = π/2

the fixed distance η = 0.25. It can be inferred from Fig. 19the three graphics that the larger S is, the larger the amplitudeis and there is a phase shift in the picture. Figure 20 showsus the amplitude of the flow decreases with the increase ofthe magnitude parameter M. This is because the magnitudeforce acts as a resistance to the flow. Similarity to M, thephase shift occurs with variation of fw. In the mean time, theamplitude decreases with the increase of fw.

Figures 22–25 depict the velocity, temperature and con-centration profiles at different times with the fixed values ofsome physics parameters.

674 L.-C. Zheng, et al.

Fig. 18 Time series of the flow of the velocity field f ′(η, τ) at fourdifferent distances from the surface for the time period τ ∈ [0, 10π]with fw = M = 1, S = 0.25, Du = 0.6, S r = 0.1, S c = 0.6,Pr = 0.7

Fig. 19 Time series of the flow of the velocity field f ′(η, τ) at afixed distance y = 0.25 in the first five periods τ ∈ [0, 10π] for dif-ferent values of M with fw = 1, S = 0.25, Du = 0.6, S r = 0.1,S c = 0.6, Pr = 0.7

Fig. 20 Time series of the flow of the velocity field f ′(η, τ) at afixed distance y = 0.25 in the first five periods τ ∈ [0, 10π] for dif-ferent values of S with fw = M = 1, Du = 0.6, S r = 0.1, S c = 0.6,Pr = 0.7

6 Conclusions

In this paper, we have investigated the unsteady boundaryflow over a moving oscillatory stretching surface under Soretand Dufour effects by means of HAM. The arising non-linearproblem is reduced to the linear one by means of similar-ity transformations and then solved analytically. The majorconclusions are:The velocity of fluid decreases as S (the un-steadiness parameter), M (the magnitude parameter) and fw(the injection/suction number) increases. The thermal boun-

Fig. 21 Time series of the flow of the velocity field f ′(η, τ) at afixed distance y = 0.25 in the first five periods τ ∈ [0, 10π] for dif-ferent values of fw with M = 1, S = 0.25, Du = 0.6, S r = 0.1,S c = 0.6, Pr = 0.7

Fig. 22 Velocity profiles for different times with fw = M = 1,S = 0.25, Du = 0.6, S r = 0.1, S c = 0.6, Pr = 0.7

Fig. 23 Velocity profiles for two different times with fw = 1,S = 0.25, M = 1, Du = 0.6, S r = 0.1, S c = 0.6, Pr = 0.7

Fig. 24 Temperature profiles at the five fixed times with Du = 0.6,S r = 0.1, S c = 0.6, fw = 1, S = 0.25, M = 1, Pr = 0.7

Unsteady heat and mass transfer in MHD flow over an oscillatory stretching surface with Soret and Dufour effects 675

Fig. 25 Concentration profiles at different times with Du = 0.6,S r = 0.1, Pr = 0.7, fw = 1, S = 0.25, M = 1 at τ = π/2

dary layer thickness increases with the increase of S and M,but decreases with the increase of fw and Pr (the Prandtlnumber). The concentration filed increases as S and M in-crease. However, it decreases with the increase of fw, Prand S c (the Schmidt number). In the case of Du (the Dufournumber) increases, S r (the Soret number) decreases at thesame time, the temperature field increases while the speciesfield decreases. And when S r increases (Du decreases),the temperature field decreases while concentration value in-creases.

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