Appl. Math. Mech. -Engl. Ed., 35(12), 1525–1540 (2014) DOI 10.1007/s10483-014-1888-6 c Shanghai University and Springer-Verlag Berlin Heidelberg 2014

Applied Mathematics and Mechanics (English Edition)

MHD effect of mixed convection boundary-layer flow of Powell-Eyring fluid past nonlinear stretching surface∗ S. PANIGRAHI1,2 ,

M. REZA1 ,

A. K. MISHRA1,2

(1. Department of Mathematics, National Institute of Science & Technology, Berhampur 761008, India; 2. Department of Mathematics, Berhampur University, Berhampur 760007, India)

Abstract Sufficient conditions are found for the existence of similar solutions of the mixed convection flow of a Powell-Eyring fluid over a nonlinear stretching permeable surface in the presence of magnetic field. To achieve this, one parameter linear group transformation is applied. The governing momentum and energy equations are transformed to nonlinear ordinary differential equations by use of a similarity transformation. These equations are solved by the homotopy analysis method (HAM) to obtain the approximate solutions. The effects of magnetic field, suction, and buoyancy on the Powell-Eyring fluid flow with heat transfer inside the boundary layer are analyzed. The effects of the nonNewtonian fluid (Powell-Eyring model) parameters ε and δ on the skin friction and local heat transfer coefficients for the cases of aiding and opposite flows are investigated and discussed. It is observed that the momentum boundary layer thickness increases and the thermal boundary layer thickness decreases with the increase in ε whereas the momentum boundary layer thickness decreases and thermal boundary layer thickness increases with the increase in δ for both the aiding and opposing mixed convection flows. Key words

stretching surface, non-Newtonian fluid, mixed convection

Chinese Library Classification O357.5, O361.5 2010 Mathematics Subject Classification 76W05, 76E06, 80A20, 76A05

1

Introduction

The mathematical results on the mixed convection inside the boundary layer over a continuous stretching sheet has brought the interest of many researchers for its wide range of applications in manufacturing and food processing industries. It is well-known that thermal buoyancy plays a crucial role in the heat transfer when the plate is moving vertically upward. Therefore, this topic has attracted continuous attention of researchers for theoretical studies and practical applications for more than half a century[1–9] . Theories about the related mathematical modeling mostly involve the system of nonlinear partial differential equations with boundary conditions whose exact solutions are not known. To find a similar solution, it is customary to find the proper similar variables which can reduce the partial differential equations to a system of ordinary differential equations. The group theoretic method is a well accepted technique for this purpose[11–17] . With one parameter group theory, Fan et al.[18] found a similar solution for the boundary layer flow of an electrically conducting viscous fluid along a vertically stretching porous flat plate in the presence of magnetic field along with suction or blowing. ∗ Received Feb. 5, 2014 / Revised May 6, 2014 Corresponding author S. PANIGRAHI, E-mail: [email protected]

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S. PANIGRAHI, M. REZA, and A. K. MISHRA

Many industrial processes demand extensional flows of non-Newtonian fluids, e.g., coasting, continuous stretching of plastic films, cooling of nuclear reactors, producing glass fiber, and growing crystals. Sometimes, the polymer sheet is stretched while extruded from a die. In earlier works, the exact solutions for non-Newtonian fluid problems are usually obtained based on perturbation methods. The motivation of this paper is to extend the discussion to the mixed convection flow of the non-newtonian Powell-Eyring fluid[19] . It is interesting to note that the Powell-Eyring model gives more advantage than other non-Newtonian fluids since the model is based on the kinetic theory of gas which acts like viscous fluids at high shear rates. In the past few years, the Powell-Eyring fluid model have been extensively studied[20–22] . It is remarked that these works are related with the flow of the free stream past a fixed surface. Hayat et al.[23] obtained an analytical solution of the flow and heat transfer of the Powell-Eyring fluid over a continuously moving surface with the convective boundary condition by use of the homotopy analysis method (HAM). Jalil et al.[24] obtained a similar solution by use of the scaling group transform for the flow and heat transfer of the Powell-Eyring fluid past a continuously moving permeable surface in a parallel free stream with variable temperature, and showed that a similar solution existed for the Powel-Eyring fluid flow along a moving surface when the surface and 1 free stream velocities were nonlinear of the form x 3 . Jalil et al.[25] studied the flow and heat transfer of the Powell-Eyring fluid over a permeable stretching surface. However, no literature exists on the effects of magnetic field on the mixed convection of the Powell-Eyring fluid having a nonlinear stretching vertical permeable plate. Therefore, the aim of this paper is to examine the analytic self-similar solution of the mixed convection flow of a non-Newtonian Powell-Eyring fluid over a nonlinear stretching vertical permeable surface in the presence of magnetic field. We obtain proper similar transformations to reduce the set of governing partial differential equations to ordinary differential equations with appropriate boundary conditions by using the one parameter linear group transform. The approximate analytic solution are obtained by the HAM developed by Liao[26–29] . The effects of various physical quantities such as suction, magnetic field, and buoyancy on the flow and heat transfer inside the boundary layer are analyzed.

2

Flow analysis

Consider a laminar mixed convection flow of a non-Newtonian Powell-Eyring fluid over a vertically stretching permeable surface in the presence of transverse magnetic field. The surface is stretched vertically upward along the positive x-direction, and the origin is fixed (see Fig. 1).

Fig. 1

Physical model

MHD effect of mixed convection boundary-layer flow of Powell-Eyring fluid

1527

The shear stress component τxy for the Powell-Eyring is given by[19] τxy = μ

 1 ∂u  1 ∂u + , sinh−1 ∂y β1 C1 ∂y

(1)

where μ is the coefficient of the viscosity, and β1 and C1 are the fluid material parameters. Let u(x, y) and v(x, y) be the velocity components along the x- and y-axes, respectively, and T (x, y) represent the temperature of the fluid. Then, the governing continuity equation, the boundary layer equation of momentum, and the energy can be obtained as follows: ∂u ∂v + = 0, ∂x ∂y u

(2)

∂u ∂u +v ∂x ∂y ∂2u

∂2u 1 ∂y 2  =ν 2 + ∂y ρβ1 C1 1 +  1

 − ∂u 2

σB 2 (x)u + gβ(T − T∞ ), ρ

(3)

C1 ∂y

u

∂T ∂2T ∂T +v =α 2, ∂x ∂y ∂y

(4)

where ρ, ν, and σ are the density, the kinematic viscosity, and the electrical conductivity of the fluid, respectively. α, β, g, and T∞ denote the thermal diffusivity, the volumetric coefficient of thermal expansion, the gravity, and the ambient temperature, respectively. For the no-slip condition at the surface, the velocity in the x-direction Cxm equals the moving surface velocity Uw (x), i.e., Uw (x) = Cxm . Define Vw = Dxn ,

B(x) = B0 xs ,

Tw = T∞ + Axr ,

where Uw (x) is the moving surface velocity, Vw is the transverse velocity at the wall, B(x) is the imposed magnetic flied, and Tw is the wall temperature, and A, C, D, and B0 are prescribed constants. The boundary conditions are given by 

u = Uw (x) = Cxm , u → 0,

T → T∞

v = Vw (x) = Dxn ,

T = Tw (x) = T∞ + Axr

as y → ∞,

at y = 0, (5)

where m and n are the index parameters of the velocity components, and r is the temperature at the wall, whose values will be determined later. Define ψ(x, y) as the stream function, and u=

∂ψ , ∂y

v=−

∂ψ . ∂x

Define the non-dimensional temperature θ by θ=

T − T∞ . Tw − T∞

Write the negative binomial series expansion of the terms under square root. Then, Eqs. (3),

1528

S. PANIGRAHI, M. REZA, and A. K. MISHRA

(4), and (5) become ψy ψxy − ψx ψyy = νψyyy −



∞  1 −1/2 2N σB0 2 x2s ψy 1 + gβ(Tw − T∞ )θ + ψyy , ψyyy ρ ρβ1 C1 N C 2N N =0 1

ψy ((Tw − T∞ )θ)x − ψx ((Tw − T∞ )θ)y = α((Tw − T∞ )θ)yy , 

y = 0 : ψy = Cxm , y → ∞ : ψy = 0,

3

ψx = −Dxn ,

(6) (7)

θ = 1, (8)

θ = 0.

Group theory analysis and similar solution

To find the similarity transformations corresponding to the problem, the following one parameter linear group transformation is used: ⎧ ⎨x ¯ = ap x, y¯ = aq y, ψ = ad ψ, (9) ⎩ θ¯ = θ, T − T = ae (T − T ), ∞ ∞ where a is the parameter of the group. p, q, d, and e are real constants whose values will be determined later. Substituting Eq. (9) into Eqs. (6)–(8), we get ψ y¯ψ x¯y¯ − ψ x¯ ψ y¯y¯ − νψ y¯y¯y¯ + − gβ(T w − T ∞ )θ¯ −

σB0 2 x ¯2s ψ y¯ ρ



∞  1 −1/2 2N 1 ψ y¯y¯y¯ ψ y¯y¯ ρβ1 C1 N C 2N N =0 1

= a2d−p−2q ψy ψxy − a2d−p−2q ψx ψyy − νad−3q ψyyy σB0 2 x2s ψy − ae gβ(Tw − T∞ )θ ρ

∞  1 −1/2 2N 1 ψyy , − ψyyy a(2N +1)d−(4N +3)q 2N ρβ1 C1 N C1

+ ad+2sp−q

(10)

N =0

¯ x¯ − ψ ((T w − T ∞ )θ) ¯ y¯ − α((T w − T ∞ )θ) ¯ y¯y¯ ψ y¯((T w − T ∞ )θ) x ¯ = ad+e−p−q ψy ((Tw − T∞ )θ)x − ad+e−p−q ψx ((Tw − T∞ )θ)y − ae−2q α((Tw − T∞ )θ)yy .

(11)

When y = 0, ⎧ ψ y¯ − C x ¯m = ad−q ψy − amp Cxm = 0, ⎪ ⎪ ⎪ ⎨ ψ x¯ + D¯ xn = ad−p ψx + anp Dxn = 0, ⎪ ⎪ ⎪ ⎩ T w − T ∞ − A¯ xr = ae (Tw − T∞ ) − arp Axr = 0.

(12)

MHD effect of mixed convection boundary-layer flow of Powell-Eyring fluid

1529

When y → ∞, T − T ∞ = ae (T − T∞ ) = 0.

ψ y¯ = ad−q ψy = 0,

(13)

For conformal invariance, the following relations between the parameters are required: 2d − p − 2q = d − 3q = d + 2sp − q = e = (2N + 1)d − (4N + 3)q,

(14)

d + e − p − q = e − 2q,

(15)

d − q = mp,

d − p = np,

e = rp.

(16)

Solving the above equations, we can obtain d = 2q,

p = 3q,

e = −q,

m=

1 , 3

1 s=r=n=− . 3

(17)

Substitute the values of the constants p, q, d, s, and e in Eq. (14). Then, we can see that Eq. (14) is independent of N . This shows that a similar transform exists if the stretching 1 velocity of the vertical sheet is of the form x 3 and the suction velocity and the magnetic field − 13 are of the form x . Therefore, Eq. (9) can be rewritten as x ¯ = a3q x,

ψ = a2q ψ,

y¯ = aq y,

θ¯ = θ,

T − T ∞ = a−q (T − T∞ ).

(18)

Then, it can be easily seen that 1

1

2

¯− 3 , yx− 3 = y¯x

2

ψx− 3 = ψ¯ x− 3 ,

from which we can find two absolute invariants expressed by 1

ζ = yx− 3 ,

2

f (ζ) = ψx− 3 .

(19)

To avoid the properties appearing explicitly in the coefficients of the equations, we use the following similarity transformations:   2 2Cx− 3 3 4 , ψ = f (η) Cνx 3 , θ = θ(η). η=y (20) 3ν 2 Define M=

σB02 , ρC

λ=

Agβ , C2

Pr =

ν , α

where M is the magnetic parameter, λ is the mixed convection parameter, and P r is the Prandtl number. Substituting Eq. (20) in Eqs. (3) and (4) yields f  + f f  −

3 f  f 2 3  − M f  + λθ + 2 2 2 ρC1 β1 ν 1 +

2C 3 (f  )2 3νC1 2

1  1 θ + (2f θ + f  θ) = 0. Pr 2

= 0,

(21)

(22)

The boundary condition (8) becomes  f (0) = fw ,

f  (0) = 1,

f  (∞) = 0,

θ(∞) = 0,

θ(0) = 1, (23)

1530

S. PANIGRAHI, M. REZA, and A. K. MISHRA

where fw is the suction or blowing parameter expressed by  3 . fw = D 2νC Positive fw corresponds to suction, and negative fw corresponds to blowing at the surface. λ is the physically mixed convection parameter. Positive λ corresponds the heating case, negative λ corresponds the cooling case, and zero λ corresponds to the forced convection flow case. Expanding the last terms of Eq. (21) in Taylor series, we can get 3 f 2 3 − M f  + λθ 2 2 2 4 6   3   2 1 C f C 6  f   5C 9  f   + f  1 − + 2 − + · · · = 0. 3 ρC1 β1 ν 3ν C1 6ν C1 54ν C1      3   When  fC1   1 , we can neglect the terms of O fC1 and reduce Eq. (21) to f  + f f  −

εδ 1 3 3 (1 + ε)f  + f f  − f 2 − M f  + λθ − f  f 2 = 0, 2 2 2 3

(24)

(25)

where ε and δ are the fluid parameters expressed by ε=

1 , ρC1 β1 ν

δ=

C3 . C12 ν

The skin-friction coefficient Cf and the local heat transfer Nusselt number Nux can be expressed as ν  ∂u  Cf = 2 , (26) U (x) ∂y y=0  ∂T  x Nux = . (27) (T w − T∞ ) ∂y y=0 Using the similarity transformation (20) and Eqs. (23) and (26), we can get ε 3  Re1/2 x Cf = (1 + ε)f (0) − δf (0), 6  N u = −θ (0), Re−1/2 x x

(28) (29)

where Rex is the local Reynolds number expressed by Rex =

4

xUw (x) . ν

Homotopy analysis solution

To solve Eqs. (22) and (25), we apply the HAM proposed by Liao[26–29] . For this purpose, we define two initial solutions which satisfy only the boundary condition (23) and two linear operators as follows: f0 (η) = 1 + fw − e−η , Lf =

d3 d2 + , dη 3 dη 2

θ0 (η) = e−η ,

(30)

d2 d . + dη 2 dη

(31)

Lθ =

MHD effect of mixed convection boundary-layer flow of Powell-Eyring fluid

1531

Define the nonlinear operators Nf and Nθ as follows:  ˆ 2 3ˆ 2ˆ ˆ q)) = (1 + ε) ∂ f (η; q) + fˆ(η; q) ∂ f (η; q) − 1 ∂ f (η; q) Nf (fˆ(η; q), θ(η; ∂η 3 ∂η 2 2 ∂η

ˆ q)) = Nθ (fˆ(η; q), θ(η;

εδ ∂ 3 fˆ(η; q)  ∂ 2 fˆ(η; q) 2 3 ∂ fˆ(η; q) 3 ˆ + λθ(η; q) − − M , 2 ∂η 2 3 ∂η 3 ∂η 2

(32)

ˆ q) ˆ ˆ q) 1 1 ∂ 2 θ(η; ∂ θ(η; ˆ q) ∂ f (η; q) , + θ(η; + fˆ(η; q) 2 P r ∂η ∂η 2 ∂η

(33)

where q ∈ [0, 1] is an embedding parameter. If hf and hθ denote the non-zero auxiliary parameters, then we can construct the zeroth-order deformation equations as follows: ˆ q)), (1 − q)Lf (fˆ(η; q) − f0 (η)) = hf Nf (fˆ(η; q), θ(η;

(34)

ˆ q) − θ0 (η)) = hθ Nθ (fˆ(η; q),θ(η; ˆ q)). (1 − q)Lθ (θ(η;

(35)

Then, Eq. (23) becomes fˆ(0; q) = 0,

 ∂ fˆ(η; q)   ∂η 

ˆ q) = 1, θ(0;

 ˆ q) θ(η;

 ∂ fˆ(η; q)   ∂η 

= 1,

η=0

= 0,

(36)

η→∞

= 0.

(37)

η→∞

When q = 0 and q = 1, we have fˆ(η; 0) = f0 ,

fˆ(η; 1) = f (η),

(38)

ˆ 0) = θ0 , θ(η;

ˆ 1) = θ(η). θ(η;

(39)

ˆ q) vary from When the embedding parameter q increases from 0 to 1, then fˆ(η; q) and θ(η; the initial approximations f0 (η) and θ0 (η) to the exact solution f (η) and θ(η), respectively. ˆ q) in the power series of the embedding By Taylor’s theorem, we can express fˆ(η; q) and θ(η; parameter q as follows: ∞ 

ˆ q) = f0 (η) + f(η;

fm (η)q m ,

(40)

θm (η)q m .

(41)

m=1 ∞ 

ˆ 0) = θ0 (η) + θ(η;

m=1

If hf and hθ are properly chosen such that the series converge at q = 1, then we have f (η) = f0 (η) +

∞ 

fm (η),

(42)

θm (η),

(43)

m=1

θ(η) = θ0 (η) +

∞  m=1

where

 1 ∂ m f (η; q)  fm (η) = , m! ∂q m q=0

 1 ∂ m θ(η; q)  θm (η) = . m! ∂q m q=0

(44)

1532

S. PANIGRAHI, M. REZA, and A. K. MISHRA

To obtain the mth-order deformation equation, we first differentiate the zeroth-order equations (34) and (35) m times with respect to q, then put q = 0, and finally divide the obtained results by m!. The obtained mth-order deformation equations are f Lf (fm (η) − χm fm−1 (η)) = hf Rm (η),

(45)

θ Lθ (θm (η) − χm θm−1 (η)) = hθ Rm (η)

(46)

with the boundary conditions fm (0) = 0,

 fm (0) = 0,

 fm (∞) = 0,

(47)

θm (0) = 0,

 θm (0) = 0,

 θm (∞) = 0,

(48)

where 3 3 f   (η) = (1 + ε)fm−1 − M fm−1 + λθm−1 Rm 2 2 +

−

m−1 

1   (fn fm−1−n − fn fm−1−n ) 2 n=0

m−1  n=0

θ Rm (η) =

 χm =

 εδ   fm−1−n fn−k fk , 3 n

m−1  1  1  θm−1 + (fn θm−1−n + fn θm−1−n ), Pr 2 n=0

0,

m  1,

1,

m > 1.

(49)

k=0

(50)

(51)

∗ ∗ and θm denote the special solutions of Eqs. (45) and (46). From the boundary conditions Let fm (47) and (48), the general solutions can be written as ∗ fm (η) = fm (η) + C1 + C2 η + C3 e−η ,

(52)

∗ θm (0) = θm (η) + C4 + C5 e−η ,

(53)

where C1 , C2 , C3 , C4 , and C5 are the integral constants, whose values will be determined by use of the boundary conditions (47) and (48). The symbolic computation software MATHEMATICA is used to obtain the solution of the mth-order deformation equations.

5

Results and discussion

To analyze the flow and temperature behavior inside the boundary layer, Eqs. (22) and (25) are solved analytically by the HAM. For the convergence of the HAM series solution, we plot the curves of hf and hθ in Fig. 2 for fw = ε = δ = λ = 0,

P r = 0.7.

The admissible ranges of hf and hθ are −0.8  hf  −0.4,

−1.4  hθ  −0.6.

MHD effect of mixed convection boundary-layer flow of Powell-Eyring fluid

Fig. 2

1533

hf and hθ curves for 10th-order HAM series

It is well-known that the transverse magnetic field (M ) can create Lorentz force. It is a resistive force similar to the drag force which will result in the deceleration of the flow, and increases with the increase in M . It can be clearly seen from Figs. 3 and 4 that the fluid velocity decreases while the temperature increases with the increase in M when ε = 1, δ = 1, fw = 0.1, and λ = 0.25. However, when ε = 1, δ = 1, fw = 0.1, and λ = −0.25, the fluid velocity increases while the temperature decreases with the increase in M (see Figs. 5 and 6). Figures 7 and 8 display the effects of buoyancy on the velocity and temperature distribution

Fig. 3

Variations of f  (η) for different M with positive λ

Fig. 4

Variations of θ(η) for different M with positive λ

Fig. 5

Variations of f  (η) for different M with negative λ

Fig. 6

Variations of θ(η) for different M with negative λ

1534

S. PANIGRAHI, M. REZA, and A. K. MISHRA

in the boundary layer when ε = 1.0, δ = 1.0, fw = 0.1, M = 0.25, and P r = 0.7. It can be seen that the fluid velocity increases while the temperature decreases when λ increases. If the opposing flow (λ < 0) increases, f  (η) becomes negative in the outer part of the velocity boundary layer, and this situation is valid as the buoyancy force acts on the opposite direction of the stretching motion of the surface. The fluid flow near the vertical surface is proportional to the upward stretching motion of the surface, which is opposed by the negative buoyancy force, and some part of the energy is lost. Figure 9 shows the patterns of the streamline of the fluid flow for different values of λ, and Fig. 10 displays the contour plots of the temperature distribution for different values of λ. It is observed that the thermal boundary layer decreases with the increase in λ.

Fig. 7

Variations of f  (η) for different λ

Fig. 9

Fig. 8

Variations of θ(η) for different λ

Patterns of stream function ψ(x, y) for different λ

MHD effect of mixed convection boundary-layer flow of Powell-Eyring fluid

Fig. 10

1535

Contour plots of temperature distribution T (x, y) for different values of λ when ε = 1.0, δ = 1.0, fw = 0.1, M = 0.25, P r = 0.7, C = 1, and ν = 1

The effects of the fluid parameter ε on the velocity and temperature with λ = 0.25 and λ = −0.25 when δ = 1.0, fw = 0.1, M = 0.25, P r = 0.7, C = 1, and ν = 1 are shown in Figs. 11–14. Figures 11 and 12 show the results in the case of aiding flow, i.e., λ = 0.25, and Figs. 13 and 14 show the results in the case of the opposing flow, i.e., λ = −0.25. It can be observed that when ε increases, the velocity increases and the temperature decreases inside the boundary layer. The effects of the fluid parameter δ on the velocity and temperature for the fixed values of

Fig. 11

Variations of f  (η) for different ε with positive λ

Fig. 12

Variations of θ(η) for different ε with positive λ

1536

S. PANIGRAHI, M. REZA, and A. K. MISHRA

other quantities are shown in Figs. 15–18. It can be seen that the velocity decreases while the temperature increases with the increase in δ for both the aiding flow and the opposing flow.

Fig. 13

Variations of f  (η) for different ε with negative λ

Fig. 14

Variations of θ(η) for different ε with negative λ

Fig. 15

Variations of f  (η) for different δ with positive λ

Fig. 16

Variations of θ(η) for different δ with positive λ

Fig. 17

Variations of f  (η) for different δ with negative λ

Fig. 18

Variations of θ(η) for different δ with negative λ

Table 1 displays the computed values of two important non-dimensional quantities such as the local skin friction coefficient and the local Nusselt number for different values of the physical parameters λ, fw , M , ε, and β. It can be observed that f  (0) and θ (0) increase when fw increases, f  (0) increases and θ (0) decreases when M increases, f  (0) decreases while θ (0)

MHD effect of mixed convection boundary-layer flow of Powell-Eyring fluid

1537

increases when λ increases, f  (0) decreases while θ (0) increases when ε increases, and f  (0) increases while θ (0) decreases when δ increases. Table 1 λ

Non-dimensional skin friction coefficient f  (0) and local heat transfer coefficient θ (0) fw

M

ε

δ

Pr

f  (0)

θ  (0)

0.00

0.00

0.00

0.00

0.00

0.7

−0.829 945

−0.257 306

−0.50

0.10

0.25

1.00

1.00

0.7

−1.190 600

−0.263 800

−0.25

0.10

0.25

1.00

1.00

0.7

−0.957 800

−0.300 200

0.00

0.10

0.25

1.00

1.00

0.7

−0.771 800

−0.330 700

0.25

0.10

0.25

1.00

1.00

0.7

−0.615 700

−0.349 600

0.50

0.10

0.25

1.00

1.00

0.7

−0.476 900

−0.364 500

0.25

0.10

0.00

1.00

1.00

0.7

−0.468 300

−0.367 500

0.25

0.10

0.25

1.00

1.00

0.7

−0.615 700

−0.349 600

0.25

0.10

0.50

1.00

1.00

0.7

−0.747 300

−0.335 100

−0.25

0.10

0.00

1.00

1.00

0.7

−0.814 300

−0.320 600

−0.25

0.10

0.25

1.00

1.00

0.7

−0.957 800

−0.300 200

−0.25

0.10

0.50

1.00

1.00

0.7

−1.081 200

−0.285 100

0.25

0.10

0.25

1.00

1.00

0.7

−0.615 700

−0.349 600

0.25

0.25

0.25

1.00

1.00

0.7

−0.664 800

−0.431 100

0.25

0.50

0.25

1.00

1.00

0.7

−0.752 300

−0.572 900

0.25

1.00

0.25

1.00

1.00

0.7

−0.949 400

−0.873 500

−0.25

0.10

0.25

1.00

1.00

0.7

−0.957 800

−0.300 200

−0.25

0.25

0.25

1.00

1.00

0.7

−0.994 300

−0.391 700

−0.25

0.50

0.25

1.00

1.00

0.7

−1.065 000

−0.544 400

−0.25

1.00

0.25

1.00

1.00

0.7

−1.239 000

−0.857 200

0.25

0.10

0.25

0.00

1.00

0.7

−0.817 900

−0.324 500

0.25

0.10

0.25

0.25

1.00

0.7

−0.749 700

−0.332 300

0.25

0.10

0.25

0.50

1.00

0.7

−0.695 600

−0.338 800

0.25

0.10

0.25

1.00

1.00

0.7

−0.615 700

−0.349 600

0.25

0.10

0.25

2.00

1.00

0.7

−0.513 400

−0.364 200

−0.25

0.10

0.25

0.00

1.00

0.7

−1.404 000

−0.232 400

−0.25

0.10

0.25

0.25

1.00

0.7

−1.277 900

−0.235 500

−0.25

0.10

0.25

0.5

1.00

0.7

−1.153 200

−0.257 500

−0.25

0.10

0.25

1.00

1.00

0.7

−0.957 800

−0.300 200

−0.25

0.10

0.25

2.00

1.00

0.7

−0.744 600

−0.335 700

0.25

0.10

0.25

1.00

0.00

0.7

−0.607 000

−0.350 100

0.25

0.10

0.25

1.00

0.50

0.7

−0.611 600

−0.350 000

0.25

0.10

0.25

1.00

1.00

0.7

−0.615 700

−0.349 600

0.25

0.10

0.25

1.00

2.00

0.7

−0.625 600

−0.349 000

−0.25

0.10

0.25

1.00

0.00

0.7

−0.918 900

−0.303 200

−0.25

0.10

0.25

1.00

0.50

0.7

−0.937 000

−0.301 800

−0.25

0.10

0.25

1.00

1.00

0.7

−0.957 800

−0.300 200

−0.25

0.10

0.25

1.00

2.00

0.7

−1.013 100

−0.296 700

Figures 19 and 20 show the effects of suction on the velocity and temperature profiles in the boundary layer for the aiding flow, i.e., λ > 0, and Figs. 21 and 22 show the effects of suction on the velocity and temperature profiles in the boundary layer for the opposing flow, i.e., λ < 0. It can be observed that with the use of suction, the velocity and the temperature decrease when

1538

S. PANIGRAHI, M. REZA, and A. K. MISHRA

ε = 1.0, δ = 1.0, M = 0.25, λ = 0.25, and P r = 0.7 for both the aiding flow and the opposing flow.

6

Fig. 19

Variations of f  (η) for different suction with positive λ

Fig. 20

Variation of θ(η) for different suction with positive λ

Fig. 21

Variations of f  (η) for different suction with negative λ

Fig. 22

Variations of θ(η) for different suction with negative λ

Conclusions

The similar solution of a mixed convection flow of a purely viscous non-Newtonian PowellEyring fluid over a nonlinear stretching permeable sheet is studied. The results show that a similarity solution for the mixed convective boundary-layer flow of the Powell-Eyring fluid exists only when the velocity of the vertical stretching surface depends on a nonlinear function 1 of x in the form of x 3 . One parameter linear group transformation is considered to obtain the similarity transform corresponding to the governing momentum and thermal boundary layer equations. The transformed nonlinear ordinary differential equations are solved by the HAM. The values of the local skin friction coefficient f  (0) and the local heat transfer coefficient θ (0) for a wide range of suction, M , λ, ε, and δ are computed and analyzed. From the above results and discussion, the following important observations can be concluded: (i) A similar solution exists for the stretching surface velocity Cx1/3 , the suction velocity −1/3 Dx , the surface temperature T∞ + Ax−1/3 , and the applied magnetic field B0 x−1/3 .

MHD effect of mixed convection boundary-layer flow of Powell-Eyring fluid

1539

(ii) The velocity and temperature inside the boundary layer decrease with the increase in suction. (iii) The increase in the magnetic field results in a decrease in the velocity and an increase in the temperature. (iv) For both aiding and opposing flow, an increase in buoyancy leads to an increase in the velocity and a decrease in the temperature. (v) When ε increases, the velocity increases and the temperature decreases inside the boundary layer. (vi) When δ increases, the velocity decreases and the temperature increases inside the boundary layer. (vii) The local skin friction coefficient increases when any one of ε, suction, and M increases. (viii) The local skin friction coefficient decreases when δ or the mixed convection parameter increases. (ix) The local Nusselt number is an increasing function of δ, suction, and the free convection parameter while a decreasing function of ε and M . Acknowledgements

The authors are thankful to the referees for their critical comments which help in improving the presentation of the paper. The first author acknowledges the facilities and support provided by the National Institute of Science and Technology, Berhampur. The second author acknowledges the Center for Theoretical Studies at Indian Institute of Technology, Kharagpur.

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