Axisymmetric magnetohydrodynamic flow of micropolar fluid between

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Department of Mathematics, King Saud University, P. O. Box 2455,. Riyadh 11451 ... is applied perpendicular to the planes of sheets (parallel to the z- axis).
Appl. Math. Mech. -Engl. Ed., 32(3), 361–374 (2011) DOI 10.1007/s10483-011-1421-8 c Shanghai University and Springer-Verlag Berlin Heidelberg 2011

Applied Mathematics and Mechanics (English Edition)

Axisymmetric magnetohydrodynamic flow of micropolar fluid between unsteady stretching surfaces∗ T. HAYAT1 , M. NAWAZ1 , S. OBAIDAT2 (1. Department of Mathematics, Quaid-I-Azam University 45320, Islamabad 44000, Pakistan; 2. Department of Mathematics, King Saud University, P. O. Box 2455, Riyadh 11451, Saudi Arabia)

Abstract This investigation examines the time dependent magnetohydrodynamic (MHD) flow problem of a micropolar fluid between two radially stretching sheets. Both strong and weak concentrations of microelements are taken into account. Suitable transformations are employed for the conversion of partial differential equations into ordinary differential equations. Solutions to the resulting problems are developed with a homotopy analysis method (HAM). The angular velocity, skin friction coefficient, and wall couple stress coefficient are illustrated for various parameters. Key words micropolar fluid, radial stretching, homotopy analysis solution, skin friction coefficient, wall couple stress coefficient, magnetohydrodynamic Chinese Library Classification O357.1, O361.3 2010 Mathematics Subject Classification 80A20

1

Introduction

During the past few decades, considerable progress has been made in the investigations of non-Newtonian fluids due to their applications in engineering and industry. The rheological characteristics of such fluids are very useful in describing the salient features associated with several fluids in nature like shampoo, ketchup, mud, paints, cosmetic products, etc. Classical Navier-Stokes equations are unable to describe the properties, such as micro-rotaion, spininertia, couple stress, and body torque, which are important in many fluids, for instance, polymeric liquids, liquid crystals, colloidal suspension, animal blood, and fluids containing small amount of polymeric liquids. Eringen[1] presented the theory of non-Newtonian fluids, in which micro-rotaion, spin-inertia, couple stress, and body torque were important. Such fluids are called the micropolar fluids. Steady and unsteady flows of micropolar fluids have been examined by several researchers. Gorla et al.[2] analyzed the combined convection axisymmetric stagnation point flow of a micropolar fluid. Gorla and Takhar[3] investigated the boundary layer flow of a micropolar fluid on a rotating axisymmetric surface with a concentrated heat source. Guram and Smith[4] analyzed the stagnation point flow of a micropolar fluid with strong and weak interactions. Kumari and Nath[5] presented an unsteady boundary layer flow of a micropolar fluid at a stagnation point. Abdullah and Amin[6] treated the flow of blood as a micropolar fluid through a tapered artery with a stenosis. Seddeek[7] examined the flow of a micropolar fluid ∗ Received Aug. 17, 2010 / Revised Dec. 1, 2010 Corresponding author M. NAWAZ, Ph, D., E-mail: nawaz [email protected]

362

T. HAYAT, M. NAWAZ, and S. OBAIDAT

over the continuous moving plate. Nazar et al.[8] studied the steady two-dimensional stagnation point flow of an incompressible micropolar fluid over a stretching sheet. They considered strong and weak concentrations of microelements. They used the Keller Box method for the numerical solution of the governing problem and compared the results with already published work in a limited sense. Takhar et al.[9] employed a finite element method for the flow and heat transfer in the micropolar fluid between two porous disks. Abo-Eldahab and Ghonaim[10] examined the effect of radiation on heat transfer in a micropolar fluid through a porous medium. Nazar et al.[11] studied the free convection flow on an isothermal sphere in a micropolar fluid by taking the boundary layer assumptions into account. Sahoo[12] considered the flow of an electrically conducting non-Newtonian, viscoelastic second grade fluid due to an axisymmetric stretching surface. Again, Sahoo[13] analyzed the effects of slip, viscous dissipation, and Joule heating on the magnetohydrodynamic (MHD) flow and heat transfer of a second grade fluid past a radially stretching sheet. Sahoo[12–13] also investigated the flow over a radially stretching surface. Hayat and Nawaz[14] considered heat transfer characteristics on the axisymmetric steady flow of an electrically conducting viscous fluid between two radially stretching sheets. They used the homotopy analysis method (HAM) to derive analytic solutions. Literature survey reveals that no research regarding the unsteady flow of the micropolar fluid between two surfaces stretching with a time dependent linear velocity has been discussed so far. The present investigation is an attempt in this direction. Both strong and weak concentrations of microelements are taken into account. The solutions to the governing problems are derived by the HAM. The HAM is currently very popular and has been used by many researchers for the solutions to nonlinear problems[15–32] . Finally, the graphical results are shown and analyzed in details.

2

Statement of problem

We consider the axisymmetric flow of an incompressible micropolar fluid between two parallel infinite sheets at z = ±L. Both of the sheets are subjected to the stretching uw = −1 ar (1 − bt) . The symmetric nature of the flow is considered. The magnetic field of the −1/2 form B = B 0 (1 − bt) is applied perpendicular to the planes of sheets (parallel to the zaxis). No electric field (E = 0) is applied. The magnetic Reynolds number is small, and the induced magnetic field is neglected. The physical model and the coordinate system are shown in Fig. 1. u, v, and w are components of velocity V , and N1 , N2 , and N3 are components of micro-rotation vector Ω in the radial, azimuthal, and axial directions, respectively.

Fig. 1

Geometry of problem and coordinate system

Axisymmetric magnetohydrodynamic flow of micropolar fluid

363

The fundamental equations which can govern the present flow are[1] ∇ · V = 0, ρ

(1)

dV = −∇p + (μ + k) ∇2 V +k∇ × Ω + J × B, dt

ρj

(2)

dΩ = (α + β + γ) ∇ (∇ · Ω) − γ∇ × (∇ × Ω) + k∇ × V −2kΩ, dt

J = σ (V × B) ,

(3) (4)

d where dt is the material derivative, V is the velocity field, j is the micro-inertia per unit mass, Ω is the micro-rotation vector, ρ is the fluid density, μ and k are the viscosity coefficients, and α, β, and γ are the gyroviscosity coefficients. Furthermore, μ, k, α, β, and γ satisfy the following constraints[15]:  2μ + k  0, k  0,

3α + β + γ  0,

γ  |β| .

The velocity and micro-rotion fields for the axisymmetric flow are  V = (u (r, z, t) , 0, w (r, z, t)) ,

(5)

Ω = (0, N2 (r, z, t) , 0) ,

in which u and w are the velocity components along the radial (r) and axial (z) directions, respectively. r and z are the radial and axial coordinates, respectively. t represents the time, and N2 is the azimuthal component of the micro-rotation vector Ω. Now, Eqs. (1)–(5) yield ∂u u ∂w + + = 0, ∂r r ∂z

(6)

 ∂ 2 u 1 ∂u ∂ 2 u ∂u ∂u ∂u 1 ∂p 1 u +u +w =− + (μ + k) + 2 − 2 + 2 ∂t ∂r ∂z ρ ∂r ρ ∂r r ∂r ∂z r −

k ∂N2 σB 2 − u, ρ ∂z ρ

(7)

 ∂ 2 w 1 ∂w ∂ 2 w  ∂w ∂w ∂w 1 ∂p 1 +u +w =− + (μ + k) + + ∂t ∂r ∂z ρ ∂z ρ ∂r2 r ∂r ∂z 2   k ∂N2 N2 − + , ρ ∂r r ∂N2 ∂N2 ∂N2 γ  ∂ 2 N2 ∂ 2 N2 1 ∂N2 N2  +u +w = + + − ∂t ∂r ∂z ρj ∂r2 r ∂r ∂z 2 r2   k ∂w ∂u 2N2 + − . − ρj ∂r ∂z The boundary conditions are ⎧ ⎪ ⎪ ⎨ u (r, L, t) = uw ,

w (r, L, t) = 0,

⎪ ⎪ ⎩ ∂u (r, z, t) = 0, ∂z z=0

N2 (r, L, t) = −n

w (r, 0, t) = 0,

∂u , ∂z z=L

N2 (r, 0, t) = 0,

(8)

(9)

(10)

364

T. HAYAT, M. NAWAZ, and S. OBAIDAT

where n (0  n  1) is a constant. Here, n = 0 depicts the situation when microelements at the stretching sheets are unable to rotate (Ω = 0 at stretching surfaces). This is also known as the strong concentration of the microelements. When n = 0.5, the anti-symmetric part of the stress tensor vanishes and denotes the weak concentration of microelements at the sheets. For n = 1, one has the turbulent boundary layer flow[8] . In the present flow analysis, we consider n = 0 and n = 0.5. We non-dimensionalize Eqs. (6)–(10) by utilizing the following transformations: ⎧ ar  ⎪ ⎪ ⎪ u = 1 − bt f (η) , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2aL ⎪ ⎪ f (η) , ⎨w = −√ 1 − bt ar ⎪ ⎪ ⎪ g (η) , N2 = ⎪ ⎪ L (1 − bt) ⎪ ⎪ ⎪ ⎪ z ⎪ ⎪ . ⎩η = √ L 1 − bt

(11)

Using the above transformations and eliminating the pressure gradient, we have ⎧ 1 ⎪ ⎪ (1 + K) f  − SRe (ηf  + 3f  ) + 2Ref f  − Kg  − ReM f  = 0, ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ f (1) = 0, f  (1) = 1, f (0) = 0, f  (0) = 0,  ⎪ ⎪ 1   1 ⎪ ⎪ 1 + K g + ReK (f  − 2g) − SRe (3g + ηg  ) + Re (2f g  − f  g) = 0, ⎪ ⎪ 2 2 ⎪ ⎪ ⎪ ⎩  g (1) = −nf (1) , g (0) = 0,

(12)

where the dimensionless quantities ⎧ k ⎪ ⎪ ⎪ ⎨ K = μ,

Re =

⎪ ⎪ σB02 ⎪ ⎩M = , ρa

aL2 , ν

S=

b , a

respectively, indicate the micropolar parameter K, the Reynolds number Re, the Hartman number M , and the unsteadiness parameter S. For the steady case, S = 0. Then, Eq. (6) is automatically satisfied, and the parameters are[8]

k γ = μ + j, 2

j=

ν(1 − bt) . a

Furthermore, Eqs. (7) and (8) reduce to the classical Navier-Stokes equations when K = 0. In fact, this is the situation when micro-rotation effects are rather small and do not affect the flow.

Axisymmetric magnetohydrodynamic flow of micropolar fluid

365

The skin friction coefficient Cf and wall couple stress Cg at z = L are[9] Cf =

τw 2

 ∂u

(μ + k)

=

−2

2

ρ (uw ) ρ (ar) (1 − bt) (1 + K)  = f (1) , Rer

∂z

+

∂w  ∂r z=L (13)

∂N2 ∂N2 Lγ ∂z z=L = − ∂z z=L Cg = − 2 2 ρ (uw ) ρ (ar) (1 − bt)−2  K 1+ 2 g  (1) , =− Rer Lγ

where Rer =

3

√ar ν 1−bt

(14)

denotes the local Reynold number.

Solution to the problem For the homotopy solutions, we choose the base functions

η 2n+1 , n  0

(15)

and write f (η) = g (η) =

∞  n=0 ∞ 

an η 2n+1 ,

(16)

bn η 2n+1 ,

(17)

n=0

where an and bn are the coefficients. We take the initial guesses of the form ⎧

⎪ ⎨ f0 (η) = 1 η 3 − η , 2 ⎪ ⎩ g (η) = −nf  (1) η

(18)

0

and suggest the following linear operators: ⎧ d4 f ⎪ ⎪ ⎪ ⎨ £1 (f (η)) = dη 4 , ⎪ d2 g ⎪ ⎪ ⎩ £2 (g (η)) = 2 dη

(19)

with the properties £1 (C1 + C2 η + C3 η 2 + C4 η 3 ) = 0,

(20)

£2 (C5 + C6 η) = 0,

(21)

where Ci (i = 1, 2, · · · , 6) are the constants of the integration.

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T. HAYAT, M. NAWAZ, and S. OBAIDAT

3.1 Zeroth-order deformation problems We construct the zeroth-order deformation problems as follows: (1 − q)£1 (Φ(η; q) − f0 (η)) = q1 N1 (Φ (η; q) , Ψ (η; q)) , ⎧ ⎪ ⎪ ⎪ ⎨ Φ(1; q) = 0, ⎪ ⎪ ⎪ ⎩ Φ(0; q) = 0,

∂Φ(η; q) = 1, ∂η η=1 ∂ 2 Φ(η; q) = 0, ∂η 2 η=0

(1 − q)£1 (Ψ(η; q) − g0 (η)) = q2 N2 (Ψ (η; q) , Φ (η; q)) , Ψ(1; q) = −n

∂ 2 Φ(η; q) , ∂η 2 η=1

Ψ(0; q) = 0.

(22)

(23)

(24) (25)

In the above expressions, q ∈ [0, 1] and i = 0 (i = 1, 2) are, respectively, called the embedding and convergence control parameters such that  Φ(η; 0) = f0 (η) , Ψ (η; 0) = g0 (η) , Φ(η; 1) = f (η) ,

Ψ (η; 1) = g (η) .

When q varies from 0 to 1, Φ(η; q) approaches from f0 (η) to f (η), and Ψ(η; q) approaches from g0 (η) to g (η). The nonlinear operators Ni (i = 1, 2) can be given by  ∂ 3 Φ(η; q) ∂ 4 Φ(η; q) 1 ∂ 2 Φ(η; q)  N1 (Φ(η; q), Ψ (η; q)) = (1 + K) SRe η − + 3 ∂η 4 2 ∂η 3 ∂η 2 + 2ReΦ(η; q)

∂ 3 Φ(η; q) ∂ 2 Ψ(η; q) ∂ 2 Φ(η; q) − K − ReM , ∂η 3 ∂η 2 ∂η 2

(26)

  1  ∂ 2 Ψ(η; q) ∂ 2 Φ(η; q)  N2 (Ψ(η; q), Φ (η; q)) = 1 + K − KRe 2Ψ(η; q) − 2 ∂η 2 ∂η 2  ∂Ψ(η; q)  1 − SRe 3Ψ(η; q) + η 2 ∂η  ∂Φ(η; q)  ∂Ψ(η; q) − Ψ(η; q) . + Re 2Φ(η; q) ∂η ∂η

(27)

Expanding Φ(η; q) and Ψ(η; q) in the Taylor series, we have Φ(η; q) = f0 (η) +

∞ 

fm (η) q m ,

(28)

gm (η) q m ,

(29)

m=1

Ψ(η; q) = g0 (η) +

∞  m=1

1 ∂ m Φ(η; q) , m! ∂η m q=0 1 ∂ m Ψ(η; q) gm (η) = . m! ∂η m q=0

fm (η) =

(30) (31)

Axisymmetric magnetohydrodynamic flow of micropolar fluid

3.2 Higher-order deformation problems The mth-order deformation problems are given by  £1 (fm (η) − χm fm−1 (η)) = 1 R1m (fm−1 (η)) , fm (1) = 0, 

fm (0) = 0,

 fm (0) = 0,

£2 (gm (η) − χm gm−1 (η)) = 2 R2m (gm−1 (η)) , gm (1) = 0, 

χm = where

 fm (1) = 0,

gm (0) = 0,

0,

m  1,

1,

m > 1,

⎧ ⎪ ⎪ R1m (fm (η), gm (η)) = ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ R1m (gm (η), fm (η)) = ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

367

(32)

(33)

(34)

 1    − ReM fm−1 (1 + K) fm−1 − SRe 3fm−1 + ηfm−1 2 m−1   + 2Re fn fm−1−n − Kgm−1 , n=0



 1   1 + K gm−1 + ReK fm−1 − 2gm−1 2

1  − SRe 3gm−1 + ηgm−1 2 m−1 

 + Re 2fn gm−1−n − fn gm−1−n . n=0

The solution expressions for mth-order deformation problems are ∗ fm (η) = fm (η) + C1m + C2m η + C3m η 2 + C4m η 2 ,

(35)

∗ (η) + C5m + C6m η, gm (η) = gm

(36)

where Cim (i = 1, 2, · · · , 6) are determined by the boundary conditions given in Eqs. (32) and ∗ ∗ (33), and fm (η) and gm (η) indicate the particular solutions to Eqs. (32) and (33), respectively.

4

Convergence of homotopy solutions

The series solutions are obtained by the HAM convergence control parameters (auxiliary parameters). The auxiliary parameters are employed in the adjustment of convergence regions of the derived series solutions. To examine the effects of auxiliary parameters, we plot the i (i = 1, 2) curves in Figs. 2 and 3. It can be noted that our series solutions converge when −1.2  1 and 2  −0.45. However, whole forthcoming analyses have been carried out for the fixed value −0.8 for 1 and 2 . Table 1 shows the convergence of derived series solutions. This table shows that convergence is achieved at the 20th-order of approximations up to 12 decimal places.

5

Results and discussion

In this section, we examine the influence of the Hartman number M , the Reynold number Re, the unsteadiness parameter S, and the micropolar parameter K on the radial component f  (η), and the axial component f (η) of the velocity and azimuthal component g (η) of the angular velocity.

368

T. HAYAT, M. NAWAZ, and S. OBAIDAT

1 curve of f  (1) with M = Re = 1 and S = K = 0.2

Fig. 2

Table 1

2 curve of g  (1) with M = Re = 1 and S = K = 0.2

Convergence of HAM solutions with Re, M = 2, S, K = 0.2, n = 0.5, and 1 , 2 = −0.8

Order of approximation 1

Fig. 3

f  (1)

g  (1)

3.950 857 142 86

–3.575 428 571 43

5

3.886 969 275 34

–3.605 904 554 66

10

3.887 104 492 42

–3.606 052 732 54

15

3.887 104 582 28

–3.606 052 850 00

20

3.887 104 582 33

–3.606 052 850 11

25

3.887 104 582 33

–3.606 052 850 11

30

3.887 104 582 33

–3.606 052 850 11

35

3.887 104 582 33

–3.606 052 850 11

The effects of M on f (η) and f  (η) are shown in Figs. 4 and 5. From these figures, it can be seen that the magnitudes of f (η) and f  (η) decrease with an increase in M. This is due to the fact that the Lorentz force retards the flow in both radial and axial directions.

Fig. 4

Influence of M on f (η) with Re = 3, K = 0.5, S = 0.5, and n = 0.5

Fig. 5

Influence of M on f  (η) with Re = 2.5, K = 0.5, S = 0.2, and n = 0.5

Figures 6 and 7 illustrate the effect of magnetic field on the dimensionless angular velocity g (η) for both the cases n = 0 (strong concentration of microelements) and n = 0.5 (weak concentration of microelements). We observe from these figures that for the situation when microelements are able to rotate at the stretching sheet (n = 0.5), the effects of M on g (η) is opposite to those of M on g (η) when n = 0 (when microelements are unable to rotate at the stretching sheet).

Axisymmetric magnetohydrodynamic flow of micropolar fluid

Fig. 6

Influence of M on g (η) with S = 0.2, Re = 1, K = 0.5, and n = 0.5

Fig. 7

369

Influence of M on g (η) with S = 0.2, Re = 1, K = 0.5, and n = 0

The magnitudes of f (η) and f  (η) are the decreasing functions of Re (see Figs. 8 and 9). It is found from Fig. 10 that for n = 0.5, Re is more influential on g (η) in the vicinity of the stretching sheet than on g (η) away from the stretching sheet. It is noted from Fig. 11 that for n = 0, there is an increase in g (η) when Re increases.

Fig. 8

Influence of Re on f (η) with M = 2, K = 0.5, S = 0.2, and n = 0.5

Fig. 9

Fig. 10

Influence of Re on g(η) with M = 3, K = 0.5, S = 0.2, and n = 0.5

Fig. 11

Influence of Re on f  (η) with M = 3, K = 0.5, S = 0.2, and n = 0.5

Influence of Re on g(η) with M = 3, K = 0.5, S = 0.2, and n = 0

Upon making an increase in K, there is an increase in the magnitude of the dimensionless axial component f (η) and the radial component f  (η) as shown in Figs. 12 and 13. Figure 14 depicts that the magnitude of the angular velocity g (η) decreases by increasing the micropolar parameter K when n = 0.5. However, it increases when n = 0, see Fig. 15.

370

Fig. 12

Fig. 14

T. HAYAT, M. NAWAZ, and S. OBAIDAT

Influence of K on f (η) with M = 2, Re = 3, S = 0.2, and n = 0.5

Influence of K on g(η) with M = 2, Re = 1, S = 0.5, and n = 0.5

Fig. 13

Fig. 15

Influence of K on f  (η) with M = Re = 4, S = 0.2, and n = 0.5

Influence of K on g(η) with M = 2, Re = 1, S = 0.5, and n = 0

The effects of the unsteadiness parameter S on f (η), f  (η), and g (η) are shown in Figs. 16–19. Figures 16 and 17 reveal that the magnitudes of f (η) and f  (η) are the increasing functions of the unsteadiness parameter S whereas g (η) decreases with an increase in S for both cases n = 0.5 and n = 0 (see Figs. 18 and 19).

Fig. 16

Influence of S on f (η) with M = Re = 1, and n = 0.5

Fig. 17

Influence of S on f  (η) with M = Re = 4, and n = 0.5

Table 2 represents the variation of the skin friction coefficients for the steady flow (S = 0) and the unsteady flow (S = 0) for the following two cases. (i) Microelements close to the stretching sheet are unable to rotate (n = 0), i.e., strong concentration of microelements; (ii) Microelements close to the stretching sheet are able to rotate (n = 0.5, i.e., weak concentration of microelements).

Axisymmetric magnetohydrodynamic flow of micropolar fluid

Fig. 18

Influence of S on g(η) with M = 4, Re = 3, K = 0.5, and n = 0.5

Fig. 19

371

Influence of S on g(η) with M = Re = 3, K = 0.5, and n = 0

This table indicates that the skin friction coefficient Rer Cf is an increasing function of M, Re, and K. Furthermore, the skin friction coefficient Rer Cf for the unsteady flow (S = 0) is larger than that for the steady flow (S = 0). The skin friction coefficient for the case of strong concentration of microelements (n = 0) is higher than that when n = 0.5 (weak concentration). The skin friction coefficient Rer Cf for magnetohydrodynamic flow (M = 0) is larger than that for the hydrodynamic flow (M = 0). The skin friction coefficient Rer Cf increases when Re increases. Furthermore, Rer Cf for the case of the dominant inertial force (Re > 1) is larger than that when viscous force is dominant (Re < 1). The skin friction coefficient Rer Cf is an increasing function of the micropolar parameter K. Obviously, the skin friction coefficient Rer Cf for non-Newtonian fluid (K = 0) is larger than that for the viscous fluid (K = 0). Table 2

Numerical values of skin friction coefficient Rer Cf Rer Cf

M

Re

K

S = 0 (steady flow)

S = 0.2 (unsteady flow)

n=0

n = 0.5

n=0

n = 0.5

0.0

2.0

0.2

3.824 730 202 38

3.787 266 918 78

3.991 203 589 26

3.942 399 350 89

1.0

2.0

0.2

4.204 454 230 54

4.169 930 620 67

4.364 616 303 21

4.319 074 412 17

2.0

2.0

0.2

4.552 210 644 58

4.520 128 178 77

4.707 324 965 12

4.664 525 498 79

3.0

2.0

0.2

4.873 952 292 20

4.843 930 679 76

5.024 943 926 74

4.984 485 546 25

2.0

0.5

0.2

3.849 616 513 65

3.840 084 532 69

3.889 364 888 51

3.876 475 010 00

2.0

1.5

0.2

4.325 559 753 65

4.300 118 946 60

4.442 916 612 55

4.408 832 700 45

2.0

2.5

0.2

4.771 684 251 32

4.733 682 288 48

4.963 867 905 99

4.913 371 653 56

2.0

3.5

0.2

5.190 461 303 92

5.142 413 975 60

5.454 879 450 55

5.391 4814 31 78

2.0

2.0

0.0

3.951 059 166 98

3.951 059 166 98

4.104 398 089 67

4.104 398 089 67

2.0

2.0

0.1

4.254 815 226 04

4.241 207 712 27

4.408 963 488 44

4.389 546 179 93

2.0

2.0

0.2

4.552 210 644 58

4.520 128 178 77

4.707 324 965 12

4.664 525 498 79

2.0

2.0

0.3

4.844 354 453 13

4.789 883 318 62

5.000 515 621 61

4.931 124 431 78

Table 3 is constructed for the influence of physical parameters on the wall couple stress coefficient Rer Cg . This table indicates that the wall couple stress increases by increasing M , Re, and K for both steady (S = 0) and unsteady (S = 0) flows. The wall couple stress Rer Cg for steady flows (S = 0) is larger than that for unsteady flows (S = 0) when n = 0, whereas wall couple stress coefficient Rer Cg for unsteady flows is larger in comparison with Rer Cg for the steady flow when n = 0.5. From Table 3, one can see that for K = 0 (Newtonian fluid), Rer Cg = 0 for both steady (S = 0) and unsteady (S = 0) flows when n = 0. This is due to

372

T. HAYAT, M. NAWAZ, and S. OBAIDAT

the fact that for K = 0, Eqs. (7) and (8) reduce to classical Navier-Stokes equations which are unable to exhibit the property like couple stress. Table 3

Numerical values of couple stress coefficient Rer Cg Rer Cg

M

Re

K

S = 0 (steady flow)

S = 0.2 (unsteady flow)

n=0

n = 0.5

n=0

n = 0.5

0.0

2.0

0.2

0.387 278 980 109

3.262 953 247 28

0.383 145 930 553

3.688 737 851 10

1.0

2.0

0.2

0.387 543 251 367

3.407 709 588 77

0.383 603 165 498

3.833 134 884 47

2.0

2.0

0.2

0.387 797 110 015

3.541 403 629 83

0.384 023 949 443

3.966 658 135 13

3.0

2.0

0.2

0.388 039 304 423

3.666 081 339 07

0.384 412 418 868

4.091 282 931 10

2.0

0.5

0.2

0.099 220 968 475

2.151 034 362 28

0.098 934 723 438

2.264 765 356 84

2.0

1.5

0.2

0.293 069 065 641

3.095 143 186 66

0.290 798 273 461

3.420 856 571 47

2.0

2.5

0.2

0.481 151 680 563

3.972 378 971 88

0.475 653 829 432

4.493 614 049 36

2.0

3.5

0.2

0.664 043 350 895

4.793 426 729 77

0.654 723 551 751

5.497 746 427 37

2.0

2.0

0.0

0.000 000 000 000

2.679 778 360 57

0.000 000 000 000

3.136 805 493 37

2.0

2.0

0.2

0.197 854 021 227

3.119 117 059 25

0.195 587 393 503

3.558 478 312 59

2.0

2.0

0.4

0.387 797 110 015

3.541 403 629 83

0.384 023 949 443

3.966 658 135 13

2.0

2.0

0.6

0.571 285 510 760

3.950 187 243 78

0.566 534 046 078

4.363 971 237 75

6

Conclusions

This work investigates the flow of an incompressible micropolar fluid between radially stretching sheets. Convergent series solutions are derived through the homotopy analysis. The following observations have been noted from the performed analysis method: (i) The effect of the micropolar parameter K on f (η) and f  (η) is opposite to that of M and Re on f (η) and f  (η). (ii) The effect of the unsteadiness parameter S on f (η), f  (η), and g (η) is similar in qualitative sense. (iii) The variations of M and K on the dimensionless angular velocity g (η) for the case of strong concentration (n = 0) is opposite to that on the dimensionless angular velocity g (η) for the case of weak concentration (n = 0.5). However, Re and S have similar effects on g (η) when n = 0 and n = 0.5. (iv) The boundary layer increases when M increases. (v) The skin friction coefficient Rer Cf and the couple stress coefficient Rer Cg are the increasing functions of Re, M , S, and K when n = 0 and n = 0.5. (vi) The skin friction coefficient Rer Cf for n = 0.5 is less than that when n = 0. (vii) The skin friction coefficient Rer Cf for the MHD flows (M = 0) is larger when compared with that for the hydrodynamic flow M = 0. (viii) The skin friction coefficient Rer Cf in a steady flow (S = 0) is smaller than that in an unsteady flow (S = 0). (ix) The wall couple stress Rer Cg for an unsteady flow (S = 0) is smaller than that for the steady flow (S = 0) when n = 0. The wall couple stress Rer Cg for the unsteady flow is larger than the wall couple stress Rer Cg in the steady flow when n=0.5. Acknowledgements We express our sincere thanks to the Higher Education Commission (HEC) of Pakistan for the financial support of the present research. The first author (as a visiting professor) also thanks the King Saud University (KSU) of Saudi Arabia for the support under project number KSU-VPP-103. Authors are also thankful to the reviewers for their useful suggestions.

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