Numerical investigation of the flow of a micropolar fluid through a porous channel with expanding or contracting walls

Propulsion and Power Research 2014;3(3):133–142

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ORIGINAL ARTICLE

Numerical investigation of the flow of a micropolar fluid through a porous channel with expanding or contracting walls M.T. Darvishia, F. Khania,n, F.G. Awadb, A.A. Khidirb, P. Sibandab Department of Mathematics, Razi University, Kermanshah 67149, Iran School of Mathematical Sciences, University of KwaZulu-Natal, Private Bag X01, Scottsville, Pietermaritzburg 3209, South Africa a

b

Received 4 November 2013; accepted 28 January 2014 Available online 3 October 2014

KEYWORDS Homotopy analysis method; Spectral collocation method; Series solution; Micropolar fluid

Abstract In this paper, we study the flow of a micropolar fluid in a porous channel with expanding or contracting walls. First, we use spectral collocation method on the governing equations to obtain an initial approximation for the solution of equations. Then using the obtained initial approximation, we apply the homotopy analysis method to obtain a recursive formula for the solution. & 2014 National Laboratory for Aeronautics and Astronautics. Production and hosting by Elsevier B.V. All rights reserved.

1. Introduction Most nonlinear models of real-life problems are still very difficult to solve, either numerically or theoretically. In recent years, the investigation of the traveling wave solutions of nonlinear partial differential equations has begun to play an important role in the study of nonlinear physical phenomena. Nonlinear wave phenomena appear in various scientific and engineering fields, such as fluid mechanics, plasma physics, n

Corresponding author: Tel.: þ98 9124984928. E-mail address: [email protected] (F. Khani).

Peer review under responsibility of National Laboratory for Aeronautics and Astronautics, China.

optical fibers, biology, solid state physics, chemical kinematics, chemical physics and geochemistry. One of the most exciting advances of nonlinear science and theoretical physics has been the development of methods that look for exact solutions of nonlinear evolution equations. The availability of symbolic computational tools such as Mathematica and Maple has popularized the search for exact solutions of nonlinear equations. Several recent attempts have been made to develop new techniques for obtaining analytical solutions which reasonably approximate the exact solutions. Such techniques include the homotopy analysis method (HAM) [1–3], the Hirota method and Rimann theta function [4]. The HAM has been used successfully to solve a variety of nonlinear differential equations. However, the HAM suffers from

2212-540X & 2014 National Laboratory for Aeronautics and Astronautics. Production and hosting by Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jppr.2014.07.002

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a number of restrictive measures, such as the requirement that the solution sought ought to conform to the so-called rule of solution expression and the rule of coefficient ergodicity. In a recent study, Motsa and Sibanda et al. [5,6] proposed a spectral modification of the homotopy analysis method, the spectral-homotopy analysis method (SHAM) that seeks to remove some restrictive assumptions associated with the implementation of the standard homotopy analysis method. The aim of this paper is to find solutions of the flow of a micropolar fluid through a porous channel with expanding or contracting walls. The study of the flow of a micropolar fluid in porous media has been active filed of research. Its theory (see Eringen [7,8]) derives from the need to model the flow of fluids that contain rotating micro-constituents. The rotation and gyration of the micro-constituents has a profound effect on the hydrodynamics of the flow [9] and the usual Navier-Stokes equations cannot adequately describe the motion of such fluids. The concept of micropolar fluid has been used in the investigation of various fluids, such as the flow of low concentration suspensions, liquid crystals [10], polymeric fluids and blood [11] as well as fluids with additives and turbulent shear flows [9].

2. Mathematical formulation We consider the motion of an incompressible flow of a micropolar fluid between two contracting or expanding, porous disks and neglect the effects of body forces and body couples. Assuming the flow to be fully developed. The distance between the disks is 2aðtÞ. The disks have the same permeability and expand or contract uniformly at a time-dependent rate ̇aðtÞ. The velocity component u; w are taken to be in the r; z direction, ϕ is the microrotation component, respectively. A geometry of the problem is given in Figure 1. Under these assumptions, the continuity and the momentum equations are given by the following relations, respectively [12]: ∂u u ∂w þ þ ¼ 0; ∂r r ∂z

ð1Þ

∂u ∂u ∂u þu þw ∂t ∂r ∂z ¼


 1 ∂p μ þ κ ∂2 u 1 ∂u u ∂2 u κ ∂ϕ þ  ; þ þ  2 2 2 ρ ∂r ρ ∂r r ∂r r ∂z ρ ∂z ð2Þ

∂w ∂w ∂w þu þw ∂t ∂r ∂z

 1 ∂p μ þ κ ∂2 w 1 ∂w ∂2 w κ ∂ϕ ϕ þ þ 2 þ þ ¼ þ ; ρ ∂z ρ ∂r 2 r ∂r ∂z ρ ∂r r ð3Þ ∂ϕ ∂ϕ ∂ϕ þu þw ∂t ∂r
∂z  γ ∂2 ϕ 1 ∂ϕ ∂2 ϕ ϕ κ ∂u ∂w κ þ ¼ þ  þ ð  Þ  2 ϕ; jρ ∂r 2 r ∂r ∂z2 r 2 jρ ∂z ∂r jρ ð4Þ Where ρ is the density, and j; γ; κ are the microinertial per unit mass, spin gradient viscosity and vortex viscosity, respectively. Here γ is assumed to be  κ γ ¼ μ þ j; 2 in which μ is the dynamic viscosity and we take j ¼ a2 as the reference length. According to [13], we also assume that there is the strong concentration of microelements and the microelements close to the wall are unable to rotate. The problem has the following boundary conditions: u ¼ 0;

w ¼ 2vw ¼ Ȧa;

u ¼ 0;

w ¼  2vw ¼  Ȧa;

ϕ ¼ 0;

z ¼ aðtÞ;

ϕ ¼ 0;

z ¼  aðtÞ;

Where A ¼ 2vw =ȧ is the measure of wall permeability [14]. We introduce the following similarity transformations that are motivated by the definition of the stream function (see Si et al. [15]) 1 ∂ψ νr 1 ∂ψ 2ν ¼  2 F η ðη; tÞ; w ¼ ¼ Fðη; tÞ; u¼  ð5Þ r ∂z 2a r ∂r a νr z ð6Þ gðη; tÞ; η ¼ ; 3 a a Where F is the dimensionless velocity and g is the dimensionless microrotation velocity. Substituting Eq. (5) and Eq. (6) into Eqs. (1)–(4) and eliminating pressure, one obtains the following nonlinear partial differential equations, ϕ¼ 

Figure 1 Geometry of the problem.

ð1 þ KÞF ηηηη þ αð3F ηη þ ηF ηηη Þ  2FF ηηη  KGηη  a2 ν  1 F ηηt ¼ 0;

ð7Þ

K ð1 þ ÞGηη þ αð3G þ ηGη Þ þ KF ηη  2KG 2 þ F η G  2FGη  a2 ν  1 Gt ¼ 0;

ð8Þ

Numerical investigation of the flow of a micropolar fluid through a porous channel

Where K ¼ κ=μ and α is the wall expansion ratio defined by α ¼ aȧ=ν and the expansion ratio is positive for expansion and negative for contraction, Re ¼ avw =ν is the permeability Reynolds number and positive for injection. Let f g ; G¼ ; ð9Þ F¼ Re Re substituting Eq. (9) into Eqs. (7) and (8), we have ð1 þ KÞf ηηηη þ αð3f ηη þ ηf ηηη Þ  2Ref f ηηη  Kgηη  a2 ν  1 f ηηt ¼ 0;

ð10Þ

ð11Þ

A similar solution with respect to both space and time can be developed following the transformation described by Uchida and Aoki [16], Majdalani et al. [17] and Majdalani et al. [18], respectively. This can be accomplished by considering in the case: α is a constant and f ¼ f ðηÞ, it leads to f ηηt ¼ 0. Similarly, in the present paper we also assume that g ¼ gðηÞ. From a physical standpoint [16–18], our idealization is based on a decelerating expansion rate that follows a plausible model according to which ȧa ȧ0 a0 α¼ ¼ ¼ constant; ð12Þ ν ν where a0 and ȧ0 denote the initial channel height and expansion velocity, respectively. As a result, the rate of expansion decreases as the internal radius increases integrating Eq. (12) with respect to time, the similar solution can be achieved. The result is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a vw ð0Þ ¼ 1 þ 2ναta0 2 : : ¼ vw ðtÞ a0 Since 2vw ¼ Aȧ and A ¼ constant [17,18], then the expression for the injection velocity also can be determined. Under these provisions, the differential Eqs. (10) and (11) become ð1 þ KÞf iv þ αð3f ″ þ ηf ‴Þ Kg″  2Ref f ‴ ¼ 0;

ð13Þ

ð1 þ K=2Þg″ þ αð3g þ ηg0 Þ þ Kf ″  2Kg þ Reðf 0 g 2f g0 Þ ¼ 0;

ð14Þ

f ð 1Þ ¼  1; f ð1Þ ¼ 1;

f ð 1Þ ¼ 0

f 0 ð1Þ ¼ 0

and gð 1Þ ¼ 0;

FðηÞ ¼ f ðηÞ  f 0 ðηÞ

ð17Þ

where 1 3 f 0 ðηÞ ¼  η3 þ η 2 2

and g0 ðηÞ ¼ 1 η2 ;

ð1 þ KÞF iv þ a1 F‴ þ a2 F″ þ a3 F  KG″  2ReFF‴ ¼ ϕ1 ðηÞ;

ð18Þ

ð19Þ


 K 1þ G″ þ b1 G0 þ b2 G þ KF″ þ b3 F 0 þ b4 F 2 þ ReðF 0 G  2FG0 Þ ¼ ϕ2 ðηÞ;

ð20Þ

subject to the following boundary conditions: Fð 1Þ ¼ F 0 ð 1Þ ¼ 0 0

Fð1Þ ¼ F ð1Þ ¼ 0

and

Gð  1Þ ¼ 0;

Gð1Þ ¼ 0;

and

where a1 ¼ αη  2Ref 0 ; a2 ¼ 3α; a3 ¼  2Ref ‴0 ; b1 ¼ αη  2Ref 0 ; 2 ¼ 3α  2K þ Ref 0 ; b3 ¼ Reg0 ; b4 ¼  2Reg00 ;   ϕ1 ¼  ð1 þ KÞf iv0 þ αð3f ″0 þ ηf ‴0 Þ Kg″0  2Ref 0 f ‴0
 K ″ ϕ2 ¼  1 þ g þ αð3g0 þ ηg00 Þ þ Kf ″0  2Kg 2 0  þReðf 00 g0  2f 0 g00 Þ The initial approximation is taken to be the solution of the nonhomogeneous linear part of the governing Eqs. (19) and (20) 0

ð21Þ

þ b4 F 0 ¼ ϕ2 ðηÞ;

ð22Þ

ð1 þ KÞF iv0 þ a1 F ″0 þ a2 F ″0 þ a3 F 0  KG″0 ¼ ϕ1 ðηÞ;
 K 0 0 1þ G″0 þ b1 G0 þ b2 G0 þ KF ″0 þ b3 F 0 2

ð15Þ

0

F 0 ð 1Þ ¼ F 0 ð  1Þ ¼ 0 0

and gð1Þ ¼ 0:

GðηÞ ¼ gðηÞ g0 ðηÞ;

and

subject to the boundary conditions:

and the boundary conditions change to 0

method. This method has been applied to solve the system of differential Eqs. (13) and (14) subject to the boundary conditions Eqs. (15) and (16). Implementation of the SHAM involves simplifying the governing equations by making use of the following transformations:

are initial approximations that are chosen to satisfy the boundary conditions. Substituting Eqs. (17) and (18) in the governing Eqs. (13) and (14) gives


 K 1þ g þ αð3g þ ηgη Þ þ Kf ηη  2Kg 2 ηη þ Reðf η g  2f gη Þ  a2 ν  1 gt ¼ 0:

135

ð16Þ

3. Spectral homotopy analysis solution The spectral homotopy analysis method developed by Motsa et al. [5] blends Chebyshev pseudo-spectral collocation method with some aspects of the homotopy analysis

F 0 ð1Þ ¼ F 0 ð1Þ ¼ 0

and

and

G0 ð 1Þ ¼ 0;

G0 ð1Þ ¼ 0:

We use the Chebyshev pseudospectral method to solve Eqs. (19) and (20). The unknown functions F 0 ðηÞ and G0 ðηÞ are approximated as a truncated series of Chebyshev polynomials of the form N

F 0 ðηÞ  F N0 ðηj Þ ¼ ∑ F^ k T k ðηj Þ; k¼0 N

G0 ðηÞ  GN0 ðηj Þ ¼ ∑ G^ k T k ðηj Þ; k¼0

j ¼ 0; 1; :::; N;

ð23Þ

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M.T. Darvishi et al.

Figure 2 ℏ-curves on different order for the functions (a) f and (b) g when K ¼ 3; Re ¼ 5 and α ¼ 10.

^ k , are where T k is the k th Chebyshev polynomial, F^ k and G coefficients and η0 ; η1 ; :::; ηN are Gauss-Lobatto collocation points (see Canuto et al. [19]) defined by πj ð24Þ ηj ¼ cos ; j ¼ 0; 1; 2; :::; N: N The derivatives of the functions F 0 ðηÞ and G0 ðηÞ at the collocation points ηj are given by N ds F 0 ¼ ∑ Dskj F 0 ðηj Þ; s dη k¼0

N ds G 0 ¼ ∑ Dskj G0 ðηj Þ; s dη k¼0

ð25Þ

Where s is the order of differentiation and D is the Chebyshev spectral differentiation matrix which can be expressed as follows (see Don and Solomonoff [20]) Dkj ¼ 

1 ck ð 1Þkþj ; π π 2 cj sin 2N ðj þ kÞ sin 2N ðj  kÞ

1 Dkj ¼  2

cos πk N ; sin 2 πk N

ka0

ja k

Figure 3 Effect of α on (a) f ðηÞ and (b) f 0 ðηÞ when K ¼ 4 and Re ¼  5.

D00 ¼  DNN ¼

2N 2 þ 1 ; 6

N þ 1; …; N; 2 Here c0 ¼ cN ¼ 2 and cj ¼ 1 with 1 r jr N  1. Substituting Eqs. (23) - (25) into Eqs. (21) and (22) gives Dkj ¼  DN  k;N  j ;

k¼

AU 0 ¼ Φ; subject to the boundary conditions N

∑ D0k F 0 ðηk Þ ¼ 0;

k¼0

N

∑ DNk F 0 ðηk Þ ¼ 0;

k¼0

F 0 ðη0 Þ ¼ 0 and F 0 ðηN Þ ¼ 0; G0 ðη0 Þ ¼ 0 and G0 ðηN Þ ¼ 0; where A¼

ð1 þ KÞD4 þ a1 D3 a2 D2 þ a3

 1 þ K2 D2 þ b1 D þ b2

 KD2 KD2 þ b3 D þ b4

! ;

Numerical investigation of the flow of a micropolar fluid through a porous channel

^ þa3 þ F^  K G;
 2 ^ ^   ^ þ K F^ ^ qÞ; Gðη; qÞ ¼ 1 þ k ∂ G þ b1 ∂G þ b2 G LG Fðη; 2 2 ∂η ∂η

U 0 ¼ ½F 0 ðη0 Þ; :::; F 0 ðηN Þ; G0 ðη0 Þ; :::; G0 ðηN ÞT ; Φ ¼ ½ϕ1 ðη0 Þ; :::; ϕ1 ðηN Þ; ϕ2 ðη0 Þ; :::; ϕ2 ðηN ÞT ; ai ¼ diagð½ai ðη0 Þ; ai ðη1 Þ; :::; ai ðηN ÞÞ; bi ¼ diagð½bi ðη0 Þ; bi ðη1 Þ; :::; bi ðηN ÞÞ; where the superscript T denotes the transpose, and diagð:; …; :Þ is a diagonal matrix of size ðN þ 1Þ  ðN þ 1Þ. The values of U 0 ¼ ½U 0 ðη0 Þ; U 0 ðη1 Þ; :::; U 0 ðηN ÞT are then determined from the equation U0 ¼ A  1 Φ

ð26Þ

which provides us the initial approximation for the SHAM solution of the governing equations. We select the linear operators ^ ^ ^   ^ qÞ ¼ ð1 þ kÞ ∂ F þ a1 ∂ F þ a2 ∂ F ^ qÞ; Gðη; LF Fðη; 4 3 ∂η ∂η ∂η2 4

3

137

2

Figure 4 Effect of α on (a) f ðηÞ and (b) f 0 ðηÞ when K ¼ 1 and Re ¼  5.

þb3

∂F^ ^ þ b4 F; ∂η

^ qÞ where qA ½0; 1 is the embedding parameter, and Fðη; ^ qÞ are unknown functions. The zeroth order and Gðη; deformation equations are given by ^ qÞ F 0 ðηÞ ¼ qℏ½N F ½η; q; N G ½η; q ϕ1 ; ð1 qÞLF ½Fðη; ^ qÞ G0 ðηÞ ¼ qℏ½N F ½η; q; N G ½η; q  ϕ2 ; ð1 qÞLG ½Gðη; where ℏ is the non-zero convergence controlling auxiliary parameter and N F and N G are nonlinear operators given by N F ½Fðη; qÞ; Gðη; qÞ ¼ ð1 þ KÞ

∂4 F^ ∂3 F^ ∂2 F^ þ a1 3 þ a2 2 4 ∂η ∂η ∂η

þ a3 F^  K

∂3 G ∂3 F^  2ReF^ 3 ; 2 ∂η ∂η

Figure 5 Effect of α on gðηÞ at K ¼ 4 when (a) Re ¼  5 and (b) Re ¼ 5.

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M.T. Darvishi et al.

m1

and ^ ^ K ∂2 G ∂G ^ þ b2 G N G ½Fðη; qÞ; Gðη; qÞ ¼ ð1 þ Þ 2 þ b1 2 ∂η" ∂η # ∂2 F^ ∂F^ ∂F^ ^ ∂G^ ^ ^ þ b4 F þ Re G  2F  K 2 þ b3 ; ∂η ∂η ∂η ∂η Differentiating above equations m times with respect to q and then setting q ¼ 0 and finally dividing the resulting equations by m! yields the m th order deformation equations LF ½F m ðηÞ χ m F m  1 ðηÞ ¼ ℏRFm ; LG ½Gm ðηÞ χ m Gm  1 ðηÞ ¼ ℏRG m; subject to the boundary conditions F m ð 1Þ ¼ F 0m ð 1Þ ¼ 0; F m ð1Þ ¼ F 0m ð1Þ ¼ 0;

0

Gm ð  1Þ ¼ 0;

0

Gm ð1Þ ¼ 0;

where

 2Re ∑ ðF m F ‴m  1  n Þ ϕ1 ðηÞð1  χ m Þ; n¼0
 K 0 G Rm ¼ 1 þ G″m þ b1 Gm þ b2 Gm þ KF ″m 2 m1

0

0

0

þb3 F m þ b4 F m þ Re ∑ ðF m Gm  1  n  2F m Gm  1  n Þ n¼0

 ϕ2 ðηÞð1 χ m Þ; where ( χm ¼

0;

mr1

1;

m41

:

Applying the Chebyshev pseudo-spectral transformation on above equations gives AU m ¼ ðχ m þ ℏÞAU m  1  ℏð1 χ m ÞΦ þ ℏQm  1 ;

ð27Þ

with the boundary conditions

RFm ¼ ð1 þ KÞF ivm þ a1 F ‴m þ a2 F ″m þ a3 F m  KG″m

Figure 6 Effect of Re on (a) f ðηÞ and (b) f 0 ðηÞ when K ¼ 2 and α ¼  2.

Figure 7 Effect of Re on (a) f ðηÞ and (b) f 0 ðηÞ when K ¼ 2 and α ¼ 2.

Numerical investigation of the flow of a micropolar fluid through a porous channel

F m ðη0 Þ ¼ 0;

N

∑ D0k F m ðηj Þ ¼ 0;

k¼0

N

∑ DNk F m ðηj Þ ¼ 0;

k¼0

139

~ m1 U m ¼ ðχ m þ ℏÞA  1 AU 1 þA ℏ½Qm  1  ð1  χ m ÞΦ;

ð30Þ

ð28Þ F m ðηN Þ ¼ 0;

Gm ðη0 Þ ¼ 0;

Gm ðηN Þ ¼ 0:

ð29Þ

and U m ¼ ½F m ðη0 Þ; ::; F m ðηN Þ; Gm ðη0 Þ; :::; Gm ðηN ÞT ; m1

Q1;m  1 ¼  2Re ∑ ½F n ðDF m  1  n Þ; n¼0 m1

where Qm  1 ¼ ½Q1;m  1;

Q2;m  1 T :

Thus, starting from the initial approximation, which is obtained from Eq. (26), higher order approximations U m ðηÞ for m Z 1, can be obtained through the recursive formula (30).

Q2;m  1 ¼ Re ∑ ½ðDF n ÞGm  1  n  2F n ðDGm  1  n Þ: n¼0

The boundary conditions Eqs. (28) and (29) are implemented in matrix A on the left hand side of Eq. (27) in rows 1; 2; :::; N; N þ 1; N þ 2; :::; 2ðN þ 1Þ, respectively, as before with the above initial solution. The corresponding rows, all columns of A on the right hand side of Eq. (27) Q1 and Q2 are all set to be zero. This results in the following recursive formula for m Z 1

Figure 8 α ¼ 2.

Effect of Re on gðηÞ at K ¼ 5 when (a) α ¼  2 and (b)

4. Convergence of the SHAM solution As in the case of the standard homotopy analysis method, the convergence of the SHAM depends on a careful selection of the auxiliary parameter ℏ which controls the convergence of the series solutions. The standard way of

Figure 9 α ¼  2.

Effect of K on (a) f ðηÞ and (b) f 0 ðηÞ when Re ¼ 5 and

140

M.T. Darvishi et al.

choosing admissible values of ℏ that ensure convergence of the approximate series solution is to select a value of ℏ on the horizontal segment of the so-called ℏ-curves. Sibanda and Motsa et al. [6] have suggested that the optimal value of ℏ to use corresponds to the turning point of the second order ℏ-curve. In Figure 2, we show the ℏ curves for different orders of the SHAM approximation. The optimal value of ℏ that gave the most accurate results is the value at which the maximum or minimum of the second order SHAM ℏ-curve is located. We also observe that for higher order approximations, the length of the horizontal segment of the ℏ-curve is larger, giving a wider range of valid ℏ-values for which the method will converge. The optimum value used in subsequent calculations is ℏ ¼  1. Plots in Figure 2 show the convergence of the method with different orders. We note that there is a good convergence to the solutions series starting from the second order approximation.

Figure 10 α ¼ 2.

Effect of K on (a) f ðηÞ and (b) f 0 ðηÞ when Re ¼ 5 and

5. Results and discussion In this study we investigated the problem of micropolar fluid flow through a channel. The governing equations were solved using the spectral homotopy analysis method. The results showing the influence of our physical parameters α, K and Re on the microrotational and stream-wise velocities are presented below. Figures 3(a),(b) and 4(a)(b) show the effects of α on the stream-wise velocity field. From the figures, it is clear that the channel center gives the maximum stream-wise velocity, similar to general Poiseuille flow. In the lower half region, Figures 3(a) and 4(a), present the back flow and the upper-half region shows the forward flow. In this region increasing α leads to increase the stream-wise velocity. Figures 3(b) and 4(b) may be divided into four quarters. In the first and fourth quarters, the velocity profile address the back flow, however the second and third quarters show

Figure 11 The shear stress at the lower disk for different values of (a) α and (b) Re.

Numerical investigation of the flow of a micropolar fluid through a porous channel Table 1 The effects of the Reynolds number Re and α on skinfriction coefficient f ″ð 1Þ for K ¼ 10. Quantity

Re

α

1nd order

2nd order

bvp4c

f ″ð 1Þ

1 2 3 5

1 1 1 1

2.715602 2.753734 2.794526 2.884776

2.715602 2.753734 2.794525 2.884776

2.715602 2.753734 2.794525 2.884776

f ″ð 1Þ

1 1 1 1

1 2 3 5

2.715602 2.630445 2.544919 2.372483

2.715602 2.630445 2.544919 2.372483

2.715602 2.630445 2.544919 2.372483

the forward flow. It is clear that the stream-wise velocity increase with increasing α. Figures 5(a) and 5(b) show the effects of α on the microrotation profile when Re ¼  5 and Re ¼ 5, respectively. The microrotation velocity is symmetric about the channel center. Moreover the lower-half and upper-half regions are responsible for both the upward and downward fluid movements, respectively. We also note that increasing α leads to an increase in the magnitude of the microrotation velocity. Figure 6(a) and (b) and Figure 7(a) and (b) show the effects of the Reynolds number Re on the stream-wise velocity. Increasing Re reduces the stream-wise velocity when K is fixed for negative or positive values of α in both Figure 6(a)(b) and Figure 7(a)(b). Figure 8(a)(b) shows the effects of the Reynolds number Re and micropolar parameter K on the microrotation velocity, respectively. Increasing Re or K leads to the increasing of the microrotation velocity. The effects of K on the stream-wise velocity are shown in both Figure 9(a)(b) and Figure 10(a)(b). The velocity increases as the micropolar parameter K increases in both Figure 9(a)(b) and Figure 10(a)(b). The same results can be obtained from Figure 11(a)(b). Table 1 shows the effects of the Reynolds number Re and α on the skin-friction coefficient with K ¼ 10. The table further gives a comparison between the Matlab bvp4c and the SHAM results in terms of the accuracy and convergence of the SHAM. It is clear that there is an excellent agreement between the Matlab bvp4c solver and SHAM results up to six decimal places. Increasing Re leads to the increasing on the skin-friction coefficient, on the other hand increasing on α reduces the skin-friction coefficient.

6. Conclusion In this paper, we applied the spectral homotopy analysis collocation method to obtain a series solution for the flow of a micropolar fluid through a porous channel with expanding or contractingwalls. The numerical approach used to solve the governing equations is easy and straightforward to apply. It's accuracy has been demonstrated by comparison

141

between the SHAM solution and the inbuilt Matlab bvp4c ODE solver. The method converges rapidly to the required solution. We have showed, among other results, that the shear stress at the lower disk increases with increasing in Re and that the fluid velocity increases with the micropolar parameter K.

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