Series solution for heat transfer of continuous

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Series solution for heat transfer of continuous stretching sheet immersed in a micropolar fluid in the existence of radiation M.S. Shadloo Center of Excellence in Energy Conversion, School of Mechanical Engineering, Sharif University of Technology, Tehran, Iran

Series solution for heat transfer

289 Received 1 January 2011 Revised 26 May 2011 Accepted 11 June 2011

A. Kimiaeifar Department of Mechanical Engineering, Aalborg University, Aalborg, Denmark, and

D. Bagheri Polymer Engineering Department, Amirkabir University of Technology, Tehran, Iran Abstract Purpose – The purpose of this paper is to study a two-dimensional steady convective flow of a micropolar fluid over a stretching sheet in the presence of radiation with constant temperature. Design/methodology/approach – The corresponding momentum, microrotation and energy equations are analytically solved using homotopy analysis method (HAM). Findings – To validate the method, investigate the accuracy and convergence of the results, a comparison with existing numerical and experimental results is done for several cases. Finally, by using the obtained analytical solution, for the skin-friction coefficient and the local Nusselt number as well as the temperature, velocity and angular velocity, profiles are obtained for different values of the constant parameters, such as Prandtl number, material, boundary and radiation parameter. Originality/value – In this paper, a series solution is presented for the first time. Keywords Flow, Convection, Heat transfer, Micropolar fluid, Stretching sheet, Radiation, Homotopy analysis method Paper type Research paper

1. Introduction Micropolar fluids are characterized by microstructures belonging to a class of fluids with nonsymmetrical stress tensor referred to as polar fluids (Lukaszewicz, 1999). The theory of micropolar fluids, which was initially introduced and investigated by Eringen (1966), takes into account fluids consisting of randomly oriented particles suspended in a viscous medium. Since this theory may be applied to explain a wide variety of industrial and engineering fluid flows such as polymeric fluids, liquid crystals, paints, human and animal blood, colloidal fluids, etc. the dynamics of micropolar fluids has become a popular area of research. Extensive reviews of the theory and its applications can be found in the review article by Ariman et al. (1973) and more recently in Eringen’s (2001) book.

International Journal of Numerical Methods for Heat & Fluid Flow Vol. 23 No. 2, 2013 pp. 289-304 q Emerald Group Publishing Limited 0961-5539 DOI 10.1108/09615531311293470

HFF 23,2

290

Most scientific problems in fluid mechanics are inherently non-linear and, except for a limited number of cases, most of them do not have analytical solutions. Accordingly, the non-linear equations are usually solved using other methods, including numerical techniques or by using analytical perturbation methods. In the numerical methods, stability and convergence should be considered so as to avoid divergence or inappropriate results. Using an analytical perturbation method, a small characteristic parameter should be introduced into the governing equations. One of the semi-exact methods which does not need small/large parameters is called the homotopy analysis method (HAM), first proposed by Liao (1992). This method has already been applied successfully to solve many problems in fluid mechanics (Kimiaeifar et al., 2009; Moghimi et al., 2010; Rahimpour et al., 2009; Sohouli et al., 2010; Chowdhury et al., 2007). In this method it is possible to adjust and control the convergent region, and this is the most important feature of this technique in comparison to other techniques. The main goal of the present study is to use HAM to find an analytical solution to the problem of stretching sheet immersed in micropolar fluid in the presence of radiation effect. For this reason, the local similarity solutions will be obtained and then the series solution will be first computed using HAM and then its convergence will be discussed in detail. Afterward, the results for the skin-friction coefficient and the local Nusselt number as well as the temperature, velocity and angular velocity profiles will be calculated and discussed from the physical point of view. 2. Mathematical model Consider a steady laminar two-dimensional thermal boundary layer flow over a stretching sheet, kept at constant temperature Tw, in a viscous incompressible micropolar fluid. The x-axis is taken along the plate and y-axis is normal to the plate. It is assumed that the surface is stretched in the x-direction such that the x-component of the velocity varies linearly along it, i.e. Uw ¼ ax, where a is an arbitrary constant and a . 0. Taking into account the thermal radiation term in the energy equation, the governing equations of continuity, momentum, angular momentum and energy, respectively, are given by Nazar et al. (2004):

› u ›y þ ¼ 0; ›x ›y ›u ›u k ›2 u k ›N ¼ nþ ; þ u þy ›x ›y r ›y 2 r ›y ›N ›N g ›2 N k ›u u þy ¼ 2N þ ; 2 ›x ›y ›y rj › y 2 rj u

›T ›T k ›2 T 1 › qr þy ¼ ; 2 ›x ›y r cP › y 2 r cP › y

ð1Þ ð2Þ ð3Þ ð4Þ

where u and v are the velocity components along the x- and y-axes, N is the microrotation or angular velocity with a direction of rotation in the x-y plane, T is the fluid temperature, n is the kinematic viscosity, r is the density and j, g and k are the microinertia per unit mass, spin gradient viscosity and vortex viscosity, respectively,

which are assumed to be constant. Here j ¼ n/a is taken as a reference length and g is assumed to be given by Ahmadi (1976):

g ¼ mð1 þ K=2Þj;

ð5Þ

where m is the dynamic viscosity and K ¼ k=m is the dimensionless viscosity ratio and is called the material parameter. It is worth mentioning here that relation (5) is invoked to allow equations (1)-(4) to predict the correct behaviour in the limiting case when microstructure effects become negligible, and the microrotation, N, reduces to the angular velocity (Yu¨cel, 1989). The appropriate boundary conditions can be written as follows: u ¼ U w;

y ¼ 0;

u ! 0;

N ¼ 2m

N ! 0;

›u ; ›y

T ¼ Tw

T ! T1

at

as y ! 1:

y ¼ 0;

ð6Þ ð7Þ

Here the subscripts w and 1 show the wall surface and the ambient flow properties, respectively, and m is the boundary parameter with the value of 0 # m # 1. By using Rosseland’s approximation, the radiative heat flux is given by Brewster (1992): qr ¼ 2

4s * ›T 4 ; 3k * ›y

ð8Þ

where s * and k * are the Stephan-Boltzman constant and the mean absorption coefficient, respectively. We assume the differences within the flow are such that T 4 can be expressed as a linear function of temperature. Expanding T 4 in a Taylor series about T1 and neglecting higher order terms thus: T 4 < 4T 13 T 2 3T 14 :

ð9Þ

By substituting equations (8) and (9) in equation (4), one obtains: u

›T ›T ›2 T þy ¼ að1 þ N R Þ 2 ; ›x ›y ›y

ð10Þ

where a ¼ k=ðrcP Þ is the thermal diffusivity, N R ¼ 16s * T 31 =ð3kk * Þ is the radiation parameter (Datti et al., 2004), and k is the thermal conductivity. Introducing the stream function c (x, y) such that: u¼

›c ›y

and


y ¼2

›c ; ›x

ð11Þ

where:

c ðx; yÞ ¼ ðnxU w Þ0:5 f ðhÞ;

ð12Þ

0:5 Uw h¼ y; nx

ð13Þ

and the similarity variable:

291

HFF 23,2

together with: N ¼ Uw

uð hÞ ¼

292

Uw nx

0:5 hðhÞ;

ð14Þ

T 2 T1 : Tw 2 T1

ð15Þ

Note that the definitions of c, u and v in the above expressions satisfy the continuity equation (1) identically. Substituting equations (12) through (15) into equations (2), (3) and (10), the resulting non-linear ordinary differential equations are: ð1 þ KÞf 000 þ ff 00 2 f 02 þ Kh 0 ¼ 0; K 00 1þ h þ fh 0 2 f 0 h 2 Kð2h þ f 00 Þ ¼ 0; 2 1 ð1 þ N R Þu 00 þ f u 0 ¼ 0; Pr

ð16Þ ð17Þ ð18Þ

with the following reduced boundary conditions: f ð0Þ ¼ 0; 0

f 0 ð0Þ ¼ 1;

f ðhÞ ! 0;

hð0Þ ¼ 2mf 00 ð0Þ;

hðhÞ ! 0;

uð0Þ ¼ 1;

uðhÞ ! 0 as h ! 1:

ð19Þ ð20Þ

In the above equations, primes denote differentiation with respect to h and Pr is the dimensionless Prandtl number and is equal to: Pr ¼

n : a

ð21Þ

It is noted that K ¼ 0 is corresponded to viscous fluid case, which is studied before without consideration of the thermal radiation effect (Grubka and Bobba, 1985). The physical quantities of interest are the skin-friction coefficient Cf and the local Nusselt number Nux, which are defined, respectively, as:

tw ; rU 2w xqw ; Nux ¼ kðT w 2 T 1 Þ Cf ¼

ð22Þ ð23Þ

where the wall shear stress tw and the surface heat flux qw for micropolar boundary layer flow are given by: ›u tw ¼ ðm þ kÞ þ kN ; ð24Þ ›y y¼0 ›T ; ð25Þ qw ¼ 2k ›y y¼0

Substituting equations (14) and (15) into equations (22) through (25), we get: C f Re x0:5 ¼ ð1 þ KÞf 00 ð0Þ þ Khð0Þ; Nux Re x0:5

¼ 2u 0 ð0Þ;

ð26Þ ð27Þ

293

where Rex ¼ U w x=n is the local Reynolds number. 3. Application of HAM The governing equations for heat transfer of continuous stretching sheet immersed in a micropolar fluid are expressed by equations (16) through (18). Nonlinear operators are defined as follows: ›3 f ðh; qÞ ›hðh; qÞ ›f ðh; qÞ 2 ›2 f ðh; qÞ N f ½ f ðh; qÞ ¼ ð1 þ KÞ þ K 2 þf ð h ; qÞ ; ð28Þ ›h 3 ›h ›h ›h 2 2 K ›2 hðh; qÞ › f ðh; qÞ 2K þ 2hðh; qÞ N h ½hðh; qÞ ¼ 1 þ ›h 2 ›h 2 2 ð29Þ ›f ðh; qÞ ›hðh; qÞ 2 hðh; qÞ þ f ðh; qÞ ; ›h ›h N u ½uðh; qÞ ¼

1 ›2 u ðh; qÞ ›u ðh; qÞ ð1 þ N R Þ þ f ðh; qÞ ; ›h 2 ›h Pr

ð30Þ

where q [ [0, 1] is the embedding parameter. As the embedding parameter increases from 0 to 1, the parameters expressed by U(h; q), V(h; q) and Y(h; q) vary from the initial guess, U0(h), V0(h) and Y0(h), to the exact solution, U(h), V(h) and Y(h): f ðh; 0Þ ¼ U 0 ðhÞ;

f ðh; 1Þ ¼ U ðhÞ;

ð31Þ

h ðh; 0Þ ¼ V 0 ðhÞ;

h ðh; 1Þ ¼ V ðhÞ;

ð32Þ

u ðh; 0Þ ¼ Y 0 ðhÞ;

u ðh; 1Þ ¼ Y ðhÞ:

ð33Þ

Expanding f (h;q), h(h;q) and u(h;q) in Taylor series with respect to q results in: f ðh; qÞ ¼ U 0 ðhÞ þ

1 X

U m ðhÞq m ;

ð34Þ

V m ðhÞq m ;

ð35Þ

Y m ðhÞq m ;

ð36Þ

m¼1

hðh; qÞ ¼ V 0 ðhÞ þ

1 X m¼1

u ðh; qÞ ¼ Y 0 ðhÞ þ

1 X m¼1

where: U m ðhÞ ¼

1 ›m f ðh; qÞ ; m! ›q m q¼0


ð37Þ

HFF 23,2

294

1 ›m hðh; qÞ V m ð hÞ ¼ ; m! ›q m q¼0

ð38Þ

1 ›m u ðh; qÞ : Y m ðhÞ ¼ m! ›q m q¼0

ð39Þ

HAM can be expressed by many different base functions (Liao, 1992), according to the governing equations; it is straightforward to use a base function in the form of: 1 X 1 X bpm h p e 2mh ; ð40Þ U ð hÞ ¼ m¼1 p¼1

V ðhÞ ¼

1 X 1 X

kpm h p e 2mh ;

ð41Þ

dpm h p e 2mh ;

ð42Þ

m¼1 p¼1

Y ð hÞ ¼

1 X 1 X m¼1 p¼1

bpm, kpm, and dpm are the coefficients to be determined. When the base function is selected, the auxiliary functions Hf (h), Hh(h) and Hu (h), initial approximations U0(h), V0(h) and Y0(h) and the auxiliary linear operators Lf, Lh and Lu must be chosen in such a way that the corresponding high-order deformation equations have solutions with the functional form similar to the base functions. It is worth mentioning that the presence of expressions such as hsin(mh) prevents the convergence of the analytical solution. This method is referred to as the rule of solution expression (Liao, 1992). The linear operators Lf, Lh and Lu are chosen through: Lf ½ f ðh; qÞ ¼

›3 f ðh; qÞ ›f ðh; qÞ þ ; ›h 3 ›h

ð43Þ

Lh ½hðh; qÞ ¼

›2 hðh; qÞ ›hðh; qÞ þ ; ›h 2 ›h

ð44Þ

Lu ½uðh; qÞ ¼

›2 uðh; qÞ ›u ðh; qÞ þ : ›h 2 ›h

ð45Þ

Equations (43) through (45) result in: Lf ½c1 þ c2 e h þ c3 e 2h ¼ 0;

ð46Þ

Lh ½c4 þ c5 e 2h ¼ 0;

ð47Þ

Lu ½c6 þ c7 e

2h

¼ 0;

ð48Þ

According to the rule of solution expression and the initial conditions, the initial approximations U0, V0 and Y0, as well as the integral constants, c1 through c7, are expressed as: U 0 ðhÞ ¼ c1 þ c2 e h þ c3 e 2h ;

c1 ¼ 0;

c2 ¼ 0;

c3 ¼ h ;

ð49Þ

V 0 ð h Þ ¼ c 4 þ c5 e 2 h ;

c4 ¼ 0;

Y 0 ðhÞ ¼ c6 þ c7 e 2h ;

c6 ¼ 0;

c5 ¼ 2m;

ð50Þ

c7 ¼ 1;

ð51Þ


The zeroth order deformation equation and their boundary condition for f(h), h(h), and u(h) are: ð1 2 qÞLf ½f ðh; qÞ 2 U 0 ðhÞ ¼ q"f H f ðhÞN f ½ f ðh; qÞ;

ð52Þ

ð1 2 qÞLh ½hðh; qÞ 2 V 0 ðhÞ ¼ q"h H h ðhÞN h ½hðh; qÞ;

ð53Þ

ð1 2 qÞLu ½uðh; qÞ 2 Y 0 ðhÞ ¼ q"u H u ðhÞN u ½uðh; qÞ:

ð54Þ

According to the rule of solution expression and from equations (40) through (42), the auxiliary functions Hf (h), Hh(h) and Hu(h) can be chosen as follows: H f ðhÞ ¼ h p e 2mh ;

ð55Þ

H h ðhÞ ¼ h p e 2mh ;

ð56Þ

H u ðhÞ ¼ h p e 2mh :

ð57Þ

Differentiating equations (52) through (54), m times, with respect to the embedding parameter q and then setting q ¼ 0 in the final expression and dividing it by m!, are reduced to: Z h 1 U m ðhÞ ¼ xm U m21 ðhÞ þ " H f ðhÞRm ðU m21 Þd h 2 0 ð58Þ Z 1 2q h q q 2h ; H f ðhÞe Rm ðU m21 Þdh þ c1 þ c2 e þ c3 e 2 e 2 0 Z hZ m V m ðhÞ ¼ xm V m21 ðhÞ þ " H f ðhÞe h Rm ðV m21 Þdhdm þ c4 þ c5 e 2h ; ð59Þ 0

Y m ðhÞ ¼ xm Y m21 ðhÞ þ "

0

Z hZ 0

m

H u ðhÞe h Rm ðY m21 Þdhdm þ c4 þ c5 e 2h :

ð60Þ

0

Equations (58)-(60) is the mth order deformation equation for f (h), h(h), and u (h), where: d 3 U m21 ðhÞ dV m21 ðhÞ þK þ Gcx Rex V ðhÞ 3 dh dh 21 X Rex dU m21 ðhÞ m dU z ðhÞ dU m212z ðhÞ 2 2 M Rex þ 2 K dh dh dh z¼0

Rm ðU m21 Þ ¼ ð1 þ KÞ

þ

m 21 X z¼0

U z ðhÞ

d 2 U m212z ðhÞ ; dh 2

ð61Þ

295

K d 2 V m21 ðhÞ d 2 U m21 ðhÞ Rm ðV m21 Þ ¼ 1 þ þ K Sr þ 2V m21 ðhÞ 2 dh 2 dh 2

HFF 23,2

2

m21 X z¼0

296 Rm ðY m21 Þ ¼

21 X dU m212z ðhÞ m dV m212z ðhÞ V z ðhÞ þ U z ð hÞ ; dh dh z¼0

21 X 1 d 2 Y m21 ðhÞ m dY m212z ðhÞ ð1 þ N R Þ þ U z ð hÞ ; Pr dh 2 dh z¼0

ð62Þ

ð63Þ

and: (

xm ¼

0; m # 1 1; m . 1

:

ð64Þ

The rate of convergence can be increased when suitable values are selected for m and p. According to the rule of solution expression, the suitable values for m and p are {p ¼ 0, m ¼ 1}. Consequently, the corresponding auxiliary functions were determined as Hf (h) ¼ Hh(h) ¼ Hu(h) ¼ e 2 h. As a result of this selection, the first and second terms of the solution’s series are as follows: U 0 ð hÞ ¼ he 2 h ; U 1 ð hÞ ¼

h ð9K 2 12Km 2 9he 2h 2 5e 2h 12 þ 3e 2h h 2 2 9e 2h K h þ 3e 2h K h 2 þ · · ·Þ Y 0 ðhÞ ¼ e 2h ;

1 he 2h ð218m 2 81Km þ 8K 36 2 24me 2h þ 36Kme 2h þ 12Ke 2h h þ · · ·Þ

Y 1 ðhÞ ¼ 2

V 0 ðhÞ ¼ 2me 2h ; V 1 ðhÞ ¼

1 0:0104166he 2h ð232e 2h 2 32N R e 2h Pr þ 12e 22h h Pr þ 32 þ 32N R þ · · ·Þ

ð65Þ ð66Þ ð67Þ ð68Þ ð69Þ ð70Þ

4. Accuracy and convergence The analytical solution should converge. It should be noted that the auxiliary parameter ", as pointed out by Liao (2003), controls the convergence and accuracy of the solution series. In order to define a region such that the solution series is independent on ", "-curves are plotted as shown in Figure 1. The region where the distribution of f, f0 , f00 , h, h0 , u, u versus " is a horizontal line is known as the convergence region for the corresponding function. It is seen that the interval between 2 0.2 and 2 0.9 can be chosen as the convergence region.


f(0.5), h (0.5), θ (0.5)

1

297

0.5

0 θ (0.5) h (0.5) –0.5

f (0.5)

Figure 1. "-curves to show the convergence region

–1 –1

–0.5

0

h

In Table I, a comparison has been made with previous works to show the influence of iterations order in accuracy. It can be easily seen that, by increasing the number of iterations in series solution, the accuracy increases which is in very good agreement with other works. 5. Results and discussion Figure 2 shows the velocity profiles for various values of K when m ¼ 0.5, Pr ¼ 1 and NR ¼ 1. From this figure it is obvious that the velocity profiles increase as values of K increase. It can also be noticed that the velocity gradient at the surface f 00 (0) decreases (in absolute sense) as K increases. From this, it can be concluded that micropolar fluids exhibit drag reduction compared to viscous fluids. The negative velocity gradient at the surface, f 00 (0), as shown in Figure 2 reveals that a drag force is exerted by the stretching sheet on the fluid. This outcome is expected, because the development of the boundary layer is only induced by velocity profile. Figure 3 shows the variations of the angular velocity profile with respect to K when m ¼ 0.5, Pr ¼ 1 and NR ¼ 1 parameter. It can be seen that as K increases,

Pr 0.01 0.72 1 3 10 100

Grubka and Bobba (1985)

Ali (1994)

Chen (1998)

Present with sixth order

Present with 15th order

0.0099 0.4631 0.582 1.1652 2.308 7.7657

– 0.4617 0.5801 1.1599 2.296 –

0.00991 0.46315 0.58199 1.16523 2.30796 7.76536

0.0089 0.4595 0.5793 1.16381 2.2998 7.7532

0.00991 0.04631 0.0582 1.1652 2.30791 7.76532

Table I. Values of 2 u 0 (0) for various values of Pr when m ¼ 0.5, NR ¼ 0 and K ¼ 0

HFF 23,2

298

Figure 2. Variation of velocity profile f 0 (h) with h at different values of K

Figure 3. Angular velocity h(h) for various values of K

the profiles decrease at the beginning (near the surface) and then (as h increases) increase smoothly. The effect of K parameter over the temperature distribution when m ¼ 0.5, Pr ¼ 1 and NR ¼ 1 is shown in Figure 4. It is clear that the temperature is reduced due to the increase in K values. This observation is in agreement with the results presented in Table III, which shows that as the value of K increases, the values of skin-friction factors decreases (in absolute sense) and subsequently a decrease in the temperature


299

Figure 4. Temperature profiles u(h) for various values of K

profiles occurs. It can also be found that the heat transfer rate at the surface is increased as K increases. This observation is in agreement with the results of Table II. Most of the previous investigations have been done based on considering the boundary parameter m ¼ 0 or m ¼ 0.5 (m ¼ 0 represents concentrated particles flows in which the elements close to the surface are unable to rotate and m ¼ 0.5 indicates the vanishing of asymmetric part of the stress tensor). In the present work, more values of m have been considered. The effect of boundary parameter m on velocity profiles is Pr

NR

M

K

Nux = Re 0:5 x

1

1

0.5

1

1

1

0 0.5 1 5 1

0 0.2 0.4 0.6 0.8 1 0.5

0 1 2.5 5 10 1

0.41181 0.42723 0.47041 0.52179 0.57507 0.45513 0.45217 0.44869 0.4446 0.43975 0.43378 0.61332 0.50777 0.42723 0.34031 0.36031 0.42723 0.68679 0.90555 1.11815

0.5 1 2.5 5 10

0.5

1

1

Table II. Values of 2 u 0 (0) for various values of Pr, NR, m and K

HFF 23,2

300

Figure 5. Variation of velocity profile f 0 (h) with h at different values of m

Figure 6. Temperature profiles u (h) for various values of m

shown in Figure 5 when K, Pr and NR are fixed to unity. It can be seen that increasing in the values of m leads to decrease in the velocity profiles. It can also be found that the velocity gradients at surface are larger for larger values of m. When, different trends are observed for the effect of boundary parameter m on the temperature profile and also on the heat transfer rate at the surface as shown in Figure 6. These observations are in agreement with the results presented in Tables II and III. Furthermore, the effect of m on the temperature profiles in comparison with effects of K, NR and Pr is slight.

Figure 7 shows larger angular velocity for larger m values. Also, the couple stress h(0) is more dominant for larger values of m, as it can be seen in this figure. Figure 8 shows the temperature profiles for various values of NR when m ¼ 0.5, K ¼ 1, Pr ¼ 1. It can be observed that the temperature distribution increases in the flow region as values of NR increase. This is due to the heat energy stored in the liquid because of frictional heating. It is also seen that the temperature gradient at the surface u 0 (0) decreases (in absolute sense) as NR increases. This observation is evidence that values of 2 u 0 (0) are lower for NR . 0 compared to NR ¼ 0 as it presented in Table II. Thus, the heat transfer rate at the surface is lower in the presence of radiation. However, Figure 9 shows a different trend when the effect of Pr on the temperature distribution was explored as m and K values were kept constant at 0.5 and 1, respectively, and NR ¼ 1. From Figure 9 it can be seen that increasing Pr leads to decrease the temperature distribution which implies that thermal boundary layer thickness decreases and on the contrary, leads to an increase in the heat transfer rate at the surface. M

K

C f Re 0:5 x

0.5

0 1 2.5 5 10 1

2 8.0252 2 6.5356 2 5.1566 2 4.2871 2 3.3208 2 10.264 2 8.9227 2 7.3884 2 5.6153 2 3.5205 2 1.0118

0 0.2 0.4 0.6 0.8 1


301

Table III. Values of skin-friction factor when Pr ¼ 1 and NR ¼ 1 for various values of m and K

Figure 7. Angular velocity h(h) for various values of m

HFF 23,2

302

Figure 8. Temperature profiles u(h) for various values of NR

Figure 9. Temperature profiles u(h) for various values of Pr

6. Conclusions In this paper we have studied the problem of steady two-dimensional heat transfer of continuous stretching sheet immersed in a micropolar fluid in the existence of radiation. Using existence similarity transformations the governing equations have been transformed into non-linear ordinary differential equations and solved analytically by using HAM. The obtained results were validated against previous works. Effects of the various parameters such as Prandtl number, the material parameter, boundary parameter and radiation parameter on the temperature, velocity and angular velocity profiles, as well as the skin-friction coefficient and the local Nusselt number were examined. The following conclusions can be drawn as a result of the HAM calculations:

.

.

.

An increase in the value of K leads to a decrease of the angular velocity profiles, the wall temperature and the skin-friction coefficient; however, the velocity profile and the local Nusselt number are increasing. An increase in the value of m leads to an increase of the angular velocity profiles, the wall temperature, and a decrease of the velocity profile, the local Nusselt number and the skin-friction coefficient. An increase in the value of NR leads to a reduction of the local Nusselt number due to the wall temperature enhancement, while the trend is opposite for Pr number.

References Ahmadi, G. (1976), “Self-similar solution of incompressible micropolar boundary layer flow over a semi-infinite plate”, International Journal of Engineering Science, Vol. 14, pp. 639-46. Ali, M.E. (1994), “Heat transfer characteristics of a continuous stretching surface”, Heat Mass Transfer, Vol. 29, pp. 227-34. Ariman, T., Turk, M.A. and Sylvester, N.D. (1973), “Microcontinuum fluid mechanics – review”, International Journal of Engineering Science, Vol. 11, pp. 905-30. Brewster, M.Q. (1992), Thermal Radiative Transfer Properties, Wiley, New York, NY. Chen, C.H. (1998), “Laminar mixed convection adjacent to vertical, continuously stretching sheets”, Heat Mass Transfer, Vol. 33, pp. 471-6. Chowdhury, M.S.H., Hashim, I. and Abdulaziz, O. (2007), “Comparison of homotopy analysis method and homotopy-perturbation method for purely nonlinear fin-type problems”, Commun Nonlinear Science Numerical Simulation, Vol. 14 No. 2, pp. 371-8. Datti, P.S., Prasad, K.V., Abel, M.S. and Joshi, A. (2004), “MHD viscoelastic fluid flow over a non-isothermal stretching sheet”, International Journal of Engineering Science, Vol. 42, pp. 935-46. Eringen, A.C. (1966), “Theory of micropolar fluids”, Journal of Math. Mech., Vol. 16, pp. 1-18. Eringen, A.C. (2001), Microcontinuum Field Theories, II. Fluent Media, Springer, New York, NY. Grubka, L.J. and Bobba, K.M. (1985), “Heat transfer characteristics of a continuous, stretching surface with variable temperature”, ASME Journal of Heat Transfer, Vol. 107, pp. 248-50. Kimiaeifar, A., Bagheri, G.H., Rahimpour, M. and Mehrabian, M.A. (2009), “Analytical solution of two-dimensional stagnation flow towards a shrinking sheet by means of homotopy analysis method”, Journal of Process Mechanical Engineering, Vol. 223 No. 3, pp. 133-43. Liao, S.J. (1992), “The proposed homotopy analysis technique for the solution of nonlinear problems”, PhD thesis, Shanghai Jiao Tong University, Shanghai. Liao, S.J. (2003), Beyond Perturbation: Introduction to Homotopy Analysis Method, Chapman & Hall, Boca Raton, FL. Lukaszewicz, G. (1999), Micropolar Fluids: Theory and Application, Birkha¨user, Basel. Moghimi, M., Kimiaeifar, A., Rahimpour, M. and Bagheri, G.H. (2010), “Homotopy analysis method for Marangoni mixed convection boundary layer flow”, Journal of Mechanical Engineering Science, Vol. 224 No. 6, pp. 1193-202. Nazar, R., Amin, N. and Pop, I. (2004), “Stagnation point flow of a micropolar fluid towards a stretching sheet”, International Journal of Non-Linear Mechanics, Vol. 39, pp. 1227-35.


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Rahimpour, M., Mohebpour, S.R., Kimiaeifar, A. and Bagheri, G.H. (2009), “On the analytical solution of axisymmetric stagnation flow towards a shrinking sheet”, International Journal of Mechanics, Vol. 2 No. 1, pp. 1-10. Sohouli, A.R., Famouri, M., Kimiaeifar, A. and Domairry, G. (2010), “Application of homotopy analysis method for natural convection of Darcian fluid about a vertical full cone embedded in pours media prescribed surface heat flux”, Communication in Nonlinear Science and Numerical Simulation, Vol. 15 No. 7, pp. 1691-9. Yu¨cel, A. (1989), “Mixed convection in micropolar fluid flow over a horizontal plate with surface mass transfer”, International Journal of Engineering Science, Vol. 27, pp. 1593-602. Corresponding author A. Kimiaeifar can be contacted at: [email protected]

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