17

CHAPTER-2

MIXED CONVECTION FLOW OF MICROPOLAR FLUID

OVER A POROUS SHRINKING SHEET WITH THERMAL

RADIATION

2.1 INTRODUCTION

Boundary layer flow of incompressible fluid over a shrinking sheet has attracted the

interest of many researchers due to its applications in polymeric materials processing.

Shrinking sheet flows are important in the manufacture of certain polymers and high-

performance materials for aerospace coatings, as documented by Baird and Baird [21].

Further discussions of shrinking sheet flows and their importance in the processing of

various non-Newtonian materials have been provided by Zhong et al. [214] for ceramic

suspensions, Parng and Yang [155] for super-plastic polymeric sheets, Gupta and Ward

[87] for thermal shrinking of polythene sheets and Cheremisinoff [51] for viscoelastic

membranes used in petroleum applications.

Many theoretical and numerical studies of such flows have been reported. Ishak et

al. [115] investigated the stagnation point flow of micropolar fluid over a shrinking sheet.

The flow and heat transfer over a shrinking sheet immersed in a micropolar fluid was

considered by Yacob and Ishak [209]. Das [59] investigated the slip effects on MHD

mixed convection flow of a micropolar fluid towards a shrinking vertical sheet.

Bhattacharyya and Layek [39] considered suction/blowing effects on stagnation-point flow

towards a shrinking sheet with thermal radiation. Oblique stagnation-point flow towards a

shrinking sheet with thermal radiation was examined by Mahapatra et al. [132]. Ahmad et

al. [3] studied the effect of thermal radiation on MHD axisymmetric stagnation point flow

and heat transfer of micropolar fluid over a shrinking sheet.

The objective of the present study is to analyze the steady boundary layer mixed

convection flow and heat transfer of micropolar fluid with suction over a shrinking sheet in

the presence of thermal radiation. This flow regime arises in the mechanical process

engineering systems including shrink packaging, shrink wrapping, shrink film and

18

temperature controlling of final products. By suitable similarity transformations the

governing partial differential equations are transformed into a set of coupled nonlinear

ordinary differential equations. These equations are solved numerically subject to

physically-realistic boundary conditions using a variational formulation of the finite

element method. The results are presented graphically for velocity, microrotation and

temperature functions with the various values of physical parameters such as suction,

radiation and buoyancy parameters. Additionally, the skin friction coefficient, local couple

stress and the local Nusselt number have also been computed.

2.2 MATHEMATICAL MODEL

Let us consider a steady two-dimensional laminar flow of an incompressible

micropolar fluid of temperature T driven by a porous shrinking sheet with prescribed

surface heat flux. It is assumed that the velocity )(xU and the surface heat flux )(xq of

the sheet vary proportional to the distance x from the fixed point on the sheet, i.e.

xaxU )( and bxxq )( , where a and b are constants. The axisx is taken along the

sheet and the axisy is normal to it. A uniform transpiration (suction) velocity wV is

applied normal to the sheet. The micropolar fluid is assumed to be a gray, emitting and

absorbing, but non-scattering medium. The micropolar fluid also has constant properties

except for the density changes which produce a thermal buoyancy force. The flow

configuration and the coordinate system are shown in the Fig. 2.1.

Figure 2.1: Physical model and coordinate system

ux,

T

O

wV

T)(xq)(xU

eg

vy,

19

The governing boundary layer equations for the flow problem are as follows

Mass Conservation (Continuity)

,0

y

v

x

u (2.1)

Linear Momentum Conservation

),(2

2

TTg

y

NS

y

uS

y

uv

x

uu e (2.2)

Angular Momentum (Micro-rotation) Conservation

,22

2

y

uNS

y

N

y

Nv

x

Nuj (2.3)

Energy Conservation

,1

2

2

y

q

cy

T

cy

Tv

x

Tu r

pp

(2.4)

with the boundary conditions

,0at,50,,)(

y

q

y

T

y

u.NVvxUu w

(2.5a)

.as,0,0 yTTNu (2.5b)

At the sheet, the boundary condition for angular momentum (microrotation) implies that

the microrotation is equal to the fluid vorticity. In accordance with this, in the

neighborhood of a rigid boundary, the effect of microstructure is negligible since the

suspended particles cannot get closer to the boundary than their radius. Hence in the

neighborhood of the boundary, the only rotation is due to fluid shear and therefore, the

gyration vector must be equal to the fluid vorticity. The spin gradient viscosity is given

by ,2/ jS where aj / is the reference length. This assumption is invoked to

allow the field of equations to predict the correct behavior in the limiting case when the

microstructure effects become negligible and the total spin N reduces to the angular

velocity as suggested by Ahmadi [4].

The radiative heat flux rq is defined by the Rosseland diffusion approximation, which is

valid for boundary layer flows [30], as given below

,3

4 4

1

1

y

T

kqr

(2.6)

20

where 1 and 1k are the Stefan–Boltzmann constant and the Rosseland mean absorption

coefficient, respectively. It is assumed that the temperature differences within the flow are

sufficiently small such that 4T may be expressed as a linear function of temperature by

expanding 4T in a Taylor series about T and neglecting higher-order terms. Thus

.34 434

TTTT (2.7)

By using (2.6) and (2.7), energy eqn. (2.4) reduces to

.3

162

2

1

3

1

2

2

y

T

kc

T

y

T

cy

Tv

x

Tu

pp

(2.8)

The velocity components u and v can be expressed in terms of the stream function as

follows

yu

, .

xv

(2.9)

So the continuity eqn. (2.1) is satisfied automatically. Using the similarity transformations

,ya

,fxa ,

3

gxa

N

,

a

q

TT (2.10)

the governing partial differential eqns. (2.2), (2.3) and (2.8) reduce to the following system

of coupled nonlinear ordinary differential equations

,0)1(2

gKffffK (2.11)

,02)2

1( fgKgfgfgK

(2.12)

,0Pr3

41

f

R (2.13)

and the boundary conditions transform to

0,at1,5.0,1, fgff (2.14a)

.as0,0,0 gf (2.14b)

Here prime denotes the differentiation with respect to and

SK the micropolar coupling constant parameter,

2/5Re xxGr buoyancy parameter,

25 xbgGr ex local Grashof number,

2Re xax local Reynolds number,

21

3

11 4 TkR radiation parameter,

pcPr Prandtl number and

aVw transpiration parameter.

In the present study only suction is considered at the sheet for which > 0. The

engineering parameters of relevance to materials processing are the skin friction

coefficient, local couple stress and the local Nusselt number, which are defined

respectively as

2

2

UC w

f

,

U

MM w

x

, )(

TT

xqNu

w

w

x

, (2.15)

where the wall shear stress w , plate couple stress wM and the heat flux q are given by

,

0

y

w SNy

uS ,

0

y

wy

NM .

3

4

0

4

1

1

0

yy

wy

T

ky

Tq

(2.16)

Using the similarity transformations (2.10), we obtain

,02Re2/1

fKC xf 05.01 gKM x

and

.0

1

3

41

Re2/1

R

Nu

x

x (2.17)

2.3 FINITE ELEMENT SOLUTION

The set of differential eqns. (2.11)-(2.13) with the boundary conditions (2.14) has

been solved numerically by using finite element method [170]. In order to apply finite

element method first we assume

.hf (2.18)

With this substitution, the eqns. (2.11)-(2.13) become

,0)1( 2 gKhhfhK (2.19)

,02)2

1( hgKhggfgK

(2.20)

,0Pr3

41

f

R (2.21)

and the corresponding boundary conditions reduce to

0,at1,5.0,1, hghf (2.22a)

22

.as0,0,0 gh (2.22b)

It has been observed that for large value of ),8( there is no appreciable change in the

results. Therefore, for the computation purpose infinity has been fixed at 8. The whole

domain is divided into a set of n line elements of width nhe 8 .

2.3.1 VARIATIONAL FORMULATION

The variational form associated with eqns. (2.18)-(2.21) over a two-noded element

1, ee is given by

,01

1 e

e

dhfw

(2.23)

,0)1(1

2

2 e

e

dgKhhfhKw

(2.24)

,02)2

1(1

3

e

e

dhgKhggfgK

w

(2.25)

,0Pr3

41

1

4

dfR

we

e

(2.26)

where 321 ,, www and 4w are weight functions which may be viewed as the variation in

ghf ,, and respectively.

2.3.2 FINITE ELEMENT FORMULATION

The finite element model can be obtained from eqns. (2.23)-(2.26) by substituting

finite element approximations of the form

2

1

,j

jjff

2

1

,j

jjhh

2

1

,j

jjgg

2

1

,j

jj (2.27)

with ),21,(4321 iwwww i and 1 and 2 are the shape functions for a

typical element 1, ee which are taken as

,1

1

1

ee

e

,1

2

ee

e

.1 ee (2.28)

The finite element model of the equations thus formed can be expressed in the form

23

,

}{

}{

}{

}{

}{

}{

}{

}{

][][][][

][][][][

][][][][

][][][][

4

3

2

1

44434241

34333231

24232221

14131211

b

b

b

b

g

h

f

KKKK

KKKK

KKKK

KKKK

(2.29)

where ][ nmK and ][ mb )43,2,1,,( nm are the matrices of order 2×2 and 2×1

respectively. All these matrices are defined as follows

,1

11

dd

dK

e

e

j

iji

,1

21

dKe

e

jiji

04131 jiji KK , ,021 jiK

,)1(111

22

dhdd

dfd

d

d

d

dKK

e

e

e

e

e

e

ji

j

i

jiji

,1

23

dd

dKK

e

e

j

iji

,1

24

dKe

e

jiji

,031 jiK ,1

23

dd

dKK

e

e

j

iji

,22

11111

33

dKdhdd

dfd

d

d

d

dKK

e

e

e

e

e

e

e

e

jiji

j

i

jiji

,043 jiK ,0342414 jijiji KKK

,Pr3

41

11

44

dd

dfd

d

d

d

d

RK

e

e

e

e

j

i

jiji

(2.30)

and ,01 ib ,)1(

1

2

e

e

d

dhKb ii

,)2

1(

1

3

e

e

d

dgKb ii

1

3

414

e

e

d

d

Rb ii

(2.31)

where

2

1i

iiff and

2

1i

iihh are assumed to be known.

After assembly of the element equations, a system of algebraic nonlinear equations

is obtained, which is solved iteratively. The function f and h are assumed to be known

and are used for linearizing the system. The velocity, microrotation and temperature

functions are set equal to 1.0 for the first iteration and global equations are solved for the

nodal values of these functions. This process is repeated until the desired accuracy of four

significant figures is obtained. We have carried out calculations for n 20, 40, ..., 160 and

the final results are reported for n 160 only which are shown in table 2.1 (a).

24

For the case of viscous fluid )0( K and in the absence of buoyancy force

),0( the exact solution for )(f subject to the boundary conditions (2.22) is given as

follows

,exp1 zzf (2.32)

where ,242 z and is identical to that obtained by Fang and Zhang [77] with

0M and by Yacob and Ishak [209] with 0K . The comparison of the flow velocity

f obtained by finite element method and by analytical method from (2.32) is

tabulated in table 2.1 (b). It is demonstrated from the table that the numerical results so

obtained are in full agreement with the analytical results and thus confirm the validity of

the present FEM computational solutions.

Table 2.1 (a): Convergence of results with the variation of number of elements

)5,1,3,1Pr,1( RK

Number of elements )2.1(h )2.1(g )2.1(

20 0.26279 -0.05336 0.16341

40 0.25982 -0.07246 0.16903

60 0.25973 -0.07666 0.17000

80 0.25976 -0.07823 0.17032

100 0.25980 -0.07898 0.17047

120 0.25982 -0.07940 0.17055

140 0.25984 -0.07965 0.17059

160 0.25985 -0.07982 0.17062

Table 2.1 (b): Comparison of the flow velocity f obtained by analytical method and FEM

)1,1Pr,3,0,0( RK

f

Analytical Method FEM

0 -1 -1

1 -0.07295 -0.07290

2 -0.00532 -0.00531

3 -0.00039 -0.00039

4 -0.00003 -0.00003

5, 6, 7, 8 0.00000 0.00000

25

2.4 RESULTS AND DISCUSSION

The numerical computations are performed for various values of suction parameter

, the radiation parameter R and the buoyancy parameter . The other parameters such

as Prandtl number Pr and coupling constant parameter K are kept fixed at 1.0. The results

so obtained, are presented in Figs. 2.2-2.10 and the corresponding values are shown in

tables 2.2-2.10. The skin friction coefficient, local couple stress and local Nusselt number

have also been computed for these parameters and are tabulated in tables 2.11-2.13. Higher

values of suction parameter are taken so as to sustain steady flow near the sheet by

confining the generated vorticity inside the boundary layer.

Some interesting observations can be made for velocity from Figs. 2.2 (a)-2.2 (f).

For 0 considered by Bhattacharyya and Layek [39] and Muhaimin et al. [145],

increase in increases the velocity. Our Fig. 2.2 (a) completely agrees with them. But as

increases, the effects of increase in are different in the regions close to the boundary

and away from it as depicted in Figs. 2.2 (b) through 2.2 (f). Near the boundary, velocity

increases while away from it the effect is just opposite. For large , the effect of is

negligible near the boundary. The suction (mass removal from the boundary layer) effect in

fact induces flow reversal very close to the sheet, as indicated by negative values of the

velocity in this region.

Fig. 2.3 depicts the variation of microrotation i.e. micro-element angular velocity,

with the suction parameter. It is observed that near the sheet surface the microrotation

increases with the increase in . The negative values of microrotation show the reverse

rotation only near the boundary. Thus the reverse rotation can be reduced by increasing the

suction. Further from the boundary (sheet surface) the reverse trend is observed i.e.

increasing suction acts to decrease microrotation. However positive values for

microrotation are sustained all the way to the free stream indicating that there is no reversal

in spin further from the wall. This trend has also been identified by Bhattacharyya and

Layek [39].

The effect of suction on the temperature is depicted in Fig. 2.4. Temperature is

observed to decrease as suction increases. Micropolar fluid particles near the heated

surface absorb heat from the sheet via thermal conduction, due to which the temperature of

the fluid is larger near the sheet. As suction is applied, these fluid particles are removed

from the sheet and as a consequence the temperature of the fluid falls. The suction

parameter provides an effective means of controlling the flow and heat transfer

26

characteristics. This has also been documented by Bhargava et al. [37] and furthermore by

Yao et al. [211]. Confidence in the present FEM computations is therefore high.

Fig. 2.5 illustrates that the velocity of the micropolar fluid is reduced with

increasing radiation parameter R . 3

11 4 TkR represents the relative contribution of

thermal conduction heat transfer to thermal radiation heat transfer. It is sometimes referred

to as the Boltzmann-Rosseland number in the literature [30]. Large R values will therefore

imply weaker thermal radiation contribution and vice versa for low R values. When R is

unity both modes of heat transfer are expected to have the same contribution. An increase

in R will imply greater thermal conduction contribution and causes flow reversal near the

sheet surface. Flow velocity is therefore maximized with lowest R values, for which

thermal radiation has a greater effect. The maximum velocity computed corresponds to

5.0R . In light of this, high-temperature materials processing operations are found to

benefit from thermal radiation which tends to oppose flow reversal and sustains a more

stable flow regime in shrinking sheets.

From Fig. 2.6 it is observed that the microrotation increases as R increases in the

vicinity of the sheet. The strong microrotation reversal near the sheet surface is

progressively reduced with increasing R i.e. with weaker thermal radiation contribution.

After a small distance from the sheet the effect is reversed and thereafter micro-rotation is

found to be more positively affected by the stronger thermal radiation case )5.0( R .

However microrotation magnitudes are observed to be much lower as we progress from the

sheet surface towards the free stream.

Fig. 2.7 illustrates the influence of the radiation parameter on the temperature

distribution. Temperature is very strongly increased with a decrease in R . As elaborated

earlier, thermal radiation flux has a progressively greater effect as the value of R is

reduced, with a simultaneous decrease in thermal conduction contribution. With lower R

values, therefore the thermal radiative flux supplements energy in the boundary layer and

this elevates temperatures. The concomitant acceleration in the flow (Fig. 2.5) implies that

radiation has a very prominent and beneficial effect on the dynamics of the shrinking sheet

and the temperature in the vicinity of the sheet, which aids in manufacturing control.

Fig. 2.8 shows that the velocity increases with the increase in buoyancy parameter

. This parameter, 2/5Re xxGr and larger values correspond to greater thermal

buoyancy force. This acts to aid momentum development in the boundary layer regime and

27

effectively accelerates the flow. Near the sheet the flow is strongly reversed. The

maximum velocity computed is associated therefore with the strongest buoyancy parameter

case i.e. 10 . A velocity peak is computed near the sheet for 10,7,5,3 but

vanishes for the weakest buoyancy case of 1 (buoyancy and viscous force equal).

Fig. 2.9 represents the microrotation distribution with the variation of buoyancy

parameter . As increases the microrotation decreases markedly near the boundary.

Further from the surface, the opposite behaviour is observed and micro-rotation is found to

be enhanced with increasing buoyancy parameter. However with further progression into

the boundary layer there is vanishing in micro-rotation.

Fig. 2.10 shows that temperature decreases with increase in buoyancy parameter .

Peak temperature always arises at the sheet surface. The profiles decay smoothly from the

wall to the free stream. In the boundary layer, the influence of buoyancy is found to be

strongest at intermediate distances from the wall. The presence of buoyancy accelerates the

flow i.e. enhances viscous diffusion but inhibits thermal diffusion in the boundary layer i.e.

heat is diffused less effectively. This manifests in suppression in temperature values in the

micropolar fluid. This behaviour has also been observed by Hayat et al. [91] and

furthermore by Ishak et al. [111] both studies concerning micropolar flows. In fact the

depression in temperatures under buoyancy forces is also observed experimentally in many

other non-Newtonian free convection flows (plastics) as elaborated by Baird and Baird

[21]. We further note that in Figs. 2.8-2.10, 1R indicating that thermal conduction and

thermal radiation have an equal contribution and also 3 corresponding to strong

suction at the wall.

Table 2.11 gives skin friction for 1,1Pr K and for different values of , R and

. It is clear that the skin friction decreases numerically with increase in both suction

parameter and radiation parameter R , while it increases with an increase in buoyancy

parameter . The positive values of the skin friction indicate that the fluid exert a drag

force on the sheet. Thus the skin friction can be effectively reduced by introducing the

suction and radiation. From table 2.12 couple stress i.e. microrotation gradient, is found to

decrease with increase in radiation whereas it increases with a rise in the suction and

buoyancy parameters. Therefore a fast rate of micro-element rotation can be achieved by

the suction and buoyancy effects. Also the rotation of the microparticles can be reduced by

increasing the radiation parameter (decreasing thermal radiative flux contribution). It is

clear from table 2.13 that the heat transfer rate increases with the increase in suction,

28

buoyancy and radiation parameters. Thus by applying suction, radiation and buoyancy

parameters the heating of the sheet can be controlled in actual manufacturing operations.

2.5 CONCLUSIONS

The present study has addressed theoretically and numerically the steady flow and

heat transfer of an incompressible micropolar fluid over a porous shrinking sheet in the

presence of thermal radiation. Using a similarity transformation, the governing partial

differential equations have been normalized into a set of nonlinear, coupled, multi-degree

ordinary differential equations. A robust, validated variational finite element method

(FEM) has been employed to solve the resulting well-posed two-point boundary value

problem. Numerical results obtained have clearly demonstrated that the skin friction can be

reduced effectively by imposing suction and decreasing radiation contribution. It has also

been observed that a fast rate of cooling can be achieved with judicious selection of

suction, radiation and buoyancy parameters. The results of this investigation play an

important role in thermal control of the synthesis of packaging units such as shrink

wrapping, bundle wrapping, shrink packaging and shrink film.

29

Table 2.2 (a): Velocity distribution for different )0,1( R

2.5 3.0 3.5 4.0

0 -1 -1 -1 -1

1 -0.36677 -0.20646 -0.13529 -0.09183

2 -0.13432 -0.04261 -0.01830 -0.00843

3 -0.04902 -0.00879 -0.00247 -0.00077

4 -0.01773 -0.00181 -0.00033 -0.00007

5 -0.00626 -0.00037 -0.00005 -0.00001

6 -0.00206 -0.00007 -0.00001 0.00000

7 -0.00054 -0.00001 0.00000 0.00000

8 0 0 0 0

-1

-0.8

-0.6

-0.4

-0.2

0

0 1 2 3 4 5 6 7 8

λ=2.5, 3, 3.5, 4f '

η

Figure 2.2 (a): Velocity distribution for different )0,1( R

30

Table 2.2 (b): Velocity distribution for different )5.0,1( R

2.5 3.0 3.5 4.0

0 -1 -1 -1 -1

1 -0.12483 -0.11507 -0.09145 -0.06845

2 0.02699 0.00428 -0.00070 -0.00108

3 0.03085 0.00947 0.00297 0.00102

4 0.01707 0.00458 0.00120 0.00033

5 0.00759 0.00172 0.00036 0.00008

6 0.00283 0.00055 0.00009 0.00002

7 0.00077 0.00013 0.00002 0.00000

8 0 0 0 0

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0 1 2 3 4 5 6 7 8

λ=2.5, 3, 3.5, 4

f '

η

Figure 2.2 (b): Velocity distribution for different )5.0,1( R

31

Table 2.2 (c): Velocity distribution for different )1,1( R

2.5 3.0 3.5 4.0

0 -1 -1 -1 -1

1 -0.01333 -0.04698 -0.05290 -0.04648

2 0.08533 0.03552 0.01391 0.00560

3 0.05220 0.02014 0.00720 0.00260

4 0.02342 0.00781 0.00231 0.00067

5 0.00909 0.00262 0.00064 0.00015

6 0.00306 0.00077 0.00016 0.00003

7 0.00077 0.00017 0.00003 0.00000

8 0 0 0 0

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0 1 2 3 4 5 6 7 8

f '

λ=2.5, 3, 3.5, 4

η

Figure 2.2 (c): Velocity distribution for different )1,1( R

32

Table 2.2 (d): Velocity distribution for different )2,1( R

2.5 3.0 3.5 4.0

0 -1 -1 -1 -1

1 0.13551 0.05714 0.01329 -0.00609

2 0.15074 0.07791 0.03721 0.01737

3 0.07088 0.03261 0.01340 0.00525

4 0.02700 0.01094 0.00380 0.00122

5 0.00919 0.00330 0.00097 0.00026

6 0.00277 0.00089 0.00023 0.00005

7 0.00063 0.00019 0.00004 0.00001

8 0 0 0 0

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0 1 2 3 4 5 6 7 8

λ=2.5, 3, 3.5, 4

η

f '

Figure 2.2 (d): Velocity distribution for different )2,1( R

33

Table 2.2 (e): Velocity distribution for different )3,1( R

2.5 3.0 3.5 4.0

0 -1 -1 -1 -1

1 0.24293 0.13827 0.06948 0.03048

2 0.19027 0.10688 0.05534 0.02744

3 0.07915 0.03971 0.01773 0.00737

4 0.02741 0.01228 0.00470 0.00162

5 0.00858 0.00346 0.00115 0.00033

6 0.00241 0.00088 0.00026 0.00006

7 0.00052 0.00018 0.00005 0.00001

8 0 0 0 0

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0 1 2 3 4 5 6 7 8

λ=2.5, 3, 3.5, 4

η

f '

Figure 2.2 (e): Velocity distribution for different )3,1( R

34

Table 2.2 (f): Velocity distribution for different )5,1( R

2.0 2.5 3.0 3.5 4.0

0 -1 -1 -1 -1 -1

1 0.57163 0.40208 0.26450 0.16275 0.09501

2 0.36054 0.23855 0.14564 0.08230 0.04393

3 0.13869 0.08555 0.04720 0.02330 0.01054

4 0.04494 0.02608 0.01308 0.00566 0.00217

5 0.01322 0.00728 0.00334 0.00128 0.00042

6 0.00350 0.00184 0.00078 0.00027 0.00008

7 0.00071 0.00036 0.00014 0.00004 0.00001

8 0 0 0 0 0

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0 1 2 3 4 5 6 7 8

λ=2, 2.5, 3, 3.5, 4

ηf '

Figure 2.2 (f): Velocity distribution for different )5,1( R

35

Table 2.3: Microrotation distribution for different )5,1( R

2.0 2.5 3.0 3.5 4.0

0 -2.07754 -1.97512 -1.89600 -1.85179 -1.84993

1 -0.25692 -0.20273 -0.15984 -0.12653 -0.10048

2 0.03901 0.02755 0.01731 0.00955 0.00467

3 0.03480 0.02231 0.01276 0.00647 0.00297

4 0.01415 0.00842 0.00433 0.00191 0.00074

5 0.00468 0.00262 0.00122 0.00047 0.00015

6 0.00138 0.00073 0.00031 0.00011 0.00003

7 0.00033 0.00017 0.00007 0.00002 0.00000

8 0 0 0 0 0

-2.25

-2

-1.75

-1.5

-1.25

-1

-0.75

-0.5

-0.25

0

0.25

0 1 2 3 4 5 6

λ =2, 2.5, 3, 3.5, 4g

η

Figure 2.3: Microrotation distribution for different )5,1( R

36

Table 2.4: Temperature distribution for different )5,1( R

2.0 2.5 3.0 3.5 4.0

0 1.13915 0.95378 0.79714 0.66263 0.59079

1 0.45117 0.32151 0.22232 0.14693 0.11124

2 0.16262 0.09940 0.05774 0.03107 0.02036

3 0.05581 0.02941 0.01444 0.00638 0.00364

4 0.01872 0.00854 0.00356 0.00130 0.00065

5 0.00613 0.00244 0.00087 0.00026 0.00011

6 0.00190 0.00066 0.00020 0.00005 0.00002

7 0.00048 0.00015 0.00004 0.00001 0.00000

8 0 0 0 0 0

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4 5 6 7 8η

θ

λ =2, 2.5, 3, 3.5, 4

Figure 2.4: Temperature distribution for different )5,1( R

37

Table 2.5: Velocity distribution for different R )5,3(

R

0.5 0.7 1.0 1.5 2.0

0 -1 -1 -1 -1 -1

1 0.60817 0.42092 0.26450 0.13449 0.06818

2 0.31800 0.22062 0.14564 0.08778 0.05985

3 0.11801 0.07583 0.04720 0.02767 0.01907

4 0.03990 0.02311 0.01308 0.00714 0.00481

5 0.01268 0.00659 0.00334 0.00166 0.00108

6 0.00362 0.00171 0.00078 0.00035 0.00022

7 0.00076 0.00034 0.00014 0.00006 0.00004

8 0 0 0 0 0

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

0 1 2 3 4 5 6 7 8

R =0.5, 0.7, 1, 1.5, 2

f '

η

Figure 2.5: Velocity distribution for different R )5,3(

38

Table 2.6: Microrotation distribution for different R )5,3(

R

0.5 0.7 1.0 1.5 2.0

0 -2.60834 -2.22224 -1.89600 -1.62062 -1.47779

1 -0.14703 -0.15426 -0.15984 -0.16385 -0.16554

2 0.04720 0.03178 0.01731 0.00417 -0.00292

3 0.02564 0.01874 0.01276 0.00758 0.00486

4 0.00949 0.00654 0.00433 0.00266 0.00185

5 0.00319 0.00200 0.00122 0.00072 0.00050

6 0.00099 0.00056 0.00031 0.00017 0.00012

7 0.00024 0.00013 0.00007 0.00003 0.00002

8 0 0 0 0 0

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

0 1 2 3 4 5 6

g

η

R =0.5, 0.7, 1, 1.5, 2

Figure 2.6: Microrotation distribution for different R )5,3(

39

Table 2.7: Temperature distribution for different R )5,3(

R

0.5 0.7 1.0 1.5 2.0

0 1.13915 0.95378 0.79714 0.66263 0.59079

1 0.45117 0.32151 0.22232 0.14693 0.11124

2 0.16262 0.09940 0.05774 0.03107 0.02036

3 0.05581 0.02941 0.01444 0.00638 0.00364

4 0.01872 0.00854 0.00356 0.00130 0.00065

5 0.00613 0.00244 0.00087 0.00026 0.00011

6 0.00190 0.00066 0.00020 0.00005 0.00002

7 0.00048 0.00015 0.00004 0.00001 0.00000

8 0 0 0 0 0

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4 5 6 7 8

θ

η

R =0.5, 0.7, 1, 1.5, 2

Figure 2.7: Temperature distribution for different R )5,3(

40

Table 2.8: Velocity distribution for different )1,3( R

1 3 5 7 10

0 -1 -1 -1 -1 -1

1 -0.04698 0.13827 0.26450 0.36319 0.48160

2 0.03552 0.10688 0.14564 0.17111 0.19660

3 0.02014 0.03971 0.04720 0.05065 0.05264

4 0.00781 0.01228 0.01308 0.01297 0.01229

5 0.00262 0.00346 0.00334 0.00308 0.00268

6 0.00077 0.00088 0.00078 0.00067 0.00054

7 0.00017 0.00018 0.00014 0.00012 0.00009

8 0 0 0 0 0

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0 1 2 3 4 5 6 7 8η

f '

σ =1, 3, 5, 7, 10

Figure 2.8: Velocity distribution for different )1,3( R

41

Table 2.9: Microrotation distribution for different )1,3( R

1 3 5 7 10

0 -1.10625 -1.54750 -1.89600 -2.19849 -2.60003

1 -0.16357 -0.16221 -0.15984 -0.15717 -0.15298

2 -0.01390 0.00582 0.01731 0.02523 0.03352

3 0.00264 0.00976 0.01276 0.01429 0.01536

4 0.00209 0.00391 0.00433 0.00437 0.00420

5 0.00087 0.00124 0.00122 0.00113 0.00099

6 0.00030 0.00035 0.00031 0.00027 0.00021

7 0.00008 0.00008 0.00007 0.00005 0.00004

8 0 0 0 0 0

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

0 1 2 3 4 5 6

g

η

σ =1, 3, 5, 7, 10

Figure 2.9: Microrotation distribution for different )1,3( R

42

Table 2.10: Temperature distribution for different )1,3( R

1 3 5 7 10

0 0.85365 0.81859 0.79714 0.78137 0.76336

1 0.27252 0.24107 0.22232 0.20881 0.19369

2 0.08728 0.06811 0.05774 0.05076 0.04348

3 0.02764 0.01871 0.01444 0.01181 0.00928

4 0.00867 0.00508 0.00356 0.00271 0.00195

5 0.00267 0.00136 0.00087 0.00061 0.00041

6 0.00078 0.00035 0.00020 0.00013 0.00008

7 0.00019 0.00007 0.00004 0.00003 0.00001

8 0 0 0 0 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 1 2 3 4 5 6 7 8

σ =1, 3, 5, 7, 10

θ

η

Figure 2.10: Temperature distribution for different )1,3( R

43

Table 2.11: The skin friction coefficient )0(f for different values of , R and 1,1Pr K

R =1, =5 = 3, =5 ,3 R =1

)0(f R )0(f )0(f

2.0 4.15509 0.5 5.21669 1 2.21250

2.5 3.95024 0.7 4.44449 3 3.09500

3.0 3.79200 1.0 3.79200 5 3.79200

3.5 3.70358 1.5 3.24125 7 4.39699

4.0 3.69985 2.0 2.95557 10 5.20005

Table 2.12: The local couple stress 0g for different values of , R and 1,1Pr K

R =1, =5 = 3, =5 ,3 R =1

)0(g R )0(g )0(g

2.0 3.21206 0.5 5.62872 1 1.95611

2.5 3.50852 0.7 4.65042 3 2.99345

3.0 3.84318 1.0 3.84318 5 3.84318

3.5 4.25235 1.5 3.17565 7 4.59885

4.0 4.78018 2.0 2.83447 10 5.62690

Table 2.13: The local Nusselt number )0(/1 for different values of , R and 1,1Pr K

R =1, =5 = 3, =5 ,3 R =1

)0(/1 R )0(/1 )0(/1

2.0 0.91434 0.5 0.87785 1 1.17145

2.5 1.07588 0.7 1.04846 3 1.22162

3.0 1.25449 1.0 1.25449 5 1.25449

3.5 1.44778 1.5 1.50914 7 1.27980

4.0 1.65203 2.0 1.69266 10 1.30999