MHD and Radiation Effects on Mixed Convection Unsteady Flow of Micropolar Fluid Over A Stretching Sheet

MHD and Radiation Effects on Mixed Convection Unsteady Flow of Micropolar Fluid Over A Stretching Sheet

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Available online at www.sciencedirect.com

ScienceDirect Procedia Computer Science 57 (2015) 65 – 76

2013 International Conference on Computational Science

MHD AND RADIATION EFFECTS ON MIXED CONVECTION UNSTEADY FLOW OF MICROPOLAR FLUID OVER A STRETCHING SHEET K. Govardhan1, G. Nagaraju1, K.KALADHAR2 and M.Balasiddulu1 Department of Engineering Mathematics, Gitam University, Hyderabad-502329, India. Department of Mathematics,NIT pudacherry

Abstract This present paper is concerned with the study of the magnetohydrodyamics (MHD) effects on mixed convection flow of an incompressible micropolar fluid over a stretching sheet in case of unsteady flow. Energy equation takes into account of thermal radiation. The stretching velocity is assumed to vary linearly with the distance along the sheet. Two equal and opposite forces are assumed to be impulsively applied along axial direction. The governing non-linear equations and their associated boundary conditions are first cast into a dimensionless form using local non-similarity transformations. The resulting equations are solved numerically using the Adams-Predictor Corrector method. A representative set of numerical results is displaced graphically to illustrate the influence of various physical parameters on velocity, microrotation profiles as well as the skin friction coefficient. It is found that there is a smooth transition from small-time solution to the large-time solution. © 2015 byby Elsevier B.V. This is an open access article under the CC BY-NC-ND license 2013 The TheAuthors. Authors.Published Published Elsevier B.V. (http://creativecommons.org/licenses/by-nc-nd/4.0/). Selection and/or peer-review under responsibility of the organizers of the 2013 International Conference on Computational Peer-review under responsibility of organizing committee of the 3rd International Conference on Recent Trends in Computing 2015 Science (ICRTC-2015) Keywords: Unsteady flow, micro polar fluid, stretching surface, skin friction, MHD.

1. Introduction: The boundary layer flow on a continuous stretching sheet has attracted considerable attention during the last few decades due to its numerous applications in industrial manufacturing processes such as hot rolling, wire drawing, glass-fiber and paper production, drawing of plastic films, metal and polymer extrusion and metal spinning. The development of boundary layer flow induced solely by a stretching sheet was first studied by Crane (1970), who found an exact solution for the flow field. This problem was then extended by Gupta and Gupta (1977) to a permeable surface. The flow problem due to a linearly stretching sheet belongs to a class of exact solutions of the Navier-Stokes equations. Thus, the exact solutions reported by Crane (1970) and Gupta and Gupta (1977) are also the exact solutions to the Navier-Stokes equations. Dinarvand (2008) offered a

reliable treatment for viscous flow over a non-linearly stretching sheet in presence of a chemical reaction and under influence of a magnetic field using HAM. Mehmood and Ali (2008) obtained the HAM solution for the three-dimensional viscous flow and heat transfer over a stretching flat surface. A numerical study on flow and heat transfer of a nano fluid over a nonlinearly stretching

1877-0509 © 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of organizing committee of the 3rd International Conference on Recent Trends in Computing 2015 (ICRTC-2015) doi:10.1016/j.procs.2015.07.366

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sheet has been presented by Rana and Bhargava (2012). Most recently, Malvandi et al. (2014) considered the slip effects on unsteady stagnation point flow of a nano fluid over a stretching sheet. The flow properties of certain fluids like polymeric fluids, muds, colloidal fluids, animal blood, fluids containing additives, ferro fluids etc. cannot be explained properly by the Navier ̢ Stokes equations of Newtonian and non-Newtonian fluids theory. Because of this, many constitutive models have been suggested by several researchers. Among these models, the micro fluid model proposed by Eringen (1964) has attracted considerable attentions. The micropolar fluid theory describes the flow of a class of non-Newtonian fluids that are endowed with microinertia. This allows the fluid to withstand stress and body couples. In view of applications several authors (Sajid et al., 2009; Si et al., 2010; Sibanda and Awad, 2010; Srinivasacharya and RamReddy, 2011) presented the nature and applications of micro polar fluids in different geometries with the effects of various emerging parameters. The study of magneto-hydrodynamic (MHD) flow of an electrically conducting fluid is of considerable interest in modern metallurgical and metal-working processes. Some important applications of radiative heat transfer include MHD accelerators, high temperature plasmas, power generation systems and cooling of nuclear reactors. Many processes in engineering areas occur at high temperatures and knowledge of radiation heat transfer becomes very important for the design of pertinent equipment (Seddeek, 2002). In controlling momentum and heat transfers in the boundary layer flow of different fluids over a stretching sheet, applied magnetic field may play an important role (Turkyilmazoglu, 2012). Fathizadeh et al. (2013) presented an effective modification on MHD viscous flow over a stretching sheet by using HPM. Satya Narayana et al. (2013) studied the effects of Hall current and radiation absorption on MHD micropolar fluid in a rotating system. Most recently, Mabood et al. (2014) have reported the influence of MHD flow over exponential radiating stretching sheet with HAM Solution. Many of the problems in the literature deal with steady flow of Newtonian and non-Newtonian fluids with radiation and MHD effects but much attention is not given to Magnetohydrodynamics and thermal radiation effects on unsteady non- Newtonian fluids. The aim of the present paper is therefore, the analysis of unsteady radiation effects on mixed convection unsteady flow of micropolar fluid flow over a stretching sheet. In addition the sheet is placed in a magnetic field and also stretched in its own plane. The nonlinear governing equations and their associated boundary conditions are initially cast into dimensionless form by similarity transformations. A numerical solution is obtained for the governing equations using the Adams predictor-corrector method. In order to get a clear insight of the physical problem, behavior of the emerging flow parameters on the velocity and temperature are displayed through graphical illustrations. Nomenclature u , v velocity components x, y cartesian co-ordinates q Radiative heat flux n constant j micro-inertia

N K B0 M

micro rotation matériel parameter applied magnetic field

magnetic field parameter electrical conductivity T temperature KC coefficient of thermal conductivity

σ

δ micropolar heat conduction coefficient G Grashof number Pr Prandtl number

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ȕ coefficient of volume expansion j microinertia ȡ density ȝ coefficient of dynamic viscosity k coefficient of gyroviscosity Ȗ coefficient of couple stress C Cp specific heat at constant pressure T’ temperature of the free stream fluid Tw temperature of the sheet f,g non-dimensional velocity and microrotation components ș non-dimensional temperature

δ*

heat conduction parameter R radiation parameter Rex local Reynolds number Į thermal diffusivity Į* heat generation parameter

1.1. Mathematical formulation Consider the unsteady Magnetohydrodynamic flow of incompressible micropolar fluid along a streaching sheet with radiation effects. The incompressible micropolar fluid flow in the region y > 0 driven by a plane surface located at y = 0 with a fixed end at x = 0 is considered. It is assumed that the surface is stretched in the x − direction such that temperature and the x − component of the velocity varies linearly along the plate, i.e.

Tw ( x ) = T∞ + ax

and

u w ( x) = cx respectively, where a and c are arbitrary positive

constants. The wall is maintained at a constant temperature Tw and the free stream fluid temperature is maintained at T’. The effects of viscous dissipation are assumed to be negligible. This configuration is assumed to be placed under a transversely applied uniform magnetic field of strength B0. With the above assumptions, the equations governing the unsteady flow of an incompressible micropolar fluid and under the usual MHD approximations along with the Boussinesq approximation are:

∂u ∂v + =0 ∂x ∂y

(1)

∂u ∂u ∂ u § μ + k · ∂ 2 u k ∂ N σ B 2 0 + ρgβ (T − T ) +u +v =¨ + − u T ∞ ¸ ρ ∂t ∂x ∂y © ρ ¹ ∂y 2 ρ ∂y

(2)

§ ∂u · ¨ 2 N + ∂y ¸ © ¹ K C ∂ 2T γ0 1 ∂q δ § ∂T ∂N ∂T ∂N · ∂T ∂T ∂T +u +v = − + − (T − T∞ ) ¨ ¸+ 2 ρ C p ∂y ρ C p ∂y ρ C p © ∂x ∂y ∂t ∂x ∂y ∂y ∂x ¹ ρ C p

γ ∂2N κ ∂N ∂N ∂N +u +v = − 2 ρ j ∂y ρ j ∂t ∂x ∂y

where the term γ 0

(3) (4)

(T − T∞ ) in equation (4) represents the amount of heat generated (heat source) per unit time, per

unit volume. Here

is the spin gradient viscosity,

is the microinertia density, k

Coefficient of gyroviscosity KC Coefficient of thermal conductivity. The above equations (1)-(4) are subject to the following initial and boundary conditions:

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t ≤ 0 : u = v = N = 0 , for any x, y . ∂u , T = Tw, at y = 0 ∂y u → 0 , T → T∞ and N → 0 as y → ∞ . (5) where n is a constant and it varies from 0 to 1. The case with n = 0 , indicates that N = 0 at the wall i.e., in the t > 0 : v = 0, u = u w ( x) = cx, N = −n

concentrated particle flows; the microelements close to the wall surface are unable to rotate. This case is known as the strong concentration of microelements in the fluid. The case n = 1 / 2 indicates the vanishing of anti– symmetric part of the stress tensor and denotes weak concentration of microelements. The case n = 1 is used for the modeling of turbulent boundary layer flows. We shall consider here both cases of n = 0 and n = 1 / 2 . By using the Rosseland approximation, the radiative heat flux q is given by

q=−

4σ s ∂ T 4 3k e ∂y

(6)

Where ıs is the Stefan-Boltzmann constant and ke is the mean absorption coeƥcient. It should be noted that by using the Rosseland approximation, the present analysis is limited to optically thick fluids. If the temperature deˆerence within the flow is sufficiently small then (6) can be linearized by expanding T4 in terms of Taylor series about T’ and after neglecting higher order terms T4 is of the form (7) T 4 ≅ 4 T ∞3T − 3 T ∞4 In view of (6) and (7), (4) reduces to

γ ∂T ∂T ∂T ∂ 2T δ § ∂T ∂N ∂T ∂N · +u +v = α (1 + R ) 2 + − + 0 ( T − T∞ ) ∂t ∂x ∂y ∂y ρ C p ¨© ∂ x ∂ y ∂ y ∂ x ¹¸ ρ C p

(8)

Where α = K c is the thermal diffusivity and R = 1 6σ s T ∞ is the radiation parameter 3

ρcp

3K cke In view of the continuity equation (1), we introduce the ψ stream function by ∂ψ ∂ψ and v = − u= ∂x ∂y

(9)

substituting (9) in (2)-(4) and then using the following transformations

ψ = (cv )1/ 2 ξ 1/ 2 x f (ξ ,η ), N = (c / v )1/ 2 ξ −1/ 2 cx g (ξ ,η ), η = (c / v )1 / 2ξ −1 / 2 y, ξ = 1 − e −τ ,τ = ct , θ (ξ , η ) = T − T ∞

(10)

Tw − T ∞ we get the following non-linear system of differential equations

(1 + K ) f ′′′ + (1 − ξ ) K § ¨1 + 2 © §1+ R ¨ © Pr

η

2

f ′′ + ξ

( ff ′′ −

f ′ 2 − M f ′ ) + K g ′ + G rθ = ξ (1 − ξ

)

∂f ′ ∂ξ

η ∂g · ′′ §1 · ¸ g + (1 − ξ ) ¨ g + g ′ ¸ + ξ ( fg ′ − f g′ ) − K ξ ( 2 g + f ′′ ) = ξ (1 − ξ ) ∂ξ 2 ¹ ©2 ¹ η ∂θ · * ∗ ¸ θ ′′ + 2 (1 − ξ ) θ ′ + ξ ( f θ ′ − f ′θ ) + δ (θ g ′ − θ ′ g ) + α ξ θ = ξ ∂ ξ (1 − ξ ) ¹

where K =

k

μ

(11) (12) (13)

is the material parameter, P r = μ c p is the prandtl number, δ * = c δ is the heat conduction

Kc

μcp

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parameter and

α* =

γ0 ρ cc p

is the heat generation parameter.

The boundary conditions given in equation (5) becomes

f (ξ , 0 ) = 0, f ′(ξ , 0 ) = 1, g (ξ , 0 ) = −n f ′′ (ξ , 0 ) , θ (ξ ,0) = 1

f ′(ξ , ∞ ) = 0, g (ξ , ∞ ) = 0. θ (ξ , ∞ ) = 0

The physical quantity of interest in this problem is the skin friction coefficient

τw

Cf =

ρ uw 2 / 2 where τ w is the skin friction and it is given by ª

º

∂u

+ kN » τ w = «( μ + k ) . ∂y ¬ ¼ y =0

(14)

C f , which is defined as (15)

(16)

From the above, friction factor can be written as

C f Re x1/ 2 = ξ −1/ 2 ª¬1 + (1 − n ) K º¼ f ′′ (ξ , 0 ) .

(17)

In the forgoing analysis some particular cases of this problem are discribed. A. Early Unsteady Flow For early unsteady flow i.e., 0 < τ << 1, we have ξ ≈ 0 , hence equations (11)-(13) reduce in the leading order approximation to

(1 + K ) f ′′′ +

η

(18) f ′′ + K g ′ = 0 2 K · η 1 § (19) ¨1 + ¸ g ′′ + g ′ + g = 0 2 2 ¹ 2 © §1+ R · ∗ (20) ¨ Pr ¸ θ ′′ − ( δ g )θ ′ = 0 © ¹ and the boundary conditions given in equation (14) become f ( 0 ) = 0 , f ′ ( 0 ) = 1, g ( 0 ) = − n f ′′ ( 0 ) , f ′ ( ∞ ) = 0 , g ( ∞ ) = 0 , θ ( 0 ) = 1, θ ( ∞ ) = 0 (21) B. Final steady- state Flow For this case with ξ = 1 and equations (11) - (13) take the following form:

(1 + K ) f ′′′ + f

f ′′ − f ′2 − Mf ′ + Kg ′ = 0

K· § ¨1 + ¸ g ′′ + f g ′ − f ′ g − K (2 g + f ′′) = 0 2¹ © §1+ R · ∗ ∗ ¨ Pr ¸ θ ′′ + f − δ g θ ′ + α θ = 0 © ¹

(

)

(22) (23) (24)

Subject to the boundary conditions given in (14). 1.2. Method of Solution To solve the equations (11)-(13), the system is converted to five first order equations at

ξ + Δξ ,

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( y1 →

f , y 4 → g ), y1′ = y 2 , y 2′ = y3 ,

y3′ = {ξ (1 − ξ ) y2 (ξ + Δξ ) − y2 (ξ ) − (1 − ξ ) (η / 2) y3 + ξ ( y1 y3 − y 2 2 − M * y 2 ) − ky5}/(1 + k ) ,

y 4′ = y5 and

y5′ = {ξ (1 − ξ ) ( y4 (ξ + Δξ ) − y5 ) / Δξ − (1 − ξ )( y4 / 2 + η y5 / 2 ) − ξ ( y1 y5 − y2 y4 )}/ (1 + k / 2 ) .

The early unsteady flow is obtained by solving these equations with ξ = 0 . For ξ > 0 , the above

equations reflect a fully implicit scheme with respect to and y 5 (ξ ,0 ) =

∞) . ∂y 2 ∂ y 4 ∂ y 2 ∂y 4 To solve α and β by Newton–Raphson method, and are required at η = η max and , , ∂α ∂ α ∂ β ∂β these quantities are obtained by the following solutions, namely, y1′ = y 2 , y 2′ = y3 , y3′ = {ξ (1 − ξ ) ∂y2 (ξ + Δξ ) / Δξ − (1 − ξ )η y3 / 2 + ξ ( y1Y3 + Y1 y3 − 2 y2Y2 − M * y2 ) − ky5}/ (1 + k ) y 4′ = y5

β

, the above system is solved until η max (≈

ξ . In both cases, assuming y 3 (ξ ,0 ) = α

and y5′ = {ξ (1 − ξ ) y4 (ξ + Δξ ) / Δξ − (1 − ξ )( y4 / 2 + η y5 / 2 ) + ξ ( y1Y5 + Y1 y5 − y2Y1 − Y2 y1 )}/ (1 + k / 2 ) It is assumed that at the initial time

y1 ( 0 ) = y 2 ( 0 ) = 0, y3 ( 0 ) = 0, y 4 ( 0 ) = − n , y 5 (0) = 0 and at

another time with y1 (0) = y2 (0) = y3 (0) = y4 (1) = 0, y5 (0 ) = 1. This procedure converging in three iterations and gives correct values of α and β . This system of ordinary differential equations is solved by Adams predictor- corrector method of fourth order. Accuracy is ensured by solving with different values of Δ ξ , η max , Δη . 1.2. Results and Discussion The transformed equations (11)-(13) satisfying the boundary conditions (14) were solved numerically using the above said method. Influence of various flow parameters on skin friction, the velocity distribution and microrotation distribution are shown graphically and analyzed in the subsequent paragraphs. To validate the present numerical results, present results have been compared with the special case of Nazar et al. (2004) and is shown in Table 1. It can be observed that the results are very good agreement. Therefore, the developed code can be used with great confidence to study the present problem. Here the fully unsteady boundary layer equations are solved until the steady state solutions are reached. The velocity distribution of initial flow ( ξ = 0 ) and unsteady flow (0 < ξ ≤ 1) for various values of

K with n = 0 and n=1/2 is shown in Figs. 1 and 2 respectively. From these figures it is observed that the velocity increases as an increase in the parameter K. This is because of the velocity boundary layer thickness increases with the increasing of K, in both the cases n = 0 and n=1/2. Figure 3 and 4 represents the final

(

)

steady state flow ξ = 1 when n = 0 and n=1/2. It can be seen from these figures that the velocity increases with the increasing values of K. It is evident from the above that the boundary layer thickness increases with K. The velocity distribution of the fully developed unsteady flow 0 < ξ < 1 and final steady state

(

)

flow (ξ = 1) are presented in the Figs. 5 and 6 at n = 0 and n=1/2 respectively. It is clear from these figures that the velocity decreases with the increase of [. Here the profile corresponding to increasing of

ξ ( 0 < ξ ≤ 1)

approaches the final steady profile corresponding to ξ = 1 . It is also observed that there is a

K. Govardhan et al. / Procedia Computer Science 57 (2015) 65 – 76

smooth transition from small time solution

(ξ ≈ 0) to large time solution (ξ = 1) .

The effect of magnetic parameter M on the velocity profile f ′(η ) is shown in figure 7 at final steady

flow (ξ = 1) with K = 1 and n = 0 . It is obvious that existence of magnetic field M decreases the flow velocity. It is a known fact that the application of a uniform magnetic field normal to the flow direction gives rise to a force called Lorentz force. This force has tendency to slow down the velocity of the fluid. Figure 8 represents the velocity distribution for various values of Grashof number Gr . It can be observed that as the value of Gr increases, the dimensionless velocity rises. Figure 9 represents the microrotation distribution for various values of K , n in case of steady state

(ξ = 1) increases with the increase of the material parameter K . The microrotation distribution of final steady state flow (ξ = 1) with

flow. It can be seen from Fig. 9 that the microrotation distribution of final steady flow

n = 1 / 2 is shown in Fig.10. It is clear that the microrotation decreases as K increases in the vicinity of the plate where as it increases as one moves away from it. Figure 11 represents the microrotation distribution of fully developed unsteady flow at n = 0 and K = 1 for 0 < ξ ≤ 1 . It is noticed that the microrotation distribution as a parabolic distribution and increases with the increase of ξ .

Figure 12 depicts that the microrotation distribution of early unsteady flow ( 0 < ξ ≤ 1) for various K values with n = 1 / 2 . It is obvious that the microrotation distribution decreases as K increase near to the plate but reverse phenomena is observed as one moves away from the plate. The microrotation distribution of fully developed unsteady flow when K = 1 , n = 1 / 2 and for the flow from unsteady to steady state flow is shown in Figure 13. It is seen that the microrotation distribution increases near the plate while, the reverse trend is observed far away from the plate with the increase of ξ . The magnetic field effect on the microrotation distribution is plotted in Figure 14. It can be seen from this figure that the microrotation decreases as magnetic effect increases near to the plate but reverse trend can be seen when the it is away from the plate. Since the magnetic field effect accelerates the microrotation distribution near the plate, where as it de accelerates this distribution far away from the plate. So the magnetic field effect is more on microrotation distribution when far away from the plate. The effect of Gr on microrotation is shown in figure 15. It is obvious that the microrotation increases with an increase in Gr .

α ∗ and δ ∗ when K = 1 ∗ and η = 1 / 2 . It is noted that the temperature profile increases with the increasing of α and also temperature ∗ increases as an increase in δ . The variation of skin friction coefficient with ξ at various values of K is drawn when n = 0 in Fig. 18 and n = 1 / 2 in Fig. 19. It can be seen from these figures that the friction factor decreases as K increases. Figures 16 and 17 represent the temperature distribution for different

1.3. Conclusions In this study, a boundary layer analysis for magnetohydrodyamics (MHD) and radiation effects on mixed convection flow of incompressible micropolar fluid over a stretching sheet in case of unsteady flow has been presented. Using the local non-similarity transformations, the governing equations are transformed into a set of non similar parabolic equations where numerical solution has been presented for a wide range of parameters. The main findings are summarized as follows: ¾ In case of unsteady and steady, the dimensionless velocity increases as an increase in the parameter K in both the cases of n = 0 and n=1/2.

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¾ ¾

The velocity profile decreases with the increase of [ when n = 0 and n=1/2. As magnetic parameter M increases, flow velocity decreases in case of steady flow. The dimensionless velocity rises as Gr number increases. The microrotation distribution of final steady flow (at [= 1) increases with the increase of the material parameter K in both the cases of n = 0 and n=1/2. The microrotation distribution as a parabolic distribution and increases with the increase of [. The microrotation profile decreases as K increase near to the plate but reverse phenomena is observed as one moves away from the plate in the case of unsteady flow. The microrotation distribution increases near the plate while, the reverse trend is observed far away from the plate with the increase of [. The magnetic field effect is more on microrotation distribution when far away from the plate. As an increase in Gr leads to increase in microrotation distribution.

¾ ¾

The dimensionless temperature increases with the increasing of The friction factor decreases as K increases.

¾ ¾ ¾ ¾ ¾ ¾ ¾

α ∗ and δ ∗ .

Table 1: Comparison of skin friction coefficient for flow of a micropolar fluid towards a stretching sheet (Nazar et al. 2004) K

n=0 Nazar et al

n=1/2 Present

Nazar et al

present

0

-1.0000

-1.0043

-1.0000

-1.0043

1

-1.3679

-1.3952

-1.2247

-1.2400

2

-1.6213

-1.6635

-1.4142

-1.4532

4

-2.0042

-2.0092

-2.0042

-1.8105

Fig.1. Velocity distribution of initial flow (ξ = 0) and early unsteady flow ( 0 < ξ << 1) for various K with n = 0 .

Fig. 2. Velocity distribution of initial flow ([ = 0) and early unsteady flow ( 0 < [ << 1) for various K with n = 1/ 2.

K. Govardhan et al. / Procedia Computer Science 57 (2015) 65 – 76

Fig.3. Velocity distribution of final steady- state flow (ξ = 1) for various K with n=0.

73

Fig.4. Velocity distribution of final steady-state flow (ξ = 1) for various K with n=1/2.

Fig.5. Velocity distribution of fully developed unsteady flow for K=1 when n = 0 .

Fig.6. Velocity distribution of fully developed unsteady flow for K=1 when n = 1 / 2 .

Fig.7.Velocity distribution of final steady-state flow ([=1) for various M with n = 0 and K=1.

Fig.8.Velocity distribution of final steady-state flow ([=1) for various Gr with n = 0 and K=1.

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Fig.9.Micro rotation distribution of final steady - state flow (ξ = 1) for various K when n=0.

Fig.10. Micro rotation distribution of final steady- state flow (ξ = 1) for various K when n=1/2.

Fig.11. Micro rotation distribution of fully developed unsteady flow for n = 0 and K=1.

Fig.12. Micro rotation distribution of early unsteady flow ( 0 < ξ << 1) for various K with n = 1 / 2 .

Fig.13.Microrotation distribution of unsteady flow for n = 1 / 2 and K = 1 .

Fig.14. Micro rotation distribution of final steady-state flow (ξ = 1 ) for various M when K=1, n = 0 .

fully

developed

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Fig.15.Microrotation distribution of unsteady flow for n = 1 / 2 and K = 1 .

fully

developed

Fig.17.Temparaturer Profile ș for different δ ∗ parameter for K = 1 and η = 1 / 2 .

Fig.16.Temparaturer Profile θ α ∗ parameter for K = 1 and η = 1 / 2 .

different

Fig.18. Variation of [ on skin friction coefficient for various K with n = 0 .

Fig.19. Variation with ξ of the skin friction coefficient for various K with n = 1 / 2 .

for

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