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Alexandria Engineering Journal (2015) 54, 223-232

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Alexandria University Alexandria Engineering Journal

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ORIGINAL ARTICLE

Numerical investigation on heat and mass transfer c^Ma* effect of micropolar fluid over a stretching sheet through porous media

B. Mohanty a, S.R. Mishra a'*, H.B. Pattanayak b

a Department of Mathematics, Institute of Technical Education and Research, Siksha 'O' Anusandhan University, Khandagiri, Bhubaneswar 751030, Orissa, India b Department of Mathematics, KIIT University, Bhubaneswar, India

Received 3 June 2014; revised 12 February 2015; accepted 10 March 2015 Available online 31 March 2015

KEYWORDS

Micropolar; Mixed convection; Heat transfer; Mass transfer; Runge-Kutta scheme; Shooting technique

Abstract The present paper deals with the study of unsteady heat and mass transfer characteristics of a viscous incompressible electrically conducting micropolar fluid. The flow past over a stretching sheet through a porous medium in the presence of viscous dissipation. A uniform magnetic field is applied transversely to the direction of the flow. Similarity transformations are used to convert the governing time dependent non-linear boundary layer equations into a system of non-linear ordinary differential equations that are solved numerically by Runge-Kutta fourth order method with a shooting technique. The influence of unsteady parameter (A), Eckert number (Ec), porous parameter (Kp), Prandtl number (Pr), Schmidt number (Sc) on velocity, temperature and concentration profiles are shown graphically. The buoyancy force retards the fluid near the velocity boundary layer in case of opposing flow and is favorable for assisting flow. In case of assisting flow, the absence of porous matrix enhances the flow. The impact of physical parameters on skin friction co-efficient, wall couple stress and the local Nusselt number and Sherwood number are shown in tabular form. © 2015 Faculty of Engineering, Alexandria University. Production and hosting 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/).

1. Introduction

The boundary layer flow, heat and mass transfer in a quiescent Newtonian and non-Newtonian fluid driven by a continuous stretching sheet are of significance in a number of industrial engineering processes such as the drawing of a polymer sheet or filaments extruded continuously from a die, the cooling of

* Corresponding author.

E-mail address: satyaranjan_mshr@yahoo.co.in (S.R. Mishra).

Peer review under responsibility of Faculty of Engineering, Alexandria

University.

a metallic plate in a bath, the aerodynamic extrusion of plastic sheets, the continuous casting, rolling, annealing and thinning of copper wires, the wires and fiber coating, etc. The final product of desired characteristics depends on the rate of cooling in the process and the process of stretching. Mohammadi and Nourazar [1] studied on the insertion of a thin gas layer in micro cylindrical Couette flows involving power-law liquids. The analytical solution for two-phase flow between two rotating cylinders filled with power-law liquid and a micro layer of gas has been investigated by Mohammadi et al. [2]. The dynamics of the boundary layer flow over a stretching surface originated from the pioneering work of Crane [3]. Later on,

http://dx.doi.org/10.1016/j.aej.2015.03.010

1110-0168 © 2015 Faculty of Engineering, Alexandria University. Production and hosting 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/).

Nomenclature

A unsteadiness parameter T non-dimensional temperature

a positive constant with dimension per time t non-dimensional time

B non-dimensional parameter T T x wall temperature of the fluid

Bo applied magnetic field T1 temperature of the fluid far away from the sheet

b constant with dimension temperature over length Ux sheet velocity

C concentration of the solute (u, v) velocity components

Cfx local skin friction co-efficient (x, y) cartesian co-ordinates

CP specific heat at constant pressure

Csx local couple stress co-efficient Greek symbol

CX concentration of the solute at the sheet a stretching rate

C concentration of the solute far from the sheet ao thermal diffusivity

D molecular diffusivity b coefficient of thermal expansion

Ec Eckert number b coefficient of concentration expansion

e positive constant D micropolar parameter

f dimensionless stream function t kinematics coefficient of viscosity

g acceleration due to gravity K kinematics micro-rotation viscosity

Gc solutal Grashof number k0 non-dimensional material parameter

Gr thermal Grashof number l coefficient of viscosity

h non-dimensional variable c spin gradient viscosity

j micro inertia density r electrical conductivity

Kp local porous parameter p density of the fluid

k thermal conductivity sw wall shear stress

Kp permeability of the porous medium / non-dimensional concentration

M magnetic field parameter h non-dimensional temperature

Mwx local wall couple stress g similarity variable

N micro-rotation component

Nux local Nusselt number Subscripts

Pr Prandtl number x condition at wall

Rex local Reynold number i condition at free stream

Sc Schmidt number

Shx local Sherwood number

various aspects of the problem have been investigated such as Gupta and Gupta [4], Chen and Char [5], and Dutta et al. [6] extended the work of Crane [3] by including the effect of heat and mass transfer analysis under different physical situations.

Micropolar fluids are fluids with microstructure and asymmetrical stress tensor. Physically, they represent fluids consisting of randomly oriented particles suspended in a viscous medium. These types of fluids are used in analyzing liquid crystals, animal blood, fluid flowing in brain, exotic lubricants, the flow of colloidal suspensions, etc. The theory of micropolar fluids, is first proposed by Eringen [7,8]. In this theory the local effects arising from the microstructure and the intrinsic motion of the fluid elements are taken into account. The comprehensive literature on micropolar fluids, thermomicropolar fluids and their applications in engineering and technology was presented by Ariman et al. [9], Prathap Kumar et al. [10]. Kelson and Desseaux [11] studied the effect of surface conditions on the micropolar flow driven by a porous stretching sheet. Srinivasacharya et al. [12] analyzed the unsteady flow of micropolar fluid between two parallel porous plates. Bhargava et al. [13] investigated by using a finite element method the flow of a mixed convection micropolar fluid driven by a porous stretching sheet with uniform suction.

Gorla and Nakamura [14] discussed the combined convection from a rotating cone to micropolar fluids with an arbitrary variation of surface temperature. Takhar et al. [15] examined the bouncy effects in a forced flow in the three dimensional non-steady motion of an incompressible micropolar fluid in the vicinity of the forward stagnation point of a blunt nosed body. Ibrahim et al. [16] discussed the case of mixed convection flow of a micropolar fluid past a semi-infinite, steadily moving porous plate with varying suction velocity normal to the plate in the presence of thermal radiation and viscous dissipation. Damseh Rebhi et al. [17] have investigated natural convection heat and mass transfer adjacent to a continuously moving vertical porous infinite plate for incompressible, micropolar fluid in the presence of heat generation or absorption effects and a first-order chemical reaction. Ali and Magyari [18] have studied the unsteady fluid and heat flow by a submerged stretching surface while its steady motion is slowed down gradually. Mukhopadhyay [19] extended it by assuming the viscosity and thermal diffusivity are linear functions of temperature and studied unsteady mix convection boundary layer flow of an incompressible viscous liquid through porous medium along a permeable surface, and the thermal radiation effect on heat transfer was also considered.

Nadeem et al. [22] have been analyzed the heat transfer analysis of water-based nanofluid over an exponentially stretching sheet. The nanofluid flow over an unsteady stretching surface in the presence of thermal radiation was examined by Das et al. [23]. The fluid flow past over a stretching sheet has been studied by many authors [24-27]. Yacos et al. [28] have been investigated melting heat transfer in boundary layer stagnation-point flow toward a stretching/shrinking sheet in a micropolar fluid. The mixed convection flow of a micropolar fluid from an unsteady stretching surface with viscous dissipation has been proposed by El-Aziz [29]. Mahmood et al. [30] analyzed the non-orthogonal stagnation point flow of a micropolar second-grade fluid toward a stretching surface with heat transfer. The micropolar fluid flow toward a stretching/ shrinking sheet in a porous medium with suction was observed by Rosali et al. [31]. Heat and mass transfer on MHD flow of a viscoelastic fluid through porous media over a shrinking sheet was investigated by Bhukta et al. [32]. Mahmood and Waheed [33] have been proposed the MHD flow and heat transfer of a micropolar fluid over a stretching surface with heat generation (absorption) and slip velocity. The boundary layer flow of hyperbolic tangent fluid over a vertical exponentially stretching cylinder was studied by Naseer et al. [34].

The problem on mass transfer of a magnetohydrodynamics unsteady mixed convection flow of a micropolar fluid from a stretching surface through porous media has remained unexplored. So the main objective of this paper was to extend the work of Abd El-Aziz [20] in three directions: (i) to consider MHD micropolar fluid from a stretching sheet, (ii) to consider mass transfer effect, (iii) to include porous media. The governing equations of the flow are solved numerically, and the effects of various flow parameters on the flow field have been discussed.

2. Mathematical model

Consider an unsteady two-dimensional, mixed convection boundary layer flow and heat transfer of a viscous, incompressible, micropolar fluid past a semi-infinite permeable stretching sheet embedded in a homogenous porous medium coinciding with the plane y — 0, then the flow is occupied above the sheet y > 0. A schematic representation of the physical model and coordinates system is depicted in Fig. 1 [20]. The continuous sheet moves in its own plane with a velocity Uw — y0j7, and has temperature distribution

TW TOC +

(1-at)'

which varies with the distance x from the slot

Concentration Boundary layer Thermal Boundary layer Velocity Boundary layer

and time coordinate t. An external variable magnetic field B0 is applied along the positive y-direction. Magnetic field is sufficiently weak to ignore magnetic induction effects i.e. magnetic Reynolds number is small. The physical properties of the fluid are assumed to be uniform, isotropic, and constant. The governing equations of the flow in two dimensions are as follows: Equation of continuity:

du dv o

Equation of momentum:

du du du dt dx dy

l + k\ du

P ) dy2 + gß(T- Ti

j dN PJ dy gß(C -

Equation of angular momentum:

PA öT

c02 - k{2n dy

Equation of energy:

dT dT dT_ d2T fi + j (du\ 2 dt dx dy 0 dy2

Equation of concentration:

dÇ u@€ dÇ_ tfÇ dt dx dy dy2

The associated boundary conditions are

u = Uw(x, t), v = 0, N = 0, T = Tw(x, t) at y = 0, i

u ! 0, N ! 0, T-

as y !i

Transformation variables are as follows:

Í ■s

u(1 - at)

■y-, w —

(1 - at)

C — +

u(1 - at)

-3xh(g), T — Ti + bX 2 9(g)

(1 - at)2

(1 - at)

■/(g)

The physical stream function y; t) which automatically assures mass conservation is given in Eq. (1). The velocity components are readily obtained as

dw Tj jf , dw I ta) ,

u = dy = Ufw, v = ~dx

The mathematical problem defined in Eqs. (2)-(5) transforms into the following set of ordinary differential equations and their associated boundary conditions:

Figure 1 Flow geometry.

(1 + D)f" + ff -(f)2 - A (2f + f) + Atí + Gr 9 + Gc/ - m*f — 0

ko h '' + fh - fh - - (3h + gh' ) - AB(2h + f') = 0

P h + fh' - f 0 - A (40 + g0' ) + Ec(1 + A)f ')2 = 0

ir: /'' + f/' - f / - ^ (4/ + g/') = o

f(0) = 0, f (0) = 1, h(0)=0, 0(0) = l, /(0) = 1 (15)

f (i) = 0, h(i) = 0, 0(i) = 0, /(i) = 0, A = a, A = *, m* = M + f, Gr = ¡b- = Grx/ReX,

a7 * 7 Kp1 r a2 x' x^

Grx = ¡b(Tw - Tœ)x3/u2, Rex = Uwx/o, a0 = -C,

G: = ¡0e = G?, Gcx = gb(Cw - Ci)x3/o2, k0 = l,

_ p(1-at) _

"" Jf, Pr = ^, Ec =

-p(Ti T i)

m _ rB2<1-at) _ p(l-a')

prime indicates differentiation with respect to g.

The local skin friction coefficient, couple stress coefficient, local rate of heat and mass transfer coefficients are defined respectively, as follows:

Cfx = = 2(1 + A)Rf '(0) h (0)

yaVw '

sx u(1 - at)

-x l @T

Shx — —

(Tx - Ti) ôy x l dC

Cx - C» \ ôy

= -VRx0' (0)

= -vR"u' (0)

boundary conditions (15) and (16) can be solved by using semi-analytic method [36,37].

4. Results and discussion

In this paper, the effects of micropolar fluid on unsteady MHD free convection and mass transfer through a porous medium have been investigated. Numerical computations for velocity, microrotation, temperature and concentration are obtained by using Runge-Kutta fourth order with shooting technique. The results of the study are compared with the result of Abd El-Aziz [20] for A — 0 (steady case) and A — 0 (unsteady case) with Gc. Also for a Newtonian fluid (D — 0), the result is compared with the result of Grubka and Bobba [21]. The present study is in good agreement with the previous study. It is noteworthy that when Gr — Gc — Ec — D — 0 in the momentum equation the present problem reduces to Ali and Magyari [18].

The discussion is made for various values of the pertinent parameter such as Gr, Gc, Ec, D on the velocity profile (Figs. 2-6) and angular velocity profiles (Figs. 7-12) such as Gr, Gc, the buoyancy parameter, Pr, Prandtl number, Ec, Eckert number, D, micropolar parameter, A, unsteadiness parameter and B — 0.1 are taken throughout the discussion. It is noticed that, buoyancy force acts in the direction of main stream and fluid accelerated in the manner of a favorable pressure gradient for Gr > 0, Gc > 0 (assisting flow). Also buoyancy force retards the fluid at the boundary layer for Gr < 0, Gc < 0 (opposing flow).

Figs. 2 and 3 present the effect of Gr on velocity profile with the absence of Gc(Gc — 0) and presence of Gc(Gr — 10) respectively with the fixed value of D — 0.5, Pr — 0.72, k0 — 0.3, Ec — 0, B — 0.1, M — 0, Kp — 100. From Fig. 2, it is observed that the hydrodynamic fluid in the absence of porous matrix (Kp — 100) and viscous dissipation (Ec — 0) in case of assisting flow (Gr > 0) enhance the flow near the boundary layer for both steady (A — 0) and unsteady (A — 0.4) cases, whereas reverse trend occurs in case of opposing flow (Gc < 0). The buoyancy force retards the velocity profile. Compared to these free forced convection (Gr — 0), the result is well agreed with that of Abd El-Aziz [20].

3. Numerical solution

The set of non-linear coupled differential Eqs. (11)-(14) subject to the boundary conditions (15) and (16) constitute a two-point boundary value problem. In order to solve these equations numerically we follow most efficient numerical shooting technique with fourth-order Runge-Kutta scheme. In this method it is most important to choose the appropriate finite values of g !i. To select some initial guess values are taken and solve the problem with some particular set of parameters to obtain f '(0), h' (0) and h' (0). The solution process is repeated with another large value of g1 until two successive values of f'(0), h' (0) and h' (0) differ only after desired digit signifying the limit of the boundary along g. The last value of g1 is chosen as appropriate value of limit g for that particular set of parameters. The resulting differential equations can be integrated by fourth order Runge-Kutta scheme. The above procedure is repeated until we get the results up to the desired degree of accuracy 10~6. In future, the present work can be extended by using temperature-dependent thermal conductivity [35]. Further, Eqs. (11)-(14) with the

Figure 2 Velocity profile with Gc = 0.

Pr=0.72,A=0.5,A,0=0.3,Ec=0,B=0.1,M=0,Kp=100

1 2 3 4 5 6

P =0.72,X„=0.3,B=0.1,M=2,K =0.5,G =1 ,A=1 r 0 p r

Figure 3 Velocity profile with Gc.

Figure 6 Velocity profile for various values of Ec and Gc.

E = -2,0,2 c

0.7 h V

0.61- \

Curve M Kp

I 0 100

II 0 0.5

III,VII 2 100

IV ,V III 2 0.5

V 5 100

VI 5 0.5

Figure 4 Velocity profile for various values of M, Kp and A.

Curve D Gr Gc

I 0 1 0

II 0 1 1

III,VII 1 1 0

IV,VIII 1 1 1

V 1 -0.5 0

VI 1 -0.5 -0.5

A=0 A=0.4

VII,VIII

Figure 5 Velocity profile for various values of D, Gr, Gc and A.

From Fig. 3 it is noticed that for the higher value of mass buoyancy (Gc — 10) for both (Gr > 0, Gr < 0) there is a pick in velocity profile near the wall and for Gr — 10, the axial velocity decreases with increase in A.

Fig. 4 reveals the effect of magnetic parameter in both the absence of porous matrix (Kp — 100) and presence of porous matrix (Kp — 0.5) with fixed values of parameters D — 0.5, Pr — 0.72, k0 — 0.3, Ec — 0.01, B — 0.1, Gr — 1, Gc — 1. The most important consideration of the present study is the saturated porous media. Porous media are widely used to insulate a heated body to maintain its temperature. These are considered to be useful in diminishing the natural free convection which would otherwise occur inversely on the stretching surface. It is observed that presence of magnetic field (M — 2, 5) produces Lorentz force which resists the motion of the fluid in both the absence/presence of porous matrix. Further, it is to note that when M — 2, 5, Kp — 100 maximum velocity occurs near the boundary layer which gradually decreases with increase in g within the flow domain 0 < g < 6.

The effect of micropolar parameter, D, thermal and mass buoyancy parameter (Gr, Gc) in the presence of viscous dissipation (C — 0.01), on velocity profile is exhibited in Fig. 5. It is seen that velocity profile is asymptotic in nature. It is significant within the layer 0 < g < 6. There is a sharp fall in velocity within the layer g < 1. Also for Gr < 0 (opposing flow) velocity near the boundary layer becomes thinner and thinner (from the curve V and VI). The result is in good agreement with the result of Abd El-Aziz [20].

Fig. 6 shows the velocity profile with the effect of Ec and Gc taking the parameter D — 1, Pr — 0.72, k0 — 0.3, B — 0.1, Gr — 1, M — 2, Kp — 0.5 as fixed. Viscous dissipation produces heat due to drag between the fluid particles, which leads to increase in fluid temperature. It is observed that the interaction of viscous heating and buoyancy force causes an increase in fluid velocity when —0.5 6 Gc 6 0.5 (both assisting and opposing). The positive Eckert number (Ec > 0) and positive Gc(Gc > 0, assisting flow) give rise to maximum fluid velocity and negative Eckert number (Ec < 0) and negative Gc(Gc < 0, opposing flow) have a reverse effect. The result of previous author Abd El-Aziz [20] is in good agreement with that of

present study for Ec — 0 and Gc — 0 with A — 0.4. Velocity distribution is asymptotic in nature and suddenly falls near the sheet within the flow domain 0 < g < 1.8. It becomes uniform when g !i.

The angular velocity profile of the hydrodynamic flow for several values of thermal Grashof number Gr(Gr < 0, Gr — 0, Gr > 0) in both steady (A — 0) and unsteady (A — 0.4) cases are plotted in Fig. 7 with some fixed values of pertinent parameters used in the model, such as D — 0.5, Pr — 0.72, k0 — 0.3, B — 0.1, Ec — 0, M — 0, Kp — 100. The present result is in good agreement for both steadiness and unsteadiness with the previous author Abd El-Aziz [20]. It is noticed that for both A — 0 and A — 0.4 there is a pick in angular velocity near the sheet (g < 1). Further, it is noticed that an increase in Gr retards the angular velocity profile and ceases to be zero as g varies far from the sheet (g ! 1). It is clear from Fig. 7 that in steady state (A — 0), the angular velocity increases for Gr > 0 (assisting flow), whereas, there is a slow fall in angular velocity as g i.e. 18 < g < 6.

Fig. 8 presents the variation of A in both steady (A — 0) and unsteady (A — 0.4) case in the presence of Gc with high values of Gr(Gr — 10) keeping the parameters D — 0.5, Pr — 0.72, k0 — 0.3, B — 0.1, Ec — 0, M — 0, Kp — 100 as fixed. It is seen that in steady state, when Gc < 0(Gc — —0.5) a back flow is well marked (curve I) for g < 0.2 and after that angular velocity (curve I) is suddenly increased for 0 < g < 1 and then falls rapidly to maintain uniformity as g !i. Similar behavior occurs in case of high value of Gc > 0(Gc — 10). It is also seen that in case of unsteadiness A — 0.4 (with Gc — —0.5, Gc — 10), the angular velocity decreases with increase in g.

The variation of magnetic parameter in both the absence/ presence of porous matrix is well marked in Fig. 9. It is clear from the graph, angular velocity increases with increase in M. It is interesting to note that there is a point of inflexion at g — 1 .2 and a rapid fall in angular velocity is well marked in the flow domain 1.2 < g < 6. Presence of magnetic field produces a Lorentz force which usually resists the momentum field

o.oi - a

Pr=0.72,A=0.5,A,0=0.3,Ec=0,B=0.1,M=0,Kp=100 Gr=-0.5

Gr=10 ........ A=0

- A=0.4

0 1 2 3 4 5

Figure 7 Angular velocity profile with Gc = 0.

0.02 0.018 0.016 0.014 0.012 J) 0.01 0.008 0.006 0.004 0.002 0

Curve M Kp

I 0 100

II 0 0.5

III,VII 2 100

IV,VIII 2 0.5

V 5 100

VI 5 0.5

A=0.0 A=0.4

Figure 9 Angular velocity profile for various values of M, Kp and A.

-0.005

A G r G c

I 0 10 -0.5

II 0 10 10

III 0.4 10 -0.5

IV 0.4 10 10

Figure 8 Angular velocity profile with Gc.

0.02 0.01 0 -0.01 -0.02 L

VIII VII 1

II 0 1 1

III,VII 1 1 0

IV ,VIII 1 1 1

V 1 -0.5 0

VI 1 -0.5 -0.5

....... A=0

- A=0.4

Figure 10 Angular velocity profile for various values of D, Gr, Gc and A.

-0.005

in both the presence/absence of porous matrix. The reverse effect is observed as M increases. It is due to the elastic property of the micropolar fluid.

The variation of Gc in the presence/absence of Gr and D on the angular velocity profile is well marked from Fig. 10, and it is remarked that in the absence of Gc(Gc — 0) and D — 0, the result is in good agreement with the result of Abd El-Aziz [20] (curves I and II). It is remarked from curves III and IV that there is a decrease in angular velocity with increase in Gc for both steady and unsteady cases.

Fig. 11 reveals the effect of Ec on angular velocity profile. It is noticed that Ec enhances the angular velocity whereas reverse effect is noticed in case of the mass buoyancy parameter Gc.

Fig. 12 depicts the variation of the buoyancy parameter in both the absence/presence of Kp. The presence of porous matrix is beneficial to generate heat in the system whereas absence of Kp is to oppose it. It is also remarked that the

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

Figure 13 Temperature profile for various values of Pr, M, Kp and A.

Pr=0.72,A,„=0.3,B=0.1,M=2,Kp=0.5,A=1,Gr=1

Figure 11 Angular velocity profile for various values of Ec and Gc.

Curve A

0.9 0.8 0.7 0.6 S 0.5 0.4 0.3 0.2 0.1 0

0 1 2 3 4 5 6

Figure 14 Temperature profile for various values of M, Kp and D .

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

Curve Gr Gc M Kp

I 0 0 0 100

II 0 0 0 0.5

III -0.5 -0.5 1 100

IV -0.5 -0.5 1 0.5

V 0.5 0.5 1 100

VI 0.5 0.5 1 0.5

VII 10 10 1 100

VIII 10 10 1 0.5

E =0.01,A=0.5,A=0.4,P =0.72 c r

B=0.1,X=0.3

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

Curve M E c

I,II 0 0.01

III, IV 1 0.01

V,VI 1 0.02

VII,VIII 1 0.00

Kp=100 Kp=0.5

'""mil.......

Figure 12 Temperature profile for various values of Gr, Gc, M Figure 15 Temperature profile for various values of M, Kp

and Kp.

and Ec.

El-Aziz [20]. Interaction of magnetic parameter produces Lorentz force which accelerates the temperature profile at all points in the presence of porous matrix (curves I and III). High Prandtl number causes low thermal diffusivity. As a result, the temperature profile becomes thinner and thinner at thermal boundary layer where presence of porous matrix enhances it. The similar observation is encountered in the presence of porous matrix.

Fig. 14 exhibits the effect of M and D on temperature profile keeping the parameter Pr, Ec, A, B, k0 fixed. Both presence/absence of Kp affect significantly the thermal boundary layer as M increases. It is assumed that the presence of porous matrix (Kp — 0.5) enhances the thermal boundary layer whereas the absence of Kp(Kp — 0) decelerates it. There will be a small variation in temperature profile as D increases. The effect of Eckert number has no significant effect on temperature profile in both the absence/presence of porous matrix which is clearly reflected in Fig. 15 taking the fixed values of the parameters Pr, A, B, k0.

Fig. 16 illustrates the concentration profile. The effect of Schmidt number and magnetic parameter is shown with the fixed values of Pr — 0.71, D — 0.5, A — 0.4, B — 0.1, k0 — 0.3. In both the absence/presence of Kp it is observed that concentration boundary layer is lowered down as Sc increases in the presence of magnetic field and both the absence/presence of Kp. Increase in magnetic parameter enhances the concentration distribution at all points. The profile is asymptotic in nature.

Table 1 Skin friction coefficient, couple stress, rate of heat and mass transfer coefficient with A = 0.0, D = 0.5, k0 = 0.3, B = 0.1.

Gr Gc Pr Sc M kP Ec s h (0) -h (0) (0)

0.0 0.0 0.72 0.00 0 100 0.00 -0.8186178 0.0619197 0.8560463 0.1666667

0.1 0.0 0.72 0.00 0 100 0.00 -0.7802563 0.0600337 0.8651445 0.1666667

0.1 0.0 0.72 0.22 0 100 0.00 -0.7802563 0.0600337 0.8651445 0.4225224

0.1 0.0 0.72 0.22 1 100 0.00 -1.1178125 0.0755773 0.7847779 0.3786616

0.1 0.0 0.72 0.22 1 0.5 0.00 -1.597829 0.0928029 0.6839179 0.3339347

0.1 0.0 0.72 0.22 1 100 0.01 -1.117757 0.0755729 0.779998 0.3786759

0.1 0.0 0.72 0.22 1 0.5 0.01 -1.5977795 0.0927992 0.6765172 0.3339465

0.1 0.1 7.00 0.22 1 100 0.01 -1.0930755 0.0741295 3.0204614 0.3836196

0.5 0.1 0.72 0.22 1 100 0.01 -0.9398687 0.0666536 0.8292215 0.4049599

0.1 0.1 0.72 0.22 1 100 0.01 -1.0679285 0.7241227 0.7985819 0.3889113

0.5 0.1 0.72 0.22 1 0.5 0.01 -1.4465139 0.0854006 0.7257612 0.3558373

Table 2 Skin friction coefficient, couple stress, rate of heat and mass transfer coefficient with A = 0.4, D = 0.5, k0 = 0.3, B = 0.1.

Gr Gc Pr Sc M kp Ec s h (0) -h (0) (0)

0.0 0.0 0.72 0.00 0 100 0.00 -0.9281049 0.0524019 1.1259186 0.1666667

0.1 0.0 0.72 0.00 0 100 0.00 -0.8964873 0.0511817 1.1297997 0.1666667

0.1 0.0 0.72 0.22 0 100 0.00 -0.8964873 0.0511817 1.1297997 0.5846887

0.1 0.0 0.72 0.22 1 100 0.00 -1.207273 0.0633876 1.0903837 0.5589641

0.1 0.0 0.72 0.22 1 0.5 0.00 -1.6639832 0.0777144 1.0422961 0.5313147

0.1 0.0 0.72 0.22 1 100 0.01 -1.2072344 0.0633852 1.0858018 0.5589694

0.1 0.0 0.72 0.22 1 0.5 0.01 -1.6639471 0.0777124 1.035344 0.531319

0.1 0.1 7.00 0.22 1 100 0.01 -1.1855281 0.0624408 3.7297431 0.5611786

0.5 0.1 0.72 0.22 1 100 0.01 -1.0565654 0.0574921 1.105791 0.5713894

0.1 0.1 0.72 0.22 1 100 0.01 -1.1655573 0.0614006 1.092518 0.5634518

0.5 0.1 0.72 0.22 1 0.5 0.01 -1.5346593 0.0729635 1.0522201 0.5410338

thermal and mass buoyancy parameter enhance the thermal boundary layer at all points.

The variation of Pr, M and Kp is well marked in Fig. 13 keeping the other parameters Sc, D, B, k.0, Gc, Gr, Ec as fixed. In the steady state (A — 0), absence of Lorentz force, porous matrix enhances the boundary layer (curves I and II). The result is in good agreement with the result of Abd

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

P =0.71 ,E =0.01 ,A=0.5,A=0.4

Curve M S c

I,II 0 0.22

III, IV 1 0.22

V,VI 1 0.60

Figure 16 Concentration profile for various values of M, Kp

and Sc.

The Numerical computations of local skin friction coefficient (/"(0)), wall couple stress (hh (0)), local heat transfer coefficient (—h (0)) and the local mass transfer coefficient (—/' (0)) are obtained for both steady state solution (A — 0) and unsteady solution (A — 0.4) and presented in tabular form in Tables 1 and 2 respectively keeping the parameters D — 0.5, k0 — 0.3, B — 0.1 as fixed. The computation is made with the different numeric value of the pertinent parameters involved in the flow problem. The present study is compared by withdrawing the parameters used in the study and it is in good agreement with the result of Abd El-Aziz [20].

It is clear from Table 1 that, in steady state (A — 0) the skin friction, heat transfer and mass transfer coefficient decrease in the presence of porous matrix whereas the opposite effect is remarked for couple stress coefficient (hh(0)). It is due to the interaction of porous matrix in the two dimensional micropolar fluid flow. It is widely used to insulate a heated body to maintain its temperature. Further, these are considered to be useful in diminishing the natural free convection which would otherwise occur inversely on the stretching surface. It is interesting to note that there will be no significant effect encountered for the Eckert number in f(0), hi(0) and —/(0) but the heat transfer coefficient decreases. In the absence of unsteadiness parameter the contribution of magnetic number is noticeable. The couple stress coefficient increases as the magnetic parameter increases, but the reverse effect is observed for skin friction, heat transfer and mass transfer coefficient in the absence of porous matrix. The influence of Schmidt number on the mass transfer coefficient is noticeable for larger value. On the other hand, the thermal buoyancy has no significant effect on both couple stress and mass transfer coefficient but it is remarked that the skin friction coefficient and heat transfer coefficient decrease respectively as Gr increases. The mass buoyancy has no significant effect on couple stress but skin friction, heat and mass transfer coefficients increase as Gc increases. The similar observation is made for the increasing value of the unsteadiness parameter.

5. Conclusions

From the above discussions we conclude the following:

• Buoyancy force retards the fluid near the velocity boundary layer in case of opposing flow and is favorable for assisting flow.

• Absence of porous matrix and viscous dissipation in case of assisting flow enhances the flow.

• Viscous dissipation produces heat due to drag between the fluid particles, which causes an increase in fluid temperature.

• Presence of magnetic field produces a Lorentz force which usually resists the momentum field in both the presence/absence of porous matrix but the reverse effect is remarked in case of angular momentum.

• Increase in Gr retards the angular velocity profile and ceases to be zero.

• Presence of porous matrix is beneficial to generate heat in the system whereas absence of Kp is to oppose it.

• High Prandtl number causes low thermal diffusivity.

• Thermal buoyancy has no significant effect on both couple stress and mass transfer coefficient.

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