Meccanica (2012) 47:277–291 DOI 10.1007/s11012-011-9435-z

MHD free-convective flow of micropolar and Newtonian fluids through porous medium in a vertical channel Navin Kumar · Sandeep Gupta

Received: 3 March 2010 / Accepted: 13 May 2011 / Published online: 23 June 2011 © Springer Science+Business Media B.V. 2011

Abstract We have studied the fully-developed freeconvective flow of an electrically conducting fluid in a vertical channel occupied by porous medium under the influence of transverse magnetic field. The internal prefecture of the channel is divided into two regions; one region filled with micropolar fluid and the other region with a Newtonian fluid or both the regions filled by Newtonian fluids. Analytical solutions of the governing equations of fluid flow are found to be in excellent agreement with analytical prediction. Analytical results for the details of the velocity, micro-rotation velocity and temperature fields are shown through graphs for various values of physical parameters. It is noticed that Newtonian fluids prop up the linear velocity of the fluid in contrast to micropolar fluid. Also the skin friction coefficient at both the walls is derived and its numerical values are offered through tables. Keywords MHD · Free-convection · Micropolar fluid · Porous medium

N. Kumar () · S. Gupta Department of Mathematics, National Defence Academy, Pune 23, Maharashtra, India e-mail: [email protected] S. Gupta e-mail: [email protected]

1 Introduction The minuscule effects arising from the local structure and micro-motions of the fluid elements have been taken into account in the theory of micropolar fluid, which was first introduced and formulated by Eringen [1, 2]. His theory of micropolar fluids has opened up new areas in research in the physics of fluid flow. According to him, a simple micro fluid is a fluent medium whose properties and behavior are affected by the local motions of the material particles contains in each of its volume elements; such a fluid possesses local inertia. Physically, they represent the fluids consisting of randomly oriented particles suspended in a viscous medium. Synovial fluid is a good example of micropolar fluids. The theory is expected to provide a mathematical model, which can be used to describe the behavior of non-Newtonian fluids such as polymeric fluids, liquid crystals, paints, animal blood, colloidal fluids, Ferro-liquids, etc., for which the classical Navier-Stokes theory is inadequate. There exist numerous approaches to explain the technicalities of fluids with a substructure. Ericksen [3, 4] derived the field equations for the fluid flow by taking into account the presence of substructures which have been experimentally shown by Hoyt and Fabula [5] and Vogel and Patterson [6]. They explain that fluids containing small amount of polymeric stabilizers display a diminution in skin friction. Widespread appraises of the theory and application can be found in the review

278

articles given by Ariman et al. [7, 8] and the recent books written by Lukaszewicz [9] and Eringen [10]. Numerous studies of external convective flows of micropolar fluids have been made due to its wide applications, which are found in various types of engineering problems, such as plasma studies, nuclear reactors, oil exploration, geothermal energy extractions, soil physics, geo-hydrology, filtration of solids from liquids, chemical engineering, and air conditioning of a room. Several authors have investigated the flows of micropolar fluids on flat plate (Ahmadi [11], Jena and Mathur [12], Ÿucel [13], Rahman et al. [14–18], Mukhopadhyay and Layek [19], Pal and Mondal [20], Elgazery and Abd Elazem [21], Sahoo [22] and Kumar and Gupta [23]) or on regular surfaces (Balram and Sastry [24], Lien et al. [25, 26]) or on stretching sheet (Ishak [27]). Fully developed free convection of a micropolar fluid in a vertical channel was studied by Chamkha et al. [28]. Cheng [29] investigated the fully developed natural convection heat and mass transfer of a micropolar fluid in a vertical channel with asymmetric wall temperatures and concentrations. Pulsatile magneto-biofluid flow and mass transfer in a non-Darcian porous medium channel have been considered by Bhargava et al. [30]. Oscillatory Couette flow in the presence of an inclined magnetic field was premeditated by Guria et al. [31]. Effects of thermal radiation and space porosity on MHD mixed convection flow in a vertical channel using HAM has been studied by Muthuraj and Srinivas [32]. Sajid et al. [33] investigated the fully developed mixed convection flow of a viscoelastic fluid between permeable parallel vertical plates. Mixed convective heat and mass transfer in a vertical wavy channel with traveling thermal waves and porous medium has been studied by Muthuraj and Srinivas [34]. Srinivas et al. [35] formulated the mixed convective heat and mass transfer in an asymmetric channel with peristalsis. The flow and heat transfer aspects of immiscible fluids through a porous medium is of special importance in the petroleum extraction and transport. For example, the reservoir rock of an oil field always contains several immiscible fluids in its pores. Part of the pore volume is occupied by water and the rest may be occupied either by oil or gas or both. Crude oils often contain dissolved gases, which may be released into the reservoir rock when the pressure decreases. These examples show the importance of knowledge of the laws governing immiscible multi-phase flows for

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proper understanding of the process involved. In the modeling of such problems, the presence of a second immiscible fluid phase adds a number of complexities as to the nature of interacting transport phenomena and interface conditions between the phases. Two-fluid flow and heat transfer has also to its importance in chemical and nuclear industries, aerodynamics. Magnetohydrodynamic heat transfer in twophase flow between parallel plates has been investigated Lohsasbi and Sahai [36]. Malashetty and Leela [37] have studied the Hartmann flow characteristic of two fluids in horizontal channel. The study of twophase flow and heat transfer in an inclined channel has been made by Malashetty and Umavathi [38] and Malashetty et al. [39, 40]. Kumar and Gupta [41] considered the unsteady MHD and heat transfer of two viscous immiscible fluids through a porous medium in a horizontal channel. Analytic solution of natural convection flow of a non-Newtonian fluid between two vertical flat plates using HAM has been studied by Ziabakhsh and Domairry [42]. Fully-developed freeconvective flow of micropolar and viscous fluids in a vertical channel was investigated by Kumar et al. [43]. Muthuraj and Srinivas [44] extended this work through porous medium under the action magnetic field using HAM. In view of the above, the fully-developed MHD free-convective flow of two viscous immiscible fluids (micropolar fluid and Newtonian fluid) through porous medium confined in a vertical channel with asymmetric wall temperature distribution has been considered in this present paper.

2 Formulation of the problem Consider two-dimensional free-convective flow of two viscous immiscible incompressible electrically conducting fluids through a porous medium bounded by two infinite vertical parallel walls, under the action of uniform magnetic field applied normal to the direction of flow. The wall are placed in the planes y ∗ = −h1 and y ∗ = h2 and maintained at constant temperatures Tw∗1 and Tw∗2 respectively, with Tw∗2 ≤ Tw∗1 . The walls are extending in x ∗ -and z∗ -directions and y ∗ -direction is taken normal to it. The micropolar fluid having the density ρ1 , viscosity μ1 , vortex viscosity k, thermal conductivity κ1 , thermal expansion coefficient β1 is flowing in Region-I (−h1 ≤ y ∗ ≤ 0) and the Region-II

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279

Mass Balance Equation ∂u∗i =0 ⇒ ∂x ∗

u∗i = u∗i (y).

(1)

Region-I: Linear Momentum Balance Equation (μ1 + k) −

Fig. 1 Geometrical configuration

(0 ≤ y ∗ ≤ h2 ) is occupied by Newtonian fluid of density ρ2 , viscosity μ2 , thermal conductivity κ2 , thermal expansion coefficient β2 . The fluids are assumed to have unvarying properties apart from the density in the buoyancy force term, i.e. gravitational heat convection causes the fluid flow inside the vertical channel. The walls of the channel are assumed to be isothermal. All transport properties of both the fluids are assumed to be constant. It should be stated here that the micropolar and Newtonian fluids are immiscible in nature, i.e. no such mixing is possible and therefore the constitutive equations for micropolar and Newtonian fluids are different. For the generalization of this model, we can take any micropolar fluid and Newtonian fluid. The flow in both the regions of the channel is assumed to be fully developed. The model of porous medium is based on the Darcy’s law and so the porous material is simply defined by one parameter; the intrinsic permeability. Thus the flow of an electrically conducting fluid arises due to buoyancy force in the channel occupied with porous medium in the presence of magnetic field applied normal to the flow is given by Navier-Stokes equations of viscous incompressible fluid incorporating with the Buoyancy force, Darcy’s law and Maxwell’s equations of electromagnetism. To describe the flow of micropolar fluid in Region-I, the angular momentum equation is also taken into consideration. The temperature in the fluid flowing through the porous medium is governed by the Energy conservation equation on disregarding the heat generated by the viscous forces in the fluid. Under the above assumptions, the governing equations of fluid flow are:

d 2 u∗1 + ρ1 gβ1 (T1∗ − T0∗ ) dy ∗2

μ1 ∗ dω∗ 2 ∗ u − σ B u = −k , 1 0 1 K∗ dy ∗

Angular Momentum Balance Equation   du∗ d 2 ω∗ γ ∗2 − k 2ω∗ + ∗1 = 0, dy dy

(2)

(3)

Energy Conservation Equation d 2 T1∗ = 0, dy ∗2

(4)

Region-II: Linear Momentum Balance Equation μ2

d 2 u∗2 μ2 + ρ2 gβ2 (T2∗ − T0∗ ) − ∗ u∗2 − σ B02 u∗2 = 0, ∗2 K dy (5)

Energy Conservation Equation d 2 T2∗ = 0, dy ∗2

(6)

where ω∗ is the component of micro-rotation vector normal to the plane x ∗ y ∗ , g the acceleration due to gravity, σ the coefficient of electrical conductivity, B0 the coefficient of electromagnetic field, K ∗ the permeability of porous medium and γ the spin gradient viscosity. The second term on the LHS of momentum equations (2) and (5) denote buoyancy effects; the third term is the bulk matrix linear resistance, i.e. Darcy term and the fourth term denotes the magnetic effect. The appropriate boundary and interface conditions are required for solving the above set of differential equations from (2) to (6). The first two boundary conditions on velocity are the no-slip boundary conditions which is required that the component of velocity along x ∗ -direction must vanishes at the wall. Next condition is obtained by presuming the continuity of velocity and the last four conditions are obtained from

280

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the equality of stresses at the interface and constant cell rotational velocity at the interface as considered by Ariman et al. [7]. Thus, the appropriate boundary and interface conditions on velocity in the mathematical form are u∗1 (−h1 ) = 0, u∗2 (h2 ) = 0, u∗1 (0) = u∗2 (0), du∗1 du∗2 ∗ + kω = μ , 2 dy ∗ dy ∗

(7)

du∗1 = −ω∗ , dy ∗ dω∗ = 0 at y ∗ = 0. dy ∗ The boundary conditions for temperature field at the wall and interface are given by T1∗ (−h1 ) = Tw∗1 , T1∗ (0) = T2∗ (0) κ1

dT1∗ dT2∗ = κ 2 dy ∗ dy ∗

We introduce the following non-dimensional quantities u∗ y∗ , ui = i , hi U0 ∗ ∗ T − T0 h1 ∗ Ti = i , ω= ω , T U0

yi =

σ B02 h21 K∗ 2 , M = , μ1 h21 gβ1 T h31 U0 h1 , Re = , Gr = 2 ν1 ν1 Gr k , k1 = GR = , Re μ1 K=

ω∗ (−h1 ) = 0, (μ1 + k)

3 Method of solution

T2∗ (h2 ) = Tw∗2 , and

(8)

at y ∗ = 0.

Following Ahmadi [11], Kumar and Gupta [23], Kumar et al. [43], Kline [45], Rees and Bassom [46], Gorla [47], Rees and Pop [48], the spin-gradient viscosity γ which gives some relationship between the coefficients of micro inertia, is given by     k k1 γ = μ1 + j = μ1 1 + j, 2 2

(9)

where j is the micro-inertia density and k1 (= k/μ1 ), is the micropolar fluid material parameter. We perceived that k1 = 0 is corresponds to the NewtonianNewtonian fluids system. This assumption is invoked to allow the field equations to predict the correct behavior in the limiting case when the microstructure effects become negligible and the total spin reduces to the angular velocity (see Yücel [13]).

(10)

where M is the Hartmann number or magnetic parameter, Gr the Grashof number, Re the Reynolds number, GR the ratio of Grashof number to Reynolds number, j = h21 the characteristic length and T the characteristic temperature which is defined as T = Tw∗1 − Tw∗2 if Tw∗1 > Tw∗2 . Using (10), the governing equations (2) to (6) in non-dimensional form become Region-I:   d 2 u1 1 M2 + − u1 K (1 + k1 ) (1 + k1 ) dy 2 GR k1 dω =− T1 , 1 + k1 dy (1 + k1 )   du1 d 2ω 2k1 2ω + = 0, − 2 + k1 dy dy 2 +

(11) (12)

d 2 T1 = 0, dy 2

(13)

Region-II:   2 h d 2 u2 2 2 u2 = −mbρh2 GR T2 . (14) + mh − M K dy 2 d 2 T2 = 0, dy 2

(15)

along with the boundary conditions u1 (−1) = 0,

u2 (1) = 0,

u1 (0) = u2 (0),

ω(−1) = 0,

k1 du2 1 du1 + , ω= dy 1 + k1 mh(1 + k1 ) dy

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dω = 0 at y = 0, dy

du1 = −ω, dy T1 (−1) = T2 (1) =

281

Tw∗1 − T0∗ T

Tw∗2 − T0∗ T

R2 =

= m1 ,

(16)

= m2 ,

where h = hh12 , is the channel width ratio, m(= μ1 /μ2 ), the viscosity ratio, κ(= κ1 /κ2 ), the thermal conductivity ratio, ρ(= ρ1 /ρ2 ), the density ratio and b(= β2 /β1 ), the thermal expansion ratio. On solving coupled linear differential equations from (11) to (15) under the boundary and interface conditions (16), we have the solutions T1 (y) = c1 y + c2 ,

(17)

T2 (y) = c3 y + c4 ,

(18)

L3 c 1 c11 y+ , L1 2L4

(19)

u2 (y) = c5 ery + c6 e−ry mbρh2 GR (c3 y + c4 ), r2

(20)

ω(y) = c7 eR1 y + c8 e−R1 y + c9 eR2 y + c10 e−R2 y −

L3 c 1 , 2L1

(21)

where  h2 + mh2 M 2 , K   M2 1 L1 = + , K(1 + k1 ) (1 + k1 ) r=

L3 =

GR , (1 + k1 )

L8 =

R2 2 − 2L4 R2

L4 =

k1 , (2 + k1 )

L5 = L1 + 4L4 − 2L2 L4 ,     L + L2 − 16L L  5 2 4 5 , R1 = 2

L2 =

For a Newtonian fluid (k1 = 0), the solution of (11) and (14) using boundary and interface conditions (16) are 



u1 (y) = c7 er y + c8 e−r y + u2 (y) = c5 ery + c6 e−ry +

GR (c1 y + c2 ), r 2

(22)

mbρh2 GR (c3 y + c4 ), (23) r2

 where r  = ( K1 + M 2 ) and c5 to c8 are constants of integration, not included here for the sake of brevity.

4 Skin-friction coefficient

u1 (r) = c7 L7 eR1 y − c8 L7 e−R1 y + c9 L8 eR2 y

+

R1 2 − , 2L4 R1

,

Limiting case

at y = 0,

− c10 L8 e−R2 y +

2

and c1 to c11 are constants of integration, not included here for the sake of brevity.

T1 (0) = T2 (0), dT1 1 dT2 = dy hκ dy

L7 =

    L − L2 − 16L L  5 2 4 5

k1 , (1 + k1 )

The skin-friction coefficient (Cf ) at both plates for micropolar-Newtonian fluids system is given by  ∗ ∂u1 2 (μ + k) (Cf )y ∗ =−h1 = 1 2 ∂y ∗ y ∗ =−h1 ρ 1 U0

+ k(ω∗ )y ∗ =−h1 , (24) (Cf )y ∗ =h2

 ∗

∂u2 2 μ2 . = ∂y ∗ y ∗ =h2 ρ2 U02

Using (10), (24) and (25) become   ∂u1 2 (Cf )y=−1 = , (1 + k1 ) Re ∂y y=−1   ∂u2 2ρ (Cf )y=1 = . Re hm ∂y y=1

(25)

(26) (27)

Limiting case For a Newtonian fluid (k1 = 0), the skin-friction coefficient (Cf ) at both plates is given by  ∗

∂u1 2 μ1 , (28) (Cf )y ∗ =−h1 = ∂y ∗ y ∗ =−h1 ρ1 U02

282

(Cf )y ∗ =h2 =

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 ∗

∂u2 2 μ . 2 ∂y ∗ y ∗ =h2 ρ2 U02

Using (10), equations (28) and (29) become   2 ∂u1 (Cf )y=−1 = , Re ∂y y=1   ∂u2 2ρ (Cf )y=1 = . Re hm ∂y y=1

(29)

(30) (31)

5 Results and discussion In the present study, an analytical solution is obtained for the velocity profiles, micro-rotational velocity profiles, temperature profiles and skin-friction coefficient for the MHD free convective flow through porous medium of micropolar fluid and Newtonian fluid in a vertical channel. The results are given to carry out a parametric study showing influences of the nondimensional parameters on the flow field and results are presented through graph by assuming that at second wall, the temperature is alike of mean temperature i.e. Tw∗2  T0∗ , so that m1 → 1 and m2 → 0. The numerical results for the absence of Darcy term (nonporous region) and magnetic field (M = 0) have been similar with those reported by Kumar et al. [43]. The effect of GR, the ratio of Grashof to Reynolds number on linear velocity and micro-rotation velocity is shown through Figs. 2 to 4. It is discerned from Fig. 2 that the maximum fluid velocity appears at the centerline of the channel, i.e. surrounding the interface, and it increases with increase of GR caused by dominance of buoyancy forces over the ratio of inertial forces to viscous forces. The effect of GR is still sustained but the endorsement is high for NewtonianNewtonian fluids system as compared to micropolarNewtonian fluids system as seen in Fig. 3 as compared to Fig. 2. It can be seen from the Fig. 4 that an increase in the strength of buoyancy forces lessens the magnitude of micro-rotation velocity. The effect of the viscosity ratio m on the linear velocity and micro-rotation velocity are presented through Figs. 5 to 7. Figure 5 depicts that, as the viscosity ratio increases, the linear velocity initially increases but for large values of m, it decreases for micropolar fluid whereas it increases continuously for Newtonian fluid. Figure 6 shows that the linear velocity increases with the increase of viscosity ratio, but

Fig. 2 Velocity distribution for different values of GR, when k1 = 1, K = 3, M = 2, m = 1, ρ = 1, h = 1, b = 1, κ = 1

Fig. 3 Velocity distribution for different values of GR, when k1 = 0, K = 3, M = 2, m = 1, ρ = 1, h = 1, b = 1, κ = 1

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Fig. 4 Micro-rotational velocity distribution for different values of GR, when k1 = 1, K = 3, M = 2, m = 1, ρ = 1, h = 1, b = 1, κ = 1

Fig. 5 Velocity distribution for different values of viscosity ratio m, when k1 = 1, K = 3, M = 2, GR = 5, ρ = 1, h = 1, b = 1, κ = 1

283

Fig. 6 Velocity distribution for different values of viscosity ratio m, when k1 = 0, K = 3, M = 2, GR = 5, ρ = 1, h = 1, b = 1, κ = 1

Fig. 7 Micro-rotational velocity distribution for different values of viscosity ratio m, when k1 = 1, K = 3, M = 2, GR = 5, ρ = 1, h = 1, b = 1, κ = 1

284

the magnitude of promotion is hefty for NewtonianNewtonian fluids system as compared to micropolarNewtonian fluids system. The effect of viscosity ratio on micro-rotational velocity is observed from Fig. 7. It is noticed from figure that initially micro-rotational velocity decreases but for large value of viscosity ratio, it increases continuously. Figures 8 to 11 reveal the effect of the channel width ratio h on the linear velocity, micro-rotational velocity and temperature field. An increment in channel width ratio increases the magnitude of linear velocity for both the system (micropolar-Newtonian (Fig. 8) and Newtonian-Newtonian (Fig. 9) fluids system). It is also found that the fluid velocity attains its maximum velocity at the centerline of the channel as seen in Fig. 8 whereas Fig. 9 shows that fluid velocity attains its maximum value at surrounding the interface. Figure 10 depicts that the micro-rotational velocity decreases due to the increase in width ratio. The increment in channel width ratio h props up the temperature field as seen in Fig. 11. Figures 12 to 15 present the variations of linear velocity, micro-rotational velocity and temperature field with conductivity ratio κ. An increment in the conductivity ratio increases the linear velocity for micropolarNewtonian fluids system as observed in Fig. 12. It is noticed from Fig. 13 that the increment in conductivity ratio is found to increase the linear velocity for Newtonian-Newtonian fluids system. It is also observed that the magnitude of promotion is large for micropolar fluid in comparison to Newtonian fluid. Figure 14 shows the effect of the conductivity ratio κ on micro-rotational velocity. It is found from the figure that the micro-rotational velocity diminishes due to increase the value of this parameter. The magnitude of temperature is found to increase with an increase of conductivity ratio as seen in Fig. 15. The effect of the micropolar fluid material parameter k1 on the linear velocity and micro-rotation velocity is presented through Figs. 16 to 17. It is seen from Fig. 16 that the linear velocity decreases as k1 increases. It is also noticed from this figure that reduction in magnitude of linear velocity is maximum for micropolar fluid. The fact is in agreement with the previous results as reported by Chamkha et al. [28] for one fluid model considering only micropolar fluid. Figure 17 shows that the micro-rotational velocity reduces with an increase in the fluid material parameter.

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Fig. 8 Velocity distribution for different values of viscosity ratio h, when k1 = 1, K = 3, M = 2, GR = 5, ρ = 1, m = 1, b = 1, κ = 1

Fig. 9 Velocity distribution for different values of viscosity ratio h, when k1 = 0, K = 3, M = 2, GR = 5, ρ = 1, m = 1, b = 1, κ = 1

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285

Fig. 12 Velocity distribution for different values of conductivity ratio κ, when k1 = 1, K = 3, M = 2, GR = 5, ρ = 1, m = 1, b = 1, h = 1 Fig. 10 Micro-rotational velocity distribution for different values of width ratio h, when k1 = 1, K = 3, M = 2, GR = 5, ρ = 1, m = 1, b = 1, κ = 1

Fig. 11 Temperature distribution for different values of viscosity ratio h, when k1 = 1, GR = 5, ρ = 1, m = 1, b = 1, κ = 1

Fig. 13 Velocity distribution for different values of conductivity ratio κ, when k1 = 0, K = 3, M = 2, GR = 5, ρ = 1, m = 1, b = 1, h = 1

286

Fig. 14 Micro-rotational velocity distribution for different values of conductivity ratio κ, when k1 = 1, K = 3, M = 2, GR = 5, ρ = 1, m = 1, b = 1, h = 1

Fig. 15 Temperature distribution for different values of conductivity ratio κ, when k1 = 1, GR = 5, ρ = 1, m = 1, b = 1, h=1

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Fig. 16 Velocity distribution for different values of material parameter k1 , when K = 3, M = 2, GR = 5, ρ = 1, m = 1, b = 1, h = 1, κ = 1

Fig. 17 Micro-rotational velocity distribution for different values of material parameter k1 , when K = 3, M = 2, GR = 5, ρ = 1, m = 1, b = 1, h = 1, κ = 1

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Fig. 18 Velocity distribution for different values of permeability parameter K, when k1 = 1, M = 2, GR = 5, ρ = 1, m = 1, b = 1, h = 1, κ = 1

Figures 18 to 20 display the effect of the permeability parameter K on the linear velocity and microrotational velocity. It is concluded from the Fig. 18 that an increase in the permeability parameter promotes the linear velocity and attains its maximum value at the centerline of the channel. It is also observed from the figure that maximum change is around the centerline of the channel. Figure 19 reveals that if the micropolar fluid is replaced by the Newtonian fluid, the effect of permeability parameter is still maintain, but the achievement of maximum value of linear velocity shifted to the Region-I. Also, the magnitude of promotion is large for Newtonian-Newtonian fluids system as compared to micropolar-Newtonian fluids system. It is noticed from the Fig. 20 that an increment of permeability parameter reduces the magnitude of microrotation velocity. Also, the reduction is negligible near the wall of Region-I. Figures 21 to 23 show the effect of intensity of magnetic field on linear velocity and micro-rotational velocity. An increase of the intensity of magnetic field reduces the linear velocity as seen in Fig. 21. Also, the maximum velocity attains at the centerline of the channel initially, but shifted to close of the left wall of Region-I for higher values of magnetic parameter. If the micropolar fluid is replaced by the Newtonian

287

Fig. 19 Velocity distribution for different values of permeability parameter K, when k1 = 0, M = 2, GR = 5, ρ = 1, m = 1, b = 1, h = 1, κ = 1

Fig. 20 Micro-rotational velocity distribution for different values of permeability parameter K, when k1 = 1, M = 2, GR = 5, ρ = 1, m = 1, b = 1, h = 1, κ = 1

288

Fig. 21 Velocity distribution for different values of magnetic parameter M, when k1 = 1, K = 3, GR = 5, ρ = 1, m = 1, b = 1, h = 1, κ = 1

fluid, the effect of intensity of magnetic field is still retain as seen in Fig. 22, but the magnitude of promotion is large for viscous-viscous fluids system compared to micropolar-viscous fluids system. Also, the maximum value of the velocity achieve in Region-I. Figure 23 depicts that an increase in magnetic field supports the micro-rotational velocity. In micropolar- Newtonian fluids system, it is observed from Table 1 that the coefficient of skin friction at the wall (y = −1), increases due to the increase in micropolar fluid material parameter, permeability parameter, viscosity ratio, channel width ratio or conductivity ratio; while decreases with the increase of magnetic parameter or Reynolds number. At the wall (y = 1), the coefficient of skin friction increases due to the increase in micropolar fluid material parameter, magnetic parameter, Reynolds number, viscosity ratio or channel width ratio whereas it decreases with the increase of permeability parameter or conductivity ratio. Table 2 shows that if the micropolar fluid is replaced by the Newtonian fluid, the effect of governing parameters on skin friction coefficient is still retained, but the magnitude of promotion is large for micropolar-

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Fig. 22 Velocity distribution for different values of magnetic parameter M, when k1 = 0, K = 3, GR = 5, ρ = 1, m = 1, b = 1, h = 1, κ = 1

Fig. 23 Micro-rotational velocity distribution for different values of magnetic parameter M, when k1 = 1, K = 3, GR = 5, ρ = 1, m = 1, b = 1, h = 1, κ = 1

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289

Table 1 Coefficient of skin-friction at both the plates in micropolar-Newtonian fluids system for various values of physical parameters with ρ = 1, b = 2 k1

K

Re

M

m

h

κ

(Cf )y=−1

(Cf )y=1

1

3

2

1

5

1

1

4.50667

−0.23826

2

3

2

1

5

1

1

5.67311

−0.23757

1

6

2

1

5

1

1

4.50751

−0.24009

1

3

3

1

5

1

1

3.05068

−0.10993

1

3

2

2

5

1

1

2.25333

−0.11913

1

3

2

1

10

1

1

4.65101

−0.12317

1

3

2

1

5

2

1

6.09211

−0.16385

1

3

2

1

5

1

2

5.40779

−0.31749

Table 2 Coefficient of skin-friction at both the plates in Newtonian-Newtonian fluids system for various values of physical parameters with ρ = 1, b = 2 K

M

Re

m

h

κ

(Cf )y=−1

(Cf )y=1

3

2

1

5

1

1

3.73572

−0.24191

6

2

1

5

1

1

3.78907

−0.24422

3

3

1

5

1

1

2.76266

−0.11036

3

2

2

5

1

1

1.86786

−0.12096

3

2

1

10

1

1

3.75986

−0.1236

3

2

1

5

2

1

4.04518

−0.16385

3

2

1

5

1

2

4.01663

−0.31947

Newtonian fluids system compared to Newtonian– Newtonian fluids system.

6 Conclusions The motion of electrically conducting fluid causes due to influence of buoyancy forces of micropolarNewtonian fluids through porous medium in a vertical channel under the action of magnetic field is discussed in this present paper. The following facts have been reported: 1. The effects of the parameter GR, the ratio of Grashof number to Reynolds number, channel width ratio, conductivity ratio and permeability parameter prop up the linear velocity whereas the micropolar fluid material parameter and intensity of magnetic field hold backed the velocity. 2. The effect of the intensity of magnetic field sustains the micro-rotational velocity whereas the Grashof to Reynolds number ratio, channel width ratio, conductivity ratio, micropolar fluid material parameter

and permeability parameter repressed the microrotational velocity. 3. The temperature field sponsors by width and conductivity ratio parameters. 4. The skin friction coefficient at both the walls endorses by the micropolar fluid material parameter, viscosity or channel width ratios.

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