Scholarly article on topic 'Perturbation analysis of magnetohydrodynamics oscillatory flow on convective-radiative heat and mass transfer of micropolar fluid in a porous medium with chemical reaction'

Perturbation analysis of magnetohydrodynamics oscillatory flow on convective-radiative heat and mass transfer of micropolar fluid in a porous medium with chemical reaction Academic research paper on "Physical sciences"

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{"Chemical reaction" / "Heat and mass transfer" / "Micropolar fluid" / Magneto-hydrodynamics / "Porous medium" / "Thermal radiation" / "Viscous dissipation"}

Abstract of research paper on Physical sciences, author of scientific article — Dulal Pal, Sukanta Biswas

Abstract This paper deals with the perturbation analysis of mixed convection heat and mass transfer of an oscillatory viscous electrically conducting micropolar fluid over an infinite moving permeable plate embedded in a saturated porous medium in the presence of transverse magnetic field. Analytical solutions are obtained for the governing basic equations. The effects of permeability, chemical reaction, viscous dissipation, magnetic field parameter and thermal radiation on the velocity distribution, micro-rotation, skin friction and wall couple stress coefficients are analyzed in detail. The results indicate that the effect of increasing the chemical reaction has a tendency to decrease the skin friction coefficient at the wall, while opposite trend is seen by increasing the permeability parameter of the porous medium. Also micro-rotational velocity distribution increases with an increase in the magnetic field parameter.

Academic research paper on topic "Perturbation analysis of magnetohydrodynamics oscillatory flow on convective-radiative heat and mass transfer of micropolar fluid in a porous medium with chemical reaction"

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Engineering Science and Technology, an International Journal

journal homepage: http://www.elsevier.com/locate/jestch

Full Length Article

Perturbation analysis of magnetohydrodynamics oscillatory flow on convective-radiative heat and mass transfer of micropolar fluid in a porous medium with chemical reaction

Dulal Pala*, Sukanta Biswasb

a Department of Mathematics, Visva-Bharati University, Siksha-Bhavana, Santiniketan, West Bengal 731 235, India b Bhedia High School, Bhedia, Burdwan, West Bengal 713 126, India

ARTICLE INFO ABSTRACT

Article history: This paper deals with the perturbation analysis of mixed convection heat and mass transfer of an oscil-

Received 7 fetirnaiy 2015 latory viscous electrically conducting micropolar fluid over an infinite moving permeable plate embedded

Received in revised form in a saturated porous medium in the presence of transverse magnetic field. Analytical solutions are ob-

20 August2015 tained for the governing basic equations. The effects of permeability, chemical reaction, viscous dissipation, Accepted 2 September 2015 -^i, , ■ i i-i--,- ■ ■ ,-r-

Available online magnetic field parameter and thermal radiation on the velocity distribution, micro-rotation, skin friction and wall couple stress coefficients are analyzed in detail. The results indicate that the effect of increasing

~ "J the chemical reaction has a tendency to decrease the skin friction coefficient at the wall, while opposite Keywords: J rr

Chemical reaction trend is seen by increasing the permeability parameter of the porous medium. Also micro-rotational ve-

Heat and mass transfer locity distribution increases with an increase in the magnetic field parameter.

Micropolar fluid Copyright © 2015 The Authors. Production and hosting by Elsevier B.V. on behalf of Karabuk University. Magneto-hydrodynamics This is an open access article under the CC BY-NC-ND license Porous medium (http://creativecommons.org/licenses/by-nc-nd/4.0/). Thermal radiation Viscous dissipation

1. Introduction

There has been a considerable amount of interest among researchers to study the convective flow with simultaneous heat and mass transfer under the influence of a magnetic field and chemical reaction as such processes exist in many branches of science and technology. In many industries application of such type of problems are seen in the chemical industry, cooling of nuclear reactors and magnetohydrodynamic (MHD) power generators. The study of MHD mixed convection heat and mass transfer with chemical reaction are of great importance to engineers and scientists because of its almost universal occurrence in many branches of science and engineering, and hence received a considerable amount of attention in recent years. The study of heat transfer in porous medium in the presence of chemical reaction has important engineering applications e.g. oxidation of solid materials, synthesis of ceramic materials and tubular reactors. There are two types of reactions such as (i) homogeneous reaction and (ii) heterogeneous reaction. A homogeneous reaction occurs uniformly throughout the given phase, whereas heterogeneous reaction takes place in a restricted region

* Corresponding author. Tel.: +91 3463 261029, fax: +91 3463 261029. E-mail address: dulalp123@rediffmail.com (D. Pal). Peer review under responsibility of Karabuk University.

or within the boundary of a phase. The effects of a chemical reaction depend greatly on whether the reaction is heterogeneous or homogeneous. A chemical reaction is said to be first-order if the rate of reaction is directly proportional to the concentration itself. In many industrial processes involving flow and mass transfer over a moving surface, the diffusing species can be generated or absorbed due to some kind of chemical reaction with the ambient fluid which can greatly affect the flow and hence the properties and quality of the final product. These processes take place in numerous industrial applications, such as in the polymer production and in manufacturing of ceramics or glassware, and food processing chemical reaction that occurs between a foreign mass and a fluid in which a plate is moving. Ibrahim et al. [1] studied the effect of chemical reaction and thermal radiation absorption on the unsteady MHD free convection flow past a semi-infinite vertical permeable moving plate with heat source and suction. They found that the velocity profiles and concentration profile increased due to a decrease in the chemical reaction parameter. Al-Odat and Al-Azab [2] studied the influence of chemical reaction on transient MHD free convection over a moving vertical plate. They found that the velocity as well as concentration decrease with increasing the chemical reaction parameter. Seddeek et al. [3] examined the effects of chemical reaction and variable viscosity on hydromagnetic mixed convection heat and mass transfer for Hiemenz flow through porous media with radiation, and they found that the local

http://dx.doi.org/10.1016/jjestch.2015.09.003

2215-0986/Copyright © 2015 The Authors. Production and hosting by Elsevier B.V. on behalf of Karabuk University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

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Sherwood number significantly increases with the chemical reaction parameter. Pal and Talukdar [4] used perturbation analysis to study unsteady magnetohydrodynamic convective heat and mass transfer in a boundary layer slip flow past a vertical permeable plate with thermal radiation and chemical reaction. They solved the nonlinear coupled partial differential equations by perturbation technique and found that the velocity as well as concentration decrease with increasing the chemical reaction parameter. Hsiao [5] studied the heat and mass transfer on MHD mixed convection of viscoelastic fluid past a stretching sheet with Ohmic dissipation. Rout et al. [6] studied the effect of thermal radiation and chemical reaction on double diffusive natural convective MHD flow through a porous medium. They considered destructive reaction in this paper, i.e., when chemical reaction parameter is positive then the concentration decreases due to the contribution of mass diffusion in concentration equation, while in the generative reaction, i.e., when chemical reaction parameter is negative, they observed the reverse effect. They further found that with increasing chemical reaction parameter there is substantial increase in the temperature profile.

Mixed convection in porous media has gained significant attention for its importance in engineering applications such as in the design of nuclear reactor, ceramics processing, geothermal systems, solid matrix heat exchangers, thermal insulations, crude oil drilling and compact heat exchanges etc. Convection in porous media can also be applied to underground coal gasification, ground water hydrology, iron blast furnaces, wall cooled catalytic reactors, energy efficient drying processes, cooling of nuclear fuel in shipping flasks, cooling of electronic equipments and natural convection in the Earth's crust. The fundamental problem of flow through and past porous media has been studied extensively over the past few years both theoretically and experimentally. Modather et al. [7] analytically studied MHD heat and mass transfer oscillatory flow of a micropolar fluid over a vertical permeable plate in a porous medium, and the results indicate that increasing the permeability parameter produces an increasing effect on the skin friction coefficient and the couple stress coefficient at the wall. El-Hakiem [8] examined the MHD oscillatory flow on free convection-radiation through a porous medium with constant suction velocity. They observed that the velocity increases when the permeability of the porous medium is increased. Pal and Talukdar [9] investigated the buoyancy and chemical reaction effects on MHD mixed convection heat and mass transfer in a porous medium with thermal radiation and Ohmic heating, and they concluded that the presence of porous medium increases the skin friction coefficient, whereas the effect of increasing the value of porous permeability decrease the value of the local Nusselt number. Sugunamma et al. [10] studied the inclined magnetic field and chemical reaction effects on flow over a semi-infinite vertical porous plate through porous medium. They solved the non-linear and coupled governing equations by adopting a perturbative series expansion about a small parameter, s, and they observed that the velocity gradient at the surface increases with a decrease in the porosity parameter. Acharya et al. [11] analyzed the free convective fluctuating MHD flow through porous media past a vertical porous plate with variable temperature and heat source, and they recorded that the presence of porous media has no significant contribution to the flow characteristics whereas viscous dissipation compensates for the heating and cooling of the plate due to convective current.

The study of free convection flow with magnetic field plays a major role in liquid metals, electrolytes and ionized gases and thermal physics of hydromagnetic problems with mass transfer have enormous applications in power engineering. Prasad et al. [12] examined the influence of internal heat generation/absorption, thermal radiation, magnetic field, variable fluid property and viscous dissipation on heat transfer characteristics of a Maxwell fluid over a stretching sheet, and they have pointed out that the horizontal velocity decreases with an increase in the magnetic field parameter.

They concluded that this is due to the fact that the transverse magnetic field has a tendency to create a drag like force, known as the Lorentz force to resist the flow. Vija et al. [13] studied the effects of induced magnetic field and viscous dissipation on MHD mixed convective flow past a vertical plate in the presence of thermal radiation. They found that the values of induced magnetic field remained negative, i.e. induced magnetic flux reversal arises for all distances in the boundary layer.

The effects of radiation on MHD flow and heat transfer problems have become industrially more important. The thermal radiation effects become intensified at high absolute temperature levels due to basic difference between radiation and the convection and conduction energy-exchange mechanisms. Many engineering processes occur at high temperatures and hence the knowledge of thermal radiation heat transfer is essential for designing appropriate equipments such as nuclear power plants, gas turbines and various propulsion devices for aircraft, missiles and satellites. When radiative heat transfer takes place in the electrically conducting fluid, it is ionized due to the high operating temperature. In view of these, many authors have made contributions to the study of fluid flow with thermal radiation. Hsiao [14] analyzed heat and mass transfer of micropolar fluid flow in the presence of thermal radiation past a nonlinearly stretching sheet. Shateyi et al. [15] investigated the effects of thermal radiation, Hall currents, Soret and Dufour number on MHD mixed convection flow over a vertical surface in porous media, and they found that the fluid temperature increases due to an increase in the thermal radiation. Also they found that the concentration decreases as the radiation parameter value is increased. Pal and Mondal [16] examined the effects of Soret Dufour, chemical reaction and thermal radiation on MHD non-Darcy unsteady mixed convective heat and mass transfer over a stretching sheet, and they concluded that the effect of thermal radiation is to increase temperature in the thermal boundary layer. The effects of thermal radiation and viscous dissipation on MHD heat and mass diffusion flow past an oscillating vertical plate embedded in a porous medium with variable surface conditions were studied by Kishore et al. [17]. Thermal radiation effects on MHD con-vective flow over a plate in a porous medium was studied by Karthikeyan et al. [18] by using perturbation technique, and it was observed that the increase in the radiation parameter implies the decrease in the boundary layer thickness and enhances the rate of heat transfer. Hossain and Samand [19] examined the heat and mass transfer of a MHD free convection flow along a stretching sheet with chemical reaction, thermal radiation and heat generation in the presence of magnetic field. They conclude that the concentration profiles increase as the values of the radiation parameter is increased. Recently, Hsiao [20] performed an analysis to study the combined effects mixed convection and thermal radiation in nanofluid with multimedia physical features.

Viscous dissipation plays an important role in changing the temperature distribution, just like an energy source, which affects the heat transfer rates considerably. In fact, the shear stresses can induce a considerable amount of heat generation. El-Aziz [21] investigated the mixed convection flow of a micropolar fluid from an unsteady stretching surface with viscous dissipation. He concluded that the viscous dissipation produces heat due to drag between the fluid particles and this extra heat causes an increase of the initial fluid temperature. This increase in the temperature causes an increase in the buoyant force. Kishore et al. [22] studied the influence of chemical reaction and viscous dissipation on unsteady MHD free convection flow past an exponentially accelerated vertical plate with variable surface conditions. They examined that the increase in the viscous dissipation enhanced the fluid temperature, also the rate of heat transfer fell with increasing the Eckert number. Singh and Singh [23] analyzed MHD flow with viscous dissipation and chemical reaction over a stretching porous plate in

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porous medium. They drew a conclusion that the effect of viscous dissipation on the temperature distribution is insignificant.

The theory of micropolar fluids was originally developed by Eringen [24] which has become a popular field of research in recent times. The concept of micropolar fluid deals with a class of fluids that exhibits certain microscopic effects arising from the micromotions of the fluid elements. These fluids contain dilute suspension of rigid macromolecules with individual motions that support stress and body moments are influenced by spin inertia. Micropolar fluids are those which contain micro-constituents that can undergo rotation, which can affect the hydrodynamics of the flow so that it can be distinctly non-Newtonian. Thus, micropolar fluids are those consisting of randomly oriented particles suspended in a viscous fluid, which can undergo a rotation. Haque et al. [25] analyzed the micropolar fluid behaviors on steady MHD free convection and mass transfer flow with constant heat and mass fluxes, Joule heating and viscous dissipation. It was concluded by them that the motion of micropolar fluid is more for lighter particles and air than heavier particles and water, respectively and the angular motion of microlpolar fluid is greater for heavier particles and water than lighter particles and air, respectively also the micropolar fluid temperature is more for air than water. Sudheer Babu et al. [26] studied the mass transfer effects on unsteady MHD convection flow of micropolar fluid past a vertical moving porous plate through a porous medium with viscous dissipation. Gupta et al. [27] investigated unsteady mixed convection flow of micropolar fluid over a porous shrinking sheet using finite element method.

Different authors studied mass transfer with or without radiation and viscous dissipation effects on the flow past oscillating vertical plate by considering different surface conditions, but the study on the effects of magnetic field on free convection heat and mass transfer with thermal radiation, viscous dissipation, chemical reaction and variable surface conditions in flow through an oscillating plate has not been found in literature and hence the motivation to undertake the present study. Thus the present investigation is concerned with the study of combined effects of magnetic field and first-order chemical reaction in two-dimensional MHD flow, heat and mass transfer of a viscous incompressible fluid past a permeable vertical plate embedded in a porous medium in the presence of viscous dissipation and thermal radiation using perturbation technique. The effects of various physical parameters on the velocity, temperature and concentration profiles as well as on local skin friction co-efficient, Sherwood number and local Nusselt number are shown graphically. Validation of the analysis has been performed by comparing the present results with those of Modather et al. [7].

2. Formulation of the problem

We consider mixed convection and diffusion mass transfer of a viscous incompressible electrically conducting micropolar fluid over an infinite vertical porous moving permeable plate embedded in a porous medium in the presence of viscous dissipation, thermal radiation, heat source/sink and chemical reaction. In Cartesian coordinate system, x-axis is measured along the plate and y-axis normal to the plate. A constant magnetic field is applied in y-direction of strength B0. It is assumed that the magnetic field is of small intensity so that the induced magnetic field is negligible in comparison to the applied magnetic field. The Joule heating is negligible as the term due to electric dissipation is neglected in the energy equation. The fluid is considered to be gray absorbing emitting or radiating but not scatting medium. The Rosseland approximation is used to describe the radiative heat flux which is negligible in the x-direction in comparison to y-direction. The plate moves continuously with uniform velocity up in its own plane. It is assumed that the tem-

perature of the surface is held uniform at Tw while the ambient temperature takes the constants value T„ so that Tw > T„. The species concentration at the surface is maintained uniformly at Cw and that of the ambient fluid is taken as C«,. First-order chemical reaction is considered in this paper since the rate of reaction is directly proportional to the concentration difference which is associated with the concentration of the species C in the solutal boundary layer and the ambient fluid concentration C, [7,13]. The chemical reaction is assumed to be irreversible. Under these assumptions, the boundary layer equations of motion, energy and mass-diffusion under the influence of uniform transverse magnetic field and Ohmic dissipation in the presence of heat source or sink, viscous dissipation and thermal radiation are as follows:

du* dt*

+ V^r = (v + v + 2vr^- + gfr (T - TJ

+ gPc (C - CJu* - u*,

prl-+ v

PJ 1 dt* dy'

- + v*

dt* dy*

1 dqr * y ( du* PCp dy* + pcp Uy*

+ v*— = D + Y1*(C - C J, dt* dy* dy*2 '

where (u*, v*) are the component of the velocity at any point (x*, y*); m* is the component of the angular velocity normal to the x*y* plane; T is the temperature of the fluid; and C is the mass concentration of the species in the flow p, v, vr, g, ¡¡T, [SC, a, K, j*, y, a, D, and y* are the density, kinematic viscosity, kinematic rotational viscosity, acceleration due to gravity, coefficient of volumetric thermal expansion of the fluid, coefficient of volumetric mass expansion of the fluid, electrical conductivity of the fluid, permeability of the medium, microinertia per unit mass, spin gradient viscosity, thermal diffusivity, molecular diffusivity, and the dimensional chemical reaction parameter, respectively. It is important to note that the change in the concentration of species gives rise to the solutal buoyancy so there is direct connection between Eq. (2) and Eq. (5) hence these equations are coupled.

The appropriate boundary conditions for the problem are:

u* = u„

du* ' dy*'

= T_+e(T„- T_ )e"v,

C = C_+e(Ca- C_ )en't" aty* = 0,

u* ^ 0, a* ^ 0, T ^ T„, C ^ C„ asy*

The following comment should be made about the boundary condition used for the microrotation term: when n1 = 0, we obtain from the boundary condition stated in Eq. (6), for the microrotation, o* = 0. This represents the case of concentrated particle flows in which the microelements close to the wall are not able to rotate. The case corresponding to n1 = 0.5 results in the vanishing of the antisymmetric part of the stress tensor and represents weak concentrations. When n1 = 1 then this is the case of turbulent boundary layer flows. Further, when n1 = 0, the particles are not free to rotate near the surface. However, when n1 = 0.5 or n1 = 1.0, the microrotation term gets augmented and induces flow enhancement.

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u( y, t) = Uo( y) + ee"%(y) + O(£2)

From the Eq. (1), it is clear that the suction velocity normal to the plate is a function of time only, which is considered to be in the form [1,10,18]:

v* = -(1 + eAen'<")Vo,

where A is a real positive constant, eA small less than unity and V0 is a scale of suction velocity which has a non-zero positive constant.

The radiative heat flux term by using the Rosseland approximation [14] is given by

q' = -

4a* dT*4 3K' dy' '

(( y, t) = ( (y) + eent( (y) + O(e2),

9( y, t) = Oo(y) + een%(y) + O(e2), (14)

0( y, t) = fa(y) + eentfa(y) + O(e2).

After substituting Eq. (14) into Eqs. (9) to (13), we have

(1 + ß)u0+u0 -1M + Iß uo = -GrTÖü -Grcfo - 2ßco0, (15)

We assume that the temperature difference within the flow are sufficiently small such that T*4 may be expressed as a linear function of the temperature. This is accomplished by expanding T*4 in a Taylor series about T* and neglecting higher (second order onwards) order term [12], we get

T*4 ~ 4t*3>t*_3T*4

It is convenient to employ the following dimensionless variables:

u' = UoU, v' = VoV, y* = —y, Up * = UoUp, Vo

(=UV Wt

t' = — t, T - T„ = (Ta- T„ )9, C - C. = (Ca- c.

•* v2 . __ v _ v .. oBo2v j' = —tJ, Pr = —, Sc = —, M =—V, J Vo2J' a D pVo2 '

n' = — n,

vgßT (Ta- T.

G vgßc(Ca-C.) v ( A^ .' ui'(1 + ß

Grc = UV2 , V = (ß + 2 J = ßJ ^ 2J'

A vr „ KUoVo2

ß = :- = -, K2 =

2 + ß ' VVo2

Cp (Ta- T. )'

ß w 4a'T' v = —, Nr = -

With the help of Eq. (8), Eqs. (1)-(7) reduced to the following:

du ,1 . nn du ... . d2u 0dw n n _ .. 1 + ß -— (1 + eAent ) — = (l + ß)—r + 2ß— + GrT9 + Grc$-Mu —tt^- u,

((l + eAent ) (1 dt 'dy n dy2

d9 (1 + eAent)d9 = 1 (1 + 4Nr!" + E (du __(1 + eAe )___jdyr + Ec^dy

d<P -(1 + eAen-

dt --------9y Sc y +

(9) (10)

(11) (12)

with the following dimensionless boundary conditions du

u = Un

co = -n-i

9 = 1 + een

--1 + eent aty = o,

u ^ o, (o^ o, 9^ o, o as y

To solve Eqs. (9)-(12) subject to the boundary conditions Eq. (13) we may use the following linear transformations for low value of £ [18,26]:

(1 + p)u{'+ ui -^M + ^-nju, = -Au0 - GrTe1 -Grcfa - 2PK (16)

(Û0+V(Q'o= 0, (17)

co{' + nco{ - nnco-t =-Aria>0, (18)

(3 + 4Nr)90' + 3Pre0 + 3PrEcu'2 = 0, (19)

(3 + 4Nr)e{' + 3Pre; -3nPre, = -3APreo -6PrEcu0K (20)

$ + Sc^0 +7iScfo = 0, (21)

+ Sc$ + (y, -ri)Scfo = -ASc^O, (22) with the following boundary conditions: u0 = Up, u, = 0, CO" = -n,u', co, = -n,u1,

Go = 1, ei = 1, fa = 1, fa = 1 aty = 0,

u0 = u1 =a0 =œ1 =G0 = G1 =fa = fa = 0 as y (23)

To solve the nonlinear coupled Eqs. (15)-(22), we assume that the viscous dissipation parameter (Eckert number Ec) is small, so we can write the asymptotic expansion as follows

Uo (y) = U"1 (y) + EcU"2 (y) + O(E2 ), U1 (y) = uu(y) + EcUn(y) + O(E2), Co (y) = c (y) + Ec (y) + O(EC2),

C (y) = C (y) + Ecn(y) + O(E2 ),

G"(y) = G"1 (y) + EGo2(y) + O(E2), (24)

Gt(y) = Gu(y)+EcGn(y)+O(E2), My) = M(y) + Ecfa2(y) + O(E2), My) = 0u(y) + Ecfa2(y) + O(E2),

Substituting Eq. (24) into Eqs. (15)-(22), we obtain the following sequence of approximations for O(O) of Ec:

(1 + ß)uo\ + uo1 -|M + ^KÂ I uo1 = -Grr9o1 - Gcfa1 - 2ßmoh

(1 + ß)uo'2 + %2 -1 M + I uo2 = -G9 - Gcfa,2 - 2ßa>o2,

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a01 +n®0i = 0, (27)

®0'2 +n®02 = 0, (28)

(3 + 4Nr)001 + 3P0 = 0, (29)

(3 + 4Nr)00'2 + 3PrB'02 =-3Pr u012, (30)

001 + Scfai +7iScfoi = 0, (31)

002 + Sc002 + YiSc002 = 0, (32)

subject to the boundary conditions:

u» = Up, u02 = 0, ®0i = -nu0i, ®02 = -niu02, 001 = 1, 002 = 0, 001 = 1, 002 = 0 at y = 0,

u» = u02 =®01 = ®02 = 001 = 002 =001 = 002 = 0 OS y ^ (33) Also we get the following equations for O(1) of Ec:

(1 + p)uH + uÎ1 -1M + "K^-n ju„ = -Au01 -GrT0u -GrC0u - 2®,

(1 + j8)uf + uÎ2-| M + n |un--

-Au02 - GrT02 - GrC012 - 2®,

af + na{1 - nna 1 = - An®01,

a" + na{2 - nna12 = -An®02,

(3 + 4Nr)0f + 3P0 1 - 3nPr01 1 = -3 APr00 1 ,

(3 + 4Nr)0"2 + 3Pr02 - 3nPr0 2 = -3APr002 - 6Pru0 1 u' 1 ,

01 + Sc01 + (y - n)Sc01 1 =-ASc001 ,

012 + SC012 + (Y1 - n)Sc012 =-ASC002,

subject to the boundary conditions:

un = 0, u12 = 0, au = -n1u{1, a12 =-n1u[2,

011 = 1, 012 = 0, 011 = 1, 012 = 0 aty = 0,

un = u12 =®11 = :

11 =0i2 = 0 as y

Solving Eqs. (25)-(32) under the boundary conditions Eq. (33) and Eqs. (34)-(41) under the boundary conditions Eq. (42) and substituting into Eqs. (24) and (14), we obtain the temperature, angular velocity and velocity profiles of all the flow respectively as follows:

u( y, t) = a1e-h2y + a2e-h1y + a3e-ny + a4e-h3y + Ec (a16e-h2y + a17e-2h3y + a18e-2h2y + a19e-(h2+h3) y + a20e-2hy + a21e-2ny + a22e-(h,+n) y + a23e-(h,+hз) y + a24e-(h3+n) y + a25e-(h,+h') y + a26e-(h2+n) y + a27e-ny + a28e-h3y ) + eent (b5e-"3y + b6e-"2y + b7e-h1y + b8e-ny + b9e-h5y + b10e-h2y + bue-h4y + bne-h1y + bne-h6y + bue-ny + b15e-hiy + Ec (e1e-h3y + e2e-hsy + e3e-(h3+h7) y + e4e-(h3+hs) y

+ ese-

(h3+h4)y

+ e6e-(h3+hs) y + e7e-(h2+h7) y + e8e-(h2+h5) y + e9e-(h2+h4) y

+ e10e-(h2+h6) y + ene-(h1+h7) y + e12e-(h,+hs) y + e13e-(h+h4) y + e14e-(h1+h6) y + e15e-(h7+n) y + e16e" (h5+n) y + e17e~(h4+n) y + e18e-(h6+n) y + e19e-h6y + e20e-h7y + d1e-h2y + d2e-2h3y + d3e-2h2y + d4e"2hy + d5e-2ny + d6e-ny + d7e-("2+h3) y + d8e-(h1+n) y + d9e-(h1+h3) y + d10e-(h1+h2) y + dne-(h2+n) y + d12e-(h3+n) y ))

®( y, t) = a29e-ny + Eca30e-ny + eent (ene-h6y + b3e-"y + Ec (e22e-h6y + b4e-"y))

0( y, t) = e-h2y + Ec (a5e-2h3 y + a6e-2h2y + a7e-(h2+h3) y + a8e-2h1y + a9e-2ny + a10e- (h1+n) y + aue-(h1+h3) y + a12e-(h3+n) y + a13e - (h1+h2) y + a14e-(h2+n) y + a15e-h2y ) + eent ((1 - b1)e-h5) y + b2e-h2) y + Ec (b16e-h2y + b27e-(h3+h7)y + b32e-(h3+"5)y + b34e-(h3+h4)y + b36e-(h3+h6) y + b3Se-(h2+h7) y + b43e-(h2+h5) y + b45e-(h2+h4) y + b47e-(h2+h6) y + b49e-(h2+h7) y + b54e-(h1+hs) y + b56e-(h1+h4) y + b58e-(h1+h6) y + b60e-(h7+n) y + b65e-(hs+n) y + b67e-(h4+n) y + b69e-(h6+n) y + b71e-h5y + d13e-2h2y + d14e-2h1y + d15e-2h3y + d16e-2ny + d17e-(h1+n) y + d18e-(h2+n) y + d19e-(h3+n) y + d20e-(h2+h3) y + d21e-(h1+h3) y + d22e-(h1+h2) y ))

0( y, t) = e-h1y + eent ((1 - b1)e-h4y + b1e-h1y )

where af, bf, cf, dj, and hi are provided in the Appendix.

The local skin friction coefficient, local wall couple stress coefficient, local Nusselt number, and local Sherwood number are important physical quantities for this type of heat and mass transfer problem which are defined below.

The wall shear stress may be written as

= G" + A)

= pU0V0[1 + (1- n1)^]u ' (0)

Therefore, the local skin-friction factor is given by

= 2[1 + (1-n1)p]W (0).

The wall couple stress may be written as:

M® = y

Therefore, the local couple stress coefficient is given by

C®= ^^ = a ' (0).

YU0V02

The rate of heat transfer at the surface in terms of the local Nusselt number can be written as:

Nux = x

(dT/ dy*

Using Eqs. (8) and (45) in Eq. (47), we get NuxRe-1 =-0 ' (0),

where Rex =-is the local Reynolds number.

The rate of mass transfer at the surface in terms of the local Sherwood number is given by:

Shx = x

0C/3y*

Using Eqs. (8) and (47) in Eq. (48), we get ShxRe-1 =-0 ' (0).

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3. Results and discussions

To assess the physical properties of the problem, the effects of various parameters like Eckert number, local Grashof number, permeability parameter, magnetic field parameter, Prandtl number, thermal radiation parameter, chemical reaction parameter, viscosity parameter and Schmidt number, velocity component in x-direction, component of angular velocity, temperature of the fluid, concentration of species, skin friction coefficient, Nusselt number and Sherwood number are analyzed. Numerical evaluations of the analytical solutions were performed and the results are presented in graphical and tabular forms. The values of the physical parameters such as Nr, a etc. are taken from the literature [2,7,9]. The variation in velocity profile with y for various values in thermal Grashof numbers are shown in Fig. 1. This figure reflects that with increase in GrT there is increase in fluid velocity due to enhancement of the buoyancy force. The curves show that the velocity starts from a minimum value on the surface and increases till it attains a peak value near the plate, then it starts decreasing till the end of the boundary layer. The positive value of GrT indicates the cooling of the plate and it is observed that velocity increases rapidly near the wall of the plate and then decays to the free stream velocity. When GrT is negative, the plate becomes hotter and there is retardation in the fluid velocity. Thus the thermal Grashof number GrT signifies the relative effect of the thermal buoyancy force to the viscous hydrodynamic force in the boundary layer. As expected, it is observed that there is a rise in the velocity as GrT increases. The solutal Grashof number GrC defines the ratio of the species buoyancy force to the viscous hydrodynamic force. As expected, the fluid velocity increases as Grashof number GrC increases as seen in Fig. 2.

The effects of first order irreversible chemical reaction parameter on the velocity profiles are studied in Fig. 3. It is seen that the velocity profile increases with an increase in the chemical reaction parameter. Also, it is observed that when the chemical reaction parameter is negative (i.e. generative reaction) there is an increase in the fluid flow velocity. In fact, as chemical reaction increases there is a considerable amount of reduction in the velocity profiles, and the presence of the peak indicates that the maximum value of the velocity occurs in the fluid close to the surface but not at the surface. For positive chemical reaction parameter (i.e. destructive reaction), there is reduction in the fluid flow velocity. Fig. 4 depicts the variation of the microrotational velocity distribution with per-

Fig. 2. Velocity profiles vs. y for different values of Grc.

meability parameter of the porous medium for different time t. It is clearly observed that there is a decrease in the microrotational velocity distribution with an increase in the value of the permeability parameter. The physical reason behind this decrease is due to the fact that velocity of the fluid dominates over microrotational velocity distribution with an increase in the permeability parameter. Thus the higher the value of the porous permeability the lower the microrotational velocity, which results in increasing the velocity of the fluid.

It is observed from Fig. 5 that the decrease in the value of a is very rapid for the higher value of Eckert number. It is to be noted that the parameter n1 is associated with the boundary condition Eq. (23) and that it physically reduces to the concentration of the microelement at the plate. Further, it is noticed that this initial boundary condition for microrotational corresponds to the vanishing of the anti-symmetric part of the stress tensor and related to the weak concentration of the micro-elements of the micropolar fluid. Thus it is clear that greater n1 decreases the value of the microrotational velocity distribution rapidly. The effects of micro-rotational

GrT=-0.5,0,2,5,10

t=1 p=1 r=0.1 Pr=1 Sc=2 M=4 Y=0.1 k-5 Ec=0.01

0.370 -\ 0.368 0.366 H 0.364 0.362 -0.360 0.358 0.356 -0.354

p=1,Pr=1,Sc=2,M=2,y=1 K=5,Nr=0.1,Ec=0.1

Fig. 1. Velocity profiles vs. y for different values of GrT.

Fig. 3. Velocity profiles vs. t for different values of chemical reaction parameter.

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velocity distribution against y for different value of Eckert number are shown in Fig. 6. This figure depicts that the micro-rotational velocity distribution decreases with an increase in the Eckert number whereas its value increases with an increase in y. Physically, this is due to the fact that increase in the Eckert number increases dissipation of energy by viscous force and hence thereby decreases the micro-rotational velocity by increasing the Eckert number.

Fig. 7 depicts the variation of micro-rotational velocity distribution against y for different values of the magnetic field parameter. Here, we observed that the micro-rotational velocity distribution increases with an increase in the magnetic field parameter and it becomes zero as y tends to ^ for all the values of the magnetic field parameter. Further, it is observed that for higher values of y, all the curves of a coincide at that point (i.e. y = 8). Physically, an increase in the strength of the magnetic field enhances the microrotational velocity. The variation of microrotational velocity distribution for various values of the chemical reaction parameter are shown in Fig. 8. From this figure we see that microrotational velocity distribution decreases by increasing the value of the chem-

ical reaction parameter. The effect of permeability parameter on temperature profile is shown in Fig. 9 with time. From this figure, it is observed that the temperature increases with an increase in the permeability parameter K2 for all time t. This is due to the fact that as the velocity increases there is an increase in the viscous dissipation as it adds energy to the fluid and hence temperature of the fluid increases with the permeability parameter K2. Also, the effects of GrT on temperature profile are shown in Fig. 10. For the positive value of GrT, i.e., for externally cooled plate the temperature increases, but for the negative value of the GrT, i.e., for externally heated plate, the temperature reduces.

Fig. 11 illustrates the effect of thermal radiation on the temperature distribution with y for different values of the radiation parameter. For larger thermal radiation parameter the thermal boundary layer is thicker, so the temperature profile increases. It is also seen from this figure that the temperature distribution increases with an increase in the radiation parameter and decreases

3 -0.15

n.,=0,0.25,0.5,1,1.5

[=1,P^1,Sc=2, M=2,y1=0.1 K2=5,Nr=0.1,Ec=0.01

0.2 0.1 0.0-0.1 ■ -0.2 -0.3 -0.4 -0.5 -0.6 -0.7

M=0,1,2,3,4

[=1 t=1 Pr=1 Sc=2 y,=0.1 k2=5 Nr=0.1 Ec=0.01

Fig. 5. Plot of micro-rotational velocity distribution with y for different values of

Fig. 7. Microrotational velocity profile vs. y for different values of magnetic field parameter M.

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Fig. 8. Micro-rotational velocity distribution vs. time t for different values of chemical reaction parameter Fig. 10. Variation of temperature profiles with y for different values of GrT.

with an increase in y and the value reaches zero as y ^ ^ matching the boundary condition at that point. Fig. 12 is drawn to study the variations of temperature distribution against y for different value of Eckert number. In this figure we can see that the temperature distribution increases with an increase in the Eckert number and temperature decreases steeply with an increase in y. Physically, the Eckert number (Ec) expresses the relationship between the kinetic energy in the flow and the enthalpy. It embodies the conversion of kinetic energy into internal energy by work done against the viscous fluid stresses. Greater viscous dissipative heat causes a rise in the temperature as well as the velocity. Thus increase in the value of the Eckert number indicates that there is an increase in the viscous dissipation and thereby increases the temperature of the fluid as the dissipative force adds energy to the fluid.

Fig. 13 depicts the variation of temperature profiles for different values of the chemical reaction parameter. It is seen from this figure that the temperature profile increases with an increase in the chemical reaction parameter. The effect of skin-frictional coefficient against t for different value of chemical reaction parameter are depicted in Fig. 14. This figure shows that the local skin-

Fig. 9. Variation of temperature profiles with time t different value of K2.

frictional coefficient increases with increases in the chemical reaction parameter.

Fig. 15 depicts the variation of skin-frictional coefficient against time t for different value of the Eckert number. From this figure, it is observed that the skin-frictional coefficient increases with an increase in the Eckert number due to increase in the viscous dissipative forces. Fig. 16 depicts the variation of skin-frictional coefficient against t for different values of magnetic field parameter. It is seen from this figure that skin-frictional coefficient decreases with an increase in the magnetic field parameter. Also, it is observed from this figure that there is an increase in the value of the skin-friction coefficient with an increase in time and the rate of increase become rapid for large values of t. The profiles of the local couple stress coefficient at the wall against time t for different value of Eckert number are shown in Fig. 17. It is seen from this figure that by increasing the Eckert number there is remarkable increase in the local couple stress coefficient at the wall due to increase in the dissipa-tive forces exerted by viscosity of the fluid.

Fig. 11. Variation of temperature profiles with y for different value of radiation parameter Nr.

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Fig. 12. Temperature profile vs. y for different values of Eckert number Ec.

Fig. 14. Variation of Cf with t for different values of chemical reaction parameter.

The effect of local couple stress coefficient at the wall against t for different values of the permeability parameter are shown in Fig. 18. It is seen from this figure that increase in the permeability parameter increases the local couple stress coefficient at the wall due to increase in the porosity of the porous medium. Fig. 19 depicts the variation of concentration at the wall against t for different value of magnetic field parameter. In this figure it is observed that the local couple stress coefficient at the wall decreases by increasing the magnetic field parameter and it rapidly decreases for large values of t. This causes the local couple stress coefficient buoyancy effects to decrease, yielding a reduction in the fluid velocity. The reduction in the concentration profiles is accompanied by simultaneous reduction in the momentum and concentration boundary layers thickness.

Fig. 20 depicts the variation of Nusselt number against t for different values of the Eckert number. We observe that the Nusselt number decreases with an increase in the Eckert number due to the fact that increasing the values of the Eckert number generates heat in the fluid due to frictional heating. Thus the effect of increasing Ec is to enhance the temperature and thereby increases the Nusselt

number. The plot of Nusselt number against time t for different value of magnetic field parameter is shown in Fig. 21. It is seen from this figure that the local Nusselt number increases significantly for small values of the magnetic field parameter and the effect become insignificant for large values in the magnetic field parameter.

Fig. 22 depicts the variation of Sherwood number against t for different values of the Schmidt number. It is well known that the Schmidt number Sc embodies the ratio of the momentum to the mass diffusivity. The Schmidt number which quantifies the relative effectiveness of momentum and mass transport by diffusion in the hydrodynamic (velocity) and concentration (species) boundary layers. As the Schmidt number increases the concentration decreases. This causes the concentration buoyancy effects to decrease, yielding a reduction in the fluid velocity. The reduction in the velocity and concentration profiles is accompanied by simultaneous reduction in the velocity and concentration boundary layers. Thus it is observed from this figure that the Sherwood number increases by increasing the Schmidt number.

We have compared the result of Nusselt number and Sherwood number for different values of t with Modather et al. [7] in Table 1,

0.41840-

0.41838-

0.41836-

0.41834-

0.41832-

0.41830-

Y =0,0.1,0.2,0.3 /

ß=1,Pr=1,Sc=2,M=2,y=1 K2=5,Nr=0.1 ,Ec=0.01

Fig. 13. Variation of temperature profile with t for different value of chemical reaction parameter.

0.276 -

0.272 -

0.268 -

Ec=0.01,0.05,0.1

ß=1 Nr=0.1 Pr=1 Sc=2 M=2

Fig. 15. Variation of Cf with t for different values of Eckert number Ec.

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Y,=0.1

Nr=0.1 Ec=0.01

0.035 0.030 0.025 -0.020 -0.015 0.0100.005 -0.000 -0.005 --0.010-0.015

Nr=0.1,Pr=1,Sc=2,M=2 y=0.1,p=1,Ec=0.01

Fig. 16. Variation of Cf with t for different values of M.

Fig. 18. Variation of C' with time for different values of K2.

in the absence of thermal radiation parameter and viscous dissipation effects. Numerical values of local skin friction factor, local couple stress coefficient at the wall, Nusselt number and Sherwood number for different values of the parameters Sc, Pr, K2, M, Nr and Ec are shown in Table 2 by keeping the values of other parameters constant. We have taken the values of the parameter £ = 0.01, n = 0.1, n1 = 0.5, GrT = 2, GrC = 1, Up = 0.5, A = 0.1, for all the figures and tables presented in this paper. It is observed from this table that the effect of increase in the value of Sc, Pr, and M is to decrease Cf and C'w whereas reverse effect is found for NuxRe_1. Further, it is noticed that the effect of increase in the value of K2, Nr, and Ec is to increase Cf and C'w whereas reverse effect is seen for NuxRe_1. Also, the effect of increasing the value of Schmidt number Sc is found to be very prominent on Sherwood number ShxRe_1, i.e. an increasing trend is found on ShxRe_1 by increasing the value of the Schmidt number.

4. Conclusions

In this paper the effects of chemical reaction and viscous dissipation on MHD free convection flow in the presence of heat source

or sink and thermal radiation with variable surface temperature and concentration have been studied analytically. Perturbation method is employed to solve the governing equations of the flow. From the present investigation, the following conclusions have been drawn:

(i) The temperature distribution increases with an increase in the Eckert number, radiation parameter, thermal Grashof number and permeability parameter.

(ii) The micro-rotational velocity distribution decreases with an increase in chemical reaction parameter, permeability parameter and Eckert number but increases with the increase in the magnetic field parameter.

(iii) There are increases in the skin frictional coefficient with increases in the Eckert number and chemical reaction parameter.

(iv) When there is increase in the magnetic field parameter there is decrease in skin friction coefficient.

(v) The value of the Nusselt number decreases with an increase in the Eckert number, and increases with an increase in the magnetic field parameter.

(vi) When increasing the value of Schmidt number there is increase in the Sherwood number.

0.0310

0.0308

0.0306

0.0304

0.0302

0.0300

P=1 Nr=0.1 Pr=1 Sc=2 M=2

0.55 0.500.450.40 0.35 0.300.250.20 0.15 0.100.05 0.00 -0.05 -0.10

Nr=0.1

Ec=0.01

-1— 10

—1— 20

—I— 30

—I— 40

Fig. 17. Variation of C' with t for different values of Eckert number.

Fig. 19. Plot of C' with t for different values of M.

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(£ 0.892

0.891 -

Ec=0.01,0.05,0.07,0.1

Nr=0.1

Fig. 20. Plot of NuxRex' with time for different values of Eckert number.

0) DÎ

2 H 1 0

y=0,[=1,PM,M=2,y =0.1 K =5,Nr=0.1 ,Ec=0.01

Sc=0.5,1,2,3

Fig. 22. Plot of Sh^Re-1 with t for different values of Sc.

(vii) Local couple stress coefficient at the wall decreases with an increase in the values of the magnetic field parameter and it increases with an increase in Eckert number and permeability parameter in the specified range. Thus we can conclude that the physical parameters play an important role on the rate of heat and mass transfer over a vertical permeable plate embedded in a porous medium.

Acknowledgments

We thank the reviewers for their constructive comments that have led to a definite improvement in the paper.

Appendix

1 + . 1

3Pr 3 + 4Nr

2(1+ [) Sc

1 + ,/1 + 41 M + (1 + [)

1 + , 1

4(y- n)

3Pr + ^19Pr2 +12nPr (3 + 4Nr) ' 2(3 + 4Nr) '

h6=n\1+. 1+^

1 + A 11 + 4(1 + [)| M + n

2(1 + [

(1 + [)h2 -h2 -| M +

0.89140

0.89138-

0.89136-

0.89134-

0.89132

0.89130

M=1,2,3,4 ,,r ' ..

—I-""

Fig. 21. Plot of NuxRewith time for different values of M.

(1 + [)h2 - h -| M +1+[

Table 1

Comparison of the present result of NuxReand ShxRe-1 with Modather et al. [7] for different values of t when Pr = 1.0,Sc = 2.0,Nr = 0.0,Ec = 0.0, A = 0.0, P= 1.0, M = 2.0, £ = 0.01, n1 = 0.5, n = 0.1, GrT = 2, GrC = 1, K2 = 5, y = 0.1, Up = 0.5 .

Modather et al. [7] results

t NuxRe-1 ShxRe-1

Present results

NuxRe-1

ShxRe-1

0 1.00887 1.91217 1.0088730 1.9121732

1 1.00981 1.91404 1.0098062 1.9140395

3 1.43567 1.91838 1.0119773 1.9183817

5 1.01463 1.92369 1.0146291 1.9236853

10 1.02412 1.94267 1.0241193 1.9426657

20 1.06556 2.02555 1.0655630 2.0255532

30 1.17822 2.25086 1.1782186 2.2508645

40 1.48445 2.86332 1.4844484 2.8633242

50 2.31687 4.52816 2.3168676 4.5281625

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Effects of various values of fluid properties on the coefficients of skin friction coefficient, couple stress coefficient, Nusselt number and Sherwood number.

Sc Pr K2 M Nr Ec Cf CC NuxRe- ShxRe-1

2 1 5 2 0.1 0.001 0.2707133 0.0301309 0.8914258 1.9158002

1 0.6682574 0.0745723 0.8914186 0.8975272

0.5 1.1339602 0.1264654 0.8914083 0.3664133

2 3 -0.8333037 -0.0927355 2.6738615 1.9158002

5 -1.2005236 -0.1335068 4.4563727 1.9158002

1 3 0.0745329 0.0083268 0.8914279 1.9158002

1 -0.6985520 -0.0776972 0.8914325 1.9158002

5 3 -0.3851199 -0.0427542 0.8914313 1.9158002

4 0.7 -0.8853874 -0.0985380 0.8914328 1.9158002

2 0.3 0.5051470 0.0560755 0.7216151 1.9158002

0.5 0.6920367 0.0770952 0.6061646 1.9158002

0.1 0.05 0.2717460 0.0301607 0.8912639 1.9158002

0.07 0.2723846 0.0301891 0.8911055 1.9158002

0.1 0.2734951 0.0302488 0.8907710 1.9158002

(1+PW2-n-| m +

a4 = Up - a1 - a2 - a3 a5

1 + P K 2

(1+PW-n-\ m+

1+P K 2

-3PrEch2a2

4(3 + 4Nr)h2 - 6Prh3 -3PrEch2a?

a9 ■■

-6PrEca1a4h2h3

(3 + 4Nr)(h2 + h3)2 -3Pr(h2 + h3)

-3PrEch?a2 4(3 + 4r)h - 6Prh1

-3PrEcn2a22 4(3 + 4Nr)n2 - 6Prn

-6PrEca2a3nh1

(3 + 4Nr)(h1 + n)2 - 3Pr(h1 +n)

-6PrE ca2a4hh ' (3 + 4Nr)(h1 + h3)2 -3Pr(h1 + h3)

-6PrEca3a4nh3 '(3 + 4Nr)(h3 + n)2 - 3Pr(h3 +n

-6PrEca1a2h1h2 '' (3 + 4Nr)(h1 + h2)2 -3Pr(h1 + h2)

-6PrEca1a3nh2

(J+^Nrjh + n^-SPrih^+n

-a5 - a6 - a7 - as - a9 - aw - au - au - au - au

_-GrTa15_

(1 + P)hl - h2 M + ^

4(1 + p)h2 -2h3-I M + 1+f

4(1 +Ph -2h2-| M + 1P

-GrTa7

(1 + P)(h2 + h3)2 -(h2 + h3)-I M +

1 + P K 2

-GrTa8

4(1 + P)h2 -2h1 -I M +

1 + P K 2

_ GrTa9_

4(1 + P)n2 -2n-[M +

__ GrTa10

(1+P)(h+n)2 - (h+n)-| m+

1+P K 2

-GrTa11

(1 + P)(h + h3)2 -(h1 + h3)-I M +

1+P K 2

-GrTa12

(1+P)h + n)2 - h + n)-I M+

-GrTa13

1+P K 2

(1 + P)(h + h2)2 -(h + h2)- M +

1+P K 2

_-GrTa14_

(1 + P)(h2 +n -(h2 +n)M +

(1+PW-n-|M+^-K+P

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(1+2)n2-n-\M + ^

ds = 5,fc3

Û28 029 Oso

= -016 - 017 - 018 - ai9 - a20 - a21 - a22 - 023 - 024 - 025 - a26 - a27

= n1 (h3a4 + h2a1 + h1a2 + na3) = c2

= ni(a2sh3 + ai6h2 + 2a^h3 + 2^2 + ai9(h2 + h3)

+ 2a20h1 + 2a21n + a22(h1 + n) + a23(h1 + h3)

- a24 (h3 + n) + a25 (hi + h2) + a26 (h2 + n) - a27n) = ni[h3(-ai6 - ai7 - ai8 -ai9 - a2o -a2i - a22 - a23 - a24 -a25 -a26)

+ a16h2 + 2a17h3 + 2a18h2 + a19(h2 + h3) + 2a20h1 + 2a21n + a22(h1 +n) + a23 (hi + h3 ) + a24 (h3 + n) + a25(hi + h2 ) + a26 (h2 + n)]

(i+^)h62 - h6-( m+M-n

:_2&k_

(1 + ^ - h6-( M + n

bu =_2^

14 (i+p)n2-n-{M+1+2-n

bi5 =-b5 - b6 - b7 - b8 - b9 - b10 - bu - b12 - b13 - b14

3APra15h2

1+ n1^2(h3 -n)

Ah1Sc h2 - h1Sc + Sc(y - n)

3Ah2Pr (3 + 4Nr)h22 - 3pr - 3nPr

Ana30 n

(3 + 4Nr)h22 - 3Prh2 - 3nPr

_6 APra5h3_

4(3 + 4Nr)h32 - 6Prh3 - 3nPr

_6APrah_

4(3 + 4Nr)h2 - 6Prh2 - 3nPr

_3APra7(h2 + h3)_

(3 + 4Nr)(h2 + h3)2 - 3Pr(h2 + h3) - 3nPr

_6APra8h_

4(3 + 4Nr)h2 - 6Prh1 - 3nPr

6APra9n

4(3 + 4Nr)n2 - 6Prn- 3nPr

(! + № - ,3-( M+111. -) b22 3APra10(h1 + n)

Aha (3 + 4Nr)(h1 + n)2 - 3Pr(h1 + n) - 3nPr

(1+P)M - hM + 11+f. -) b23 3APraii(hi + h3)

(3 + 4Nr)(h1 + h3 )2 - 3Pr (h1 + h3) - 3nPr

Aha b24 3APra12(h3 + n)

(1+2)h2 - -h,-f M + i±t- (3 + 4Nr)(h3 + n)2 - 3Pr(h3 + n) - 3nPr

2 b25 3APra13(h1 + h2)

Ana3 (3 + 4Nr)(h1 + h2)2 - 3Pr (h1 + h2 ) - 3nPr

(1+^)n2 - -n-( m+ b26 3APra14(h2 +n)

-GrT (1- b2) (3 + 4Nr)(h2 + n)2 - 3Pr(h2 + n) - 3nPr

(1 + £)h2 - M -n) b27 -6Pra4b15h3h7

(3 + 4Nr)(h3 + h7)2 - 3Pr (h3 + h7) - 3nPr

-GrTb2 b28 -6Pra4b5h32

= (1 + P)h2 - h^ - (« + ^ - n) 4(3 + 4Nr)h32 - 6Prh3 - 3nPr

b29 -6Pra4b6h2h3

-G* (1-É1) (3 + 4Nr)(h2 + h3)2 - 3Pr (h2 + h3) - 3nPr

1= (1 + ^)h4- - h.-( m+1+2 - n) b30 -6Pra4b7h1h3

-Gcb (3 + 4Nr)(h1 + h3)2 - 3Pr (h1 + h3) - 3nPr

2= (1 + flh,2- - + - n) b31 -6Pra4b8h3n

(3 + 4Nr)(h3 + n)2 - 3Pr(h3 + n) - 3nPr

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-6Pra4b9h3h5 b52 -6Pra2b7h2

(3 + 4Nr)(h3 + h5)2 -3Pr(h3 + h5) - 3nPr 4(3 + 4Nr)h2 - 6Prh1 - 3nPr

-6Pra4b10h2h3 b53 -6Pra2b8h1n

(3 + 4Nr)(h2 + h3)2 -3Pr(h2 + h3) - 3nPr (3 + 4Nr)(h1 + n)2 - 3Pr(h + n) - 3nPr

-6Pra4b11h3h4 b54 -6Pra2b9h1h5

(3 + 4Nr)(h3 + h4)2 -3Pr(h3 + h4) - 3nPr (3 + 4Nr)(hi + hs)2 -3Pr(hi + hs) - 3nPr

-6Pra4b12h,h3 (3 + 4Nr)(h + h3)2 -3Pr(h + h3)- - 3nPr b55 -6Pra2biohh2

(3 + 4Nr)(h + h2)2 -3Pr(h + h2) - 3nPr

-6Pra4b13h3h6 (3 + 4Nr)(h3 + h6)2 -3Pr(h3 + h6) - 3nPr b56 -6Pra2b11h1h4

(3 + 4Nr)(h + h4)2 -3Pr(h + h4) - 3nPr

-6Pra4b14h3n -6Pra2b12h2 4(3 + 4Nr)h2 - 6Prh - 3nPr

(3 + 4Nr)(h3 + n)2 - 3Pr(h3 + n) - 3nPr b57

-6Pra1b15h2h7

(3 + 4Nr)(h2 + h7)2 -3Pr(h2 + h7) - 3nPr b58 -6Pra2b13h1h6

(3 + 4Nr)(h + h6)2 -3Pr(h + h6) - 3nPr

-6Pra1b5h2h3

(3 + 4Nr)(h2 + h3)2 -3Pr(h2 + h3) - 3nPr b59 -6Pra2b14h1n

-6Pra1b6h2 (3 + 4Nr)(h +n)2 - 3Pr(h + n) - 3nPr

4(3 + 4Nr)h2 - 6Prh2 - 3nPr b60 -6Pra3bi5h7n

-6Pra1b7h1h2 (3 + 4Nr)(h7 + n)2 -3Pr(h7 + n)- 3nPr

(3 + 4Nr)(h1 + h2)2 -3Pr(h1 + h2)- - 3nPr b61 -6Pra3b5h3n

-6Pra1b8h2n (3 + 4Nr)(h3 + n)2 - 3Pr(h3 + n) - 3nPr

(3 + 4Nr)(h2 + n)2 - 3Pr(h2 + n) - 3nPr b62 -6Pra3b6h2n

-6Pra1b9h2h5 (3 + 4Nr)(h2 + n)2 - 3Pr(h2 +n) - ■3nPr

(3 + 4Nr)(h2 + hs)2 -3Prh + h5) - 3nPr b63 -6Pra3b7hn

-6Pra,b10h2 (3 + 4Nr)(h + n)2 - 3Pr(hi +n) - 3nPr

4(3 + 4Nr)h2 - 6Prh2 - 3nPr

b64 -6Pra3b8n2

-6Pra1b11h2h4 4(3 + 4Nr)n2 - 6Prn - 3nPr

(3 + 4Nr)(h2 + h4)2 -3Pr(h2 + h4) - 3nPr

b65 -6Pra3b9h5n

-6Pra1b12h1h2 (3 + 4Nr)(h1 + h2)2 -3Pr(h + h2)- - 3nPr (3 + 4Nr)(hs + n)2 - 3Pr(hs + n) - 3nPr

-6Pra1b13h2h6 (3 + 4Nr)(h2 + h6)2 -3Pr(h2 + h6) b66 -6Pra3b10h2n

- 3nPr (3 + 4Nr)(h2 + n)2 - 3Pr(h2 + n) - ■3nPr

-6Pra1b14h2n (3 + 4Nr)(h2 +n)2 - 3Pr(h2 +n) - b67 -6Pra3b11h4n

3nPr (3 + 4Nr)(h4 + n)2 - 3Pr(h4 + n) - -3nPr

-6Pra2b15h1h7 (3 + 4Nr)(h1 + h7)2 -3Pr(h1 + h7)- b68 -6Pra3b12h1n

- 3nPr (3 + 4Nr)(h +n)2 - 3Pr(h + n) - 3nPr

-6Pra2b5h1h3 b -6Pra3b13h6n

(3 + 4Nr)(hi + h3)2 -3Pr(hi + h3)- - 3nPr (3 + 4Nr)(h6 + n)2 -3Pr(h6 + n) - -3nPr

-6Pra2b6h1h2 b70 -6Pra3b14n2

(3 + 4Nr)(h1 + h2)2 -3Pr(h + h2)- -3nPr 4(3 + 4Nr)n2 -6Prn-3nPr

ARTICLE IN PRESS

b71 = -b16 - b17 - b18 - b19 - b20 - b21 - b22 - b23 - b24 - b25 - b26

- b27 - b28 - b29 - b30 - b31 - b32 - b33 - b34 - b35 - b36 - b37

- b38 - b39 - b40 - b41 - b42 - b43 - b44 - b45 - b46 - b47 - b48

- b49 - b50 - b51 - b52 - b53 - b54 - b55 - b56 - b57 - b58 - b59

- b60 - b61 - b62 - b63 - b64 - b65 - b66 - b67 - b68 - b69 - b70

D. Pal, S. Biswas/Engineering Science and Technology, an International Journal ■■ (2015) ■■-■■

-GrTb16

Aa28h3

(1 + 2)hf -h3-I M + 1+2-n

(1 + i)h2 -h2 -\ M + 1+2-n

2Aa17h3

4(1 + 2)h32 -2h3-| M +1+2-n

2Aa18h2

4(1 + 2)h2 - 2h2-|M +1+2-n

Aa19(h2 + h3)

76 (1 + 2)(h2 + h3)2 - (h2 + h3)-|( M + 1+2-n

2Aa20h1

4(1 + 2)h? -2h1 -| M + 1+2-n

2Aa21n

4(1 + 2)n2 -2n-\ M +1+2-n

b =_Aa22(hj +n)_

79 (1 + 2)(h1 +n)2 - (h + n) -|( M + 1+2-n

(1 + 2)h2 -h2 -\ M + 1+2-n

._-Griby.

(1 + 2)h32 - h3 -( M +

1 + 2 K 2

-Grib (

88 4(1 + i)h2 -2h2-(m + 1+2-n

Grlb19

(1 + i)(h2 + h3)2 - (h2 + h3)-| M + 1+2-n

_ Grlb20_

4(1 + j3)h? - 2h1 -( M + 1+2-n

b =_ Grlb21_

4(1 + ¿3)n2 - 2n - (M + ^ - n

~Gribx

(1+2)(h1 + n)2-(h1 + n)-| M+1+2-n

__ Grlb23_

(1 + 2)(h1 + h3)2 - (h1 + h3)-( M + 1+2-n

-Grlb2.

94 (1+2)(h3 + n)2 -(h3 + n)-(M+1+2-n

Aa23(h1 + h3)

(1 + 2)(h1 + h3)2-(h1 + h3)-| M + 1+2-n

—Grib?>

(1 + 2)(h1 + h2)2 -(h1 + h2)-\ M + 1+2-n

Aa24(h3 +n)

(1 + 2)(h3 + n)2 -(h3 + n)-I M + 1+2-n

-GrIb2l

(1+i)(h2+n)2 - (h+n)-I m+1+2-n

b =_Aa25(hj + h2)_

82 (1 + 2)(h1 + h2)2 - (h + h2) -|( M + 1+2-n

-Grlb2

(1 + 2)(h3 + h7)2 -(h3 + h7)-I M + 1+2-n

Aa26(h2 +n)

(1 + 2)(h +n)2-(h + n)M + 1+2-n

-GrIb2i

4(1 + 2)h3 -2h3-| M +1+2-n

(1 + 2)n2-n-|M + 1+2-n

-GrIb2'

(1 + i)(h2 + h3)2 - (h2 + h3)M + 1K-2-n

b =_ Grlb71_

85 (1 + 2)h2 -h5-(m + 1K-2-n

b =_ Grlb30_

100 (1 + 2)(h1 + h3)2 -(h1 + h3)-(m + 1+2-n

ARTICLE IN PRESS

D. Pal, S. Biswas/Engineering Science and Technology, an International Journal ■■ (2015) ■■-■■

b101 ="

—Grib3

(1+2)(h3 + n)2 - (h3 + n)-\ M+1+2-n

b116 =-

Grlb4l

(1 + 2)(h1 + h2)2 -(h + h2)-| M + 1+2-n

b102 =

~GrIb3'.

(1 + 2)(h3 + h5)2 -(h3 + h5)-I M + 1K2-n

-Gribf

(1 + 2)(h2 + h6)2 - (h2 + h6)-I M + 1+2-n

b =_ Grlb33_

103 (1 + 2)(h + h3)2 -(h2 + h3)-(m+1+2-n

b =_ Grlb48_

118 (1 + 2)(h2 + n)2 -(h2 + n)-(m + n

-Grib3,

(1+2)(h3 + h4)2 - (h3 + h4)-| M+1+2-n

b„9 =-

~GrIb4'.

(1 + 2)(h1 + h7)2 -(h1 + h7)-\ M + 1+2-n

b105 =-

-GrIb3

(1 + 2)(h1 + h3)2 -(h + h3)-| M+1+2-n

b120 = "

Grlb5i

(1 + 2)(h1 + h3)2 -(h1 + h3)-| M + 1+2-n

bog =_-Grib36_

106 (1 + 2)(h3 + h6)2 -(h3 + he)-(M +n

b =_ Grlb51_

121 (1 + 2)(h1 + h2)2 - (h + h2)-(M + 1+2-n

b107 =

-GrIb3

(1+2)(h3 + n)2 - (h3 + n)-| M+1+2-n

-Grib5:

4(1 + 2)h2 -2h1 -| M + 1+2-n

b108 =

-GrIb3i

(1 + 2)(h2 + h7)2 - (h2 + h,)M + 1K2-n

-GrIb5

(1 + 2)(h1 + n)2 -(h1 + n)-| M+1+2-n

b =_ Grlb39_

109 (1 + 2)(h2 + h3)2 -(h2 + h3)-|(M + M-n

b =_ Grlb54_

124 (1 + 2)(h1 + h5)2 -(h1 + h5)-|(M + 1+2-n

-Gribft

4(1 + 2)h2 - 2h2-|M + 1+2-n

-Grlb5'

(1 + 2)(h1 + h2)2 -(h1 + h2)-\ M + 1+2-n

(1 + 2)(h1 + h2)2 - (h + h2)M + 1+2-n

b126 =-

Grlb5f

(1 + 2)(h1 + h4)2 -(h1 + h4)-| M+1+2-n

b =_ Grlb42_

112 (1+2)(h2+n)2 -(h2+n)-(M+1+2-n

-Grib5

127 4(1 + 2)h2 - 2h1 -( M + 1+2- n

b113 =-

(1 + 2)(h + h5)2 - (h2 + h5) -1 M +1+2 -- n

Grlb5i

(1 + 2)(h1 + h2)2 -(h + h2)-| M+1+2-n

-Grib4.

4(1 + 2)h22 -2h2-| M + 1+2-n

b129 = "

Grlb5!

(1+2)(h1 + n)2 -(h1 + n)M+1+2-n

b =_ Grlb45_

115 (1 + 2)(h + h4)2 -(h2 + h4)-(m +1+2-n

b =_~GrIb60_

130 (1+2)(h7+n)2 -(h7+n)-(M+1+2-n

ARTICLE IN PRESS

-GrTbc

D. Pal, S. Biswas/Engineering Science and Technology, an International Journal ■■ (2015) ■■-■■

bi45 = -b4 +nibU3h7 +nibUil\ +nibU6 = k4

(1 + P)(h3 +n)2 - (h3 +n) - GrTb62 ( M +10-n] b146

(1 + P)(h2 +n)2 (h2 +n) - GrTb63 ( m+1KPP-")

(1 + P)(h + n)2 - (hi + n)-|M + jP-n

bi34 =

_-Grib64_

4(1 + P)n2 - 2n-[M + l+Â-n

__ GrTb6S_

(1 + P)(h5 + n)2 -(hs + n)M+n

b136 ="

-GrTba

(1 + P)(h2 +n)2 -(h2 +n)-|M + 1+P-n

(1 + P)(h4 + n)2 -(h4 + n)-I M + n

__-GrTb6l

(1 + P)(hi +n)2 - (hi +n)-| M +

1+P K 2

b139 ="

(1+P)(h6 + n)2 - (h6 + n)-I M+

1+P K 2

b140 =

_-GrTb70_

4(1 + P)n2 - 2n--1M + 1K+P - n

4(1 + P)hl - 2h6M + n

4(1 + P)hl -2h6M + i+A-n

din = ^4

4(1 +PW-n-| M + ^A-n

bl44 =

b72 - b74 - b7S - b76 - b77 - b78 - b79 - b80 - b81 - b82 - b83

- b84 - b85 - b86 - b87 - b88 - b89 - b90 - b91 - b92 - b93 - b94

- b95 - b96 - b97 - b98 - b99 - b100 - b101 - b102 - b103 - b104

- b10S - b106 - b107 - b108 - b109 - b110 - b111 - b112 - b113 - b114

- b11S - b116 - b117 - b118 - b119 - b120 - b121 - b122 - b123 - b124

- b12S - b126 - b127 - b128 - b129 - b130 - b131 - b132 - b133

- b134 - b135 - b136 - b137 - b138 - b139 - b140 - b141 - b142

-b3 + n (b1h + b5h3 + b6h2 + b7h1 +b8n + b9h5 + bwh2 + buh4 + bnh + buh6 + bun) = k3

+ b79(h1 +n) + b80(h1 + h3) + b81(h3 +n) + b82(h1 + h2) + b83 (h2 +n) + b84n + b8shs + b86h2 + 2b87h,3 + 2b88h2 + b89 h + h3) + 2b90h + 2b91n + b92 (h +n) + b93(h1 + h3) + b94 h + n) + b95(h + h2) + b96 (h2 +n) + b97(h3 + h7) + 2b98h3 + b99(h2 + h3) + b100 (h1 + h3) + b101 (h3 +n) + b102 (h3 + h5) + b103 (h2 + h3) + b^h + h4) + bW5(h1 + h3) + b106(h3 + he) + b107(h3 +n) + bmh + h7) + b109(h2 + h3) + 2bU0h2 + bm(h1 + h2) + bU2h +n) + bmh + h5) + 2bU4h2 + bU5(h2 + h4) + bU6(h1 + h2) + bmh + h6) + b^fa + h3) + 2bU0h2 + bm(h1 + h2) + bU2(h2 +n) + bmh + h5) + 2bU4h2 + bn5(h2 + h4) + bu6(h + h2) + bn7(h2 + hi) + bm(h2 +n) + bU9(h + h7) + b120(h1 + h3) + bmh + h2) + 2bU2h + bU3(h +n) + b124(h1 + h5) + bush + h2) + bmh + h4) + 2b127h1 + b128(h1 + h6) + b129 (h +n) + b130 (h7 +n) + b131 (h3 + n)

+ b132 (h2 +n) + b133 (h + n) + 2b134n + b13s(hs + n) + b136 (h2 +n + b137 (h4 +n + b139(h6 + n) + 2b140n + b142n)

d1 = b73 + b86

d2 = b74 + b87 + b98

d3 = b7S + b88 + bU0 + bU4 d4 = b77 + b90 + b122 + b127 dS = b78 + b91 + b134 + b140

d6 = b84 + b142

d7 = b76 + b89 + b99 + b103 + b109

d8 = b79 + b92 + b123 + b129 + bU3 + bU8

d9 = b80 + b93 + bm0 + bms + b120

d10 = b82 + b9s + bm + bU6 + b121 + b12S

d11 = b83 + b96 + bU2 + bn8 + b132 + b136

d12 = b81 + b94 + bW1 + bm7 + b131

d13 = b18 + b40 + b 44

d14 = b20 + bs2 + bs7

d1s = b17 + b28

du = b21 + b64 + b70

d17 = b22 + bs3 + bs9 + b63 + b68

d18 = b26 + b42 + b 48 + b62 + bœ

d19 = b24 + b31 + b37 + bm

d20 = b19 + b29 + b33 + b39

d21 = b23 + b30 + b3s + bs0

d22 = b2S + b41 + b 46 + bs1 + bss

e1 = b72< e2 = b8S< e3 = b97< e4 = b102< eS = b104> e6 = b106> e7 = b108t

ARTICLE IN PRESS

D. Pal, S. Biswas/Engineering Science and Technology, an International Journal ■■ (2015) I

e8 = b113. e9 = b115> e10 = b117> e11 = b119> e12 = b124. e13 = b126. e14 = b128» e15 = b130. e16 = b135» e17 = b137. e18 = b139. e19 = b141. e20 = b143> e21 = b144» e22 = b145

Nomenclature

Grc j*

Tw t* t

Vo v* v

small real positive constant

externally imposed transverse magnetic field strength, gaussmeter

concentration of the fluid, mol m-3

free stream concentration, mol m-3

frictional coefficient

concentration at the wall, mol m-3

specific heat at constant pressure, Jkg1 K-

molecular diffusivity, m2 s-1

Eckert number

local Grashof number

acceleration due to gravity, ms-1

thermal Grashof number

mass Grashof number

microinertia per unit mass, m2

dimensionless microinertia per unit mass

permeability parameter, m2

mass absorption coefficient, m-1 dimensionless permeability parameter magnetic field parameter frequency parameter, hertz

parameter related to micro-gyration vector and share stress local Nusselt number Prandtl number

thermal radiative heat flux, Wm~2 local Reynolds number radiation parameter Schmidt number Sherwood number temperature of the fluid, K free stream temperature, K temperature at the wall, K dimensional time, s dimensionless time

velocity component in x-direction, ms-1 dimensionless velocity component in x-direction free stream velocity, ms-1

uniform velocity of the fluid in its own plane, ms'1 dimensionless velocity of the plate scale of suction velocity at the plate, ms-1 velocity component in y-direction, ms-1 dimensionless velocity component in y-direction distance along and perpendicular to the plate, respectively, m

Greek symbols

a thermal diffusivity, m2 s-1

P dimensionless viscosity ratio

Pc coefficient of mass expansion of the fluid, K-1

pT coefficient of thermal expansion of the fluid, K-1

Y spin gradient viscosity, kg m s-1

Y* chemical reaction parameter

Y1 dimensionless chemical reaction parameter

£ small positive quantity

9 dimensionless temperature of the fluid n similarity variable / coefficient of viscosity v kinematic viscosity, m2 s-1 vr kinematic rotational viscosity, m2 s-1 p density of the fluid, kg m~3 a electrical conductivity of the fluid, S mo* StephanBoltzmann constant, W m-2 K-4 a* component of angular velocity, m2 s-2 a dimensionless component of angular velocity A coefficient of vortex viscosity, kg m-1 s-1 k thermal conductivity of the fluid, W mK-1 X heat generation/absorption parameter

Subscripts

differentiation with respect to y

References

[5 [6 [7 [8 [9

[10 [11 [12

[14 [15

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