Combined effects of Joule heating and chemical reaction on unsteady magnetohydrodynamic mixed convection of a viscous dissipating fluid over a vertical plate in porous media with thermal radiation

Combined effects of Joule heating and chemical reaction on unsteady magnetohydrodynamic mixed convection of a viscous dissipating fluid over a vertical plate in porous media with thermal radiation

Mathematical and Computer Modelling 54 (2011) 3016–3036 Contents lists available at SciVerse ScienceDirect Mathematical and Computer Modelling journ...

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Mathematical and Computer Modelling 54 (2011) 3016–3036

Contents lists available at SciVerse ScienceDirect

Mathematical and Computer Modelling journal homepage: www.elsevier.com/locate/mcm

Combined effects of Joule heating and chemical reaction on unsteady magnetohydrodynamic mixed convection of a viscous dissipating fluid over a vertical plate in porous media with thermal radiation Dulal Pal a,∗ , Babulal Talukdar b a

Department of Mathematics, Visva-Bharati University, Santiniketan, West Bengal-731 235, India

b

Department of Mathematics, Gobindapur High School, Murshidabad-742 213, West Bengal, India

article

info

Article history: Received 28 February 2011 Received in revised form 24 July 2011 Accepted 25 July 2011 Keywords: Porous medium Mass transfer Magnetohydrodynamics Heat transfer Thermal radiation Chemical reaction

abstract This paper deals with the interaction of convection and thermal radiation on an unsteady hydromagnetic heat and mass transfer for a viscous fluid past a semi-infinite vertical moving plate embedded in a porous media in the presence of heat absorption and firstorder chemical reaction of the species. The fluid is considered to be a gray, absorbing– emitting but non-scattering medium, and the Cogley–Vincent–Gilles formulation is adopted to simulate the radiation component of heat transfer (Cogley et al. (1968) [25]). The plate moves with a constant velocity in the direction of fluid flow while the free stream velocity is assumed to increase exponentially. A uniform transverse magnetic field is applied to the porous surface which absorbs the fluid with suction velocity varying with time. Analytical perturbation solutions are obtained for the velocity, temperature and concentration fields as well as for the skin friction coefficient, Nusselt number and Sherwood number. Results are presented graphically and in tabulated forms to study the effects of various physical parameters. The computed results are in good agreement with the earlier published results. © 2011 Elsevier Ltd. All rights reserved.

1. Introduction The problem of boundary layer flow over a continuously moving solid surface is an important type of flow occurring in many industrial processes, such as heat-treated materials traveling between a feed roll and a wind-up roll or materials manufactured by extrusion, glass fiber and paper production. In these cases, the final product of desired characteristics depends greatly on the rate of cooling of the stretched materials. Convection of a heated or cooled vertical plate is one of the fundamental problems in heat and mass transfer studies in recent times. If the existing free convection is accompanied by an external flow, the combined mode of free and forced convection exists, which is commonly known as mixed convection. At the same time the study of magnetohydrodynamics (MHD) is important in many engineering applications such as, in MHD power generators, cooling of nuclear reactors, the boundary layer control in aerodynamics and crystal growth. Transport processes in porous media play a significant role in various applications, such as thermal insulation, energy conservation, petroleum industries, solid matrix heat exchangers, geothermal engineering, chemical catalytic reactors, and underground disposal of nuclear waste materials. In many transport processes in nature and in industrial applications, the heat and mass transfer with variable viscosity is a consequence of buoyancy effects caused by the diffusion of heat and chemical species. The study of such processes is useful for improving a number of chemical technologies, such as polymer production and food processing. In nature, the presence of pure air or water is impossible. Some foreign mass may be presented either naturally



Corresponding author. Tel.: +91 3463 261029; fax: +91 3463 261029. E-mail address: [email protected] (D. Pal).

0895-7177/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2011.07.030

D. Pal, B. Talukdar / Mathematical and Computer Modelling 54 (2011) 3016–3036

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or mixed with air or water. A large amount of research work has been reported in this field. Particularly, the study of heat and mass transfer with chemical reactions is of considerable importance in the chemical and hydrometallurgical industries. Heat and mass transfer on continuously moving or a stretching sheet has many practical applications in electrochemistry and polymer processing [1–3], where the object, after passing through a die, enters the fluid for cooling below a certain temperature. The mixed convection flow over a vertical plate embedded in a porous medium contains a species slightly soluble in the fluid. The presence of a foreign mass in fluid causes some kind of chemical reactions. In most of the chemical reaction cases, the reaction rate depends on the concentration of the species itself. A chemical reaction is said to be firstorder it the reaction rate is directly proportional to the concentration itself. In this paper, first-order chemical reaction is considered, which takes place in the fluid. The moving plate is not soluble in the fluid. The concentration of the species at the plate surface is higher than the solubility of species in the fluid far away from the plate (i.e. free stream concentration). The study of combined heat and mass transfer problems with chemical reaction are of important in many processes such as drying, evaporation at the surface of water body, energy transfer in a wet cooling tower, solidification of binary alloy, dispersion of dissolved materials, drying and dehydration operations in chemical food processing plants, and combustion of atomized liquid fuels. Since the pioneering work of Sakiadis [4], various aspects of the problem have been investigated by many authors. Danberg and Fansler [5] obtained the solution of the boundary layer flow past a wall that is stretched with a speed proportional to the distance along the wall. Kim [6] looked at the unsteady MHD convective heat transfer past a semiinfinite vertical porous moving plate with variable suction. EL-Kabeir et al. [7] applied the group transformation method for solving the combined convection problem to an unsteady two-dimensional laminar boundary-layer flow of a viscous, incompressible and electrically-conducting fluid over a vertical continuous moving plate embedded in a fluid-saturated porous medium in the presence of a uniform transverse magnetic field. Hassanien and Obied [8] looked at the oscillatory two-dimensional free convection and mass transfer flow of a viscous incompressible and electrically conducting fluid through a porous medium with variable permeability in the presence of transverse magnetic field. Siddiqa et al. [9] presented an investigation which deals with the study of laminar natural convection flow of a viscous fluid over a semi-infinite flat plate inclined at a small angle to the horizontal plane in the presence of internal heat generation and variable viscosity. Saeid [10] tackled the problem of mixed convection flow along a vertical plate subjected to time-periodic surface temperature oscillations. Yildirm and Sezer [11] analyzed the effects of partial slip on the peristaltic flow of a MHD Newtonian fluid in an asymmetric channel using homotopy perturbation method. In all the previous investigations, the effects of radiation on the flow and heat transfer have not been studied. It is well known that radiative heat transfer flow is very important in manufacturing industries for the design of reliable equipments, nuclear plants, gas turbines and various propulsion devices for aircraft, missiles, satellites and space vehicles. Also, the effects of thermal radiation on forced and free convection flow are important in the content of space technology and process involving high temperature. Plumb et al. [12] was the first to examine the effect of horizontal cross-flow and radiation on natural convection from vertical heated surface in a saturated porous media. Keeping in mind some specific industrial applications such as a polymer processing technology, numerous attempts have been made to analyze the effect of transverse magnetic field on a boundary layer flow characteristics. Mansour and El-Shaer [13] analyzed the effects of thermal radiation on magnetohydrodynamic natural convection flows in a fluid-saturated porous media. Pal [14] studied heat and mass transfer in stagnation-point flow toward a stretching sheet in the presence of buoyancy force and thermal radiation. Vajravelu and Rollins [15] studied heat transfer in electrically conducting fluid over a stretching sheet by taking into account of magnetic field only. Molla et al. [16] studied the effect of thermal radiation on a steady two-dimensional natural convection laminar flow of viscous incompressible optically thick fluid along a vertical flat plate with streamwise sinusoidal surface temperature. Abo-Eldahab and El-Gendy [17] investigated the problem of free convection heat transfer characteristics in an electrically conducting fluid near an isothermal sheet to study the combined effect of buoyancy and radiation in the presence of uniform transverse magnetic field. In all the above studies, the effects of both the viscous and Joule heating were neglected since they were of the same order as well as negligibly small [18]. The effect which bears a great importance on heat transfer is viscous dissipation. When the viscosity of the fluid is high, the dissipation term becomes important. For many cases, such as polymer processing which is operated at a very high temperature, viscous dissipation cannot be neglected. El-Amin [19] studied the combined effect of viscous dissipation and Joule heating on MHD forced convection flow over a non-isothermal horizontal cylinder embedded in a fluid saturated porous medium. Recently, Abo-Eldahab and El-Aziz [20] studied the effect of Ohmic heating on mixed convection boundary layer flow of a micropolar fluid from a rotating cone with power-law variation in surface temperature. Abel et al. [21] investigated on momentum and heat transfer characteristics in an incompressible electrically conducting viscoelastic boundary layer flow over a linear stretching sheet in the presence of viscous and Ohmic heating. Pal and Mondal [22] examined the effect of thermal radiation on MHD non-Darcy flow and heat transfer over a stretching sheet in presence of Ohmic dissipation. Aydin and Kaya [23] numerically analyzed the problem of steady laminar MHD mixed convection heat transfer about a vertical plate by taking into account of Ohmic heating and viscous dissipation effects. Pal and Talukdar [24] presented perturbation analysis of unsteady MHD mixed convective heat and mass transfer in a boundary layer flow with thermal radiation and chemical reaction effects. The purpose of the present paper is to study the effect of thermal radiation on unsteady magnetohydrodynamic flow past an infinite moving vertical porous plate in the presence of first-order chemical reaction, viscous and Ohmic heating. The flow in the porous medium deals with the analysis in which the differential equations governing the motion is based on

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D. Pal, B. Talukdar / Mathematical and Computer Modelling 54 (2011) 3016–3036

Fig. 1. Physical configuration of the problem.

Darcy’s law, which accounts for the drag exerted by the porous medium. The classical model introduced by Cogley et al. [25] is used for the radiation effects as it has the merit of simplicity and enables us to introduce linear term in temperature in the analysis for optically thin media. Section 2 involves with the governing equations in non-dimensional form. In Section 3, the solution methodology of the basic equations derived in Section 2 are presented. Section 4 is devoted to the quantitative discussion of the results obtained. The equations governing the flow, heat and mass transfer are solved analytically using perturbation technique. The present results are compared with those available in the literature and the numerical solutions of the present problem using Runge–Kutta Fehlberg method with shooting technique. 2. Formulation of the problem Consider unsteady MHD mixed convection flow from a permeable semi-infinite vertical plate embedded in a porous medium in the presence of thermal radiation, viscous and Joule heating effects. Time dependent suction velocity is imposed on the plate surface. A uniform magnetic field is applied in the transverse direction in the presence of radiative heat transfer. The x∗ -axis is taken along the vertical infinite plate, which is the direction of the flow and y∗ -axis is taken normal to the plate (see Fig. 1). We assume that no electric field is present and induced magnetic fields are negligible. The porous medium is assumed to be uniform, isotropic and in thermal equilibrium with the plate. The plate is having variable temperature and concentration which varies with time. All fluid properties are assumed to be constant except the density in the body force term of the linear momentum balance. Under the Boussinesq and boundary layer approximations the governing equations for this problem can be written as

∂v ∗ =0 ∂ y∗

(1)

 σ B20 ν + ∗ u∗ + g βT (T ∗ − T∞ ) + g βC (C ∗ − C∞ ) ρ K    2 ∗ ∂ 2T ∗ ν ∂ u∗ 1 ∂ q∗r ∂T ∗ Q0 σ B20 ν ∗ ∂T ∗2 ∗ + v = α − ( T − T ) + + u + − ∞ ∂t∗ ∂ y∗ ρ cp ρ cp cp K ∗ cp ∂ y∗ ρ cp ∂ y∗ ∂ y∗ 2 ∂ u∗ ∂ u∗ 1 ∂ p∗ ∂ 2 u∗ + v∗ ∗ = − + ν ∗2 − ∗ ∗ ∂t ∂y ρ ∂x ∂y



∗ ∂C∗ ∂ 2C ∗ ∗ ∂C − R(C ∗ − C∞ ) + v = D ∂t∗ ∂ y∗ ∂ y∗ 2

(2)

(3)

(4)

where, x∗ , y∗ are the dimensional distances along and perpendicular to the plate, respectively. u∗ and v ∗ are the components of dimensional velocities along x∗ and y∗ directions, respectively. g is the gravitational acceleration, T ∗ is the dimensional temperature of the fluid near the plate, T∞ is the free stream dimensional temperature, K ∗ is the permeability of the porous medium, C ∗ is the dimensional concentration, C∞ is the free stream dimensional concentration. βT and βC are the thermal and concentration expansion coefficients, respectively. α is the fluid thermal diffusivity, p∗ is the pressure, cp is the specific heat of constant pressure, B0 is the magnetic field coefficient, µ is viscosity of the fluid, q∗r is the radiative heat flux, ρ is the density, σ is the magnetic permeability of the fluid, ν(=µ/ρ) is the kinematic viscosity, D is the molecular diffusivity, Q0 is the dimensional heat absorption coefficient and R is the chemical reaction parameter. The fourth and fifth terms on RHS of the momentum Eq. (2) denote the thermal and concentration buoyancy effects, respectively. Also, third and fourth terms on the RHS of energy Eq. (3) represents the viscous and Joule heating effects, respectively. The second and fifth term on the RHS of Eq. (3) denote the inclusion of the effect of heat absorption and thermal radiation effects, respectively.

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The radiative heat flux is given by [25] as

∂ q∗r = 4(T ∗ − T∞ )I ′ ∂ y∗ ∞ ∂e where I ′ = 0 Kλw ∂ Tb∗λ dλ, Kλw is the absorption coefficient at the wall and ebλ is Planck’s function.

(5)

The advantages and limitations of the Cogley–Vincent–Gilles [25] formulation, which is to used to simulate the radiation component of heat transfer, are (i) it does not require an extra transport equation for the incident radiation, and (ii) it can only be used for an optically thin, near-equilibrium and non-gray gas. Cogley model is well suited for (i) surface-to-surface radiant heating or cooling, (ii) Coupled radiation, convection, and/or conduction heat transfer and (iii) Radiation in glass processing, glass fiber drawing, and ceramic processing. Under these assumptions, the appropriate boundary conditions for the velocity, temperature and concentration fields are ∗t∗

u∗ = u∗p ,

T ∗ = Tw + ϵ(Tw − T∞ )en ∗t∗

∗ u∗ → U∞ = U0 (1 + ϵ en

),

,

C ∗ = Cw + ϵ(Cw − C∞ )en

T ∗ → T∞ ,

C ∗ → C∞

∗t∗

at y∗ = 0

(6)

as y∗ → ∞

(7)

where up , Tw and Cw are the wall dimensional velocity, concentration and temperature, respectively. is the free stream dimensional velocity. U0 and n∗ are constants. It is clear from Eq. (1) that the suction velocity at the plate surface is a function of time only. Assuming that it takes the following exponential form: ∗

∗ U∞

∗t∗

v ∗ = −V0 (1 + ϵ Aen

)

(8)

where A is a real positive constant, ϵ and ϵ A are small quantities less than unity, and V0 is a scale of suction velocity which has non-zero positive constant. Outside the boundary layer, Eq. (2) gives



1 dp∗

ρ dx∗

=

∗ dU∞

dt

 +

 σ B20 ν ∗ + ∗ U∞ . ρ K

(9)

Following dimensionless variables are used: u=

u∗ U0

,

v=

K ∗ V02

K =

ν

2

n ν

,

v∗ V0

t =



n= M =

Gr =

V02

,

Ec =

,

η=

t ∗ V0 2

ν

ν θ=

U02 cp (Tw − T∞ )

Rν σ B20 ν , γ = 2, ρ V02 V0 ν g βT (Tw − T∞ ) U0 V02

,

V0 y ∗

,

U∞ = T ∗ − T∞ , Tw − T∞

,

Sc =

ν

∗ U∞

U0

,

C =

Pr =

νρ cp ν = , κ α

,

F =

4ν I ′

ρ cp V02 ν g βC (Cw − C∞ ) and Gc = . 2 D

up =

u∗p U0

,

C ∗ − C∞ , Cw − C∞

φ=

Q0 ν , ρ cp V02

(10)

,

U0 V0

Using Eqs. (5), (8)–(10), Eqs. (2)–(4) reduce to the following dimensionless form:

∂u ∂u dU∞ ∂ 2u − (1 + ϵ Aent ) = + N (U∞ − u) + 2 + Gr θ + Gc C ∂t ∂η dt ∂η  2 ∂θ 1 ∂ 2θ ∂u nt ∂θ 2 − (1 + ϵ Ae ) = − φθ + NEc u + Ec − Fθ ∂t ∂η Pr ∂η2 ∂η

(11)

(12)

∂C ∂C 1 ∂ 2C − (1 + ϵ Aent ) = − γC (13) ∂t ∂η Sc ∂η2   where N = M + K1 and Gr is the thermal Grashof number, Gc is the solutal Grashof number, Pr is the Prandtl number, M is the magnetic field parameter, F is the radiation parameter, Sc is the Schmidt number, φ is the heat source parameter, and γ is the chemical reaction parameter and Ec is the Eckert number and K is a permeability parameter of the porous medium, as defined in (10).

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The corresponding boundary conditions (6) and (7) become u = up ,

θ = 1 + ϵ ent ,

u → U∞ = 1 + ϵ ent ,

C = 1 + ϵ ent

θ → 0,

at η = 0

(14)

as η → ∞.

C →0

(15)

3. Method of solution In order to reduce the above system of partial differential equations to a system of ordinary differential equations in dimensionless form, we represent the velocity, temperature and concentration as u = u0 (η) + ϵ ent u1 (η) + O(ϵ 2 ),

(16)

θ = θ0 (η) + ϵ e θ1 (η) + O(ϵ ),

(17)

C = C0 (η) + ϵ e C1 (η) + O(ϵ ).

(18)

nt

2

nt

2

Substituting these Eqs. (16)–(18) into Eqs. (11)–(13) and equating the harmonic and non-harmonic terms, and neglecting the higher order terms of O(ϵ 2 ), we get the following pairs of equations for (u0 , θ0 , C0 ) and (u1 , θ1 , C1 ). u′′0 + u′0 − Nu0 = −N − Gr θ0 − Gc C0

(19)

u1 + u1 − (N + n)u1 = −(N + n) − Au0 − Gr θ1 − Gc C1 ′′





θ0 ′′ + Prθ0 ′ − Pr(F + φ)θ0 = −PrNEc u0 2 − PrEc u′0

(20)

2

(21)

θ1 + Prθ1 − Pr(F + φ + n)θ1 = −APrθ0 − 2PrNEc u0 u1 − 2PrEc u0 u1

(22)

C0 + ScC0 − Scγ C0 = 0

(23)

′′



′′









C1 + ScC1 − Sc(γ + n)C1 = −AScC0 ′′





(24)

and the corresponding boundary conditions are u0 = up ,

u1 = 0,

u0 = 1,

u1 = 1,

θ0 = 1,

θ1 = 1,

C0 = 1,

θ0 → 0,

θ1 → 0,

C0 → 0,

C1 = 1

at η = 0

C1 → 0

as η → ∞.

(25) (26)

Eqs. (19)–(24) are still coupled, but a great deal of insight into the behavior of the flow variables can be obtained if we seek an asymptotic series solution about the Eckert number Ec , which for most incompressible flows is small. We therefore expand our flow variables as u0 (η) = u01 (η) + Ec u02 + O(Ec2 ) u1 (η) = u11 (η) + Ec u12 + O(Ec2 )

θ0 (η) = θ01 (η) + Ec θ02 + O(Ec2 ) θ1 (η) = θ11 (η) + Ec θ12 + O(Ec2 )

(27)

C0 (η) = C01 (η) + Ec C02 + O(

)

C1 (η) = C11 (η) + Ec C12 + O(

).

Ec2 Ec2

Substitution of Eq. (27) into Eqs. (19)–(24) and equating to zero the coefficient of different powers of Ec and neglecting higher order terms in Ec , we obtained the following sequence of approximations u′′01 + u′01 − Nu01 = −N − Gr θ01 − Gc C01

(28)

u02 + u02 − Nu02 = −Gr θ02 − Gc C02 ′′



(29)

u11 + u11 − (N + n)u11 = −(N + n) − Au01 − Gr θ11 − Gc C11

(30)

u12 + u12 − (N + n)u12 = −Au02 − Gr θ12 − Gc C12

(31)

θ01 + Prθ01 − Pr(F + φ)θ01 = 0

(32)

′′ ′′

′′











θ02 + Prθ02 − Pr(F + φ)θ02 = − ′′



PrNu201

′2

− Pru01

(33)

θ11 + Prθ11 − Pr(F + φ + n)θ11 = −APrθ01 ′′





(34)

θ12 + Prθ12 − Pr(F + φ + n)θ12 = −APrθ02 − 2PrNu01 u11 − 2Pru01 u11

(35)

C01 + ScC01 − Scγ C01 = 0

(36)

C02 + ScC02 − Scγ C02 = 0

(37)

′′

′′ ′′



′ ′







D. Pal, B. Talukdar / Mathematical and Computer Modelling 54 (2011) 3016–3036

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′′ ′ ′ C11 + ScC11 − Sc(γ + n)C11 = −AScC01

(38)

C12 + ScC12 − Sc(γ + n)C12 = −AScC02 .

(39)

′′





The corresponding boundary conditions are u01 = up ,

θ12 = 0, u01 → 1, θ12 → 0,

u02 = 0, C01 = 1,

u11 = 0, C02 = 0,

u12 = 0,

θ01 = 1,

C11 = 1,

C12 = 0

θ02 = 0,

θ11 = 1,

at η = 0

u02 → 0,

u11 → 1,

u12 → 0,

θ01 → 0,

C01 → 0,

C02 → 0,

C11 → 0,

C12 → 0

θ02 → 0,

(40)

θ11 → 0,

as η → ∞.

(41)

We can now write the solution for the mean concentration C0 as C 0 = e− A 1 η

(42)

and for the mean temperature θ0 by neglecting higher order terms in Ec as

 θ0 = e−A2 η + Ec R12 e−A2 η − f2 e−A3 η − f3 e−2A3 η + f4 e−2A2 η  + f5 e−d1 η + f6 e−2A1 η + f8 e−A1 η + f9 e−d2 η + f10 e−d3 η

(43)

and the mean velocity u0 as u0 = 1 + B3 e−A3 η − B1 e−A2 η − B2 e−A1 η + Ec −R2 e−A2 η + R11 e−A3 η + R4 e−2A3 η



 − R5 e−2A2 η − R6 e−d1 η − R7 e−2A1 η − R8 e−A1 η − R9 e−d2 η − R10 e−d3 η .

(44)

For the transient state concentration C1 , we have C1 = A7 e−A6 η + R13 e−A1 η .

(45)

For the transient state temperature θ1 and transient state velocity u1 , by neglecting higher order terms in Ec , we have

 θ1 = A9 e−A8 η + R14 e−A2 η + Ec t27 e−A8 η + t1 e−A2 η − t2 e−A3 η − t3 e−2A3 η + t4 e−2A2 η + t5 e−d1 η + t6 e−2A1 η + t7 e−A1 η + t8 e−d2 η + t9 e−d3 η − t11 e−A10 η + t13 e−A6 η − t14 e−A12 η + t15 e−A13 η + t16 e−A14 η + t17 e−A15 η  + t18 e−A16 η + t19 e−A17 η + t20 e−A18 η − t21 e−A19 η − t22 e−A20 η + t23 e−A21 η − t24 e−A22 η − t25 e−A23 η  u1 = 1 + A11 e−A10 η + R15 e−A3 η + R16 e−A2 η + R17 e−A1 η − R18 e−A8 η − R19 e−A6 η + Ec W27 e−A10 η + W1 e−A3 η − W2 e−A2 η + W3 e−2A3 η − W4 e−2A2 η − W5 e−d1 η − W6 e−2A1 η − W7 e−A1 η − W8 e−d2 η − W9 e−d3 η − W10 e−A8 η − W13 e−A6 η + W14 e−A12 η − W15 e−A13 η − W16 e−A14 η − W17 e−A15 η − W18 e−A16 η  − W19 e−A17 η − W20 e−A18 η + W21 e−A19 η + W22 e−A20 η − W23 e−A21 η + W24 e−A22 η + W25 e−A23 η .

(46)

(47)

Using the above solutions (42)–(47) the velocity, temperature and concentration distributions in the boundary layer are obtained from (16)–(18) as follows: u(η, t ) = 1 + B3 e−A3 η − B1 e−A2 η − B2 e−A1 η + Ec −R2 e−A2 η + R11 e−A3 η + R4 e−2A3 η



 − R5 e−2A2 η − R6 e−d1 η − R7 e−2A1 η − R8 e−A1 η − R9 e−d2 η − R10 e−d3 η  + ϵ ent 1 + A11 e−A10 η + R15 e−A3 η + R16 e−A2 η + R17 e−A1 η − R18 e−A8 η  − R19 e−A6 η + Ec W27 e−A10 η + W1 e−A3 η − W2 e−A2 η + W3 e−2A3 η − W4 e−2A2 η − W5 e−d1 η − W6 e−2A1 η − W7 e−A1 η − W8 e−d2 η − W9 e−d3 η − W10 e−A8 η − W13 e−A6 η + W14 e−A12 η − W15 e−A13 η − W16 e−A14 η − W17 e−A15 η − W18 e−A16 η − W19 e−A17 η − W20 e−A18 η  + W21 e−A19 η + W22 e−A20 η − W23 e−A21 η + W24 e−A22 η + W25 e−A23 η  θ (η, t ) = e−A2 η + Ec R12 e−A2 η − f2 e−A3 η − f3 e−2A3 η + f4 e−2A2 η + f5 e−d1 η + f6 e−2A1 η + f8 e−A1 η    + f9 e−d2 η + f10 e−d3 η + ϵ ent A9 e−A8 η + R14 e−A2 η + Ec t27 e−A8 η + t1 e−A2 η − t2 e−A3 η − t3 e−2A3 η + t4 e−2A2 η + t5 e−d1 η + t6 e−2A1 η + t7 e−A1 η + t8 e−d2 η + t9 e−d3 η − t11 e−A10 η + t13 e−A6 η − t14 e−A12 η + t15 e−A13 η + t16 e−A14 η + t17 e−A15 η + t18 e−A16 η + t19 e−A17 η  + t20 e−A18 η − t21 e−A19 η − t22 e−A20 η + t23 e−A21 η − t24 e−A22 η − t25 e−A23 η   C (η, t ) = e−A1 η + ϵ ent A7 e−A6 η + R13 e−A1 η .

(48)

(49) (50)

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D. Pal, B. Talukdar / Mathematical and Computer Modelling 54 (2011) 3016–3036

The physical quantities of interest are the skin-friction coefficient, the local Nusselt number and the local Sherwood number which are defined as

 ∂ u  ρ U 0 V0 ∂η η=0  = −B3 A3 + B1 A2 + B2 A1 + Ec R2 A2 − R11 A3 − 2R4 A3 + 2R5 A2 + R6 d1 + 2R7 A1 + R8 A1 + d2 R9    + d3 R10 + ϵ ent −A10 A11 − A3 R15 − A2 R16 − A1 R17 + A8 R18 + A6 R19 + Ec −A10 W27 − A3 W1 + A2 W2 − 2A3 W3 + 2A2 W4 + d1 W5 + 2A1 W6 + A1 W7 + d2 W8 + d3 W9 + A8 W10 + A6 W13 − A12 W14 + A13 W15 + A14 W16 + A15 W17 + A16 W18 + A17 W19 + A18 W20 − A19 W21 − A20 W22 + A21 W23  − A22 W24 − A23 W25  ∂T   ∂ y∗ y∗ =0 ∂θ  1 ⇒ Nux Re− = Nux = x x Tw − T∞ ∂η η=0   = −A2 + Ec −A2 R12 + A3 f2 + 2A3 f3 − 2A2 f4 − d1 f5 − 2A1 f6 − A1 f8 − d2 f9 − d3 f10   + ϵ ent −A8 A9 − A2 R14 + Ec −A8 t27 − A2 t1 + A3 t2 + 2A3 t3 − 2A2 t4 − d1 t5 − 2A1 t6 − A1 t7 − d2 t8 − d3 t9 − A10 t11 − A6 t13 − A12 t14 − A13 t15 − A14 t16 − A15 t17 − A16 t18 − A17 t19  − A18 t20 + A19 t21 + A20 t22 − A21 t23 + A22 t24 + A23 t25  ∂C   ∂ y∗ y∗ =0 ∂ C  −1 Shx = x ⇒ Shx Rex = Cw − C∞ ∂η η=0 nt = −A1 + ϵ e (−A6 A7 − A1 R13 ) Cf x =

τw

=

(51)

(52)

(53)

where Rex = ν0 is the local Reynolds number. The constants Ai , Bi , di , fi , Ri , ti , Wi appearing in Eqs. (42)–(53) are recorded in the Appendix. V x

4. Results and discussions The non-linear coupled Eqs. (16)–(24) subject to boundary conditions (25)–(26), which describe heat and mass transfer flow past an infinite vertical plate immersed in a porous medium in the presence of thermal radiation, viscous and Joule heating under the influence of magnetic field are solved analytically by perturbation technique. In order to get physical insight into the problem, the effects of various parameters encountered in the equations of the problem are analyzed on velocity, temperature and concentration fields with the help of figures. These results show the influence of the various physical parameters such as thermal Grashof number Gr, solutal Grashof number Gc , magnetic field parameter M, Schmidt number Sc, permeability parameter K , heat absorption parameter φ , chemical reaction parameter γ and thermal radiation parameter F on the velocity, temperature and the concentration profiles. We have also analyzed the effects of various physical parameters such as magnetic field, thermal radiation, viscous and Joule heating on skin friction coefficient, local Nusselt number and local Sherwood number. We can extract interesting insights regarding the influence of all the parameters that govern this problem. The velocity profiles are plotted in Fig. 2 for various values of magnetic field parameter M and time t. It is obvious that the existence of the magnetic field is to decrease the velocity in the momentum boundary layer because the application of the transverse magnetic field results in a resisting type of force called Lorentz force, similar to drag force, that resists the fluid flow which results in reducing the velocity of the fluid in the boundary layer. Further, it is observed from this figure that as the time increases the velocity profiles also increases. The influence of the heat absorption parameter φ on the velocity when t = 2.0 is illustrated in Fig. 3. Physically, the presence of heat absorption (thermal sink) effect has the tendency to reduce the fluid velocity across the momentum boundary layer. This causes the thermal buoyancy effects to decrease which results in a net reduction in the fluid velocity. These behaviors are clearly seen close to the plate. Fig. 4 shows the velocity profiles across the boundary layer for various values of the chemical reaction parameter γ for t = 0.5. The results display that an increase in the value of γ results in decrease in the velocity profiles. Similar facts are seen in the case of velocity profiles when Schmidt number is increased for t = 1.0 as noted in Fig. 5 which shows that the effect of increasing the value of Schmidt number Sc results in decreasing the velocity distribution slowly for higher values of Sc. The effects of radiation parameter F on velocity profiles for t = 0.5 are presented in Fig. 6. From this figure we observe that, as the value of F increases, the velocity profiles decreases in the momentum boundary layer due to the fact that the momentum boundary layer thickness decreases with increase in the radiation parameter F . Fig. 7 is a plot to indicate the effect of exponential index n on velocity profile for t = 5.0. It is observed that increase in the value of n increases velocity profiles in the momentum boundary layer for a particular value of physical parameters. It is further observed that the peak velocity profile diminishes for higher value of n. Fig. 8 shows that an increase in the Prandtl number has the effect of reducing the velocity in the boundary layer due to decrease in boundary layer thickness for t = 1.0. The velocity profiles for different values of thermal Grashof number Gr and solutal Grashof number Gc are displayed in Figs. 9 and 10, respectively against

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Fig. 2. Effect of M and t on velocity profiles against spanwise coordinate η for Ec = 0.01.

Fig. 3. Effect of φ on velocity profiles against spanwise coordinate η for n = 0.1 and Ec = 0.01.

the span-wise coordinate η for t = 1.0. It is observed that an increase in the values of Gr and Gc lead to rise the values of the velocity profiles. From these figures it is observed that the peak values of velocity increases rapidly near the wall of the porous plate as Grashof number is increased which ultimately decays to the relevant free stream velocity (i.e. matching the velocity condition as η → ∞). Physically, Gr > 0 means heating of the fluid by cooling the boundary surface. Fig. 11 illustrates the variation of velocity distribution across the boundary layer for several values of plate velocity in the direction of fluid flow. Although, we have different initial plate velocities but the velocity decays to the constant value as η → ∞ for given material parameters. Further, it is observed that due to increase in the plate velocity there is increase in the velocity profiles near the porous plate and its effects diminishes away from the plate. The velocity profiles across the boundary layer are shown in Fig. 12, for various values of the permeability parameter K . It is clearly seen that as K increases the velocity profiles across the boundary layer increases since the resistance offered by the porous medium decreases as the permeability of the porous medium increases. Fig. 13 depicts the variation of heat sink parameter φ and time t on temperature profiles in the thermal boundary layer. It is observed from the figure that increasing the heat absorption parameter is to decrease the temperature in the boundary

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Fig. 4. Effect of γ on velocity profiles against spanwise coordinate η for n = 0.1 and Ec = 0.01.

Fig. 5. Effect of Sc on velocity profiles against spanwise coordinate η for n = 0.1 and Ec = 0.01.

layer. This effect is more prominent for fluid closer to the porous plate. This is due to the fact that heat from the fluid is been absorbed by the porous plate and hence higher the value of heat absorption parameter, lower the value of temperature profile in the boundary layer Further it is observed that the temperature increases with time near the porous plate whereas no effect of time is observed in free stream. Fig. 14 shows the variation of temperature profiles with η for thermal radiation parameter F . It is noted that the increase in the radiation parameter F results in decrease in the value of the temperature θ in the thermal boundary layer due to the fact that, the divergence of radiation heat flux ∂ qr /∂ y decreases as the absorption coefficient kλw at the wall increases which in turn decreases the rate of radiative heat transfer to the fluid which causes the fluid temperature to decrease. Fig. 15 depicts the plot of temperature profiles with η for various values of Prandtl number Pr. It is seen from this figure that an increase in the value of the Prandtl number leads to a decrease in the temperature due to the fact that thermal boundary layer decreases with increased value of Pr. The effect of increasing the reaction rate parameter γ on the species concentration profiles for generative chemical reaction is shown in Fig. 16. It is noticed from this graph that there is marked effect of increasing the value of the chemical reaction rate parameter γ on concentration distribution in the solutal boundary layer. It is clearly observed from this figure

D. Pal, B. Talukdar / Mathematical and Computer Modelling 54 (2011) 3016–3036

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Fig. 6. Effect of F on velocity profiles against spanwise coordinate η for n = 0.1 and Ec = 0.01.

Fig. 7. Effect of n on velocity profiles against spanwise coordinate η for up = 0.5 and Ec = 0.01.

that the value of the concentration of species (i.e. C > 1.0) at start of the boundary layer decreases till it attains the minimum value of zero at the end of the boundary layer and this trend is seen for all the values of reaction rate parameter. Further, it is observed that increasing the value of the chemical reaction rate parameter decreases the concentration of species in the boundary layer, this is due to the fact that solutal boundary layer decreases with γ . Further, it is observed that the temperature increases with increase in time which is prominently seen near the vertical plate. Similar facts are seen in the case when the value of Schmidt number is increased as noted from Fig. 17. As expected, the mass transfer decreases as Sc increases by keeping all other physical parameter fixed, i.e. an increase in the value of the Schmidt number Sc is associated with the reduction in the concentration profiles. Further it may also be observed from this figure that the effect of Schmidt number Sc on concentration distribution slowly decreases in the solutal boundary layer for higher values of Sc. Figs. 18–23 present the variation of the skin friction coefficient Cf , Nusselt number Nux and Sherwood number Shx against the suction velocity parameter A and time t for various values of K , F and γ . It has been observed from Fig. 18 that the effect of increasing the values of time t increases the value of the skin friction coefficient for all the values of suction velocity parameter A. As shown in Fig. 19, the value of the Nusselt number Nux decreases with increasing time t. Fig. 20 depicts the

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D. Pal, B. Talukdar / Mathematical and Computer Modelling 54 (2011) 3016–3036

Fig. 8. Effect of Pr on velocity profiles against spanwise coordinate η for n = 0.1 and Ec = 0.01.

Fig. 9. Effect of Gr on velocity profiles against spanwise coordinate η for n = 0.1 and Ec = 0.01.

variation of surface mass flux Shx versus the suction velocity parameter A for several values of time t. From this figure it is seen that the surface mass flux decreases with increase in time t. Further it is found that the effect of increase in the suction velocity is to decrease surface mass flux Shx for all the values of time t. Fig. 21 is plotted to show the graph of surface skin friction coefficient Cf against time t for different values of porous parameter K . From this figure it is observed that the surface skin friction coefficient increases with increase the porous parameter K and there is also increase in Cf with time t. Fig. 22 shows the variation of surface heat flux with time t for different values of radiation parameter F . The surface heat transfer decreases with increase in the radiation parameter, due to the fact that increase in the radiation parameter is accompanied by a decrease in the velocity. Further, the effect of increasing the value of time t is to decrease the surface heat flux, Nux for all the values of thermal radiation parameter F . Fig. 23 illustrates the variation of surface mass flux Shx versus the time t for several values of chemical reaction parameter γ . Computed results show that for given material parameters, the surface mass flux tends to decrease by increasing the chemical reaction rate parameter for all the values of time t. Also, this figure reveals that increasing the value of the time t results in decreased value of surface mass flux Shx for all the values of chemical reaction rate parameter.

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Fig. 10. Effect of Gc on velocity profiles against spanwise coordinate η for n = 0.1 and Ec = 0.01.

Fig. 11. Effect of up on velocity profiles against spanwise coordinate η for n = 0.1 and Ec = 0.01.

In order to verify the accuracy of the present results, we have considered the analytical solutions obtained by Kim [6] and computed these solutions for various values of the physical parameters for skin-friction coefficient and local Nusselt number which are tabulated in Table 1. It is interesting to observe from this table that the present analytical results (under some limiting conditions) are in very good agreement with the computed results obtained from the analytical solutions of Kim [6] and numerical solutions obtained using Runge–Kutta Fehlberg method with shooting technique, which clearly shows the correctness of the present analytical solutions and the computed results. The values of skin-friction coefficient, local Nusselt number and local Sherwood number for various values of time which are obtained from the present analytical solutions and numerical solutions using Runge–Kutta Fehlberg method with shooting technique are recorded in Table 2. It is observed from this table that an excellent agreement of the analytical results with the numerical results has been obtained, which clearly shows the accuracy of the present analytical results and the numerical results. Further, it is observed from this table that the skin-friction coefficient increases with increase in the value of time whereas reverse effects are found on local Nusselt number and local Sherwood number. Computed results for skin friction coefficients, local Nusselt number and local Sherwood number are tabulated in Table 3 for various values of ϵ and t. It is clearly seen that the series converges as the

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D. Pal, B. Talukdar / Mathematical and Computer Modelling 54 (2011) 3016–3036

Fig. 12. Effect of K on velocity profiles against spanwise coordinate η for n = 0.1 and Ec = 0.01.

Fig. 13. Effect of φ and t on temperature profiles against spanwise coordinate η for Ec = 0.01.

Table 1 Comparison of analytical and numerical results (Gc = 0.0, Ec = 0.0, γ = 0.0, F = 0.0, φ = 0.00001) with those of Kim [6] with different values of Pr for Cfx , Nux /Rex with Gr = 2.0, M = 2.0, K = 0.5, Sc = 0.00001, up = 0.5, n = 0.1, t = 1.0, ϵ = 0.001, A = 0.05. Pr

0.7 1.0 3.0 5.0

Kim [6] results

Analytical results

Numerical results

Cfx

Nux /Rex

Cfx

Nux /Rex

Cfx

Nux /Rex

2.1689 2.0652 1.7226 1.5888

−0.7009 −1.0013 −3.0036 −5.0059

2.1689 2.0652 1.7226 1.5888

−0.7009 −1.0013 −3.0036 −5.0059

2.1686 2.0649 1.7222 1.5885

−0.7008 −1.0011 −3.0033 −5.0054

values of ϵ is made smaller and smaller. Thus for the present work the value ϵ = 0.001 or 0.005 is sufficient which provides accurate results as observed from Table 3.

D. Pal, B. Talukdar / Mathematical and Computer Modelling 54 (2011) 3016–3036

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Fig. 14. Effect of F on temperature profiles against spanwise coordinate η for n = 0.1 and Ec = 0.01.

Fig. 15. Effect of Pr on temperature profiles against spanwise coordinate η for n = 0.1 and Ec = 0.01.

Table 2 Comparison of analytical and numerical results for different values of t on Cfx , Nux /Rex and Shx /Rex for Gr = 2.0, Gc = 2.0, Sc = 0.6, M = 0.5, φ = 0.3, Pr = 0.7, K = 0.5, γ = 0.1, F = 0.5, up = 0.5, n = 0.1, Ec = 0.01, ϵ = 0.001 and A = 0.05. t

1.0 3.0 5.0 10.0 20.0

Analytical results

Numerical results

Cfx

Nux /Rex

Shx /Rex

Cfx

Nux /Rex

Shx /Rex

2.9878 2.9887 2.9899 2.9941 3.0125

−1.1652 −1.1656 −1.1661 −1.1677 −1.1746

−0.6882 −0.6884 −0.6886 −0.6894 −0.6931

3.0651 3.0734 3.0817 3.1023 3.1436

−1.1130 −1.1152 −1.1174 −1.1230 −1.1342

−0.6881 −0.6897 −0.6912 −0.6951 −0.7030

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D. Pal, B. Talukdar / Mathematical and Computer Modelling 54 (2011) 3016–3036

Fig. 16. Effect of γ on concentration profiles against spanwise coordinate η for n = 0.1 and Ec = 0.01.

Fig. 17. Effect of Sc on concentration profiles against spanwise coordinate η for n = 0.1 and Ec = 0.01. Table 3 Series convergence test with variations in ϵ and t on Cfx , Nux /Rex , Shx /Rex for Gr = 2.0, Gc = 2.0, Sc = 0.6, M = 0.5, φ = 0.3, Pr = 0.7, K = 0.5, γ = 0.1, F = 0.5, up = 0.5, n = 0.1, Ec = 0.01, and A = 0.05. t 1.0

20.0

ϵ

Cfx

Nux /Rex

Shx /Rex

0.01 0.005 0.001 0.0001 0.01 0.005 0.001 0.0001

3.0269 3.0052 2.9878 2.9838 3.2744 3.1289 3.0125 2.9863

−1.1801 −1.1718 −1.1652 −1.1638 −1.2738 −1.2187 −1.1746 −1.1647

−0.6959 −0.6916 −0.6882 −0.6874 −0.7451 −0.7162 −0.6931 −0.6879

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Fig. 18. Variation of the surface skin friction with the suction velocity parameter A for various values of t for n = 0.1 and Ec = 0.01.

Fig. 19. Variation of the surface heat flux with the suction velocity parameter A for various values of t for n = 0.1 and Ec = 0.01.

5. Conclusions We have examined theoretically the problem of an unsteady, incompressible MHD mixed convection boundary layer flow past a moving vertical plate whose velocity is maintained constant by considering radiation effects in the presence of viscous and Joule heating. The computed values obtained from analytical solutions of the velocity, temperature, concentration fields as well as for the skin friction coefficient, Nusselt number and Sherwood number are presented graphically and in tabular form. After a suitable transformation, the governing partial differential equations were transformed to ordinary differential equations. These equations were solved analytically by using perturbation technique. Thus we conclude the following after analyzing the graphs: 1. The velocity decreases with increase in the values of chemical reaction parameter, heat source parameter, magnetic parameter, Schmidt number, Prandtl number and radiation parameter, whereas reverse trend is seen with increasing the exponential index parameter, thermal and solutal Grashof numbers.

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Fig. 20. Variation of the surface mass flux with the suction velocity parameter A for various values of t for n = 0.1 and Ec = 0.01.

Fig. 21. Variation of the surface skin friction with the time dependent parameter t for various values of porous parameter K for n = 0.1 and Ec = 0.01.

2. The velocity, temperature and concentration increase with increase in time, and the impact of the porous medium on velocity profiles is significant due to increase in the porous parameter in the hydrodynamic boundary layer. 3. The temperature decreases with increase in the values of thermal radiation parameter, heat absorption parameter and Prandtl number. 4. The concentration decreases with increase in the values of the chemical reaction parameter and Schmidt number. 5. The value of the local skin-friction coefficient decreases with increase in the suction velocity parameter whereas reverse effect is seen by increasing porous parameter and time. 6. The value of the local Nusselt number decreases as the suction velocity parameter and thermal radiation parameter increase. 7. The value of the local Sherwood number decreases with increase in suction velocity parameter, chemical reaction rate parameter and time.

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Fig. 22. Variation of the surface heat flux with the time dependent parameter t for various values of radiation parameter F for n = 0.1 and Ec = 0.01.

Fig. 23. Variation of the surface mass flux with the time dependent parameter t for various values of reaction parameter γ for n = 0.1 and Ec = 0.01.

Appendix

A1 = B1 =

Sc +



Sc2 + 4Scγ 2

Gr A22 − A2 − N

d1 = A1 + A2 , f2 =

,

,

A2 =

B2 =

A23 − PrA3 − Pr(F + φ)

,

Pr2 + 4Pr(F + φ) 2

Gc

,

,

A3 =

1+

1 + 4N 2

B3 = Up − 1.0 + B1 + B2

d3 = A1 + A3 , f3 =





A21 − A1 − N

d2 = A2 + A3 ,

2PrNB3

Pr +

f1 =

N F +φ

PrB23 (N + A23 ) 4A23 − 2PrA3 − Pr(F + φ)

,

f4 = −

PrB21 (N + A22 ) 4A22 − 2PrA2 − Pr(F + φ)

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D. Pal, B. Talukdar / Mathematical and Computer Modelling 54 (2011) 3016–3036

2PrB1 B2 (N + A1 A2 )

d21

− Prd1 − Pr(F + φ)

GrR12

R2 =

A22

− A2 − N

,

R3 =

Grf4

R5 =

4A22 − 2A2 − N Grf9

,

f6 = −

2PrNB1

Grf2 A23

− A3 − N

4A21

,

Grf5

R6 =

d21 − d1 − N

R4 =

,

Grf3 4A23

− 2A3 − N

,

Grf6

R7 =

4A21 − 2A1 − N

,

R8 =

Grf8 A21 − A1 − N

,

Grf10

, − d2 − N − d3 − N A5 = R2 − R3 − R4 + R5 + R6 + R7 + R9 + R10 , R11 = A5 + R3 ,   2 Pr + Pr2 + 4Pr(F + φ + n) Sc + Sc + 4Sc(γ + n) , A8 = , A6 = R9 =

,

,

PrB22 (N + A21 )

, f7 = 2 − 2PrA1 − Pr(F + φ) A2 − PrA2 − Pr(F + φ) 2PrNB2 2PrB1 B3 (N + A2 A3 ) 2PrB2 B3 (N + A3 A1 ) f8 = 2 , f9 = 2 , f10 = 2 A1 − PrA1 − Pr(F + φ) d2 − Prd2 − Pr(F + φ) d3 − Prd3 − Pr(F + φ) A4 = f2 + f3 − f4 − f5 − f6 − f7 − f8 − f9 − f10 , R12 = A4 + f7 , f5 = −

d22

R14 =

R10 =

d23

2 AA2 Pr

2

A22 − PrA2 − Pr(F + φ + n)



A9 = 1.0 − R14 , R15 = R18 =

A10 =

AB3 A3 A23 − A3 − (N + n) GrA9 A28 − A8 − (N + n)

A12 = A3 + A10 ,

1+

1 + 4(N + n) 2

,

R16 = −

,

R19 =

,

GrR14 + AB1 A2 A22 − A2 − (N + n) GcA7

A26 − A6 − (N + n)

A13 = A2 + A10 ,

AA1 Sc

R13 =

A21

,

,

, A7 = 1.0 − R13 , − ScA1 − Sc(γ + n) GcR13 + AB2 A1 R17 = − 2 , A1 − A1 − (N + n) A11 = −1 − R15 − R16 − R17 + R18 + R19 ,

A14 = A1 + A10 ,

A15 = A2 + A3 ,

A16 = A1 + A3 ,

A17 = A2 + A1 ,

A18 = A3 + A8 ,

A19 = A2 + A8 ,

A20 = A1 + A8 ,

A21 = A3 + A6 ,

A22 = A2 + A6 ,

A23 = A1 + A6 ,

A24 = B1 R15 − B3 R16 , S1 = A3 B3 A11 A10 ,

A25 = B2 R15 − B3 R17 , S2 = A23 B3 R15 ,

S4 = A1 A3 (B3 R17 − B2 R15 ),

S3 = A2 A3 (B3 R16 − B1 R15 ),

S5 = A3 B3 A8 R18 ,

S7 = B1 A2 A11 A10 ,

S8 =

S10 = A2 B1 A8 R18 ,

S11 = A2 B1 A6 R19 ,

S13 = B2 A21 R17 , t1 = t3 = t5 =

B1 A22 R16

,

2Pr(AA3 f3 + NB3 R15 + S2 )

− 2PrA3 − Pr(F + φ + n) APrd1 f5

,

t11 = t13 = t14 =

S15 = B2 A1 A6 R19 ,

d21

APrd2 f9 d22 − Prd2 − Pr(F + φ + n)

,

t9 =

− PrA10 − Pr(F + φ + n) 2PrNR19

A26 − PrA6 − Pr(F + φ + n)

t2 =

t4 = t6 =

2PrNA11 A210

,

,

− Prd1 − Pr(F + φ + n) 2Pr(Af8 A1 /2 − N (R17 − B2 )) t7 = 2 , A1 − PrA1 − Pr(F + φ + n) t8 =

S12 = B2 A1 A11 A10 ,

S14 = A1 B2 A8 R18 ,

A22 − PrA2 − Pr(F + φ + n)

S6 = A3 B3 A6 R19 ,

S9 = A1 A2 (B1 R17 + B2 R16 ),

Pr(AA2 A4 + AA2 f7 − 2N (R16 − B1 ))

4A23

A26 = B2 R16 + B1 R17 ,

Pr(AA3 f2 + 2N (R15 + B3 )) A23 − PrA3 − Pr(F + φ + n)

2Pr(AA2 f4 + NB1 R16 + S8 ) 4A22 − 2PrA2 − Pr(F + φ + n)

2Pr(AA1 f6 + NB2 R17 + S13 ) 4A21 − 2PrA1 − Pr(F + φ + n)

APrd3 f10 d23 − Prd3 − Pr(F + φ + n)

,

t12 =

,

t15 =

,

,

2PrNR18 A28

,

− PrA8 − Pr(F + φ + n)

,

,

2Pr(NB3 A11 + S1 ) A212 − PrA12 − Pr(F + φ + n)

2Pr(NB1 A11 + S7 ) A213 − PrA13 − Pr(F + φ + n)

,

,

D. Pal, B. Talukdar / Mathematical and Computer Modelling 54 (2011) 3016–3036

3035

2Pr(NB2 A11 + S12 )

, − PrA14 − Pr(F + φ + n) 2Pr(NA24 − S3 ) 2Pr(NA25 − S4 ) t17 = 2 , t18 = 2 , A15 − PrA15 − Pr(F + φ + n) A16 − PrA16 − Pr(F + φ + n) 2Pr(NA26 + S9 ) t19 = 2 , A17 − PrA17 − Pr(F + φ + n) 2Pr(NB3 R18 + S5 ) 2Pr(NB1 R18 + S10 ) t20 = 2 , t21 = 2 , A18 − PrA18 − Pr(F + φ + n) A19 − PrA19 − Pr(F + φ + n) 2Pr(NB2 R18 + S14 ) t22 = 2 , A20 − PrA20 − Pr(F + φ + n) 2Pr(NB3 R19 + S6 ) 2Pr(NB1 R19 + S11 ) t23 = 2 , t24 = 2 , A21 − PrA21 − Pr(F + φ + n) A22 − PrA22 − Pr(F + φ + n) 2Pr(NB2 R19 + S15 ) t25 = 2 , A23 − PrA23 − Pr(F + φ + n) t26 = −t1 + t2 + t3 − t4 − t5 − t6 − t7 − t8 − t9 + t11 − t12 − t13 + t14 − t15 − t16 − t17 − t18 − t19 − t20 + t21 + t22 − t23 + t24 + t25 , AA5 A3 + Grt2 + AA3 R3 AA2 R2 + Grt1 t27 = t26 + t12 , W1 = , W2 = 2 , A23 − A3 − (N + n) A2 − A2 − (N + n) 2AA3 R4 + Grt3 2AA2 R5 + Grt4 Ad1 R6 + Grt5 W3 = , W4 = , W5 = 2 , 4A23 − 2A3 − (N + n) 4A22 − 2A2 − (N + n) d1 − d1 − (N + n) 2AA1 R7 + Grt6 AA1 R8 + Grt7 Ad2 R9 + Grt8 W6 = , W7 = 2 , W8 = 2 , 2 4A1 − 2A1 − (N + n) A1 − A1 − (N + n) d2 − d2 − (N + n) Ad3 R10 + Grt9 Grt27 Grt11 W9 = 2 , W10 = 2 , W12 = 2 , d3 − d3 − (N + n) A8 − A8 − (N + n) A10 − A10 − (N + n) t16 =

A214

W13 = W16 = W19 = W22 = W25 =

Grt13 A26 − A6 − (N + n)

,

Grt16 A214

− A14 − (N + n) Grt19

A217

− A17 − (N + n) Grt22

A220 − A20 − (N + n) Grt25 A223 − A23 − (N + n)

W14 =

Grt14 A212 − A12 − (N + n)

,

W17 =

,

W20 =

,

W23 =

,

Grt17 A215

− A15 − (N + n) Grt20

A218

− A18 − (N + n) Grt23

A221 − A21 − (N + n)

W15 =

Grt15 A213 − A13 − (N + n)

,

W18 =

,

W21 =

,

W24 =

,

Grt18 A216

− A16 − (N + n) Grt21

A219

− A19 − (N + n) Grt24

A222 − A22 − (N + n)

, , ,

,

W26 = −W1 + W2 − W3 + W4 + W5 + W6 + W7 + W8 + W9 + W10 − W12 + W13 − W14 + W15 + W16 + W17 + W18 + W19 + W20 − W21 − W22 + W23 − W24 − W25 , W27 = W26 + W12 . References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]

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