mathematical analysis for optically thin radiating ...mathematical analysis for optically thin...

Global Journal of Pure and Applied Mathematics.

ISSN 0973-1768 Volume 13, Number 6 (2017), pp. 1777-1798

© Research India Publications

http://www.ripublication.com

Mathematical Analysis for Optically Thin Radiating/

Chemically Reacting Fluid in a Darcian Porous Regime

Nava Jyoti Hazarika1 and Sahin Ahmed2

1Department of Mathematics, Tyagbir Hem Baruah College, Jamugurihat, Sonitpur-784189, Assam, India.

2Department of Mathematics, Rajiv Gandhi University, Rono Hills, Itanagar, Arunachal Pradesh-791112, India.

Abstract

In this paper, we analyzed an unsteady MHD flow of two-dimensional,

laminar, incompressible, Newtonian, electrically-conducting and radiating

fluid along a semi-infinite vertical permeable moving plate with periodic heat

and mass transfer by taking into account the effect of viscous dissipation in

presence of chemical reaction. A uniform magnetic field is applied

transversely to the porous plate. The plate moves with a constant velocity in

the direction of the fluid flow while the free stream velocity follows an

exponentially increasing small perturbation law subject to a constant suction

velocity to the plate. The dimensionless governing equations for this

investigation are solved analytically using two-term harmonic and non-

harmonic functions. Numerical evaluation of the analytical results are

performed and graphical results for velocity, temperature and concentration

profiles within the boundary layer and the tabulated results for the Skin-

friction co-efficient, Nusselt number and Sherwood number are presented and

discussed. It is seen that, an increase in chemical reaction parameter leads to

decrease both fluid velocity as well as concentration. Moreover, the skin-

friction has been depressed by the influence of chemical reaction parameter,

where as the rate of heat transfer is escalated. The present model has several

important applications such as dispersion of chemicals contaminants,

superconvecting geothermics, geothermal energy extractions and plasma

physics.

Keywords: Thin gray gas; Dispersion of chemicals contaminants; Viscous

dissipation; MHD; Darcian regime; skin-friction.

Corresponding author: Sahin Ahmed

1778 Nava Jyoti Hazarika and Sahin Ahmed

1. INTRODUCTION

The study of heat and mass transfer to chemical reacting MHD free convection flow

with radiation effects on a vertical plate has received a growing interest during the last

decades. Accurate knowledge of the overall convection heat transfer has vital

importance in several fields such as thermal insulation, dying of porous solid

materials, heat exchangers, stream pipes, water heaters, refrigerators, electrical

conductors and industrial, geophysical and astrophysical applications such as polymer

production, manufacturing of ceramic, packed-bed catalytic reactor, food processing,

cooling of nuclear reactor, enhanced oil recovery, underground energy transport,

magnetized plasma flow, high speed plasma wind, cosmic jets and stellar system. For

some industrial application such as glass production, furnace design, propulsion

systems, plasma physics and spacecraft re-entry aerothermodynamics which operate

at higher temperatures and radiation effect can also be significant. Consolidated

effects of heat and mass transfer problems are of importance in many chemical

formulations and reactive chemicals. Therefore, considerable attention had been paid

in recent years to study the influence of the participating parameters on the velocity

fields. More such engineering application can be seeing in electrical power generation

system when the electrical energy is extracted directly from a moving conducting

fluid.

There has been a renewed interest in studying Magnetohydrodynamic (MHD) flow

and heat transfer in porous and non-porous media due to the effect of magnetic fields

on the boundary layer flow control and on the performance of many systems using

electrically conducting fluids. In addition, this type of flow finds applications in many

engineering problems such as MHD generators, plasma studies, nuclear reactors and

geothermal energy extractors. Chamkha [1] presented an unsteady MHD convective

heat and mass transfer past a semi-infinite vertical permeable moving plate with heat

absorption. An analysis of an unsteady MHD convective flow past a vertical moving

plate embedded in a porous medium in the presence of transverse magnetic field a

reported by Kim [2]. Singh [3] studied the effects of mass transfer on free convection

in MHD flow of viscous fluid. Ahmed [4] looked the effects of unsteady free

convective MHD flow through a porous medium bounded by an infinite vertical

porous plate. Raptis [5] studied mathematically the case of unsteady two-dimensional

natural convective heat transfer of an incompressible, electrically conducting viscous

fluid in a highly porous medium bound by an infinite vertical porous plate.

Soundalgekar [6] obtained approximate solutions for the two-dimensional flow an

incompressible, viscous fluid past an infinite porous vertical plate with constant

suction velocity normal to the plate, the difference between the temperature of the

plate and the free stream is moderately large causing the free convection currents.

Recently, free convective fluctuating MHD flow through porous media past a vertical

porous plate with variable temperature and heat source was studied by Acharya et al. [7]. Rao et al. [8] was discussed the heat transfer on steady MHD rotating flow through porous medium in a parallel plate channel. Pattnaik and Biswal [9] studied

the analytical solution of MHD free convective flow through porous media with time

dependent temperature and concentration. More recently, Hazarika and Ahmed [10]

Mathematical analysis for optically thin Radiating/ Chemically reacting fluid 1779

have investigated the analytical study of unsteady MHD chemically reacting fluid

over a vertical porous plate in a Darcian porous Regime. Chemical reaction effects on

MHD free convective flow through porous medium with constant suction and heat

flux has discussed by Seshaiah and Varma [11].

All the above investigations are restricted to MHD flow and heat transfer problems

only. However of the late the effects of radiation on MHD flow, heat and mass

transfer have becomes more important industrially. The radiation flows of an

electrically conducting fluid with high temperature, in the presence of magnetic fields,

are encountered in electrical power generation, astrophysical flows, solar power

technology, space vehicle re-entry, nuclear engineering applications and other

industrial areas. Radiative heat and mass transfer play an important role in

manufacturing industries for the design of fins, steel rolling, nuclear power plants, gas

turbines and various propulsion devices for aircraft, missiles, satellites and space

vehicles are examples of such engineering applications. Radiation effects on mixed

convection along an isothermal vertical plate were studied by Hossain and Takhar

[12]. Prasad et al. [13] studied the radiation and mass transfer effects on unsteady MHD free convection flow past a vertical porous plate embedded in porous medium.

Zueco and Ahmed [14] proposed the mixed convection MHD flow along a porous

plate with chemical reaction in presence of heat source. The transient MHD free

convective flow of a viscous, incompressible, electrically conducting, gray,

absorbing-emitting, but not scattering, optically thick fluid medium which occupies a

semi-infinite porous region adjacent to an infinite hot vertical plate moving with

constant velocity was presented by Ahmed and Kalita [15]. The effects of chemical

reaction as well as magnetic field on the heat and mass transfer of Newtonian two-

dimensional flow over an infinite vertical oscillating plate with variable mass

diffusion investigated by Ahmed and Kalita [16]. Recently, Ahmed [17] presented the

effects of conduction-radiation, porosity and chemical reaction on unsteady

hydromagnetic free convection flow past an impulsively started semi-infinite vertical

plate embedded in a porous medium in presence of thermal radiation. The thermal

radiation and Darcian drag force MHD unsteady thermal-convection flow past a semi-

infinite vertical plate immersed in a semi-infinite saturated porous regime with

variable surface temperature in the presence of transversal uniform magnetic field

have been discussed by Ahmed et al. [18]. Radiation and mass transfer on unsteady MHD convective flow past an infinite vertical plate in presence of Dufour and Soret

effects studied by Vedavathi et al. [19]. Ahmed et al. [20] investigated the effects of chemical reaction and viscous dissipation on MHD heat and mass transfer flow

through Perturbation method.

In all these investigations, the viscous dissipation is neglected. Gebhart [21] had

shown the importance of viscous dissipative heat in free convection flow in the case

of isothermal and constant heat flux at the plate. Soundalgekar [22] analyzed the

viscous dissipative heat on the two-dimensional unsteady free convective flow past an

infinite vertical porous plate when the temperature oscillates in time and there is

constant suction at the plate. Prasad and Reddy [23] had discussed about the Radiation

and Mass transfer effects on an unsteady MHD convection flow with viscous


dissipation. Cookey et al. [24] had investigated the influence of viscous dissipation and radiation on unsteady MHD free convection flow past an infinite heated vertical

plate in a porous medium with time dependent suction. Recently, radiation effects on

an unsteady MHD convective flow past a vertical plate in porous medium with

viscous dissipation analyzed by Gudagani et al. [25]. In this paper the effects of chemical reaction and thermal radiation of optically thin

gray gas on a mixed convective boundary layer flow of an electrically conducting

fluid over an semi-infinite porous surface embedded in a Darcian porous regime in

presence of viscous dissipative heat is investigated. The governing equations are

solved by using a regular perturbation theory.

2. MATHEMATICAL ANALYSES

In this flow model, we consider two-dimensional unsteady hydromagnetic laminar

mixed convective boundary layer flow of a viscous, incompressible, electrically

conducting and radiating fluid in an optically thin environment, past a semi-infinite

vertical permeable moving plate embedded in a Darcian porous medium, in presents

of thermal and concentration buoyancy effects with chemical reaction of first order.

The 𝑥-axis is taken in the upward direction along the plate and 𝑦-axis normal to it. A uniform magnetic field is applied in the direction perpendicular to the plate. The

transverse applied magnetic field and magnetic Reynolds number are assumed to be

very small, so that the induced magnetic field is negligible. Also, it is assumed that

there is no applied voltage, so that the electric field is absent. The concentration of the

diffusing species in the binary mixture is assumed to be very small in comparison

with the other chemical species which are present, and hence the Soret and Dufour

effects are negligible. Further, due to semi-infinite plane surface assumption, the flow

variables are functions of normal distance 𝑦 and 𝑡 only. Now, under the usual Boussinesq’s approximation, the governing boundary layer equations are:

𝜕𝑣

𝜕𝑦= 0 (1)

𝜕𝑢

𝜕𝑡+ 𝑣

𝜕𝑣

𝜕𝑦= −

1

𝜌

𝜕𝑝

𝜕𝑥+ 𝜈

𝜕2𝑢

𝜕𝑦2 + 𝑔𝛽𝑇(𝑇 − 𝑇∞) + 𝑔𝛽𝐶(𝐶 − 𝐶∞) − (

𝜈

𝜅+𝜎𝐵0

2

𝜌)𝑢 (2)

𝜕𝑇

𝜕𝑡+ 𝑣

𝜕𝑇

𝜕𝑦=

𝑘

𝜌𝑐𝑝[𝜕2𝑇

𝜕𝑦2 −

1

𝑘

𝜕𝑞

𝜕𝑦] +

𝜈

𝑐𝑝(𝜕𝑢

𝜕𝑦)

2

(3)

𝜕2𝑞

𝜕𝑦2 − 3𝛼

2𝑞 − 16𝜎∗𝛼𝑇∞3 𝜕𝑇

𝜕𝑦= 0 (4)

𝜕𝐶

𝜕𝑡+ 𝑣

𝜕𝐶

𝜕𝑦= 𝐷

𝜕2𝐶

𝜕𝑦2 − 𝐶𝑟(𝐶 − 𝐶∞) (5)


The third and fourth terms on the right hand side of momentum Eq. (2) denote the

thermal and concentration buoyancy effects respectively. The second and third terms

on right hand side of energy Eq. (3) represent the radiative heat flux and viscous

dissipation respectively. Also the second term on right hand of concentration Eq. (5)

represents the chemical reaction effect.

The permeable plate moves with a constant velocity in the direction of fluid flow and

the free steam velocity follows the exponentially increasing small perturbation law. In

addition, it is assumed that the temperature and concentration at the wall as well as the

suction velocity are exponentially varying with time. Eq. (4) is the differential

approximation for radiation and the radiative heat flux 𝑞 satisfies this non-linear differential equation.

The boundary conditions for the velocity, temperature and concentration fields are:

{𝑢 = 𝑢𝑝, 𝑇 = 𝑇𝑤 + 𝜀(𝑇𝑤 − 𝑇∞)𝑒

𝑛𝑡, 𝐶 = 𝐶𝑤 + 𝜀(𝐶𝑤 − 𝐶∞)𝑒𝑛𝑡 𝑎𝑡 𝑦 = 0

𝑢 = 𝑈∞ = 𝑈0(1 + 𝜀𝑒𝑛𝑡), 𝑇 ⟶ 𝑇∞ , 𝐶 ⟶ 𝐶∞ 𝑎𝑠 𝑦 ⟶ ∞

} (6)

It is clear from the equation (1) that the suction velocity at the plate is either a

constant or function of time only. Hence, the suction velocity normal to the plate is

assumed in the form:

𝑣 = −𝑉0(1 + 𝜀𝐴𝑒𝑛𝑡) (7)

The negative sign indicates that the suction is towards the plate.

Outside the boundary layer, Eq. (2) gives:

−1

𝜌

𝜕𝑝

𝜕𝑥= 𝑑𝑈∞

𝑑𝑡+𝜈

𝜅 𝑈∞ +

𝜎

𝜌 𝐵0

2𝑈∞ (8)

Since the medium is optically thin with relatively low density and 𝛼 ≪ 1, the radiative heat flux given by Eq. (3), in the spirit of Cogley et al. [22] becomes:

𝜕𝑞

𝜕𝑦= 4𝛼2 (𝑇 − 𝑇∞) where 𝛼

2 = ∫ 𝛿𝜆𝜕𝐵

𝜕𝑇

∞

0

, (9)

where B is Planck’s function.

In order to write the governing equations and boundary conditions in dimensionless

form, the following non-dimensional quantities are introduced.


{

𝑢 =

𝑢

𝑈0 , 𝑣 =

𝑣

𝑉0, 𝑦 =

𝑉0𝑦

𝜈 , 𝑈∞ =

𝑈∞𝑈0

,

𝑈𝑝 =𝑢𝑝

𝑈0, 𝑡 =

𝑡 𝑉02

𝜈 ,

𝜃 =𝑇 − 𝑇∞

𝑇𝑤 − 𝑇∞ , 𝜙 =

𝐶 − 𝐶∞

𝐶𝑤 − 𝐶∞ , 𝑛 =

𝑛 𝜈

𝑉02 , 𝐾 =

𝐾 𝑉02

𝜈2 , 𝐶𝑟 =

𝜈𝐶𝑟

𝑉02

𝑃𝑟 = 𝜈𝜌𝐶𝑝

𝑘, 𝑆𝑐 =

𝜈

𝐷 , 𝑀 =

𝜎𝐵02𝜈

𝜌𝑉02 , 𝐺𝑟 =

𝜈𝛽𝑇𝑔(𝑇𝑤 − 𝑇∞)

𝑈0𝑉02

,

𝐺𝑚 =𝜈𝛽𝐶𝑔(𝐶𝑤 − 𝐶∞)

𝑈0𝑉02 , 𝐸𝑐 =

𝑈02

𝐶𝑝(𝑇𝑤 − 𝑇∞), 𝑅2 =

𝛼2(𝑇𝑤 − 𝑇∞)

𝜌𝐶𝑝𝑘𝑈02

,

}

(10)

In view of Eqs. (4) and (7) –(10), Eqs. (2), (3) and (5) reduce to the following

dimensionless form:

𝜕𝑢

𝜕𝑡− (1 + 𝜀𝐴𝑒𝑛𝑡)

𝜕𝑢

𝜕𝑦=𝑑𝑈∞𝑑𝑡

+𝜕2𝑢

𝜕𝑦2+ 𝐺𝑟𝜃 + 𝐺𝑚𝜙 + 𝑁(𝑈∞ − 𝑢) (11)

𝜕𝜃


𝜕𝜃

𝜕𝑦=

1

𝑃𝑟[𝜕2𝜃

𝜕𝑦2− 𝑅2𝜃] + 𝐸𝑐 (

𝜕𝑢

𝜕𝑦)2

(12)

𝜕𝜙


𝜕𝜙

𝜕𝑦=1

𝑆𝑐

𝜕2𝜙

𝜕𝑦2− 𝐶𝑟𝜙 (13)

where 𝑁 = 𝑀 + 𝐾−1

The corresponding dimensionless boundary conditions are:

{𝑢 = 𝑈𝑝 , 𝜃 = 1 + 𝜀𝑒

𝑛𝑡, 𝜙 = 1 + 𝜀𝑒𝑛𝑡, 𝑎𝑡 𝑦 = 0

𝑢 = 𝑈∞ = 1 + 𝜀𝑒𝑛𝑡, 𝜃 ⟶ 0, 𝜙 ⟶ 0 𝑎𝑠 𝑦 ⟶ ∞

} (14)

SOLUTION OF THE PROBLEM

The Eqs. (11-13) are coupled, non-linear partial differential equations and these

cannot be solved in closed-form. However, these equations can be reduced to a set of

ordinary differential equations, which can be solved analytically. This can be done by

representing the velocity, temperature and concentration of the fluid in the

neighbourhood of the plate as:

{

𝑢(𝑦, 𝑡) = 𝑢0(𝑦) + 𝜀𝑒𝑛𝑡𝑢1(𝑦) + 0(𝜀

2) + ⋯

𝜃(𝑦, 𝑡) = 𝜃0(𝑦) + 𝜀𝑒𝑛𝑡𝜃1(𝑦) + 0(𝜀

2) + ⋯

𝜙(𝑦, 𝑡) = 𝜙0(𝑦) + 𝜀𝑒𝑛𝑡𝜙1(𝑦) + 0(𝜀

2) + ⋯

} (15)


Substituting Eq. (15) in Eqs. (11-13) and equating the harmonic and non-harmonic

terms, and neglecting the higher order terms of 0(𝜀2), we obtain:

𝑢0″(𝑦) + 𝑢0

′ (𝑦) − 𝑁𝑢0(𝑦) = −𝑁 − 𝐺𝑟𝜃0(𝑦) − 𝐺𝑚𝜙0(𝑦) (16)

𝑢1″(𝑦) + 𝑢1

′ (𝑦) − (𝑁 + 𝑛)𝑢1(𝑦)

= −(𝑁 + 𝑛) − 𝐴𝑢0′ (𝑦) − 𝐺𝑟𝜃1(𝑦) − 𝐺𝑚𝜙1(𝑦) (17)

𝜃0″(𝑦) + 𝑃𝑟 𝜃0

′(𝑦) − 𝑅2 𝜃0(𝑦) = −𝑃𝑟𝐸𝑐 [𝑢0′ (𝑦)]2 (18)

𝜃1″(𝑦) + 𝑃𝑟 𝜃1

′(𝑦) − (𝑅2 + 𝑛𝑃𝑟)𝜃1(𝑦) = −𝑃𝑟𝐴 𝜃0′(𝑦) − 2𝑃𝑟𝐸𝑐 𝑢0

′ (𝑦)𝑢1′(𝑦) (19)

𝜙0″(𝑦) + 𝑆𝑐 𝜙0

′ (𝑦) − 𝑆𝑐 𝐶𝑟𝜙0(𝑦) = 0 (20)

𝜙1″(𝑦) + 𝑆𝑐 𝜙1

′ (𝑦) − 𝑆𝑐(𝑛 + 𝐶𝑟)𝜙1(𝑦) = −𝐴𝑆𝑐 𝜙0′ (𝑦) (21)

where prime denotes ordinary differentiation with respect to y.

The corresponding boundary conditions can be written as:

{𝑢0 = 𝑈𝑝, 𝑢1 = 0, 𝜃0 = 1, 𝜃1 = 1, 𝜙0 = 1, 𝜙1 = 1 𝑎𝑡 𝑦 = 0

𝑢0 = 1, 𝑢1 = 1, 𝜃0 ⟶ 0, 𝜃1 ⟶ 0, 𝜙0 ⟶ 0, 𝜙1 ⟶ 0 𝑎𝑠 𝑦 ⟶ ∞ } (22)

The Eqs. (16) – (21) are still coupled and non-linear, whose exact solutions are not

possible. So we expand 𝑢0 , 𝑢1 , 𝜃0 , 𝜃1 , 𝜙0 , 𝜙1 in terms of 𝐸𝑐 in the following form, as the Eckert number is very small for incompressible flows.

𝐹(𝑦) = 𝐹0(𝑦) + 𝐸𝑐 𝐹1(𝑦) + 0(𝐸𝑐2) (23)

where 𝐹 stands for any 𝑢0 , 𝑢1 , 𝜃0 , 𝜃1 , 𝜙0 , 𝜙1 .

Substituting Eq. (23) in Eqs. (16) – (21), equating the co-efficient of 𝐸𝑐 to zero and neglecting the terms in 𝐸𝑐2 and higher order, we get the following equations:

The zeroth order equations are:

𝑢01″ (𝑦) + 𝑢01

′ (𝑦) − 𝑁𝑢01(𝑦) = −𝑁 − 𝐺𝑟 𝜃01(𝑦) − 𝐺𝑚 𝜙01(𝑦) (24)

𝑢02″ (𝑦) + 𝑢02

′ (𝑦) − 𝑁𝑢02(𝑦) = −𝐺𝑟 𝜃02(𝑦) − 𝐺𝑚 𝜙02(𝑦) (25)

𝜃01″ (𝑦) + 𝑃𝑟 𝜃01

′ (𝑦) − 𝑅2 𝜃01(𝑦) = 0 (26)

𝜃02″ (𝑦) + 𝑃𝑟 𝜃02

′ (𝑦) − 𝑅2 𝜃02(𝑦) = −𝑃𝑟[𝑢01′ (𝑦)]2 (27)

𝜙01″ (𝑦) + 𝑆𝑐 𝜙01

′ (𝑦) − 𝑆𝑐 𝐶𝑟 𝜙01(𝑦) = 0 (28)

𝜙02″ (𝑦) + 𝑆𝑐 𝜙02

′ (𝑦) − 𝑆𝑐 𝐶𝑟 𝜙02(𝑦) = 0 (29)


and the respective boundary conditions are:

{𝑢01 = 𝑈𝑝, 𝑢02 = 0, 𝜃01 = 1, 𝜃02 = 0, 𝜙01 = 1, 𝜙02 = 0 𝑎𝑡 𝑦 = 0

𝑢01 ⟶ 1, 𝑢02 ⟶ 0, 𝜃01 ⟶ 0, 𝜃02 ⟶ 0,𝜙01 ⟶ 0, 𝜙02 ⟶ 0 𝑎𝑡 𝑦 ⟶ ∞} (30)

The first order equations are:

𝑢11″ (𝑦) + 𝑢11

′ (𝑦) − (𝑁 + 𝑛)𝑢11(𝑦) = {−(𝑁 + 𝑛) − 𝐺𝑟 𝜃11(𝑦)

−𝐺𝑚 𝜙11(𝑦) − 𝐴 𝑢01′ (𝑦)

} (31)

𝑢12″ (𝑦) + 𝑢12

′ (𝑦) − (𝑁 + 𝑛)𝑢12(𝑦) = −𝐺𝑟 𝜃12(𝑦) − 𝐺𝑚 𝜙12(𝑦) − 𝐴 𝑢02′ (𝑦) (32)

𝜃11″ (𝑦) + 𝑃𝑟 𝜃11

′ (𝑦) − 𝑁1𝜃11(𝑦) = −𝑃𝑟𝐴 𝜃01′ (𝑦) (33)

𝜃12″ (𝑦) + 𝑃𝑟 𝜃12

′ (𝑦) − 𝑁1𝜃12(𝑦) = −𝑃𝑟𝐴 𝜃02′ (𝑦) − 2𝑃𝑟 𝑢01

′ (𝑦)𝑢11′ (𝑦) (34)

𝜙11″ (𝑦) + 𝑆𝑐 𝜙11

′ (𝑦) − 𝑆𝑐(𝑛 + 𝐶𝑟)𝜙11(𝑦) = −𝐴𝑆𝑐 𝜙01′ (𝑦) (35)

𝜙12″ (𝑦) + 𝑆𝑐 𝜙12

′ (𝑦) − 𝑆𝑐(𝑛 + 𝐶𝑟)𝜙12(𝑦) = −𝐴𝑆𝑐 𝜙02′ (𝑦) (36)

where 𝑁1 = 𝑅2 + 𝑛𝑃𝑟.

and respective boundary conditions are:

{ 𝑢11 = 0, 𝑢12 = 0, 𝜃11 = 1, 𝜃12 = 0, 𝜙11 = 1, 𝜙12 = 0 𝑎𝑡 𝑦 = 0

𝑢11 ⟶ 1, 𝑢12 ⟶ 0, 𝜃11 ⟶ 0, 𝜃12 ⟶ 0, 𝜙11 ⟶ 0, 𝜙12 ⟶ 0 𝑎𝑡 𝑦 ⟶ ∞ } (37)

Solving Eqs. (24) – (29) under the boundary conditions in Eq. (30) and Eqs. (31) -

(36) under the boundary conditions in Eq. (37) and using Eqs. (15) and (23), we

obtain the Velocity, Temperature and Concentration distributions in the boundary

layer as:

𝑢(𝑦, 𝑡) =

{

𝑃3𝑒−𝑚3𝑦 + 𝑃1𝑒

−𝑚2𝑦 + 𝑃2𝑒−𝑚1𝑦 + 1

+𝐸𝑐 { 𝐽8𝑒

−𝑚3𝑦 + 𝐽1𝑒−𝑚2𝑦 + 𝐽2𝑒

−2𝑚3𝑦 + 𝐽3𝑒−2𝑚2𝑦 + 𝐽4𝑒

−2𝑚1𝑦

+𝐽5𝑒−(𝑚2+𝑚3)𝑦 + 𝐽6𝑒

−(𝑚1+𝑚2)𝑦 + 𝐽7𝑒−(𝑚1+𝑚3)𝑦

}

+ 𝜀𝑒𝑛𝑡

[ {

𝐺6𝑒−𝑚6𝑦 + 𝐺1𝑒

−𝑚5𝑦 + 𝐺2𝑒−𝑚2𝑦 + 𝐺3𝑒

−𝑚4𝑦

+𝐺4𝑒−𝑚1𝑦 + 𝐺5𝑒

−𝑚3𝑦 + 1}

+𝐸𝑐

{

𝐿19𝑒−𝑚6𝑦 + 𝐿1𝑒

−𝑚5𝑦 + 𝐿2𝑒−𝑚2𝑦 + 𝐿3𝑒

−2𝑚3𝑦

+𝐿4𝑒−2𝑚2𝑦 + 𝐿5𝑒

−2𝑚1𝑦 + 𝐿6𝑒−(𝑚2+𝑚3)𝑦 + 𝐿7𝑒

−(𝑚1+𝑚2)𝑦

+𝐿8𝑒−(𝑚1+𝑚3)𝑦 + 𝐿9𝑒

−(𝑚3+𝑚6)𝑦 + 𝐿10𝑒−(𝑚3+𝑚5)𝑦

+𝐿11𝑒−(𝑚3+𝑚4)𝑦 + 𝐿12𝑒

−(𝑚2+𝑚6)𝑦 + 𝐿13𝑒−(𝑚2+𝑚5)𝑦

+𝐿14𝑒−(𝑚2+𝑚4)𝑦 + 𝐿15𝑒

−(𝑚1+𝑚6)𝑦 + 𝐿16𝑒−(𝑚1+𝑚5)𝑦

+𝐿17𝑒−(𝑚1+𝑚4)𝑦 + 𝐿18𝑒

−𝑚3𝑦 }

]

}


𝜃(𝑦, 𝑡) =

{

𝑒−𝑚2𝑦 + 𝐸𝑐 {

𝑆7𝑒−𝑚2𝑦 + 𝑆1𝑒

−2𝑚3𝑦 + 𝑆2𝑒−2𝑚2𝑦 + 𝑆3𝑒

−2𝑚1𝑦

+𝑆4𝑒−(𝑚2+𝑚3)𝑦 + 𝑆5𝑒

−(𝑚1+𝑚2)𝑦 + 𝑆6𝑒−(𝑚1+𝑚3)𝑦

}

+𝜀𝑒𝑛𝑡

[

{𝐷2𝑒−𝑚5𝑦 + 𝐷1𝑒

−𝑚2𝑦} +

𝐸𝑐

{

𝑅17𝑒−𝑚5𝑦 + 𝑅1𝑒

−𝑚2𝑦 + 𝑅2𝑒−2𝑚3𝑦 + 𝑅3𝑒

−2𝑚2𝑦

+𝑅4𝑒−2𝑚1𝑦 + 𝑅5𝑒

−(𝑚2+𝑚3)𝑦 + 𝑅6𝑒−(𝑚1+𝑚2)𝑦

+𝑅7𝑒−(𝑚1+𝑚3)𝑦 + 𝑅8𝑒

−(𝑚3+𝑚6)𝑦 + 𝑅9𝑒−(𝑚3+𝑚5)𝑦

+𝑅10𝑒−(𝑚3+𝑚4)𝑦 + 𝑅11𝑒

−(𝑚2+𝑚6)𝑦 +

𝑅12𝑒−(𝑚2+𝑚5)𝑦 + 𝑅13𝑒

−(𝑚2+𝑚4)𝑦 + 𝑅14𝑒−(𝑚1+𝑚6)𝑦

+𝑅15𝑒−(𝑚1+𝑚5)𝑦 + 𝑅16𝑒

−(𝑚1+𝑚4)𝑦 }

]

}

𝜙(𝑦, 𝑡) = 𝑒−𝑚1𝑦 + 𝜀𝑒𝑛𝑡{𝑍2𝑒−𝑚4𝑦 + 𝑍1𝑒

−𝑚1𝑦}

The Skin-friction, Nusselt number and Sherwood number are important physical

parameters for this type of boundary layer flow.

THE SKIN FRICTION

Knowing the velocity field, the Skin-friction at the plate can be obtained, which in

non-dimensional form is given by:

𝐶𝑓 =𝜏𝑤

𝜌𝑈0𝑉0= (

𝜕𝑢

𝜕𝑦 )𝑦=0

= ( 𝜕𝑢0𝜕𝑦

+ 𝜀𝑒𝑛𝑡𝜕𝑢1𝜕𝑦 )𝑦=0

=

[ −𝑚3𝑃3 −𝑚2𝑃1 −𝑚1𝑃2 + 𝐸𝑐 {

–𝑚3𝐽8 −𝑚2𝐽1 − 2𝑚3𝐽2 − 2𝑚2𝐽3 − 2𝑚1𝐽4 −(𝑚2 +𝑚3)𝐽5 − (𝑚1 +𝑚2)𝐽6 − (𝑚1 +𝑚3)𝐽7

}

+𝜀𝑒𝑛𝑡

[

(−𝑚6𝐺6 −𝑚5𝐺1 −𝑚2𝐺2 −𝑚4𝐺3 −𝑚1𝐺4 −𝑚3𝐺5)

+𝐸𝑐

{

−𝑚6𝐿19 −𝑚5𝐿1 −𝑚2𝐿2 − 2𝑚3𝐿3 − 2𝑚2𝐿4 − 2𝑚1𝐿5 −(𝑚2 +𝑚3)𝐿6 − (𝑚1 +𝑚2)𝐿7 − (𝑚1 +𝑚3)𝐿8

−(𝑚3 +𝑚6)𝐿9 − (𝑚3 +𝑚5)𝐿10 − (𝑚3 +𝑚4)𝐿11−(𝑚2 +𝑚6)𝐿12 − (𝑚2 +𝑚5)𝐿13 − (𝑚2 +𝑚4)𝐿14

−(𝑚1 +𝑚6)𝐿15 − (𝑚1 +𝑚5)𝐿16 − (𝑚1 +𝑚4)𝐿17 −𝑚3𝐿18}

]

]

RATE OF HEAT TRANSFER

Knowing the temperature field, the rate of heat transfer co-efficient can be obtained,

which in the non-dimensional form, in terms of the Nusselt number is given by:

𝑁𝑢 = −𝑥

(𝜕𝑇𝜕𝑦⁄)𝑦=0

𝑇𝑤 − 𝑇∞= 𝑁𝑢𝑅𝑒𝑥

−1 = −( 𝜕𝜃

𝜕𝑦 )𝑦=0

= −( 𝜕𝜃0𝜕𝑦

+ 𝜀𝑒𝑛𝑡𝜕𝜃1𝜕𝑦 )𝑦=0


= −

[ −𝑚2 + 𝐸𝑐 {

−𝑚2𝑆7 − 2𝑚3𝑆1 − 2𝑚2𝑆2 − 2𝑚1𝑆3 − (𝑚2 +𝑚3)𝑆4−(𝑚1 +𝑚2)𝑆5 − (𝑚1 +𝑚3)𝑆6

}

+𝜀𝑒𝑛𝑡

[

(−𝑚5𝐷2 −𝑚2𝐷1)

+𝐸𝑐

{

−𝑚5𝑅17 −𝑚2𝑅1 − 2𝑚3𝑅2 − 2𝑚2𝑅3 − 2𝑚1𝑅4−(𝑚2 +𝑚3)𝑅5 − (𝑚1 +𝑚2)𝑅6 − (𝑚1 +𝑚3)𝑅7−(𝑚3 +𝑚6)𝑅8 − (𝑚3 +𝑚5)𝑅9 − (𝑚3 +𝑚4)𝑅10−(𝑚2 +𝑚6)𝑅11 − (𝑚2 +𝑚5)𝑅12 − (𝑚2 +𝑚4)𝑅13−(𝑚1 +𝑚6)𝑅14 − (𝑚1 +𝑚5)𝑅15 − (𝑚1 +𝑚4)𝑅16}

]

]

where 𝑅𝑒𝑥 =𝑉0𝑥

𝜈 is the local Reynolds number.

RATE OF MASS TRANSFER

Knowing the concentration field, the rate of mass transfer co-efficient can be

obtained, which in the non-dimensional form, in terms of the Sherwood number is

given by:

𝑆ℎ = −𝑥

(𝜕𝐶𝜕𝑦⁄ )

𝑦=0

𝐶𝑤 − 𝐶∞ ,

𝑆ℎ𝑅𝑒𝑥−1 = −(

𝜕𝐶

𝜕𝑦 )𝑦=0

= −( 𝜕𝐶0𝜕𝑦

+ 𝜀𝑒𝑛𝑡𝜕𝐶1𝜕𝑦 )𝑦=0

= −[−𝑚1 + 𝜀𝑒𝑛𝑡(−𝑚4𝑍2 −𝑚1𝑍1)]

VALIDITY

When Cr = 0, the present paper reduces to the work which was done by Prasad and Reddy [23].

Table 1: Comparison of the present results with those of Prasad and Reddy [23] with

effects of Gr and Gm on Cf when Gr=2.0, Gm=1.0, Pr=0.71, Sc=0.6, M=1.0, R=0.5, K=0.5, n=0.1, Up=0.5, A=0.5, Cr=0.2, t=1.0, Ec=0.001, Ԑ=0.001.

Gr Gm Prasad and Reddy [23] Present work

Effects of

Gr on Cf Effect of Gm

on Cf Effects of Gr

on Cf Effects of

Gm on Cf

0

1

2

3

4

0

1

2

3

4

1.6877

2.0974

2.5123

2.9345

3.3660

1.9741

2.5123

3.0515

3.5918

4.1331

1.60691

2.04773

2.48857

2.92944

3.37035

2.03578

2.48857

2.94137

3.39418

3.84699


The Table 1 shows that the accuracy of the present model in comparison with the

previous model studied by Prasad and Reddy [23] and this comparison is validated the

present study.

RESULTS AND DISCUSSION

The formulation of the problem that accounts for the effects of radiation and viscous

dissipation on the flow of an incompressible viscous chemically reacting fluid along a

semi-infinite, vertically moving porous plate embedded in a porous medium in the

presence of transverse magnetic field was accomplished. Following Cogley et al. [22] approximation for the radiative heat flux in the optically thin environment, the

governing equations on the flow field were solved analytically, using a perturbation

method and the expressions for the velocity, temperature, concentration, Skin-friction,

Nusselt number and Sherwood number were obtained. In order to get a physical

insight of the problem, the above physical quantities are computed numerically for

different values of the governing parameters viz. Thermal Grashof number Gr, the Solutal Grashof number Gm, Radiation parameter R, Magnetic parameter M, Permeability parameter K, Plate velocity Up, Prandtl number Pr, Schmidt number Sc, Eckert number Ec and Chemical reaction Cr. Figure 1 shows the typical velocity profiles in the boundary layer for various values

of the thermal Grashof number. It is observed that an increase in Gr, leads to a rise in the values of the velocity due to enhancement in the buoyancy force. Here, the

positive values of Gr correspond to cooling of the plate. In addit0ion, it is observed that the velocity increases rapidly near the wall of the porous plate as Grashof number

increases and then decays to the free stream velocity. Figure 2 depicts the typical

velocity profiles in the boundary layer for distinct values of the solutal Grashof

number Gm. The velocity distribution attaints a distinctive maximum value in the region of the plate surface and then decrease properly to approach the free stream

value. As expected, the fluid velocity increases and the peak value becomes more

distinctive due to increase in the buoyancy force represented by Gm. For different values of thermal radiation parameter R on the velocity and temperature profiles are shown in Figure 3 and 4. It is noticed that an increase in the radiation

parameter results a decrease in the velocity and temperature within the boundary

layer, as well as decreased the thickness of the velocity and temperature boundary

layers.

The effect of magnetic field on velocity profiles in the boundary layer is depicted in

Figure 5. 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, which results in

reducing the velocity of the fluid in the boundary layer. Figure 6 shows the effect of

the permeability of the porous medium parameter K on the velocity distribution. It is found that the velocity increases with an increase in K.


The velocity distribution across the boundary layer for several values of plate moving

velocity Up in the direction of the fluid flow is depicted in Figure 7. Although we have different initial plate moving velocities, the velocity decreases to a constant

value for given material parameters.

Figure 8 and 9 shows the behaviour velocity and temperature for different values of

Prandtl number Pr. The numerical results show the effect of increasing values of Prandtl number results in the decreasing velocity. From Figure 9, it is observed that

an increase in the Prandtl number results a decrease in the thermal boundary layer

thickness and in general lower average temperature within the boundary layer. The

reason is that smaller values of Pr are equivalent to increase in the thermal conductivity of the fluid and therefore heat is able to diffuse away from the heated

surface more rapidly for higher values of Pr. Hence in the case of smaller Prandtl numbers as the thermal boundary layer is thicker and the rate of heat transfer is

reduced.

Figure 10 and 11 shows the effects of Schmidt number on the velocity and

concentration respectively. As the Schmidt number increases, the concentration

decreases. This causes the concentration buoyancy effects to decrease yielding a

reduction in the fluid velocity. Reductions in the velocity and concentration

distributions are accompanied by simultaneous reductions in the velocity and

concentration boundary layers.

The effects of chemical reaction on velocity and concentration are depicted by Figure

12 and 13. It is noticed that an increase in the chemical reaction parameter results a

decrease in the velocity and concentration within the boundary layer.

Table 2-5, represents the effects of Eckert number and Chemical reaction on the

velocity u, temperature Ө, Skin-friction Cf , Nusselt number Nu and Sherwood number Sh. The effects of viscous dissipation parameter i.e. the Eckert number on the velocity

and temperature are shown in Table 2 and 3. It is revealed that velocity and

temperature profiles scores grow with the increase of the Eckert number Ec. Eckert number, physically is a measure of frictional heat in the system. Hence the thermal

regime with large Ec values is subjected to rather more frictional heating causing a source of rise in the temperature. To be specific, the Eckert number Ec signifies the relative importance of viscous heating to thermal diffusion. Viscous heating may

serve as energy source to modify the temperature regime respectively. It is observed

from Table 4, when Eckert number increases the Skin-friction increases and Nusselt

number decreases. However, from Table 5, it can be seen that as the Chemical

reaction increases, the Skin-friction decreases and Sherwood number increases.


Table 2: Effects of Ec on velocity (u) when Gr=2.0, Gm=2.0, Pr=0.71, Sc=0.6, M=1.0, R=0.5, K=0.5, n=0.1, Up=0.5, A=0.5, Cr=0.2, t=1.0, Ԑ=0.001.

y Ec=0 Ec=0.1 Ec=0.2 Ec=0.3

0

1

2

3

4

5

0.5

1.33264

1.18374

1.07798

1.0318

1.01321

0.499889

1.33594

1.18577

1.07895

1.03221

1.01337

0.499779

1.33925

1.1878

1.07992

1.03262

1.01354

0.499668

1.34255

1.18983

1.08089

1.03303

1.0137


Table 3: Effects of Ec on temperature (Ө) when Gr=2.0, Gm=2.0, Pr=0.71, Sc=0.6, M=1.0, R=0.5, K=0.5, n=0.1, Up=0.5, A=0.5, Cr=0.2, t=1.0, Ԑ=0.001.

y Ec=0 Ec=0.1 Ec=0.2 Ec=0.3

0

1

2

3

4

5

1.00111

0.38005

0.144281

0.0547757

0.0207956

0.0078952

1.00111

0.386705

0.147783

0.0563569

0.0214404

0.00814729

1.00111

0.393361

0.151285

0.0579382

0.0220851

0.00839937

1.00111

0.400016

0.154787

0.0595194

0.0227298

0.00865146

Table 4: Effects of Ec on Cf and NuRex-1. Reference values in the figure 14 and 15:

Ec Cf NuRex-1

0

0.1

0.2

0.3

2.48849

2.4965

2.50451

2.51252

0.969636

0.881341

0.793046

0.70475

Table 5: Effects of Cr on Cf and NuRex-1. Reference values in the figure 12 and 13:

Cr Cf NuRex-1

0

0.3

0.6

0.9

2.51189

2.47906

2.45576

2.43759

0.800967

0.956863

1.08227

1.19017

CONCLUSIONS

The governing equations for unsteady MHD convective heat and mass transfer flow

past a semi-infinite vertical permeable moving plate embedded in a porous medium

with radiation and viscous dissipation effects were formulated .Chemical reaction

effects is also included in the present work. The plate velocity is maintained at

constant value and the flow is subjected to a transverse magnetic field. The present

investigation brings out the following conclusions of physical interest on the velocity,

temperature and concentration distribution of the flow field.

It is found that when thermal and solutal Grashof number is increased, the thermal and concentration buoyancy effects are enhanced and thus the fluid

velocity increased.

However, the presence of radiation effects caused reductions in the fluid temperature, which resulted in decrease in the fluid velocity.


It is observed that the existence of magnetic body force and chemical reaction decreases the fluid velocity.

The permeability parameter and plate velocity have the influence of increasing the fluid velocity.

As Prandtl number increased the velocity and temperature are both decreased. When Schmidt number increased, the concentration level decreased resulting

in decreased fluid velocity.

In presence of Eckert number both velocity and temperature increased.

NOMENCLATURE

𝑢 , 𝑣 Velocity components in 𝑥 , 𝑦 directions respectively,

𝑡 Time,

𝑝 Pressure,

𝑔 Acceleration due to gravity,

𝜅 Permeability of porous medium,

𝑇 Temperature of the fluid in the boundary layer,

𝑇∞ Temperature of the fluid far away from the plate,

𝐶 Species concentration in the boundary layer,

𝐶∞ Species concentration in the fluid far away from the plate,

𝐵𝑜 Magnetic induction,

𝑐𝑝 Specific heat at constant pressure,

𝑘 Thermal conductivity,

𝑞 Radiative heat flux,

𝜎∗ Stefan-Boltzmann constant,

D Mass diffusivity and

𝐶𝑟 Chemical reaction.

𝑢𝑝 Plate velocity,

𝑇𝑤 Temperature of the plate,

𝐶𝑤 Concentration of the plate,

𝑈∞ Free stream velocity,

𝑈0 Constant,

𝑛 Constant


A Real positive constant

𝑉0 Non-zero positive constant

GREEK SYMBOL

𝜌 Density,

𝛽𝑇 Thermal expansion co-efficient,

𝛽𝐶 Concentration expansion co-efficient,

𝜈 Kinematic viscosity,

𝜎 Electrical conductivity of the fluid,

𝛼 Fluid thermal diffusivity,

Ԑ small such that Ԑ ≪ 1

APPENDIX

𝑚1 =𝑆𝑐 + √1 + 4𝑆𝑐𝐶𝑟

2, 𝑚2 =

𝑃𝑟 + √𝑃𝑟2 + 4𝑅2

2, 𝑚3 =

1 + √1 + 4𝑁

2 ,

𝑚4 =𝑆𝑐 + √𝑆𝑐2 + 4𝑆𝑐(𝑛 + 𝐶𝑟)

2 ,𝑚5 =

𝑃𝑟 + √𝑃𝑟2 + 4𝑁12

,𝑚6

=1 + √1 + 4(𝑁 + 𝑛)

2 ,

𝑃1 =−𝐺𝑟

𝑚22 −𝑚2 − 𝑁

, 𝑃2 =−𝐺𝑚

𝑚12 −𝑚1 − 𝑁

, 𝑃3 = 𝑈𝑝 − 1 − 𝑃1 − 𝑃2 ,

𝐽1 =−𝐺𝑟𝑆7

𝑚22 −𝑚2 −𝑁

, 𝐽2 =−𝐺𝑟𝑆1

4𝑚32 − 2𝑚3 − 𝑁

, 𝐽3 =−𝐺𝑟𝑆2

4𝑚22 − 2𝑚2 −𝑁

,


4𝑚12 − 2𝑚1 − 𝑁

, 𝐽5 =−𝐺𝑟𝑆4

(𝑚2 +𝑚3)2 − (𝑚2 +𝑚3) − 𝑁 ,


(𝑚1 +𝑚2)2 − (𝑚1 +𝑚2) − 𝑁 , 𝐽7 =

−𝐺𝑟𝑆6(𝑚1 +𝑚3)2 − (𝑚1 +𝑚3) − 𝑁

,

𝐽8 = −(𝐽1 + 𝐽2 + 𝐽3 + 𝐽4 + 𝐽5 + 𝐽6 + 𝐽7), 𝐺1 =−𝐺𝑟𝐷2

𝑚52 −𝑚5 − (𝑁 + 𝑛)

,

𝐺2 =𝐴𝑚2𝑃1 − 𝐺𝑟𝐷1

𝑚22 −𝑚2 − (𝑁 + 𝑛)

, 𝐺3 =−𝐺𝑚𝑍2

𝑚42 −𝑚4 − (𝑁 + 𝑛)

, 𝐺4 =𝐴𝑚1𝑃2 − 𝐺𝑚𝑍1

𝑚12 −𝑚1 − (𝑁 + 𝑛)

,

𝐺5 =𝐴𝑚3𝑃3

𝑚32 −𝑚3 − (𝑁 + 𝑛)

, 𝐺6 = −(1 + 𝐺1 + 𝐺2 + 𝐺3 + 𝐺4 + 𝐺5) ,


𝐿1 =−𝐺𝑟𝑅17

𝑚52 −𝑚5 − (𝑁 + 𝑛)

, 𝐿2 =𝐴𝑚2𝐽1 − 𝐺𝑟𝑅1

𝑚22 −𝑚2 − (𝑁 + 𝑛)

, 𝐿3

=2𝐴𝑚3𝐽2 − 𝐺𝑟𝑅2

4𝑚32 − 2𝑚3 − (𝑁 + 𝑛)

,

𝐿4 =2𝐴𝑚2𝐽3 − 𝐺𝑟𝑅3

4𝑚22 − 2𝑚2 − (𝑁 + 𝑛)

, 𝐿5 =2𝐴𝑚1𝐽4 − 𝐺𝑟𝑅4

4𝑚12 − 2𝑚1 − (𝑁 + 𝑛)

,

𝐿6 =𝐴(𝑚2 +𝑚3)𝐽5 − 𝐺𝑟𝑅5

(𝑚2 +𝑚3)2 − (𝑚2 +𝑚3) − (𝑁 + 𝑛) , 𝐿7

=𝐴(𝑚1 +𝑚2)𝐽6 − 𝐺𝑟𝑅6

(𝑚1 +𝑚2)2 − (𝑚1 +𝑚2) − (𝑁 + 𝑛),

𝐿8 =𝐴(𝑚1 +𝑚3)𝐽7 − 𝐺𝑟𝑅7

(𝑚1 +𝑚3)2 − (𝑚1 +𝑚3) − (𝑁 + 𝑛), 𝐿9

=−𝐺𝑟𝑅8

(𝑚3 +𝑚6)2 − (𝑚3 +𝑚6) − (𝑁 + 𝑛) ,

𝐿10 =−𝐺𝑟𝑅9

(𝑚3 +𝑚5)2 − (𝑚3 +𝑚5) − (𝑁 + 𝑛) ,

𝐿11 =−𝐺𝑟𝑅10

(𝑚3 +𝑚4)2 − (𝑚3 +𝑚4) − (𝑁 + 𝑛) ,

𝐿12 =−𝐺𝑟𝑅11

(𝑚2 +𝑚6)2 − (𝑚2 +𝑚6) − (𝑁 + 𝑛) ,

𝐿13 =−𝐺𝑟𝑅12

(𝑚2 +𝑚5)2 − (𝑚2 +𝑚5) − (𝑁 + 𝑛) ,

𝐿14 =−𝐺𝑟𝑅13

(𝑚2 +𝑚4)2 − (𝑚2 +𝑚4) − (𝑁 + 𝑛) ,

𝐿15 =−𝐺𝑟𝑅14

(𝑚1 +𝑚6)2 − (𝑚1 +𝑚6) − (𝑁 + 𝑛) ,

𝐿16 =−𝐺𝑟𝑅15

(𝑚1 +𝑚5)2 − (𝑚1 +𝑚5) − (𝑁 + 𝑛) ,

𝐿18 =𝐴𝑚3𝐽8

𝑚32 −𝑚3 − (𝑁 + 𝑛)

, 𝐿17 =−𝐺𝑟𝑅16

(𝑚1 +𝑚4)2 − (𝑚1 +𝑚4) − (𝑁 + 𝑛) ,

𝐿19 = −(1 + 𝐿1 + 𝐿2 + 𝐿3 + 𝐿4 + 𝐿5 + 𝐿6 + 𝐿7 + 𝐿8 + 𝐿9 + 𝐿10+𝐿11 + 𝐿12 + 𝐿13 + 𝐿14 + 𝐿15 + 𝐿16 + 𝐿17 + 𝐿18

) ,

𝑆1 =−𝑃𝑟𝑚3

2𝑃32

4𝑚32 − 2𝑃𝑟𝑚3 − 𝑅2

, 𝑆2 =−𝑃𝑟𝑚2

2𝑃12

4𝑚22 − 2𝑃𝑟𝑚2 − 𝑅2

, 𝑆3 =−𝑃𝑟𝑚1

2𝑃22

4𝑚12 − 2𝑃𝑟𝑚1 − 𝑅2

,


𝑆4 =−2𝑃𝑟𝑚2𝑚3𝑃3𝑃1

(𝑚2 +𝑚3)2 − 𝑃𝑟(𝑚2 +𝑚3) − 𝑅2 , 𝑆5

=−2𝑃𝑟𝑚1𝑚2𝑃1𝑃2

(𝑚1 +𝑚2)2 − 𝑃𝑟(𝑚1 +𝑚2) − 𝑅2 ,

𝑆6 =−2𝑃𝑟𝑚3𝑚1𝑃2𝑃3

(𝑚1 +𝑚3)2 − 𝑃𝑟(𝑚1 +𝑚3) − 𝑅2 , 𝑆7 = −(𝑆1 + 𝑆2 + 𝑆3 + 𝑆4 + 𝑆5 + 𝑆6),

𝐷1 =𝑃𝑟𝐴𝑚2

𝑚22 − 𝑃𝑟𝑚2 − 𝑁1

, 𝐷2 = 1 − 𝐷1, 𝑅1 =𝑃𝑟𝐴𝑚2𝑆7

𝑚22 − 𝑃𝑟𝑚2 − 𝑁1

,

𝑅2 =2𝑃𝑟𝐴𝑚3𝑆1 − 2𝑃𝑟𝑚3

2𝐺5𝑃3

4𝑚32 − 2𝑃𝑟𝑚3 − 𝑁1

, 𝑅3 =2𝑃𝑟𝐴𝑚2𝑆2 − 2𝑃𝑟𝑚2

2𝐺2𝑃1

4𝑚22 − 2𝑃𝑟𝑚2 − 𝑁1

,

𝑅4 =2𝑃𝑟𝐴𝑚1𝑆3 − 2𝑃𝑟𝑚1

2𝐺4𝑃2

4𝑚12 − 2𝑃𝑟𝑚1 − 𝑁1

, 𝑅5

=𝑃𝑟𝐴(𝑚2 +𝑚3)𝑆4 − 2𝑃𝑟𝑚2𝑚3(𝐺2𝑃3 + 𝐺5𝑃1)

(𝑚2 +𝑚3)2 − 𝑃𝑟(𝑚2 +𝑚3) − 𝑁1 ,

𝑅6 =𝑃𝑟𝐴(𝑚1 +𝑚2)𝑆5 − 2𝑃𝑟𝑚1𝑚2(𝐺4𝑃1 + 𝐺2𝑃2)

(𝑚1 +𝑚2)2 − 𝑃𝑟(𝑚1 +𝑚2) − 𝑁1 ,

𝑅7 =𝑃𝑟𝐴(𝑚1 +𝑚3)𝑆6 − 2𝑃𝑟𝑚3𝑚1(𝐺4𝑃3 + 𝐺5𝑃2)

(𝑚1 +𝑚3)2 − 𝑃𝑟(𝑚1 +𝑚3) − 𝑁1 ,

𝑅8 =2𝑃𝑟𝑚3𝑚6𝐺6𝑃3

(𝑚3 +𝑚6)2 − 𝑃𝑟(𝑚3 +𝑚6) − 𝑁1 , 𝑅9 =

2𝑃𝑟𝑚3𝑚5𝐺1𝑃3(𝑚3 +𝑚5)2 − 𝑃𝑟(𝑚3 +𝑚5) − 𝑁1

,


(𝑚3 +𝑚4)2 − 𝑃𝑟(𝑚3 +𝑚4) − 𝑁1 , 𝑅11

=2𝑃𝑟𝑚2𝑚6𝐺6𝑃1

(𝑚2 +𝑚6)2 − 𝑃𝑟(𝑚2 +𝑚6) − 𝑁1 ,


(𝑚2 +𝑚5)2 − 𝑃𝑟(𝑚2 +𝑚5) − 𝑁1 , 𝑅13


(𝑚2 +𝑚4)2 − 𝑃𝑟(𝑚2 +𝑚4) − 𝑁1,


(𝑚1 +𝑚6)2 − 𝑃𝑟(𝑚1 +𝑚6) − 𝑁1 , 𝑅15


(𝑚1 +𝑚5)2 − 𝑃𝑟(𝑚1 +𝑚5) − 𝑁1


(𝑚1 +𝑚4)2 − 𝑃𝑟(𝑚1 +𝑚4) − 𝑁1 ,

𝑅17 = −(𝑅1 + 𝑅2 + 𝑅3 + 𝑅4 + 𝑅5 + 𝑅6 + 𝑅7 + 𝑅8 + 𝑅9

+𝑅10 + 𝑅11 + 𝑅12 + 𝑅13 + 𝑅14 + 𝑅15) ,


𝑍1 =𝐴𝑚1𝑆𝑐

𝑚12 − 𝑆𝑐 𝑚1 − 𝑆𝑐(𝑛 + 𝐶𝑟)

, 𝑍2 = 1 − 𝑍1.

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