soret and dufour effects on radiative ......convection with soret and dufour effects past a vertical...

J. Korean Soc. Ind. Appl. Math. Vol.23, No.3, 157–178, 2019 http://doi.org/10.12941/jksiam.2019.23.157

SORET AND DUFOUR EFFECTS ON RADIATIVE HYDROMAGNETIC FLOWOF A CHEMICALLY REACTING FLUID OVER AN EXPONENTIALLYACCELERATED INCLINED POROUS PLATE IN PRESENCE OF HEAT

ABSORPTION AND VISCOUS DISSIPATION

M. VENKATESWARLU 1†, P. BHASKAR 2, AND D. VENKATA LAKSHMI 3

1DEPARTMENT OF MATHEMATICS, V. R. SIDDHARTHA ENGINEERING COLLEGE, KRISHNA (DIST), A. P,INDIA, PIN: 520 007

Email address: [email protected]

2DEPARTMENT OF MATHEMATICS, SHRI VENKATESHWARA UNIVERSITY, GAJRAULA, AMROHA (DIST),U. P, INDIA, PIN: 244 236

Email address: [email protected]

3DEPARTMENT OF MATHEMATICS, BAPATLA WOMEN’S ENGINEERING COLLEGE, BAPATLA, GUNTUR

(DIST), A. P, INDIA, PIN: 522 102Email address: [email protected]

ABSTRACT. The present correspondence is conveyed on to consider the fascinating and novelcharacteristics of radiative hydromagnetic convective flow of a chemically reacting fluid overan exponentially accelerated inclined porous plate. Exact solutions for the fluid velocity, tem-perature and species concentration, under Boussinesq approximation, are obtained in closedform by the two term perturbation technique. The interesting parts of thermal dispersing out-comes are accounted in this correspondence. Graphical evaluation is appeared to depict thetrademark direct of introduced parameters on non dimensional velocity, temperature and con-centration profiles. Also, the numerical assortment for skin friction coefficient, Nusselt numberand Sherwood number is examined through tables. The certification of current examination isconfirmed by making an examination with past revelations available in composing, which setsa benchmark for utilization of computational approach.

1. INTRODUCTION

In recent years the problems of heat and mass transfer in presence of magnetic field througha porous medium have been attracted, the attention of several researchers because of their im-portant applications in engineering and technology, such as compact heat exchangers, catalyticreactors, solar power collectors, paper production, rocket boosters, geothermal systems, heatinsulation and film vaporization in combustion chambers. Li et al. [1] studied the compact heat

Received by the editors March 10 2019; Accepted September 4 2019; Published online September 25 2019.2000 Mathematics Subject Classification. 78A40, 80A32, 80A20, 76DXX.Key words and phrases. Soret effect, Dufour effect, Inclined plate, Heat absorption, Viscous dissipation.† Corresponding author.

157

158 VENKATESWARLU, BHASKAR, AND VENKATA LAKSHMI

exchangers and future applications for a new generation of high temperature solar receivers.Hunt and Tien [2] presented the non-darcian flow, heat and mass transfer in catalytic packed-bed reactors. Leccese et al. [3] considered the convective and radiative wall heat transfer inliquid rocket thrust chambers. Khairnasov and Naumova [4] presented the heat pipes applica-tion to solar energy systems.

In several heat and mass transfer problems, the Dufour and Soret effects are neglected, onthe basis that they are of a smaller order of magnitude than the effects described by Fourier’sand Fick’s laws. These effects are considered as second order phenomena and are significantin areas such as chemical reactors, drying processes, hydrology, geosciences, petrology etc.Some notable research studies on the topic are due to Postelnicu [5], Alam et al. [6], Afify [7],Venkateswarlu et al. [8], Olanrewaju and Makinde [9]. Makinde [10] studied the MHD mixedconvection with soret and dufour effects past a vertical plate embedded in a porous medium.Venkateswarlu and Padma [11] presented the unsteady MHD free convective heat and masstransfer in a boundary layer flow past a vertical permeable plate with thermal radiation andchemical reaction. Venkateswarlu et al. [12] discussed the the influence of Hall current andheat source on MHD flow of a rotating fluid in a parallel porous plate channel. Venkateswarluand Venkata Lakshmi [13] presented the Dufour and chemical reaction effects on unsteadyMHD flow of a viscous fluid in a parallel porous plate channel under the influence of slipcondition. Seth and Sarkar [14] considered the hydromagnetic natural convection flow withinduced magnetic field and nth order chemical reaction of a heat absorbing fluid past an impul-sively moving vertical plate with ramped temperature. Anjali Devi and Ganga [15] studied theeffects of viscous and joules dissipation on MHD flow, heat and mass transfer past a stretchingporous surface embedded in a porous medium.

It has been observed that combined heat and mass transfer flow with chemical reaction andthermal radiation is significant in many industrial and engineering applications, such as energytransfer in a wet cooling tower, evaporation at the surface of a water body, drying, and theflow in a desert cooler. Some of the relevant research studies are due to Hussanan et al. [16],Vedavathi et al. [17], Seth et al. [18], Venkatateswarlu et al. [19], Alam et al. [20], Tripathy etal. [21]. Venkateswarlu and Makinde [22] studied the unsteady MHD slip flow with radiativeheat and mass transfer over an inclined plate embedded in a porous medium. GnaneswaraReddy et al. [23] presented the effects of viscous dissipation and heat source on unsteadyMHD flow over a stretching sheet. Venkateswarlu et al. [24] discussed the effects of chemicalreaction and heat generation on MHD boundary layer flow of a moving vertical plate withsuction and dissipation. Malapati and Dasari [25] presented the Soret and chemical reactioneffects on the radiative MHD flow from an infinite vertical plate.

The objective of the present paper is to discuss the Soret and Dufour effects on radiativehydromagnetic flow of a chemically reacting fluid over an exponentially accelerated inclinedporous plate with heat absorption and viscous dissipation. The behaviors of fluid velocity,temperature, concentration, skin-friction coefficient, Nusselt number and Sherwood numberhas been discussed in detail for variation of thermo physical parameters. The following strategyis pursued in the rest of the paper. Section two presents the formation of the problem. The

SORET AND DUFOUR EFFECTS ON RADIATIVE HYDROMAGNETIC FLOW 159

analytical solutions are presented in section three. Results are discussed in section four andfinally section five provides a conclusion of the paper.

2. FORMATION OF THE PROBLEM

We consider an unsteady, radiative MHD natural convection flow of a viscous, incompress-ible, heat absorption and electrically conducting fluid over an exponentially accelerated in-clined porous plate in the sight of observing consequences of Soret and Dufour effects. Thephysical model and coordinate system is shown in Fig. 1. The x - coordinate be taken in ver-tically upward direction along the plate with the angle of inclination α and y - coordinate istaken normal to the plate. In the energy equation, the dissipation of viscosity is considered. Itis assumed that there exist a first order homogeneous chemical reaction with constant rate K∗

r

between the fluid and diffusing species. The thin fluid is considered for the study is gray withabsorbing/emitting radiation within a form of non- scattering medium. Initially at time t ≤ 0,both the fluid and plate are at rest with uniform temperature T∞ and uniform concentrationC∞. At time t > 0 , plate starts with exponential velocity u0 exp(iat) in x - direction. Underthese assumptions, the boundary layer equations present the fluid flow, that is the momentum,energy and solutal concentration can be written as like Malapati and Dasari [25].

Figure 1: Schematic flow of the problem.

Continuity equation:∂v

∂y= 0

Momentum equation:

∂u

∂t= ν

∂2u

∂y2− σeB

20

ρu+ gβT (T − T∞) cosα+ gβC(C − C∞) cosα− ν

K1u (2.1)


0 = −1

ρ

∂p

∂y− gβT (T − T∞) sinα− gβC(C − C∞) sinα (2.2)

Energy equation:

∂T

∂t=KT

ρcp

∂2T

∂y2+DmKT

cscp

∂2C

∂y2− Q0

ρcp(T − T∞)− 1

ρcp

∂qr∂y

+ν

cp

[∂u

∂y

]2(2.3)

Concentration equation:

∂C

∂t= Dm

∂2C

∂y2+DmKT

Tm

∂2T

∂y2−K∗

r (C − C∞) (2.4)

where u - fluid velocity in x- direction, v - fluid velocity in y- direction, t - non dimensionaltime, ν - kinematic viscosity of the fluid, σe - fluid electrical conductivity , B0 - magneticinduction, ρ - fluid density, g - acceleration due to gravity, βT - coefficient of thermal expansion,T - fluid temperature, T∞ - uniform temperature, α - angle of inclination, βC - coefficient ofconcentration expansion, C - species concentration in the fluid, C∞ - uniform concentration,K1 - permeability of porous medium, p - fluid pressure, KT - thermal diffusivity of the fluid,Dm - chemical molecular diffusivity, cp - specific heat at constant pressure, cs - concentrationsusceptibility, Q0 - dimensional heat absorption parameter, qr - radiating flux vector, Tm -mean fluid temperature and K∗

r - dimensional chemical reaction parameter respectively.The pressure gradient term −1

ρ∂p∂x is absent in Eq. (2.1) because fluid flow is induced due

to the movement of plate as well as buoyancy forces acting in x - direction. Equation (2.2)implies that fluid pressure p - varies in y - direction due to presence of buoyancy forces actingin y - direction. This means that change in fluid pressure from plate to the edge of the boundarylayer varies due to change in thermal and solutal buoyancy forces.

We should in prior warn the reader that our model is not the same as that by Malapati andDasari [25] in which the angle of inclination, Dufour effect and viscous dissipation were nottaken into account. The corresponding initial and boundary conditions of the system of partialdifferential equations are given below

t ≤ 0 : u = 0, T = T∞, C = C∞ for all y ≥ 0

t > 0 : u = uoexp(iat), T = Tw +(T−T∞)u2

0(Tw−T∞)cp

exp(int),

C = Cw +(C−C∞)u2

0(Tw−T∞)cp

exp(int) at y = 0

u→ 0, T → T∞, C → C∞ as y → ∞

where uo - characteristic velocity, a - dimensional acceleration parameter, Tw - fluid tem-

perature at the surface of the plate, Cw - species concentration at the surface of the plate and n- frequency of oscillation.

For an optically thick radiating fluid, we adopting the Rosseland approximation for radiatingflux vector as (see, Magyari and Pantokratoras [26], Venkateswarlu et al. [27])

qr = −4σ∗

3a∗∂T 4

∂y(2.5)


Here a∗ - mean absorption coefficient and σ∗ - Stefan Boltzmann constant. We assume thatthe difference between fluid temperature T and free stream temperature T∞ within the flow issufficiently small such that T 4 may be expressed as a linear function of the temperature. Thisis accomplished by expanding T 4 in Taylor series about the free stream temperature T∞ andneglecting the second and higher order terms, we have

T 4 = 4T 3∞T − 3T 4

∞ (2.6)Using Eqs. (2.5) and (2.6) in Eq. (2.3), we obtain

∂T

∂t=KT

ρcp

[1 +

16σ∗T 3∞

3a∗KT

]∂2T

∂y2+DmKT

cscp

∂2C

∂y2− Q0

ρcp(T − T∞) +

ν

cp

[∂u

∂y

]2(2.7)

We introduce the following non-dimensional variables

η =u0νy, ψ =

u

u0, ω =

ν

u20n, τ =

u20νt, b =

ν

u20a, θ =

T − T∞Tw − T∞

, ϕ =C − C∞Cw − C∞

(2.8)

It may be noticed that characteristic velocity u0 may be defined according to the non-dimensional process mentioned above as

u0 = [gβT ν(Tw − T∞)]13 (2.9)

By substituting Eqs. (2.8) and (2.9) into Eqs. (2.1), (2.4) and (2.7), we obtain the followingnon-dimensional partial differential equations

∂ψ

∂τ=∂2ψ

∂η2−[M +

1

K

]ψ + θcosα+Nϕcosα (2.10)

∂θ

∂τ=

[1 +R

Pr

]∂2θ

∂η2+Du

∂2ϕ

∂η2−Hθ + Ec

[∂ψ

∂η

]2(2.11)

∂ϕ

∂τ=

1

Sc

∂2ϕ

∂η2+ Sr

∂2θ

∂η2−Krϕ (2.12)

Here N = βC(Cw−C∞)βT (Tw−T∞) is the ratio of concentration buoyancy force and the thermal buoy-

ancy force, M =σeB2

0ν

ρu20

is the magnetic parameter, K =K1u2

0ν2

is the permeablity param-

eter, Pr =ρcpνKT

is the Prandtl number, R = 16a∗T 3∞

3a∗KTis the radiation parameter, Du =

DmKT (Cw−C∞)cscpν(Tw−T∞) is the Dufour number, H = Q0ν

ρcpu20

is the heat absorption parameter, Ec =

u20

cp(Tw−T∞) is the Eckert number, Sc = νDm

is the Schmidt number, Sr = DmKT (Tw−T∞)Tmν(Cw−C∞) is the

Soret number and Kr = νu20k∗r is the chemical reaction parameter respectively.

The corresponding initial and boundary conditions can be written as

τ ≤ 0 : ψ = 0, , θ = 0, ϕ = 0 for all η ≥ 0τ > 0 : ψ = exp(ibτ), θ = 1 + Ec exp(iωτ), ϕ = 1 + Ec exp(iωτ) at η = 0ψ → 0, θ → 0, ϕ→ 0 as η → ∞


It is now important to calculate physical quantities of primary interest, which are the localwall shear stress or skin friction coefficient, the local surface heat flux and the local surfacemass flux. Given the velocity, temperature and concentration fields in the boundary layer, theshear stress τw , the heat flux qw and mass flux jw are obtained as

τw = µ

[∂u

∂y

]y=0

(2.13)

qw = −KT

[∂T

∂y

]y=0

(2.14)

jw = −Dm

[∂C

∂y

]y=0

(2.15)

In non-dimensional form the skin-friction coefficient Cf , heat transfer coefficient Nu andmass transfer coefficient Sh are defined as

Cf =τwρu20

(2.16)

Nu =ν

uo

qwKT (Tw − T∞)

(2.17)

Sh =ν

uo

jwDm(Cw − C∞)

(2.18)

Using non-dimensional variables in Eq. (2.8) and Eqs. (2.13)-(2.15) into Eqs. (2.16)-(2.18),we obtain the physical parameters

Cf =

[∂ψ

∂η

]η=0

Nu = −[∂θ

∂η

]η=0

Sh = −[∂ϕ

∂η

]η=0

3. SOLUTION OF THE PROBLEM

Equations (2.10)-(2.12) are coupled non-linear partial differential equations and these cannotbe solved in closed form. So, we reduce these non-linear partial differential equations into aset of ordinary differential equations, which can be solved analytically. This can be done byassuming the trial solutions for the velocity, temperature and concentration of the fluid as (see,Siva Kumar et al. [28], Venkateswarlu et al. [29])

ψ(η, τ) = ψ0(η) + Ec exp(iωτ)ψ1(η) +O(Ec2) (3.1)

θ(η, τ) = θ0(η) + Ec exp(iωτ)θ1(η) +O(Ec2) (3.2)ϕ(η, τ) = ϕ0(η) + Ec exp(iωτ)ϕ1(η) +O(Ec2) (3.3)


Substituting Eqs. (3.1)-(3.3) into Eqs. (2.10)-(2.12), then equating the harmonic and non-harmonic terms and neglecting the higher order terms of 0(Ec2), we obtain

ψ′′0 − [M +

1

K]ψ0 = −[θocosα+Nϕocosα] (3.4)

ψ′′1 − [M +

1

K+ iω]ψ1 = −[θ1cosα+Nϕ1cosα] (3.5)

θ′′0 −

[HPr

1 +R

]θ0 = −

[PrDu

1 +R

]ϕ

′′0 (3.6)

θ′′1 −

[Pr(H + iω)

1 +R

]θ1 = −

[PrDu

1 +R

]ϕ

′′1 −

[PrEc

1 +R

]ψ

′2o (3.7)

ϕ′′0 − ScKrϕ0 = −ScSrθ′′0 (3.8)

ϕ′′1 − Sc[Kr + iω]ϕ1 = −ScSrθ′′1 (3.9)

where the prime denotes the ordinary differentiation with respect to η .The corresponding initial and boundary conditions can be written as

ψ0 = exp(ibτ), ψ1 = 0, θ0 = 1, θ1 = 1, ϕ0 = 1, ϕ1 = 1 at η = 0ψ0 → 0, ψ1 → 0, θ0 → 0, θ1 → 0, ϕ0 → 0, ϕ1 → 0 as η → ∞

}To slove the non linear differential equations (3.4)-(3.9), we assume that the Dufour number

is small, so we can use Du << 1 as the perturbation parameter.

ψ0(η, τ) = ψ00(η) +Du ψ01(η) +O(Du2) (3.10)

ψ1(η, τ) = ψ10(η) +Du ψ11(η) +O(Du2) (3.11)

θ0(η, τ) = θ00(η) +Du θ01(η) +O(Du2) (3.12)

θ1(η, τ) = θ10(η) +Du θ11(η) +O(Du2) (3.13)

ϕ0(η, τ) = ϕ00(η) +Du ϕ01(η) +O(Du2) (3.14)

ϕ1(η, τ) = ϕ10(η) +Du ϕ11(η) +O(Du2) (3.15)Substituting Eqs. (3.10)-(3.15) into Eqs. (3.4)-(3.9), then equating the harmonic and non-

harmonic terms and neglecting the higher order terms of 0(Du2), we obtain

ψ′′00 − [M +

1

K]ψ00 = −[θ00 cosα+Nϕ00 cosα] (3.16)

ψ′′01 − [M +

1

K]ψ01 = −[θ01 cosα+Nϕ01 cosα] (3.17)

ψ′′10 − [M +

1

K+ iω]ψ10 = −[θ10 cosα+Nϕ10 cosα] (3.18)

ψ′′11 − [M +

1

K+ iω]ψ11 = −[θ11 cosα+Nϕ11 cosα] (3.19)

θ′′00 −

[HPr

1 +R

]θ00 = 0 (3.20)


θ′′01 −

[HPr

1 +R

]θ01 = −

[Pr

1 +R

]ϕ

′′00 (3.21)

θ′′10 −

[Pr(H + iω)

1 +R

]θ10 = −

[PrEc

1 +R

]ψ

′200 (3.22)

θ′′11 −

[Pr(H + iω)

1 +R

]θ11 = −

[Pr

1 +R

]ϕ

′′10 −

[2PrEc

1 +R

]ψ

′00ψ

′01 (3.23)

ϕ′′00 − ScKrϕ00 = −ScSrθ′′00 (3.24)

ϕ′′01 − ScKrϕ01 = −ScSrθ′′01 (3.25)

ϕ′′10 − Sc[Kr + iω]ϕ10 = −ScSrθ′′10 (3.26)

ϕ′′11 − Sc[Kr + iω]ϕ11 = −ScSrθ′′11 (3.27)

The corresponding initial and boundary conditions can be written as

ψ00 = exp(ibτ), ψ01 = 0, ψ10 = 0, ψ11 = 0, θ00 = 1, θ01 = 0,θ10 = 1, θ11 = 0, ϕ00 = 1, ϕ01 = 0, ϕ10 = 1, ϕ11 = 0 at η = 0ψ00 → 0, ψ01 → 0, ψ10 → 0, ψ11 → 0, θ00 → 0, θ01 → 0,θ10 → 0, θ11 → 0, ϕ00 → 0, ϕ01 → 0, ϕ10 → 0, ϕ11 → 0 as η → ∞

(3.28)

The analytical solutions of Eqs. (3.16)-(3.27) with the boundary conditions in Eq. (3.28), aregiven by

ψ00 = A6 exp(−m1η) +A7 exp(−m2η) +A8 exp(−m3η) (3.29)ψ01 = A23 exp(−m1η)−A22 exp(−m2η) +A24 exp(−m3η) (3.30)

ψ10 = A48 exp(−m6η)−A40 exp(−m4η)−A41 exp(−m5η)

+A42 exp(−2m3η) +A43 exp(−2m1)η +A44 exp(−2m2η) +

A45 exp[−(m1 +m3)]η +A46 exp[−(m1 +m2)]η +A47 exp[−(m2 +m3)]η (3.31)

ψ11 = A78 exp(−m6η) +A69 exp(−m4η) +A71 exp(−m5η)

+A72 exp(−2m1η) +A73 exp(−2m2)η +A74 exp(−2m3η) +

A75 exp[−(m1 +m2)]η +A76 exp[−(m2 +m3)]η +A77 exp[−(m1 +m3)]η (3.32)

θ00 = exp(−m1)η (3.33)θ01 = [A9 + ηA10] exp(−m1)η −A9 exp(−m2)η (3.34)

θ10 = A31 exp(−m4η)−A26 exp(−2m1η)−A27 exp(−2m2)η

−A25 exp(−2m3η)−A29 exp[−(m1 +m2)]η −A30 exp[−(m2 +m3)]η −A28 exp[−(m1 +m3)]η (3.35)

θ11 = A57 exp(−m4η)−A49 exp(−m5η)−A50η exp(−m4η)

−A51 exp(−2m1η)−A52 exp(−2m2)η −A53 exp(−2m3η)−A54 exp[−(m1 +m2)]η −A55 exp[−(m2 +m3)]η −A56 exp[−(m1 +m3)]η (3.36)


ϕ00 = A4 exp(−m1η) +A5 exp(−m2η) (3.37)ϕ01 = A13 exp(−m1η)− [A13 +A11η] exp(−m2η) (3.38)

ϕ10 = A39 exp(−m5η)−A32 exp(−m4η) +A33 exp(−2m3η) +

A34 exp(−2m1)η +A35 exp(−2m2η) +A36 exp[−(m1 +m3)η] +

A37 exp[−(m1 +m2)η] +A38 exp[−(m2 +m3)η] (3.39)

ϕ11 = [A67 −A60η] exp(−m5η) +A59 exp(−m4η) +A61 exp(−2m1η) +

A62 exp(−2m2)η +A63 exp(−2m3η) +A64 exp[−(m1 +m2)]η −A65 exp[−(m2 +m3)]η +A66 exp[−(m1 +m3)]η (3.40)

By substituting Eqs. (3.29)-(3.40) into Eqs. (3.1)-(3.3), we obtained solutions for the fluidvelocity, temperature and concentration and are presented in the following form

ψ(η, τ) = [ψ00 +Du ψ01] + Ec exp(iωτ)[ψ10 +Du ψ11]

θ(η, τ) = [θ00 +Du θ01] + Ec exp(iωτ)[θ10 +Du θ11]

ϕ(η, τ) = [ϕ00 +Du ϕ01] + Ec exp(iωτ)[ϕ10 +Du ϕ11]

3.1 Skin friction: From the velocity field, the skin friction at the plate can be obtained, whichis given in non dimensional form as

Cf = [ψ′00 +Du ψ

′01]η=0 + Ec exp(iωτ)[ψ

′10 +Du ψ

′11]η=0

3.2 Nusselt number : From temperature field, we obtained heat transfer coefficient which isgiven in non-dimensional form as

Nu = [θ′00 +Du θ

′01]η=0 + Ec exp(iωτ)[θ

′10 +Du θ

′11]η=0

3.3 Sherwood number : From concentration field, we obtained mass transfer coefficient whichis given in non-dimensional form as

Sh = [ϕ′00 +Du ϕ

′01]η=0 + Ec exp(iωτ)[ϕ

′10 +Du ϕ

′11]η=0

4. RESULTS AND DISCUSSION

The impacts of various governing non dimensional parameters are examined, namely thebuoyancy parameterN , magnetic parameterM , permeability parameterK, angle of inclinationα, Prandtl number Pr, radiation parameter R, Dufour number Du, heat absorption parameterH , Eckert numberEc, Schmidt number Sc, Soret number Sr, chemical reaction parameterKrand acceleration parameter b in transit of flow field, fluid velocity ψ, temperature θ and solutalconcentration ϕ are studied graphically in Figs 2-21. The comparative results are reportedin Table 1. The results obtained reveals that our results and those of Malapati and Dasrari[25] are very good in agreement. The numerical values of skin friction coefficient Cf , heattransfer coefficient Nu and mass transfer coefficient Sh are presented in tables 2 and 3. Inthe present investigation following parameter values are adopted for computations: α = π

4 ,


Spanwise coordinate η

0 0.5 1 1.5 2 2.5 3 3.5 4

Vel

ocity

ψ

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

N = 0.5, 1.0, 1.5, 2.0

M = 4.0M = 5.0

Figure 2: Influence of buoyancy parameteron velocity profiles.


0 0.5 1 1.5 2 2.5 3 3.5 4

Vel

ocity

ψ

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

K = 0.1, 0.2, 0.3, 0.4

M = 4.0M = 5.0

Figure 3: Influence of permeability param-eter on velocity profiles.


0 0.5 1 1.5 2 2.5 3 3.5 4

Vel

ocity

ψ

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

α = 0, π/8, π/6, π/4

M = 4.0M = 5.0

Figure 4: Influence of angle of inclinationon velocity profiles.


0 0.5 1 1.5 2 2.5 3 3.5 4

Vel

ocity

ψ

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pr = 0.71, 1.00, 2.00, 3.00

M = 4.0M = 5.0

Figure 5: Influence of Prandtl number onvelocity profiles.

N = 2, M = 5, K = 0.5, Pr = 0.71, R = 1.0, Du = 0.1, H = 0.2, Ec = 0.2, Sc = 0.22,Sr = 1.0, Kr = 0.5, τ = 1.0, b = 0.2, ω = 1.0. Hence all the graphs and tables correspondto these values unless specifically indicated on the appropriate graph or table.

It is noticed from the Figs. 2-13 that, the fluid velocity ψ corresponding to M = 4 is higherthan the fluid velocity ψ corresponding to M = 5. Hence velocity ψ decreases on increasingthe magnetic parameter M . It is noticed from the Figs. 14-18 that, the fluid temperature θcorresponding to air (Pr = 0.71) is greater than the fluid temperature θ corresponding to elec-trolytic solution (Pr = 1.00) with the parameters R, H , Ec and Kr whereas Du has a reverse



0 0.5 1 1.5 2 2.5 3 3.5 4

Vel

ocity

ψ

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

R = 1, 2, 3, 4

M = 4.0M = 5.0

Figure 6: Influence of radiation parameteron velocity profiles.


0 0.5 1 1.5 2 2.5 3 3.5 4

Vel

ocity

ψ

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Du = 0.5, 0.6, 0.7, 0.8

M = 4.0M = 5.0

Figure 7: Influence of Dufour number onvelocity profiles.


0 0.5 1 1.5 2 2.5 3 3.5 4

Vel

ocity

ψ

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

H = 0.2, 0.4, 0.6, 0.8

M = 4.0M = 5.0

Figure 8: Influence of heat absorption pa-rameter on velocity profiles.


0 0.5 1 1.5 2 2.5 3 3.5 4

Vel

ocity

ψ

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Ec = 0.2, 0.4, 0.6, 0.8

M = 4.0M = 5.0

Figure 9: Influence of Eckert number on ve-locity profiles.

trend. It is observed from the Figs. 19-21 that, the species concentration ϕ corresponding toHydrogen (Sc = 0.22) is greater than the species concentration ϕ corresponding to Ammonia(Sc = 0.78). Figure 2 depicts the influence of buoyancy parameter N on the fluid velocity ψ.The buoyancy parameterN defines the ratio of Solutal Grashof numberGm to thermal Grashofnumber Gr. It is evident from Fig. 2. velocity ψ increases on increasing the buoyancy param-eter N . Figure 3 shows that an increase in the permeability parameter K results an increase inthe velocity component ψ. For large porosity of the medium fluid acquires more space to flowas a consequence its velocity increases. It is observed from Fig. 4 that the velocity ψ is getting



0 0.5 1 1.5 2 2.5 3 3.5 4

Vel

ocity

ψ

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Sc = 0.22, 0.30, 0.60, 0.78

M = 4.0M = 5.0

Figure 10: Influence of Schmidt number onvelocity profiles.


0 0.5 1 1.5 2 2.5 3 3.5 4

Vel

ocity

ψ

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Sr = 0.5, 1.0, 1.5, 2.0

M = 4.0M = 5.0

Figure 11: Influence of Soret number on ve-locity profiles.


0 0.5 1 1.5 2 2.5 3 3.5 4

Vel

ocity

ψ

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Kr = 1, 2, 3, 4

M = 4.0M = 5.0

Figure 12: Influence of chemical reactionparameter on velocity profiles.


0 0.5 1 1.5 2 2.5 3 3.5 4

Vel

ocity

ψ

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

b = 0.2, 0.4, 0.6, 0.8

M = 4.0M = 5.0

Figure 13: Influence of acceleration param-eter on velocity profiles.

decelerated on increasing the angle of inclination α. This is because the effect of buoyancyforce is getting reduced due to multiplication of the term cos θ in the buoyancy force term.Since cos θ assumes maximum value when θ = 0 and it is getting reduced as θ increases. Dueto this reason strength of buoyancy forces is getting reduced on increasing angle of inclinationof the plate.

Figure 5 shows the fluid velocity ψ for different estimations of the Prandtl number Pr.Prandtl number characterizes the relative adequacy of the momentum transport by dispersionin the hydrodynamic boundary layer to the essentialness transported by thermal dispersion in



0 0.5 1 1.5 2 2.5 3 3.5 4

Tem

pera

ture

θ

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

R = 1, 2, 3, 4

Pr = 0.71Pr = 1.00

Figure 14: Influence of radiation parameteron temperature profiles.


0 0.5 1 1.5 2 2.5 3 3.5 4

Tem

pera

ture

θ

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

Du = 0.5, 0.6, 0.7, 0.8

Pr = 0.71Pr = 1.00

Figure 15: Influence of Dufour number ontemperature profiles.


0 0.5 1 1.5 2 2.5 3 3.5 4

Tem

pera

ture

θ

0

0.2

0.4

0.6

0.8

1

1.2

H = 0.2, 0.4, 0.6, 0.8

Pr = 0.71Pr = 1.00

Figure 16: Influence of heat absorption pa-rameter on temperature profiles.


0 0.5 1 1.5 2 2.5 3 3.5 4

Tem

pera

ture

θ

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Ec = 0.2, 0.4, 0.6, 0.8

Pr = 0.71Pr = 1.00

Figure 17: Influence of Eckert number ontemperature profiles.

the thermal boundary layer. It is noticed that the velocity ψ decreases on increases the Prandtlnumber Pr. Figures 6 and 14 establishes that the fluid velocity ψ and temperature θ increaseson increasing the radiation parameter R. This fact is justified because the fluid temperature θincreases on increase in R as observed from Fig. 14. This implies that the fluid temperaturestrengthen the thermal buoyancy force. Hence ψ is accelerated with an increase in R. Figures7 and 15 describe the fluid velocity ψ and temperature θ for different values of Dufour numberDu. The Dufour number signifies the contribution of the concentration gradient to the thermalenergy flux in the flow. It is clear from Figs. 7 and 15 that the fluid velocity ψ decreases on



0 0.5 1 1.5 2 2.5 3 3.5 4

Tem

pera

ture

θ

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

Kr = 1, 2, 3, 4Kr = 1, 2, 3, 4

Pr = 0.71Pr = 1.00

Figure 18: Influence of chemical reactionparameter on temperature profiles.


0 0.5 1 1.5 2 2.5 3 3.5 4

Con

cent

ratio

n φ

0

0.2

0.4

0.6

0.8

1

1.2

Sr = 0.5, 1.0, 1.5, 2.0Sr = 0.5, 1.0, 1.5, 2.0

Sc = 0.22Sc = 0.78

Figure 19: Influence of Soret number onconcentration profiles.


0 0.5 1 1.5 2 2.5 3 3.5 4

Con

cent

ratio

n φ

0

0.2

0.4

0.6

0.8

1

1.2

Kr = 1, 2, 3, 4

Sc = 0.22Sc = 0.78

Figure 20: Influence of chemical reactionparameter on concentration profiles.


0 0.5 1 1.5 2 2.5 3 3.5 4

Con

cent

ratio

n φ

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Ec = 0.2, 0.4, 0.6, 0.8

Sc = 0.22Sc = 0.78

Figure 21: Influence of Eckert number onconcentration profiles.

increasing the Dufour number Du where as the fluid temperature θ increases on increasing theDufour number Du.

Figures 8 and 16 displays the impact of heat absorption parameter H on the fluid velocityψ and temperature θ. It is evident that the fluid velocity ψ and temperature θ decreases onincreasing the heat absorption parameterH . Figures 9, 17 and 21 shows the influence of Eckertnumber Ec on the fluid velocity ψ, temperature θ and concentration ϕ respectively. Eckertnumber defines the relationship between the kinematic energy in the flow and the enthalpy.It is observed that, the fluid velocity ψ, temperature θ and concentration ϕ experiences an


enhancement on increasing the Eckert numberEc . This is due to the reason that Eckert numberrepresents the conversion of kinematic energy into internal energy by work done against theviscous fluid stresses. The influence of fluid velocity ψ in presence of foreign species suchas Hydrogen (Sc = 0.22) , Helium (Sc = 0.30) , Water vapour (Sc = 0.60) , Ammonia(Sc = 0.78) is shown in Fig 10. Physically, Schmidt number signifies the relative strengthof viscosity to chemical molecular diffusivity. Therefore the Schmidt number quantifies therelative effectiveness of momentum and mass transport by diffusion in the hydrodynamic andconcentration boundary layers. This causes the concentration buoyancy effects to decrease thefluid velocity. It is observed that velocity ψ decreases on increasing Schmidt number Sc in Fig.10.

Figures 11 and 19 demonstrate the influence of Soret number Sr on the fluid velocity ψand concentration ϕ respectively. The Soret number defines the influence of the temperaturegradient inducing significant mass diffusion effects. It is noticed that the fluid velocity ψ andconcentration ϕ increases with an increase in Soret number Sr. Figures 12, 18 and 20 areplotted for different values of chemical reaction parameter Kr on the fluid velocity ψ, temper-ature θ and concentration ϕ respectively. It is observed that, the fluid velocity ψ, temperature θand concentration ϕ decreases on increasing the chemical reaction parameter Kr. This impliesthat, chemical reaction tends to reduce the fluid velocity, temperature and species concentra-tion. It is revealed from Fig. 13 that fluid velocity ψ decreases on increasing plate accelerationparameter b in the region near the plate and the effect of b is almost negligible in the regionaway from the plate.

The numerical values of skin friction coefficient Cf , heat transfer coefficient Nu and masstransfer coefficient Sh, obtained from the exact analytical solutions, are presented in tabularform in tables 1 and 2. It is clear that, the skin friction Cf increases on increasing the angleof inclination α, magnetic parameter M , Prandtl number Pr, heat absorption parameter H ,chemical reaction parameter Kr and plate acceleration parameter b whereas it decreases on in-creasing the buoyancy parameterN , permeability parameterK, radiation parameterR, Dufournumber Du, Eckert number Ec and Soret number Sr.

Heat transfer coefficient Nu decreases on increasing the buoyancy parameter N , perme-ability parameter K, Prandtl number Pr, Dufour number Du, heat absorption parameter H ,Eckert number Ec, Soret number Sr,chemical reaction parameter Kr and plate accelerationparameter b whereas it increases on increasing the angle of inclination α, Magnetic parameterM and radiation parameter R.

Mass transfer coefficient Sh increases on increasing the buoyancy parameter N , perme-ability parameter K, Prandtl number Pr, heat absorption parameter H , Soret number Sr andplate acceleration parameter b whereas it decreases on increasing the angle of inclination α,Magnetic parameter M , radiation parameter R, Dufour number Du, Eckert number Ec andchemical reaction parameter Kr.


5. CONCLUSIONS

In the present investigation we have studied analytically Soret and Dufour effects on radia-tive MHD flow of a chemically reacting fluid over an exponentially accelerated inclined porousplate with heat absorption and viscous dissipation. From the present investigation the followingconclusions can be drawn:

• Buoyancy parameterN and Eckert numberEc are tends to accelerate the fluid velocityψ whereas they have a reverse effect on the skin friction coefficient Cf .

• Angle of inclination α and plate acceleration parameter b are tends to decelerate thefluid velocity ψ whereas they have a reverse effect on the skin friction coefficient Cf .

• The fluid velecity ψ and skin friction coefficien Cf decreases with increasing valuesof Dufour number Du.

• Dufour numberDu and Eckert numberEc are tends to accelerate the fluid temperatureθ whereas they have a reverse trend on the heat transfer coefficient Nu.

• Eckert number Ec accelerates the species concentration whereas it has a reverse trendon the mass transfer coefficient Sh.

ACKNOWLEDGMENTS

The authors wish to appreciate the constructive comments and suggestions of the anonymousreviewers.

Table 1: Comparision with the skin friction coefficient values (PST) of Malapati and Dasari[25] for different values of M , N and K when α = b = Du = R = 0, ϵ = Ec = 0.001,Q = H = 5.0, Sr = 5.0 and ω = 0.1.

M N K Ref [25] Present1.0 2.0 0.5 0.0122 0.01222.0 2.0 0.5 0.4940 0.49403.0 2.0 0.5 0.8798 0.87984.0 2.0 0.5 1.2058 1.20582.0 0.2 0.5 1.6174 1.61742.0 0.4 0.5 1.4926 1.49262.0 0.6 0.5 1.3678 1.36782.0 0.8 0.5 1.2429 1.24292.0 2.0 0.1 2.5756 2.57562.0 2.0 0.3 0.9940 0.99402.0 2.0 0.5 0.4940 0.49402.0 2.0 0.7 0.2340 0.2340


Table 2: Influence of N , M , K, α, b and R on Cf , Nu and Sh.

N M K α b R Cf Nu Sh0.5 5.0 0.5 π

4 0.2 1.0 2.2356 -0.2837 -0.32361.0 5.0 0.5 π

4 0.2 1.0 2.1000 -0.2846 -0.32341.5 5.0 0.5 π

4 0.2 1.0 1.9645 -0.2854 -0.32322.0 5.0 0.5 π

4 0.2 1.0 1.8290 -0.2862 -0.32312.0 1.0 0.5 π

4 0.2 1.0 0.5529 -0.2904 -0.32242.0 2.0 0.5 π

4 0.2 1.0 0.9539 -0.2893 -0.32262.0 3.0 0.5 π

4 0.2 1.0 1.2868 -0.2882 -0.32272.0 4.0 0.5 π

4 0.2 1.0 1.5741 -0.2871 -0.32292.0 5.0 0.1 π

4 0.2 1.0 3.2981 -0.2803 -0.32422.0 5.0 0.2 π

4 0.2 1.0 2.4683 -0.2836 -0.32352.0 5.0 0.3 π

4 0.2 1.0 2.1323 -0.2850 -0.32332.0 5.0 0.4 π

4 0.2 1.0 1.9470 -0.2857 -0.32322.0 5.0 0.5 0 0.2 1.0 1.4918 -0.2879 -0.32282.0 5.0 0.5 π

8 0.2 1.0 1.5794 -0.2875 -0.32292.0 5.0 0.5 π

6 0.2 1.0 1.6459 -0.2871 -0.32302.0 5.0 0.5 π

4 0.2 1.0 1.8290 -0.2862 -0.32312.0 5.0 0.5 π

4 0.2 1.0 1.8290 -0.2862 -0.32312.0 5.0 0.5 π

4 0.4 1.0 1.8695 -0.2911 -0.32202.0 5.0 0.5 π

4 0.6 1.0 1.9523 -0.2965 -0.32092.0 5.0 0.5 π

4 0.8 1.0 2.0692 -0.3013 -0.31992.0 5.0 0.5 π

4 0.2 1.0 1.8290 -0.2862 -0.32312.0 5.0 0.5 π

4 0.2 2.0 1.8261 -0.2324 -0.32872.0 5.0 0.5 π

4 0.2 3.0 1.8241 -0.1970 -0.33242.0 5.0 0.5 π

4 0.2 4.0 1.8226 -0.1766 -0.3351


Table 3: Influence of Du, H , Sr, Kr and Ec on Cf , Nu and Sh.

Du H Sr Kr Ec Cf Nu Sh0.5 0.2 1.0 0.5 0.2 1.8263 -0.3338 -0.32740.6 0.2 1.0 0.5 0.2 1.8256 -0.3457 -0.32840.7 0.2 1.0 0.5 0.2 1.8249 -0.3577 -0.32950.8 0.2 1.0 0.5 0.2 1.8242 -0.3696 -0.33060.1 0.2 1.0 0.5 0.2 1.8290 -0.2862 -0.32310.1 0.4 1.0 0.5 0.2 1.8413 -0.4125 -0.30680.1 0.6 1.0 0.5 0.2 1.8428 -0.4935 -0.28650.1 0.8 1.0 0.5 0.2 1.8461 -0.5701 -0.27170.1 0.2 0.5 0.5 0.2 1.8319 -0.2860 -0.33500.1 0.2 1.0 0.5 0.2 1.8290 -0.2862 -0.32310.1 0.2 1.5 0.5 0.2 1.8265 -0.2863 -0.31220.1 0.2 2.0 0.5 0.2 1.8242 -0.2865 -0.30240.1 0.2 1.0 1.0 0.2 1.8541 -0.2879 -0.48070.1 0.2 1.0 2.0 0.2 1.8860 -0.2881 -0.70670.1 0.2 1.0 3.0 0.2 1.9079 -0.2906 -0.87890.1 0.2 1.0 4.0 0.2 1.9249 -0.2933 -1.02310.1 0.2 1.0 0.5 0.2 1.8290 -0.2862 -0.32310.1 0.2 1.0 0.5 0.4 1.7186 -0.3301 -0.36390.1 0.2 1.0 0.5 0.6 1.6123 -0.3768 -0.42440.1 0.2 1.0 0.5 0.8 1.5110 -0.4147 -0.4998


APPENDIX

A1 =Pr

1 +R, A2 = ScSr, A3 =

m21

m22 −m2

1

, A4 = A2A3, A5 = 1−A4,

A6 =(1 +NA4) cosα

m23 −m2

1

, A7 =NA5 cosα

m23 −m2

2

, A8 = exp(ibτ)− (A6 +A7),

A9 =A1A5m

22

m21 −m2

2

, A10 =A1A4m1

2, A11 =

A2A9m2

2, A12 = 1 +

2m1

m21 −m2

2

,

A13 =2A2A10m1 −A2A10A12m

21 −A2A9m

21

m21 −m2

2

, A14 =(A9 +NA13) cosα

m23 −m2

2

,

A15 =(A9 +NA13) cosα

m23 −m2

1

, A16 = 1 +2m2

m22 −m2

3

, A17 =NA11 cosα

m23 −m2

2

,

A18 = A16A17, A19 = 1 +2m1

m21 −m2

3

, A20 =A10 cosα

m23 −m2

1

, A21 = A19A20,

A22 = A14 +A18, A23 = A15 +A21, A24 = A22 −A23,

A25 =A1A

28m

23Ec

4m23 −m2

4

, A26 =A1A

26m

21Ec

4m21 −m2

4

, A27 =A1A

27m

22Ec

4m22 −m2

4

,

A28 =2A1A6A8m1m3Ec

(m1 +m3)2 −m24

, A29 =2A1A6A7m1m2Ec

(m1 +m2)2 −m24

,

A30 =2A1A7A8m2m3Ec

(m2 +m3)2 −m24

, A31 = 1 +A25 +A26 +A27 +A28 +A29 +A30,

A32 =A2A31m

24

m24 −m2

5

, A33 =4A2A25m

23

4m23 −m2

5

, A34 =4A2A26m

21

4m21 −m2

5

,

A35 =4A2A27m

22

4m22 −m2

5

, A36 =A2A28(m1 +m3)

2

(m1 + 3)2 −m25

, A37 =A2A29(m1 +m2)

2

(m1 + 2)2 −m25

,

A38 =A2A30(m2 +m3)

2

(m2 + 3)2 −m25

, A39 = [1 +A32]− [A33 +A34 +A35 +A36 +A37 +A38],

A40 =(A31 −NA32) cosα

m24 −m2

6

, A41 =NA39 cosα

m25 −m2

6

, A42 =(A25 −NA33) cosα

4m23 −m2

6

,

A43 =(A26 −NA34) cosα

4m21 −m2

6

, A44 =(A27 −NA35) cosα

4m22 −m2

6

, A45 =(A28 −NA36) cosα

(m1 +m3)2 −m26

,

A46 =(A29 −NA37) cosα

(m1 +m2)2 −m26

, A47 =(A30 −NA38) cosα

(m1 +m3)2 −m26

,

A48 = [A40 +A41]− [A42 +A43 +A44 +A45 +A46 +A47],


A49 =A1A39m

25

m25 −m2

4

, A50 =A1A32m4

2, A51 =

2A1m21[2A34 +A6A23Ec]

4m21 −m2

4

,

A52 =2A1m

22[2A35 −A7A22Ec]

4m22 −m2

4

, A53 =2A1m

23[2A33 +A8A24Ec]

4m23 −m2

4

,

A54 =A1A37(m1 +m2)

2 − 2A1m1m2Ec[A6A22 −A7A23]

(m1 +m2)2 −m24

,

A55 =A1A38(m2 +m3)

2 + 2A1m2m3Ec[A7A24 −A8A22]

(m2 +m3)2 −m24

,

A56 =A1A36(m1 +m3)

2 + 2A1m1m3Ec[A6A24 +A8A23]

(m1 +m3)2 −m24

,

A57 = A49 +A51 +A52 +A53 +A54 +A55 +A56,

A58 = 1 +2m4

m24 −m2

5

, A59 =A2m4[m4A50A58 −m4A57 − 2A50]

m24 −m2

5

,

A60 =A2A49m5

2, A61 =

4A2A51m21

4m21 −m2

5

, A62 =4A2A52m

22

4m22 −m2

5

,

A63 =4A2A53m

23

4m23 −m2

5

, A64 =A2A54(m1 +m2)

2

(m1 +m2)2 −m25

, A65 =A2A55(m2 +m3)

2

(m2 +m3)2 −m25

,

A66 =A2A56(m1 +m3)

2

(m1 +m3)2 −m25

, A67 = −[A59 +A61 +A62 +A63 +A64 +A65 +A66],

A68 = 1 +2m4

m24 −m2

6

, A69 =[A50A68 −A57 −NA59] cosα

m24 −m2

6

, A70 = 1 +2m5

m25 −m2

6

,

A71 =[A49 −NA67 +NA60A70] cosα

m25 −m2

6

, A72 =[A51 −NA61] cosα

4m21 −m2

6

,

A73 =[A52 −NA62] cosα

4m22 −m2

6

, A74 =[A53 −NA63] cosα

4m23 −m2

6

,

A75 =[A54 −NA64] cosα

(m1 +m2)2 −m26

, A76 =[A55 −NA65] cosα

(m2 +m3)2 −m26

, A77 =[A56 −NA66] cosα

(m1 +m3)2 −m26

,

A78 = −[A69 +A71 +A72 +A73 +A74 +A75 +A76 +A77],

m1 =√HA1, m2 =

√ScKr, m3 =

√M +

1

K, m4 =

√A1(H + iω),

m5 =√Sc(Kr + iω), m6 =

√M +

1

K+ iω


REFERENCES

[1] Q. Li, G. Flamant, X. Yuan, P. Neveu and L. Luo: Compact heat exchangers: A review and future applicationsfor a new generation of high temperature solar receivers, Renewable and Sustainable Energy Reviews, 15(2011), pp. 4855–4875.

[2] M. L. Hunt and C. L. Tien: Non-darcian flow, heat and mass transfer in catalytic packed-bed reactors, Chem-ical Engineering Science, 45 (1990), pp. 55–63.

[3] G. Leccese, D. Bianchi, B. Betti, D. Lentini and F. Nasuti: Convective and radiative wall heat transfer inliquid rocket thrust chambers, Journal of Propulsion and Power, 34 (2018), pp. 318–326.

[4] S. M. Khairnasov and A. M. Naumova: Heat pipes application to solar energy systems, Applied Solar Energy,52 (2016), pp. 47–60.

[5] A. Postelnicu: Influence of a magnetic field on heat and mass transfer by natural convection from verticalsurfaces in porous media considering Soret and Dufour effects, Int. J. Heat Mass Transfer. 47 (2004), pp.1467–1472.

[6] M. S. Alam, M. Ferdows, M. Ota and M. A. Maleque: Dufour and Soret effects on steady free convection andmass transfer flow past a semi-infinite vertical porous plate in a porous medium, Int. J. Appld Mech. Eng. 11(2006), pp.535–545.

[7] A. A. Afify: Similarity solution in MHD: Effects of thermal diffusion and diffusion thermo on free convectiveheat and mass transfer over a stretching surface considering suction or injection, Comm. Nonlinear Sci.Numer. Simu. 14 (2009), pp. 2202–2214.

[8] M. Venkateswarlu, G. V. Ramana Reddy and D. V. Lakshmi: Diffusion-thermo effects on MHD flow pastan infinite vertical porous plate in the presence of radiation and chemical reaction, International Journal ofMathematical Archive. 4 (2013), pp. 39–51.

[9] P. O. Olanrewaju and O. D. Makinde: Effects of thermal diffusion and diffusion thermo on chemically reactingMHD boundary layer flow of heat and mass transfer past a moving vertical plate with suction /injection, ArabJ. Sc. and Eng. 36 (2011), pp. 1607–1619.

[10] O. D. Makinde: On MHD mixed convection with soret and dufour effects past a vertical plate embedded in aporous medium, Latin American Applied Research . 41 (2011), pp. 63–68.

[11] M. Venkateswarlu and P. Padma: Unsteady MHD free convective heat and mass transfer in a boundary layerflow past a vertical permeable plate with thermal radiation and chemical reaction, Procedia Engineering. 127(2015), pp. 791–797.

[12] M. Venkateswarlu, G. Upender Reddy and D. Venkata Lakshmi: Influence of Hall current and heat source onMHD flow of a rotating fluid in a parallel porous plate channel, J. Korean Soc. Ind. Appl. Math. 22 (2018),pp. 217–239, 2018.

[13] M. Venkateswarlu and D. Venkata Lakshmi: Dufour and chemical reaction effects on unsteady MHD flow ofa viscous fluid in a parallel porous plate channel under the influence of slip condition, Journal of the NigerianMathematical Society. 36 (2017), pp. 369 - 397.

[14] G. S. Seth and S. Sarkar: Hydromagnetic natural convection flow with induced magnetic field and nth orderchemical reaction of a heat absorbing fluid past an impulsively moving vertical plate with ramped temperature,Bulgarian Chemical Communications. 47 (2015), pp. 66–79.

[15] S. P. Anjali Devi and B. Ganga: Effects of viscous and joules dissipation on MHD flow, heat and mass transferpast a stretching porous surface embedded in a porous medium, Nonlinear Analysis: Modelling and Control,14 (2009), pp. 303–314.

[16] A. Hussanan, M. Z. Salleh, I. Khan, R. M. Tahar and Z. Ismail: Soret effects on unteady magnetohydrodynamicmixed convection heat and mass transfer flow in a porous medium with Newtonian heating, MaejoInt. J. Sci.Technol. 9 (2015), pp.224–245.

[17] N. Vedavathi, K. Ramakrishna and K. J. Reddy: Radiation and mass transfer effects on unsteady MHD con-vective flow past an infinite vertical plate with Dufour and Soret Effects, Ain Shams Engineering Journal. 6(2015), pp. 363–371.


[18] G. S. Seth, B. Kumbhakar and S. Sarkar: Unsteady hydromagnetic natural convection flow with heat and masstransfer of a thermally radiating and chemically reactive fluid past a vertical plate with Newtonian heatingand time dependent free stream, International Journal of Heat and Technology. 32 (2014), pp. 87–94.

[19] M. Venkateswarlu, G. V. Ramana Reddy and D. V. Lakshmi: Thermal diffusion and radiation effects onunsteady MHD free convection heat and mass transfer flow past a linearly accelerated vertical porous platewith variable temperature and mass diffusion, J. Korean Soc. Ind. Appl. Math. 18 (2014), pp. 257 - 268.

[20] M. S. Alam, M. M. Rahman and M. A. Sattar: MHD free convective heat and mass transfer flow past aninclined surface with heat generation, Thammasat International Journal of Science and Technology. 11 (2006),pp. 1–8.

[21] R. S. Tripathy, G. C. Dash, S. R. Mishra and S. Baag: Chemical reaction effect on MHD free convectivesurface over a moving vertical plate through porous medium, Alexandria Engineering Journal. 54 (2015),pp.673–679.

[22] M. Venkateswarlu and O. D. Makinde: Unsteady MHD slip flow with radiative heat and mass transfer overan inclined plate embedded in a porous medium, Defect and Diffusion Forum, 384 (2018), pp. 31–48.

[23] M. Gnaneswara Reddy, P. Padma and B. Bandari Shankar: Effects of viscous dissipation and heat source onunsteady MHD flow over a stretching sheet, Ain Shams Engineering Journal, 6, (2015), pp.1195–1201.

[24] M. Venkateswarlu, G. V. Ramana Reddy and D. V. Lakshmi: Effects of chemical reaction and heat generationon MHD boundary layer flow of a moving vertical plate with suction and dissipation, Engineering Interna-tional. 1 (2013), pp. 27–38.

[25] V. Malapati and V. L. Dasari: Soret and chemical reaction effects on the radiative MHD flow from an infinitevertical plate, J. Korean Soc. Ind. Appl. Math. 21 (2017), pp. 39–61.

[26] E. Magyari and A. Pantokratoras: Note on the effect of thermal radiation in the linearized Rosseland approx-imation on the heat transfer characteristics of various boundary layer flows, Int. Commun. Heat and MassTransfer. 38 (2011), pp. 554–556.

[27] M. Venkateswarlu, O. D. Makinde and P. Rami Reddy: Influence of Hall current and thermal diffusion onradiative hydromagnetic flow of a rotating fluid in presence of heat absorption, Journal of Nanofluids, 8(2019), pp. 756–766.

[28] N. Siva Kumar, Rushi Kumar and A. G. Vijaya Kumar: Thermal diffusion and chemical reaction effects onunsteady flow past a vertical porous plate with heat source dependent in slip flow regime, Journal of NavalArchitecture and Marine Engineering. 13 (2016), pp. 51–62.

[29] M . Venkateswarlu, D. Venkata Lakshmi and G. Darmaiah: Influence of slip condition on radiative MHD flowof a viscous fluid in a parallel porous plate channel in presence of heat absorption and chemical reaction, J.Korean Soc. Ind. Appl. Math. 20 (2016), pp. 333–354.

soret and dufour effects on radiative ......convection with soret and dufour effects past a vertical...

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