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IJRAR1903634 International Journal of Research and Analytical Reviews (IJRAR) www.ijrar.org 790

MATHEMATICAL MODELLING OF MHD

NATURAL CONVECTIVE FLOW TOWARDS A

VERTICALLY INCLINED PLATE IN

PRESENCE OF RIVLIN-ERICKSEN FLUID,

THERMAL DIFFUSION, DIFFUSION

THERMO, HEAT AND MASS TRANSFER

ANALYSIS 1V. Omeshwar Reddy, 2A. Neelima and 3S. Thiagarajan

1Assistant Professor, 2Assistant Professor, 3Professor 1Department of Mathematics, TKR College of Engineering and Technology, Hyderabad, 500097, Telangana State, India.

________________________________________________________________________________________________________

Abstract: This paper focuses on the effects of thermal diffusion and diffusion thermo as well as thermal radiation, heat and mass

transfer on a two-dimensional natural convective flow of unsteady viscous, incompressible magneto hydrodynamics (MHD) non-

newtonian visco-elastic Rivlin-Ericksen Fluid over an inclined vertical plate. The constitute governing equations have been

converted into strong linear partial differential equations by employing suitable transformations and these transformed equations

are solved by the Implicit finite difference method. Graphical illustrations for velocity, temperature and concentration profiles are

developed. Physical interpretation of obtained results has been presented. The accuracy of present code is validated through

comparison with previously published results. Comparison reveals an excellent agreement with those results.

Keywords: MHD; Viscoelastic Rivlin-Ericksen fluid; Thermal Radiation; Heat and Mass transfer; Finite difference method.

_______________________________________________________________________________________________________

Nomenclature:

List of variables:

wC Concentration of the plate (3mKg )

y Dimensionless displacement ( m )

T Fluid temperature away from the plate (K)

u Velocity component in x direction ( 1sm )

x Coordinate axis along the plate ( m )

y Co-ordinate axis normal to the plate ( m )

C Fluid Concentration ( 3mKg )

T Fluid temperature )(K

wT Fluid temperature at the wall K

0B Uniform magnetic field (Tesla)

R Thermal radiation parameter

C Concentration of the fluid far away from the plate

( 3mKg )

u Fluid velocity ( 1sm )

Gc Grashof number for mass transfer

Sh The local Sherwood number coefficient

g Acceleration of gravity, 9.81(2sm )

Gr Grashof number for heat transfer

M Magnetic field parameter

Pr Prandtl number

pC Specific heat at constant pressure KKgJ 1

Nu The local Nusselt number coefficient

Re Reynolds number

mD Mass diffusivity ( sm /2

)

mT Mean fluid temperature )(K

Tk Thermal diffusion ratio

Sr Thermal diffusion parameter

Du Diffusion thermo parameter

SC Concentration susceptibility (3/ mKg )

Sc Schmidt number

D Solute mass diffusivity ( 12 sm )

Cf The local skin-friction coefficient

t Time (sec)

ek Mean absorption coefficient

oU Reference velocity (1sm )

Kr Chemical reaction parameter

Q Heat sink parameter

K Permeability parameter

rq Radiative heat flux

Greek Symbols:

Kinematic viscosity (12 sm )

Species concentration ( 3mKg )

The constant density ( 3mKg )

Volumetric coefficient of thermal expansion )( 1K

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* Volumetric Coefficient of thermal expansion with concentration ( 13 Kgm )

Fluid temperature K

w Shear stress )(2mN

Electric conductivity of the fluid )( 1ms

1 Kinematic viscoelasticity

s Stefan-Boltzmann constant

Thermal conductivity of the fluid mKW /

Angle of inclination parameter (degrees)

Superscripts:

/Dimensionless properties

Subscripts:

Free stream conditions

p Plate

w Conditions on the wall

_______________________________________________________________________________________________________ 1. Introduction:

The study of heat transfer with chemical reaction is of most realistic significance to engineers and scientists because of

its universal incidence in many branches of science and engineering. This phenomenon plays a significant role in chemical

industry, power and cooling industry for drying, evaporation, energy transfer in a cooling tower and the flow in a desert cooler,

etc. Mohamed et al. [1] studied unsteady MHD double-diffusive convection boundary-layer flow past a radiate hot vertical

surface in porous media in the presence of chemical reaction and heat sink. Devika et al. [2] studied the influence of chemical

reaction effects on MHD free convection flow in an irregular channel with porous medium. Satya Narayana and Sravanthi [3]

have analysed the influence of variable permeability on unsteady MHD convection flow past a semi-infinite inclined plate with

thermal radiation and chemical reaction. Chambre and Young [4] have analyzed a first order chemical reaction in the

neighbourhood of a horizontal plate. Kandasamy et al. [5] presented the effects of chemical reaction, heat and mass transfer along

a wedge in the presence of suction. The effects of chemical reaction, heat and mass transfer flow past an impulsively started

infinite vertical plate were investigated by Das et al. [6]. Anjali Devi and Kandasamy [7] proposed the effects of chemical

reaction, heat and mass transfer on laminar flow along a semi-infinite horizontal plate. Muthucumaraswamy [8] derived the

effects of chemical reaction, heat and mass transfer along a moving vertical surface with suction. Ramasamy et al. [9] focused the

influence of an unsteady variable viscosity and the Soret effect on the magneto hydrodynamic (MHD) mixed convection flow in

the presence of a chemical reaction over a porous wedge. Krishnendu and Layek [10] expressed the effect of radiation and

chemical reaction on MHD laminar boundary layer flow past a porous flat plate. Rout et al. [11] elucidate the influence of thermal

radiation and chemical reaction on MHD free convective flow over a vertical surface in a porous medium. Jain [12] presented the

radiation and chemical reaction effects on unsteady heat and mass transfer through an oscillating vertical plate with constant heat

rate. Farhad et al. [13] analysed the combined heat and mass transfer effects of natural convection MHD laminar flow through a

vertical plate immersed in a porous medium. Dhal and Mohanty [14] investigated the effect of chemical reaction on two-

dimensional unsteady mass transfer in a vertical plate through porous medium. Muthucumaraswamy and Meenakshisundaram

[15] investigated the effect of chemical reaction with unsteady temperature over an oscillating plate. Sudhakar et al. [16] studied

Hall Effect on an unsteady MHD flow past along a porous flat plate with thermal diffusion, diffusion thermo and chemical

reaction. Chemical reaction effect on an unsteady MHD free convection flow past an infinite vertical accelerated plate with

constant heat flux, thermal diffusion and diffusion thermo studied by Sudhakar et al. [17]. Unsteady MHD free convection flow

near on an infinite vertical plate embedded in a porous medium with chemical reaction, hall current and thermal radiation

discussed by Sarada et al. [18].

The Rivlin-Ericksen elastic-viscous fluid has relevance and importance in geophysical fluid dynamics, chemical technology and

industry. Daleep et al. [19] studied bounds for complex growth rate in thermosolutal convection in Rivlin-Ericksen viscoelastic

fluid in a porous medium. Noushima et al. [20] discussed hydro magnetic free convective Rivlin-Ericksen flow through a porous

medium with variable permeability. Rana [21] discussed thermal instability of compressible Rivlin-Efficksen rotating fluid

permeated with suspended dust particles in porous medium. Sharma et al. [22] discussed Hall effects on thermal instability of

Rivlin-Ericksen fluid. Gupta et al. [23] discussed on Rivlin-Erickson elastic-viscous fluid heated and solution from below in the

presence of compressibility, rotation and Hall currents. Uwanta et al. [24] discussed effects of mass transfer on hydro magnetic

free convective Rivlin-Ericksen flow through a porous medium with time dependent suction. Varshney et al. [25] discussed

effects of rotator Rivlin-Ericksen fluid on MHD free convective and mass transfer flow through porous medium with constants

heat and mass flux across moving plate. Magnetic field effects on transient free convection flow through porous medium past an

impulsively started vertical plate with fluctuating temperature and mass diffusion was studied by Ravikumar et al. [26]. Seth et al.

[27] considered, MHD natural convection flow with irradiative heat transfer past an impulsively moving plat with ramped wall

temperature. Effects of variable suction and thermophoresis on steady MHD combined free-forced convective heat and mass

transfer flow over a semi-infinite permeable inclined plat in the presence of thermal radiation were investigated by Alam et al.

[28]. Rajasekhar [29], Linga [30], Muthucumaraswamy [31], etc. studied various effects of heat transfer in different geometries.

In their study Raju et al. [32] considered MHD convective flow through porous medium in a horizontal channel with insulated

and impermeable bottom wall in the presence of viscous dissipation and Joule heating. Effects of Hall current and radiation

absorption on MHD micropolar fluid in a rotating system were studied by Satya Narayana et al. [33]. Mahgoub [34] investigated

forced convection heat transfer over a flat plate in a porous medium. Slip effects on MHD boundary layer flow over an

exponentially stretching sheet with suction/blowing and thermal radiation were considered by Swati [35]. Sivaraj and Rushi

Kumar [36] investigated chemically reacting dusty viscoelastic fluid flow in an irregular channel with convective boundary.




To the best of our knowledge, the problem of MHD non-newtonian visco-elastic Rivlin-Ericksen fluid flow towards a

vertically inclined plate in presence of thermal radiation, heat and mass transfer under low Reynolds number assumptions has

remained unexplored. Hence, the main objective of this study is to investigate the combined effects of chemical reaction and

radiative heat transfer flow past a vertically inclined plate filled with saturated porous medium in presence of visco-elastic Rivlin-

Ericksen fluid. The governing equations of the flow are solved analytically, and the effects of various flow parameters on the flow

field have been discussed. The format of the paper is as follows. We depict the mathematical model and argue the non-

dimensionalization of the governing equations in Section 2 Mathematical formulation of the problem, Section 3 Method of

solution, Section 4 Program code validation, Section 5 Results and discussion which contain results and discussions. Finally,

Section 6 highlights the important conclusions derived from the present study.

_____________________________________________________________________________________________________

2. Mathematical formulation: In this investigation, unsteady MHD natural convective heat and mass transfer non-newtonian viscoelastic Rivlin-

Ericksen fluid flow of a viscous, incompressible, gray, absorbing-emitting but non-scattering, optically-thick and electrically

conducting fluid occupying a vertical porous regime with constant velocity in presence of thermal radiation, heat absorption and

chemical reaction is considered. The flow configuration of the problem is presented in Fig. 1. For this investigation, let us assume

that x axis is taken along the vertical infinite porous plate in the upward direction and the y axis normal to the plate.

Initially, for time ,0t the plate and the fluid are at some temperature T in a stationary condition with the same species

concentration C at all points.

i. A constant magnetic field oB is maintained in the y direction and the plate moves uniformly along the positive

x direction with velocity 0U .

ii. The magnetic Reynolds number is so small that the induced magnetic field can be neglected. iii. At a time t > 0 a magnetic field of uniform strength is applied in the direction of y axis and the induced magnetic

field is neglected.

iv. Also no applied or polarized voltages exist so the effect of polarization of fluid is negligible. v. All the fluid properties except the density in the buoyancy force term are constants.

vi. The temperature at the surface of the plate is raised to uniform temperature wT and species concentration at the surface of

the plate is raised to uniform species concentration wC and is maintained thereafter.

vii. The viscous dissipation and Ohmic dissipation of energy are negligible.

viii. The homogeneous chemical reaction of first order with rate constant K between the diffusing species is assumed.

Fig. 1. Physical configuration of the problem

a --- Momentum boundary layer, b --- Thermal boundary layer, c --- Concentration boundary layer

Under the above foregoing assumptions and Boussinesq's approximation, the equations governing the flow and transport reduce

to the following equations:

Momentum Equation:

2

3

1

2

0*

2

2

coscosyt

uu

Ku

BCCgTTg

y

u

t

u

(1)

Energy Equation:




2

2

2

2

y

C

CC

kD

y

qTTQ

y

T

t

TC

pS

Tmrop

(2)

Species Diffusion Equation:

2 2

2 2

m Tr

m

D kC C TD k C C

t y T y

(3)

Together with initial and boundary conditions

yasCCTTu

yatCCTTUutyallforCCTTut

wwo

,,0

0,,:0&,,0:0 (4)

For an optically thick fluid, in addition to emission there is also self absorption and usually the absorption co-efficient is

wavelength dependent and large so we can adopt the Rosseland approximation for the radiative heat flux vector . Thermal

radiation is assumed to be present in the form of a unidirectional flux in the y direction i.e., rq (Transverse to the vertical

surface). By using the Rosseland approximation [40] the radiative heat flux rq is given by:

y

T

kq

e

sr

4

3

4 (5)

It should be noted that by using the Rosseland approximation, the present analysis is limited to optically thick fluids. If

temperature differences within the flow are sufficiently small, then equation (5) can be linearized by expanding 4T in Taylor

series about T which after neglecting higher order terms takes the form:

43344 344 TTTTTTTT (6) Using Eq. (5) and (6) in the last term of Eq. (2), we obtain

2

23

3

16

y

T

k

T

y

q

e

sr

(7)

Introducing (7) in the Eq. (2), the energy equation becomes

2

2

2

23

2

2

3

16

y

C

CC

kD

y

T

Ck

TTT

C

Q

y

T

Ct

T

pS

Tm

pe

s

p

o

p

(8)

Let us introduce the following non-dimensional variables and parameters:

2 2

o

2

*

3 3 2

3 22

1

2 2

U, , , , , , Re ,

( ), , Pr , , ,

16, , , ,

3

o o o

o w w o

pw w o

o o p o

m T ws or

o o e

y U t B U xT T C Cuu y t M

U T T C C U

Cg T T g C C QGr Gc Sc Q

U U D C U

D k CT UkK Kr R Du

K U U k

,

m T w

w S p w m

C D k T TSr

T T C C C C T

(9)

The above defined non-dimensionless variables in Eq. (9) into Eqs. (1), (3) and (8), and we get

2

3

2

2

coscos1

yt

uGcGru

KM

y

u

t

u (10)

2

2

2

2

Pr

1

yDuQ

y

R

t

(11)

2

2

2

21

ySrKr

ySct

(12)

with connected initial and boundary conditions

yasu

yatutyallforut

0,0,0

01,1,1:0&0,0,0:0

(13)

The Skin-friction at the plate, which in the non-dimensional form is given by

00

yo

w

y

u

vUCf

y

(14)




The rate of heat transfer coefficient, which in the non-dimensional form in terms of the Nusselt number is given by

TT

y

T

xNuw

y 0

0

1Re

yy

Nu

(15)

The rate of mass transfer coefficient, which in the non-dimensional form in terms of the Sherwood number, is given by

CC

y

C

xShw

y 0

0

1Re

yy

Sh

(16)

________________________________________________________________________________________________________ 3. Numerical Solutions by Finite Difference Method:

The non-linear momentum, energy and concentration equations given in equations (10), (11) and (12) are solved under

the appropriate initial and boundary conditions (13) by the implicit finite difference method. The transport equations (10), (11)

and (12) at the grid point (i, j) are expressed in difference form using Taylor’s expansion.

2

11

1

1

11

1

2

11

1

22

coscos12

yt

uuuuuu

GcGruK

My

uuu

t

uu

j

i

j

i

j

i

j

i

j

i

j

i

j

i

j

i

j

i

j

i

j

i

j

i

j

i

j

i

(17)

2

11

2

11

1 22

Pr

1

yDuQ

y

R

t

j

i

j

i

j

ij

i

j

i

j

i

j

i

j

i

j

i

(18)

2

11

2

11

1 221

ySrKr

ySct

j

i

j

i

j

ij

i

j

i

j

i

j

i

j

i

j

i

(19)

Where the indices i and j refer to y and t respectively. The initial and boundary conditions (13) yield.

0,0,0&01,1,1,0,0,0 000 jMjMjMjijijiiii uiatuiallforu (20) Thus the values of u, θ and ϕ at grid point t = 0 are known; hence the temperature field has been solved at time ttt ii 1

using the known values of the previous time itt for all 1........,,2,1 Ni . Then the velocity field is evaluated using the

already known values of temperature and concentration fields obtained at ttt ii 1 . These processes are repeated till the

required solution of u, θ and ϕ is gained at convergence criteria: 310,,,, numericalexact

uuabs (21)

Fig. 2. Finite difference space grid




4. Validation of Code:

Fig. 3. Comparison between present numerical results with the analytical results of Mohamed et al. [1]

for variations of Gr at Sr = Du = λ = α = 0.

For code validation purpose, we compared our numerical results with the exisistance analytical results of Mohamed et al.

[1] in Fig. 3 in the absence of non-newtonian viscoelastic Rivlin-Ericksen fluid, thermal diffusion, diffusion thermo and angle of

inclination. From this figure, it is observed that the relevant results obtained agree quantitatively with earlier results of Mohamed

et al. [1].

________________________________________________________________________________________________________

5. Results and Discussions:

Fig. 4. Gr influence on velocity profiles

Fig. 5. Gc influence on velocity profiles




In order to get a physical insight into the problem, a representative set of numerical results is shown graphically in Figure

4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12, Figure 13, Figure 14, Figure 15, Figure 16,

Figure 17, Figure 18, Figure 19, Figure 20, Figure 21, Figure 22, Figure 23 to illustrate the influence of physical parameters such

as Grashof number for heat transfer (Gr), Grashof number for mass transfer (Gc), Magnetic field parameter (M), Permeability

parameter (K), Prandtl number (Pr), Schmidt number (Sc), Thermal radiation parameter (R), Heat absorption parameter (Q),

Chemical reaction parameter (Kr), Thermal diffusion parameter (Sr), Diffusion thermo parameter (Du), Angle of inclination

parameter (α) and viscoelastic Rivlin-Ericksen fluid parameter (λ) embedded in the flow system. The Prandtl number was taken to

be Pr = 0.71 which corresponds to air. In the present study, the following default parameter values are adopted for computations:

Gr = 5.0, Gc = 5.0, K = 0.5, M = 0.5, Sc = 0.22 (Hydrogen), Q = 0.5, R = 0.5, Kr = 0.5, Sr = 0.5, Du = 0.5, α = 45o and λ = 0.5. All

graphs therefore correspond to these values unless specifically indicated in the appropriate graph.

Fig. 6. M influence on velocity profiles

Fig. 7. K influence on velocity profiles

The velocity profiles for different values of Grashof number for heat transfer Gr are described in Fig. 4. It is observed that an increase in Gr leads to a rise in the values of velocity. Here the Grashof number for heat transfer represent the effect of

free convection currents. Physically, Gr > 0 means heating of the fluid of cooling of the boundary surface, Gr < 0 means

cooling of the fluid of heating of the boundary surface and Gr = 0 corresponds the absence of free convection current.

The velocity profiles for different values of Grashof number for mass transfer Gc are described in Fig. 5. It is observed that an increase in Gc leads to a rise in the values of velocity.

Fig. 6 shows the effect of magnetic parameter M on the velocity. From this figure it is observed that velocity decreases, in both the cases of air and water, as the value of M is increased. This is due to the application of a magnetic field to an

electrically conducting fluid produces a dragline force which causes reduction in the fluid velocity.




Fig. 8. Pr influence on velocity profiles

Fig. 9. Pr influence on temperature profiles

Fig. 7 depicts the velocity profiles for various values of K. From this figure it is observed that fluid velocity increases as K increases and reaches its maximum over a very short distance from the plate and then gradually reaches to zero for both

water and air. Physically, an increase in the permeability of porous medium leads the rise in the flow of fluid through it.

When the holes of the porous medium become large, the resistance of the medium may be neglected.

Figs. 8 and 9 illustrate the velocity and temperature profiles for different values of Prandtl number. The numerical results show that the increasing values of Prandtl number, leads to velocity decreasing. From Fig. 9, the numerical results show

that, the increasing values of Prandtl number leads to a decrease in the thermal boundary layer, and in general, lower

average temperature within the boundary layer. The reason is that smaller values of Pr are equivalent to increasing 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 number, the thermal boundary layer is thicker and the rate of heat

transfer is reduced.

Fig. 10. Sc influence on velocity profiles




Fig. 11. Sc influence on concentration profiles

For various values of the Schmidt number Sc, the velocity and concentration are plotted in Figs. 10 and 11. As the Schmidt number increases, the concentration decreases. This causes the concentration buoyancy effects decrease, yielding a

reduction in the fluid velocity. Reductions in the velocity and concentration profiles are accompanied by simultaneous

reductions in the velocity and concentration boundary layers. These behaviours are evident from Figs. 10 and 11.

The effect of the thermal radiation parameter R on the velocity and temperature profiles in the boundary layer are illustrated in Figs. 12 and 13 respectively. Increasing the thermal radiation parameter R produces significant increase in the

thermal condition of the fluid and its thermal boundary layer. This increase in the fluid temperature induces more flow in

the boundary layer causing the velocity of the fluid there to increase.

Fig. 12. R influence on velocity profiles

Fig. 13. R influence on temperature profiles




Figs. 14 and 15 has been plotted to depict the variation of velocity and temperature profiles against y for different values of heat source parameter Q by fixing other physical parameters. From this Graph we observe that velocity and temperature

decrease with increase in the heat source parameter Q because when heat is absorbed, the buoyancy force decreases the

temperature profiles.

Fig. 16 displays the effect of the chemical reaction parameter Kr on the velocity profiles. As expected, the presence of the chemical reaction significantly affects the velocity profiles. It should be mentioned that the studied case is for a destructive

chemical reaction Kr. In fact, as chemical reaction Kr increases, the considerable reduction in the velocity profiles is

predicted, and the presence of the peak indicates that the maximum value of the velocity occurs in the body of the fluid

close to the surface but not at the surface.

Fig. 14. Q influence on velocity profiles

Fig. 15. Q influence on temperature profiles

Fig. 16. Kr influence on velocity profiles




Fig. 17 depicts the concentration profiles for different values of Kr, from which it is noticed that concentration decreases with an increase in chemical reaction parameter. This is due to the chemical reaction mass diffuses from higher

concentration levels to lower concentration levels.

For different values of the Diffusion thermo parameter, the velocity and temperature profiles are plotted in Figs. 18 and 19 respectively. The Diffusion thermo parameter signifies the contribution of the concentration gradients to the thermal

energy flux in the flow. It is found that an increase in the Diffusion thermo parameter causes a rise in the velocity and

temperature throughout the boundary layer.

Fig. 17. Kr influence on concentration profiles

Table-1: Numerical values of Skin-friction coefficient

Gr Gc M K Pr Sc R Q Kr Sr Du α Λ Cf

2.0 2.0 0.5 0.5 0.71 0.22 0.5 0.5 0.5 0.5 0.5 45o 0.5 - 1.3022154862

4.0 2.0 0.5 0.5 0.71 0.22 0.5 0.5 0.5 0.5 0.5 45o 0.5 - 1.2901554875

2.0 4.0 0.5 0.5 0.71 0.22 0.5 0.5 0.5 0.5 0.5 45o 0.5 - 1.2031002654

2.0 2.0 1.0 0.5 0.71 0.22 0.5 0.5 0.5 0.5 0.5 45o 0.5 - 1.4203155268

2.0 2.0 0.5 1.0 0.71 0.22 0.5 0.5 0.5 0.5 0.5 45o 0.5 - 1.2601520047

2.0 2.0 0.5 0.5 7.00 0.22 0.5 0.5 0.5 0.5 0.5 45o 0.5 - 1.3856014775

2.0 2.0 0.5 0.5 0.71 0.30 0.5 0.5 0.5 0.5 0.5 45o 0.5 - 1.3752663041

2.0 2.0 0.5 0.5 0.71 0.22 1.0 0.5 0.5 0.5 0.5 45o 0.5 - 1.2733016695

2.0 2.0 0.5 0.5 0.71 0.22 0.5 1.0 0.5 0.5 0.5 45o 0.5 - 1.3615004623

2.0 2.0 0.5 0.5 0.71 0.22 0.5 0.5 1.0 0.5 0.5 45o 0.5 - 1.3456033937

2.0 2.0 0.5 0.5 0.71 0.22 0.5 0.5 0.5 1.0 0.5 45o 0.5 - 1.2610445873

2.0 2.0 0.5 0.5 0.71 0.22 0.5 0.5 0.5 0.5 1.0 45o 0.5 - 1.2510048632

2.0 2.0 0.5 0.5 0.71 0.22 0.5 0.5 0.5 0.5 0.5 60o 0.5 - 1.3350266984

2.0 2.0 0.5 0.5 0.71 0.22 0.5 0.5 0.5 0.5 0.5 45o 1.0 - 1.3480669842

Figs. 20 and 21 depict the velocity and concentration profiles for different values of the Thermal diffusion parameter (Sr). The Thermal diffusion parameter defines the effect of the temperature gradients inducing significant mass diffusion

effects. It is noticed that an increase in the Thermal diffusion parameter results in an increase in the velocity and

concentration within the boundary layer.

The velocity profiles in the Fig. 22 shows that rate of motion is significantly reduced with increasing of viscoelastic Rivlin-Ericksen fluid parameter λ.

The effect of angle of inclination of the plate (α) on the velocity field has been illustrated in Fig. 23. It is seen that as the angle of inclination of the plate increases the velocity field decreases.

http://www.ijrar.org/http://www.sciencedirect.com/science/article/pii/S2090447914000902?np=y&npKey=78b015f6b8b69ce826965f5aa06d26d75514cb8abf4a9a50e9b0b9bcba7ebad2#f0035



Fig. 18. Du influence on velocity profiles

Fig. 19. Du influence on temperature profiles

The numerical computation of skin-friction coefficient is obtained and presented in table-1. It is observed that Magnetic field parameter, Prandtl number, Schmidt number, Chemical reaction parameter, Heat absorption parameter, Angle of

inclination parameter, Viscoelastic Rivlin-Ericksen fluid parameter decreases the skin-friction coefficient whereas it

increases due to increase in Grashof number for heat transfer, Grashof number for mass transfer, Permeability parameter,

Thermal radiation parameter, Diffusion thermo parameter and Thermal diffusion parameter.

Fig. 20. Sr influence on velocity profiles




Fig. 21. Sr influence on concentration profiles

Table-2: Numerical values of Nusselt and Sherwood numbers

Pr R Q Du Nu Sc Kr Sr Sh

0.71 0.5 0.5 0.5 1.6201135648 0.22 0.5 0.5 1.4022331521

7.00 0.5 0.5 0.5 1.5433622489 0.30 0.5 0.5 1.3544296108

0.71 1.0 0.5 0.5 1.6452300478 0.60 0.5 0.5 1.3026015832

0.71 0.5 1.0 0.5 1.5843216955 0.22 1.0 0.5 1.3620149228

0.71 0.5 0.5 1.0 1.6423018896 0.22 0.5 1.0 1.4450369557

The numerical values of Nusselt and Sherwood numbers are presented in table-2. With increase in the Thermal radiation parameter and Diffusion thermo parameter, the Nusselt number increases but for the other parameters such as Prandtl

number and Heat absorption parameter it decreases. A significant decrease is remarked in case of Sherwood number when

there is an increase in the values of the Schmidt number, Chemical reaction parameter and opposite effect is observed in

case of Thermal diffusion parameter.

Fig. 22. λ influence on velocity profiles




Fig. 23. α influence on velocity profiles

________________________________________________________________________________________________________

6. Conclusions: In this paper, the combined effects of thermal diffusion and diffusion thermo on unsteady MHD natural convection flow

past a vertically inclined plate filled in porous medium in the presence of thermal radiation, heat absorption, chemical reaction,

heat and mass transfer have been studied numerically. Finite difference method is employed to solve the governing equations of

the flow. From the present investigation, the following conclusions have been drawn:

i. The velocity profiles increases with an increase in the Grashof number for heat and mass transfer, Permeability parameter, Thermal diffusion parameter, Thermal radiation and Diffusion thermo parameters.

ii. The velocity profiles decreases with an increase in Magnetic field parameter, Chemical reaction parameter, Heat absorption parameter, Prandtl number, Schmidt number, Angle of inclination parameter and Visco-elastic Rivlin Ericksen

parameter.

iii. There are increases in the temperature profiles with increases in Thermal radiation parameter and Diffusion thermo parameter.

iv. When there is increase in Prandtl number and Heat absorption parameter there is decrease in temperature profiles. v. Concentration decreases with an increase in Schmidt number and Chemical reaction parameter. vi. When increasing the value of Thermal diffusion parameter there is increase in the Concentration boundary layer. vii. Local skin-friction increases with an increase in Grashof number for heat and mass transfer, Permeability parameter,

Thermal diffusion parameter, Thermal radiation and Diffusion thermo parameters while it decreases for rising values of

Magnetic field parameter, Chemical reaction parameter, Heat absorption parameter, Prandtl number, Schmidt number,

Angle of inclination parameter and Visco-elastic Rivlin Ericksen parameter.

viii. Nusselt number decreases with an increase in Prandtl number and Heat absorption parameter while it increases with an increase in Thermal radiation parameter and Diffusion thermo parameter..

ix. Sherwood number decreases with an increase in Schmidt number, Chemical reaction parameter while it increases with increase of Thermal diffusion parameter.

x. Finally, the present numerical results coincides with the published results of Mohamed et al. [1]. ________________________________________________________________________________________________________

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