SIMULTANEOUS EFFECTS THERMAL DIFFUSION AND DIFFUSION THERMO ON MHD NON-NEWTONIAN ... ·...

VOL. 14, NO. 10, MAY 2019 ISSN 1819-6608

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1876

SIMULTANEOUS EFFECTS THERMAL DIFFUSION AND DIFFUSION

THERMO ON MHD NON-NEWTONIAN CASSON FLUID FLOW

ALONG A VERTICALLY INCLINED PLATE IN PRESENCE

OF FREE CONVECTION AND JOULES DISSIPATION

D. V. V. Krishna Prasad

1, G. S. Krishna Chaitanya

2 and R. Srinivasa Raju

3

1Department of Mechanical Engineering, R. V. R. and J. C. College of Engineering, Guntur, Andhra Pradesh, India 2Department of Mechanical Engineering, Acharya Nagarjuna University College of Engineering, Nagarjuna Nagar, Guntur,

Andhra Pradesh, India 3Department of Mathematics, GITAM University, Hyderabad Campus, Rudraram, Telangana State, India

E-Mail: [email protected]

ABSTRACT

This paper derives numerical solutions of completely developed free convection with heat and mass transfer flow

towards a vertically inclined plate in presence of Casson fluid, thermal diffusion, diffusion thermo, heat source and porous

medium. In energy equation, the effects of viscous dissipation and Joule dissipation effects are discussed. The numerical

solution for the governing nonlinear boundary value problem is based on the numerical method scheme over the entire

range of physical parameters. The transmuted governing partial differential equations are resolved numerically by

employing finite element method. His impact of pertinent flow parameters on momentum, thermal and mass transport

behaviour including the skin-friction factor, thermal and mass transport rate are examined and published with the

assistance of graphical and tabular forms. Favourable comparisons with previously published work on various special cases

of the problem are obtained.

Keywords: thermal diffusion; diffusion thermo; casson fluid; magnetic field; free convection; joule dissipation; finite element method.

Nomenclature:

List of variables:

wC Concentration of the plate (3

mKg )

y Dimensionless displacement ( m )

T Fluid temperature away from the plate (K)

u Velocity component in x direction

(1

sm )

x Coordinate axis along the plate ( m )

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

C Fluid Concentration (3

mKg )

T Fluid temperature )(K

wT Fluid temperature at the wall K

0B Uniform magnetic field (Tesla)

C Concentration of the fluid far away from the

plate (3

mKg )

u Fluid velocity (1

sm )

Gc Grashof number for mass transfer

Sh The local Sherwood number coefficient

g Acceleration of gravity, 9.81 (2

sm )

Gr Grashof number for heat transfer 2

M Magnetic field parameter

Pr Prandtl number

pC Specific heat at constant pressure

KKgJ1

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

ov Constant Suction velocity (1

sm )

Kr Chemical reaction parameter

oQ Dimensional Heat generation parameter

Q Heat generation parameter

oK Permeability parameter

Ec Eckert number

Greek Symbols:

Kinematic viscosity (12

sm )

Species concentration (3

mKg )

mailto:[email protected]

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The constant density (3

mKg )

Volumetric coefficient of thermal

expansion )( 1K

* Volumetric Coefficient of thermal

expansion with concentration (13

Kgm )

Fluid temperature K

Electric conductivity of the fluid )( 1ms

Thermal conductivity of the fluid

mKW /

Angle of inclination parameter (degrees)

Casson fluid parameter

Superscripts:

Dimensionless properties

Subscripts: Free stream conditions

p Plate

w Conditions on the wall

1. INTRODUCTION

The theory of non-Newtonian fluid is a part of

fluid mechanics based on the continuum theory that a fluid

particle may be considered as continuous in a structure.

Pseudo plastic time independent fluid is one of the non-

newtonian fluids whose behaviour is that Viscosity

decreases with increasing velocity gradient e.g. polymer

solutions, blood, etc. Casson fluid is one of the

pseudoplastic fluids that means shear thinning fluids. At

low shear rates the shear thinning fluid is more viscous

than the Newtonian fluid, and at high shear rates it is less

viscous. So, MHD flow with Casson fluid is recently

famous. Casson [1] presented Casson fluid model for the

prediction of the flow conduct of pigment-oil suspensions.

Dash et al. [2] discussed on Casson fluid flow in a pipe

with a homogeneous porous medium. Akbar [3], [4], [5],

[6] has studied Casson Fluid flow in a Plumb

Duct/asymmetric channel. Mohyud-Din [7], [8] has

discussed on magnetic field and radiation effects on

squeezing flow of a Casson fluid between parallel plates.

Raju [9] has studied effect of induced magnetic field on

stagnation flow of a Casson fluid. Kataria [10], [11], [12]

published the work on unsteady free convective MHD

Casson/micropolar/nano fluid flow with different

boundary conditions. Recently, Makanda [13], [14] has

discussed effects on radiation as well as chemical reaction

on Casson fluid flow. Abbasi [15], [16], [17], [18], [19],

[20] has considered three dimensional MHD flow with

different fluid and different physical conditions.

The natural convection of binary fluids flow in

porous media has attracted great research interest during

the past few decades. While a good number of works have

made significant contributions for the development of the

theory, an equally good number of works have been

devoted to the numerous industrial, natural and

geophysical applications. Double-diffusive convective

flows in a differentially heated vertical annulus have been

intensively studied in relation to applications such as

oxidation of surface materials, cleaning and dying

operations, fluid storage components and energy storage in

solar ponds [21]. Consideration of two kinds of problems

concerning the convection of a binary mixture filling a

porous layer is in the literature. The first kind of problem

considers flows induced by the buoyancy forces resulting

from the imposition of both thermal and solute boundary

conditions on the layer. The second kind of problem

considers thermal convection in a binary fluid driven by

Soret-effects. For this situation, the species gradients are

not due to the imposition of solute boundary conditions.

Rather, they result from the imposition of a temperature

gradient in an otherwise uniform-concentration mixture.

This phenomenon has many applications in geophysics, oil

reservoirs, and ground water. Bahloul et al. [22] gave the

reviews of previous works done in this direction. Alloui

and Vasseur [23] studied analytically and numerically the

double-diffusive and Soret-induced natural convection in a

shallow rectangular cavity filled with a micropolar fluid.

Lakshmi Narayana et al. [24] investigated the stability of

Soret-driven thermo-solutal convection in a shallow

horizontal layer of a porous medium subjected to inclined

thermal and solutal gradients of finite magnitude

theoretically and observed that the Soret parameter has a

significant effect on convective instability. Prasad et al.

[25] considered the thermo-diffusion and diffusion-thermo

effects on MHD free convection flow past a vertical

porous plate embedded in a non-Darcian porous medium.

Ibrahim and Suneetha [26] investigated the effects of Soret

and heat source on steady MHD mixed convective heat

and mass transfer flow past an infinite vertical plate

embedded in a porous medium in the presence of chemical

reaction, viscous and Joules dissipation. Sai and Huang

[27] considered steady stagnation point flow over a flat

stretching surface in the presence of species concentration

and mass diffusion under Soret and Dufour effects.

Bhattacharya et al. [28] investigated the Soret and Dufour

effects on convective heat and mass transfer in stagnation-

point flow towards a shrinking surface by using shooting

technique. Makinde et al. [29] studied the Soret and

Dufour effects on boundary layer flow past a moving plate

with chemical reaction. Raju et al. [30], Nadeem et al.

[31] and Jayachandrababu et al. [32] analyzed the heat

transfer in the non-Newtonian fluid across the stretching

sheet by viewing parameters like Brownian motion and

thermophoresis parameters. They observed that the

decrement in the temperature distribution increases the

thermophoresis and Brownian motion lessens the rate of

heat transfer performance. With the help of homotopy

analysis method Xu et al. [33] explained the stagnation

point flow of the non-Newtonian fluids. Later on, Sajid et

al. [34] made the relative study between HAM and HPM

methods for the non-Newtonian fluid flow over a thin film

and found that HAM is the better and simple method to

guarantee the convergence of the solution series. Hayat et

al. [35] analyzed the Soret and Dufour effects on magneto

/

VOL. 14, NO. 10, MAY 2019 ISSN 1819-6608



1878

hydrodynamic flow of Casson fluid over a stretching

sheet.

Therefore our work can be considered as

extension of Ibrahim and Suneetha [26]. So Novelty of

this paper is discussion of numerical solutions using finite

element method of steady free convective Casson fluid

flow past over an inclined vertical plate in the presence of

a thermal diffusion, diffusion thermo and heat generation

with joule dissipation through porous medium. Plots for

the influence of embedded flow quantities on the velocity,

temperature and concentration are displayed and discussed

in detail. The physical quantities of the local Nusselt and

Sherwood numbers with skin-friction coefficient are

summarized in the tabular form for the engineering

parameters. We also validated the present methodology

with already existing methodologies under some limited

cases. The free convective conditions are useful in

improving the heat and mass transport phenomena.

2. MATHEMATICAL FORMULATION

In this work, the combined effects of thermal

diffusion and diffusion thermo on free convection flow of

an incompressible and electrically conducting viscous

fluid in presence of Casson fluid, heat generation,

chemical reaction and applied magnetic field. The flow

configuration of the problem is presented in Figure-1.

For this investigation, let us assume that

i. x axis is taken along the vertical infinite porous

plate in the upward direction and the y axis normal

to the plate.

ii. 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.

iii. A transverse constant magnetic field is applied, i.e. in

the direction of y axis.

iv. Since the motion is two dimensional and length of the

plate is large therefore all the physical variables are

independent of x . v. 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.

vi. A homogenous first order chemical reaction between

fluid and the species concentration is considered, in

which the rate of chemical reaction is directly

proportional to the species concentration.

vii. The magnetic Reynolds number is so small that the

induced magnetic field can be neglected.

viii. Also no applied or polarized voltages exist so the

effect of polarization of fluid is negligible.

ix. All the fluid properties except the density in the

buoyancy force term are constants.

The rheological equation of state for the Cauchy

stress tensor of Casson fluid [36] is written as

*

0 (1)

equivalently

cij

c

y

B

cij

y

B

ij

ep

ep

,2

2

,2

2

(2)

where is shear stress, 0 is Casson yield stress, is

dynamic viscosity, * is shear rate, ijijee and ije is

the thji, component of deformation rate, is the

product based on the non-Newtonian fluid, c is a critical

value of this product, B is plastic dynamic viscosity of

the non-Newtonian fluid,

2B

yp (3)

denote the yield stress of fluid. Some fluids require a

gradually increasing shear stress to maintain a constant

strain rate and are called Rheopectic, in the case of Casson

fluid (Non-Newtonian) flow where c

2

y

B

p (4)

Substituting Eq. (3) into Eq. (4), then, the

kinematic viscosity can be written as

1

1B (5)

The governing equations of continuity,

momentum, energy and mass for a flow of an electrically

conducting fluid are given by the following:

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Continuity Equation: 00

oo VVvy

v (6)

Momentum Equation:

uK

uB

CCgTTgy

u

y

uv

2

0*

2

2

coscos1

1 (7)

Energy Equation:

2

22

22

2

2

y

C

CC

kDTT

C

Qu

C

B

y

u

Cy

T

Cy

Tv

pS

Tm

p

o

p

o

pp

(8)

Species Diffusion Equation:

2

2

2

2

y

T

T

kDCCk

y

CD

y

Cv

m

Tmr

(9)

together with initial and boundary conditions

yasCCTTu

yatCCTTut

yallforCCTTut

ww

,,0

0,,0:0

,,0:0

(10)

Figure-1. Physical configuration and coordinates system.

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

layer, c --- Concentration boundary layer

Let us introduce the following non-dimensional

variables and parameters:

)(,

)(,

,,,,Pr,,)(

,,Re,,,, ,

2

2

22

2

3

*

32

2

CCT

TTkDSr

TTCC

CCkDDu

V

kKr

TTC

VEc

VC

QQ

DSc

CVKK

V

CCgGc

V

TTgGr

xV

V

BM

CC

CC

TT

TTVyy

V

uu

wm

wTm

wpS

wTm

o

r

wp

o

op

opoo

o

w

o

wo

o

o

ww

o

o

(11)

The above defined non-dimensionless variables in Eq. (11) into Eqs. (7), (8) and (9), and we get

0coscos1

1 2

2

2

GcGruKMy

u

y

uo

(12)

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1880

0PrPrPrPrPr2

222

2

2

2

yDuQuMEc

y

uEc

yy

(13)

02

2

2

2

yScSrScKr

ySc

y

(14)

with connected initial and boundary conditions

yasu

yatut

yallforut

0,0,0

01,1,0:0

0,0,0:0

(15)

The Skin-friction at the plate, which in the non-

dimensional form is given by

0

11

11

0

yo

w

y

u

VCf

y

(16)

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

y

yNu

(17)

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

y

ySh

(18)

3. NUMERICAL SOLUTIONS BY FINITE

ELEMENT METHOD

3.1 Finite element method: The finite element method

(Bathe [37] and Reddy [38]) has been employed to solve

Eqs. (12)-(14) under boundary conditions (15). Finite

element method is widely used for solving boundary value

problems. The basic concept is that the whole domain is

divided into smaller elements of finite dimensions called

“Finite Elements”. Thereafter the domain is considered as

an assemblage of these elements connected at a finite

number of joint called “Nodes”. The concept of

discretization is adopted here. Other features of the

method include seeking continuous polynomial,

approximations of the solution over each element in term

of nodal values, and assembly of element equations by

imposing the inter-element continuity of the solution and

balance of the inter-element forces. It is the most versatile

numerical technique in modern engineering analysis and

has been employed to study diverse problems in heat

transfer [39], fluid mechanics [40], chemical processing

[41], rigid body dynamics [42] and many other fields. The

method entails the following steps:

3.1.1 Finite element discretization: The whole domain is

divided into a finite number of sub-domains, which is

called the discretization of the domain. Each sub domain is

called an element. The collection of elements is called the

finite-element mesh.

3.1.2 Generation of the element equations

a) From the mesh, a typical element is isolated and the

variational formulation of the given problem over the

typical element is constructed.

b) An approximate solution of the variational problem is

assumed and the element equations are made by

substituting this solution in the above system.

c) The element matrix, which is called stiffness matrix,

is constructed by using the element interpolation

functions.

3.1.3 Assembly of the element equations: The algebraic

equations so obtained are assembled by imposing the inter

element continuity conditions. This yields a large number

of algebraic equations known as the global finite element

model, which governs the whole domain.

3.1.4 Imposition of the boundary conditions: On the

assembled equations, the Dirichlet's and Neumann

boundary conditions (15) are imposed.

3.1.5 Solution of assembled equations: The assembled

equations so obtained can be solved by any of the

numerical technique called Gauss elimination method.

3.2 Variational formulation: The variational formulation

associated with Eqs. (12)-(14) over a typical two-noded

linear element 1, ee yy is given by

0coscos1

11

2

2

1

dyGcGrNuy

u

y

uw

e

e

y

y

(19)

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1881

0PrPrPrPrPr1

2

22

2

2

2

2

dyy

DuQuMEcy

uEc

yyw

e

e

y

y

(20)

01

2

2

2

2

3

dyy

SrScScKry

Scy

we

e

y

y

(21)

Where

oKMN

12 and1w ,

2w , 3w are

arbitrary test functions and may be viewed as the variation

in and respectively. After reducing the order of

integration and non-linearity, we arrive at the following

system of equations:

0

11

coscos1

11

1

1

11111

e

e

e

e

y

y

y

yy

uw

dywGcwGruwNy

uw

y

u

y

w

(22)

0Pr

PrPr

PrPrPr

1

1

22

222

2222

e

e

e

e

y

y

y

y

ywDu

yw

dy

yy

wwDuwQ

uuMwEcy

u

y

uwEc

yw

yy

w

(23)

0

1

1

33

3333

3

e

e

e

e

y

y

y

y

ywScSr

yw

dyyy

wwSrScwScKr

ywSc

yy

w

(24)

3.3 Finite element formulation The finite element model may be obtained from

Eqs. (22) - (24) by substituting finite element

approximations of the form:

(25)

With )2,1(321 iwwwe

j , where

e

j and are the velocity, temperature and

concentration respectively at the node of typical

element 1, ee yy and are the shape functions for

this element 1, ee yy and are taken as:

ee

ee

yy

yy

1

11 and ,

1

2

ee

ee

yy

yy

1 ee yyy (26)

The finite element model of the equations for

element thus formed is given by

,u

,2

1

j

e

j

e

juu ,2

1

j

e

j

e

j

2

1j

e

j

e

j

,e

jue

j

jth ethe

i

eth

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1882

e

e

e

e

e

e

e

e

e

b

b

bu

MMM

MMM

MMMu

KKK

KKK

KKK

3

2

1

333231

232221

131211

333231

232221

131211

(27)

Where mnmnMK , and 3,2,1,,,,,, nmbanduu

meeeeeee are the set of matrices

of order 22 and 12 respectively and /' (dash) indicates

dy

d. These matrices are defined as

,1

11

11dy

yyK

e

e

y

y

e

je

iij

,1

12dyNK

e

e

y

y

e

j

e

iij

,01312 ijij MM ,03231 ijij MM

dyGcGrKe

e

y

y

e

j

e

iij

1

cos13 , ,1

11dyM

e

e

y

y

e

j

e

iij

,02321 ijij MM

,PrPr11

21dyQdyMEcK

e

i

y

y

e

j

e

i

y

y

e

j

e

jij

e

e

e

e

,

1

22dy

yyK

e

e

y

y

e

je

iij

,1

33dyM

e

e

y

y

e

j

e

iij

,PrPr11

23dy

yyDudy

yyEcK

e

j

y

y

e

ie

i

y

y

e

j

e

j

ij

e

e

e

e

,1

22dyM

e

e

y

y

e

j

e

iij

,031 ijK

,11

32dyScKrdy

yyScSrK

e

j

y

y

e

i

e

j

y

y

e

iij

e

e

e

e

,1

33dy

yyK

e

e

y

y

e

je

iij

,Pr

1

2

e

e

y

y

e

i

e

i

e

iy

Duy

b

1

3

e

e

y

y

e

i

e

i

e

iy

ScSry

b ,

1

111

e

e

y

y

e

i

e

iy

ub

The whole domain is subdivided into two noded

elements. In a nutshell, the finite element equation are

written for all elements and then on assembly of all the

element equations we obtain a matrix of order 328×328.

After applying the given boundary conditions a system of

320 equations remains for numerical solution, a process

which is successfully discharged utilizing the Gauss-

Seidel method maintaining an accuracy of 0.0005. A

convergence criterion based on the relative difference

between the current and previous iterations is employed.

When these differences satisfy the desired accuracy, the

solution is assumed to have been converged and iterative

process is terminated. The Gaussian quadrature is

implemented for solving the integrations. The code of the

algorithm has been executed in MATLAB running on a

PC. Excellent convergence was achieved for all the

results.

4. STUDY OF GRID INDEPENDENCE OF FINITE

ELEMENT METHOD

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1883

Table-1. Grid Invariance test for primary velocity, secondary velocity and temperature profiles.

Mesh (Grid) Size = 0.0001 Mesh (Grid) Size = 0.001 Mesh (Grid) Size = 0.01

u θ ϕ u θ ϕ u θ ϕ

0.0000000 1.0000000 1.0000000 0.0000000 1.0000000 1.0000000 0.0000000 1.0000000 1.0000000

0.8480260 0.1395750 0.2033457 0.8540495 0.1410919 0.2034235 0.8599686 0.1425892 0.2034936

0.7580782 0.0142992 0.0378721 0.7630903 0.0146008 0.0380398 0.7680142 0.0149034 0.0382000

0.7464278 0.0023998 0.0059360 0.7512104 0.002465 0.0060143 0.7559059 0.0025318 0.0060904

0.7450653 0.0010531 0.0007495 0.7498058 0.0010853 0.0007703 0.7544595 0.0011179 0.0007907

0.7449343 0.0009072 0.0000729 0.7496684 0.0009339 0.0000764 0.7543157 0.0009609 0.0000799

0.7449250 0.0008944 0.0000053 0.7496584 0.0009204 0.0000057 0.7543047 0.0009467 0.0000061

0.7449243 0.0008935 0.0000001 0.7496578 0.0009195 0.0000002 0.7543041 0.0009457 0.0000002

0.7448659 0.0008910 0.0000019 0.7495923 0.0009167 0.0000020 0.7542313 0.0009426 0.0000021

0.7231130 0.0007731 0.0000122 0.7270228 0.0007924 0.0000126 0.7308382 0.0008118 0.0000130

0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000

To investigate the sensitivity of the solutions to

mesh density, it was observed that in the same domain the

accuracy is not affected, even if the number of elements is

increased, by decreasing the size of the elements. This

serves only to increase the compilation times and does not

enhance in any way the accuracy of the solutions, as

shown in table 1. Thus, for computational purposes, 1000

elements were taken for presentation of the results.

Excellent convergence was achieved in the present study.

5. VALIDATION OF CODE For code validation purpose, we compared our

numerical results with the exisistance analytical results of

Ibrahim and Suneetha [26] in Tables 2-10, in the absence

of non-newtonian Casson fluid, diffusion thermo and

angle of inclination. From these tables, it is observed that

the relevant results obtained agree quantitatively with

earlier results of Ibrahim and Suneetha [26].

Table-2. Comparison of present skin-friction coefficient results with the skin-friction coefficient results of

Ibrahim and Suneetha [26] for Pr = 0.71, Ec = 0.001, Q = 0.1, Sc = 0.60, Kr = 0.1, Sr = 0.5 values.

Gr Gc M 2

Ko Present Skin-friction results Skin-friction results of Ibrahim

and Suneetha [26]

3.0 1.0 1.0 1.0 2.5344158758 2.5427

5.0 1.0 1.0 1.0 3.8045221862 3.8114

3.0 3.0 1.0 1.0 3.8422569218 3.8502

3.0 1.0 3.0 1.0 1.2150048321 1.2210

3.0 1.0 1.0 2.0 2.1296604855 2.1306

Table-3. Comparison of present Nusselt number coefficient results with the Nusselt number coefficient results of

Ibrahim and Suneetha [26] for Pr = 0.71, Ec = 0.001, Q = 0.1, Sc = 0.60, Kr = 0.1, Sr = 0.5 values.

Gr Gc M 2

Ko Present Nusselt number

coefficient results

Nusselt number coefficient results

of Ibrahim and Suneetha [26]

3.0 1.0 1.0 1.0 - 0.5744128932 - 0.5877

5.0 1.0 1.0 1.0 - 0.5799621843 - 0.5854

3.0 3.0 1.0 1.0 - 0.5766291847 - 0.5852

3.0 1.0 3.0 1.0 - 0.5744120589 - 0.5888

3.0 1.0 1.0 2.0 - 0.5741152286 - 0.5885

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Table-4. Comparison of present Sherwood number coefficient results with the Sherwood number coefficient results

of Ibrahim and Suneetha [26] for Pr = 0.71, Ec = 0.001, Q = 0.1, Sc = 0.60, Kr = 0.1, Sr = 0.5 values.

Gr Gc M 2

Ko Present Sherwood

number coefficient results

Sherwood number coefficient results of

Ibrahim and Suneetha [26]

3.0 1.0 1.0 1.0 - 0.5299854128 - 0.5338

5.0 1.0 1.0 1.0 - 0.5293315487 - 0.5345

3.0 3.0 1.0 1.0 - 0.5293315487 - 0.5345

3.0 1.0 3.0 1.0 - 0.5299564875 - 0.5335

3.0 1.0 1.0 2.0 - 0.5298813044 - 0.5336

Table-5. Comparison of present skin-friction coefficient results with the skin-friction coefficient results of

Ibrahim and Suneetha [26] for Gr = 3.0, Gc = 1.0, Ko = 1.0, Sc = 0.60, Kr = 0.1, Sr = 0.5 values.

Pr Ec Q Present Skin-friction

results

Skin-friction results of Ibrahim

and Suneetha [26]

0.71 0.001 0.100 2.5344158758 2.5427

1.00 0.001 0.100 2.2344581209 2.2469

0.71 0.002 0.100 2.5395144783 2.5481

0.71 0.001 0.005 2.5194420185 2.5250


Ibrahim and Suneetha [26] for Gr = 3.0, Gc = 1.0, Ko = 1.0, Sc = 0.60, Kr = 0.1, Sr = 0.5 values.

Pr Ec Q Present Nusselt number

coefficient results

Nusselt number coefficient results of


0.71 0.001 0.100 - 0.5744128932 - 0.5877

1.00 0.001 0.100 - 0.8744125869 - 0.8856

0.71 0.002 0.100 - 0.5741128321 - 0.5859

0.71 0.001 0.005 - 0.6488526189 - 0.6543

Table-7. Comparison of present Sherwood number coefficient results with the Sherwood number coefficient

results of Ibrahim and Suneetha [26] for Gr = 3.0, Gc = 1.0, Ko = 1.0, Sc = 0.60, Kr = 0.1, Sr = 0.5 values.

Pr Ec Q Present Sherwood number

coefficient results

Sherwood number coefficient results of


0.71 0.001 0.100 - 0.5299854128 - 0.5338

1.00 0.001 0.100 - 0.4395521863 - 0.4454

0.71 0.002 0.100 - 0.5299841207 - 0.5343

0.71 0.001 0.005 - 0.5066954128 - 0.5141

Table-8. Comparison of present skin-friction coefficient results with the skin-friction coefficient results of Ibrahim

and Suneetha [26] for Gr = 3.0, Gc = 1.0, M 2

= 1.0, Ko = 1.0, Pr = 0.71, Ec = 0.001, Q = 0.1 values.

Sc Sr Kr Present Skin-friction

results

Skin-friction results of Ibrahim

and Suneetha [26]

0.6 0.1 0.5 2.5344158758 2.5427

1.0 0.1 0.5 2.4396221077 2.4412

0.6 0.3 0.5 2.4710026344 2.4806

0.6 0.1 1.0 2.6041102500 2.6109

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1885


Ibrahim and Suneetha [26] for Gr = 3.0, Gc = 1.0, M 2 = 1.0, Ko = 1.0, Pr = 0.71, Ec = 0.001, Q = 0.1 values.

Sc Sr Kr Present Nusselt number

coefficient results

Nusselt number coefficient results of


0.6 0.1 0.5 - 0.5744128932 - 0.5877

1.0 0.1 0.5 - 0.5788621154 - 0.5880

0.6 0.3 0.5 - 0.5796620017 - 0.5879

0.6 0.1 1.0 - 0.5796621035 - 0.5875

Table-10. Comparison of present Sherwood number coefficient results with the Sherwood number coefficient results

of Ibrahim and Suneetha [26] for Gr = 3.0, Gc = 1.0, M 2 = 1.0, Ko = 1.0, Pr = 0.71, Ec = 0.001, Q = 0.1 values.

Sc Sr Kr Present Sherwood number

coefficient results

Sherwood number coefficient results

of Ibrahim and Suneetha [26]

0.6 0.1 0.5 - 0.5299854128 - 0.5338

1.0 0.1 0.5 - 0.8291154387 - 0.8373

0.6 0.3 0.5 - 0.6855200133 - 0.6912

0.6 0.1 1.0 - 0.3752210668 - 0.3804

6. RESULTS AND DISCUSSIONS

Figure-2. Gr influence on velocity profiles.

0

0.3

0.6

0.9

1.2

0 2 4 6 8 10

u

y

Gr = 3.0, 4.0, 5.0, 6.0

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1886

Figure-3. 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-2, Figure3, 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 and Figure-21

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 2

),

Permeability parameter (Ko), Prandtl number (Pr), Schmidt

number (Sc), Eckert number (Ec), Heat source parameter

(Q), Chemical reaction parameter (Kr), Thermal diffusion

parameter (Sr), Diffusion thermo parameter (Du), Angle of

inclination parameter (α) and Casson fluid parameter (γ)

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 = 3.0, Gc = 1.0, Ko = 1.0, M 2

= 1.0, Ec = 0.001, Sc = 0.22 (Hydrogen), Q = 0.1, Pr =

0.71 (Air), Kr = 0.5, Sr = 0.1, Du = 0.5, α = 45o and γ =

0.5. All graphs therefore correspond to these values unless

specifically indicated in the appropriate graph.

Figure-4. M 2

influence on velocity profiles.

0

0.3

0.6

0.9

1.2

0 2 4 6 8 10

u

y

Gc = 1.0, 2.0, 3.0, 4.0

0

0.3

0.6

0.9

1.2

0 2 4 6 8 10

u

y

M 2 = 1.0, 2.0, 3.0, 4.0

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1887

Figure-5. Ko influence on velocity profiles.

Figure-6. γ influence on velocity profiles.

Figure-7. α influence on velocity profiles.

0

0.3

0.6

0.9

1.2

0 2 4 6 8 10

u

y

Ko = 1.0, 2.0, 3.0, 4.0

0

0.3

0.6

0.9

1.2

0 2 4 6 8 10

u

y

γ = 0.5, 1.0, 1.5, 2.0

0

0.3

0.6

0.9

1.2

0 2 4 6 8 10

u

y

α = π/6, π/4, π/3, π/2

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The velocity profiles for different values of

Grashof number for heat transfer Gr are described in

Figure-2. 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 Figure-3. It is

observed that an increase in Gc leads to a rise in the values

of velocity. Figure-4 shows the effect of magnetic

parameter M 2 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 2

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. Figure-7 depicts the

velocity profiles for various values of Ko. From this figure

it is observed that fluid velocity increases as Ko 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. The

velocity profiles in the Figure-7 shows that rate of motion

is significantly reduced with increasing of Casson fluid

parameter γ. The effect of angle of inclination of the plate

on the velocity field has been illustrated in Figure-8. It is

seen that as the angle of inclination of the plate increases

the velocity field decreases.

Figure-8. Pr influence on velocity profiles.

Figure-9. Pr influence on temperature profiles.

0

0.3

0.6

0.9

1.2

0 2 4 6 8 10

u

y

Pr = 0.025, 0.71, 7.0, 11.62

0

0.5

1

0 2 4 6 8 10

θ

y

Pr = 0.025, 0.71, 7.0, 11.62

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1889

Figures 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 Figure-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. For various values of the

Schmidt number Sc, the velocity and concentration are

plotted in Figures 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

Figures 10 and 11.

Figurte-10. Sc influence on velocity profiles.

Figure-11. Sc influence on concentration profiles.

Figures 12 and 13 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. Figure-

14 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

0

0.3

0.6

0.9

1.2

0 2 4 6 8 10

u

y

Sc = 0.22, 0.30, 0.60, 0.78

0

0.5

1

0 2 4 6 8 10

ϕ

y

Sc = 0.22, 0.30, 0.60, 0.78

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1890

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. Figure-15 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.

Figure-12. Q influence on velocity profiles.

Figure-13. Q influence on temperature profiles.

For different values of the Diffusion thermo

parameter, the velocity and temperature profiles are

plotted in Figures 16 and 17 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. Figures 18 and 19 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.

0

0.3

0.6

0.9

1.2

0 2 4 6 8 10

u

y

Q = 0.1, 0.3, 0.5, 1.0

0

0.5

1

0 2 4 6 8 10

θ

y

Q = 0.1, 0.3, 5.0, 1.0

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

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1891

Figure-14. Kr influence on velocity profiles.

Figure-15. Kr influence on concentration profiles.

Figure-16. Du influence on velocity profiles.

0

0.3

0.6

0.9

1.2

0 2 4 6 8 10

u

y

Kr = 0.5, 1.0, 1.5, 2.0

0

0.5

1

0 2 4 6 8 10

ϕ

y

Kr = 0.5, 1.0, 1.5, 2.0

0

0.3

0.6

0.9

1.2

0 2 4 6 8 10

u

y

Du = 0.5, 1.0, 1.5, 2.0

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1892

Figure-17. Du influence on temperature profiles.

Figure-18. Sr influence on velocity profiles.

Figure-19. Sr influence on concentration profiles.

0

0.5

1

0 2 4 6 8 10

θ

y

Du = 0.5, 1.0, 1.5, 2.0

0

0.3

0.6

0.9

1.2

0 2 4 6 8 10

u

y

Sr = 0.1, 0.5, 0.8, 1.0

0

0.5

1

0 2 4 6 8 10

ϕ

y

Sr = 0.1, 0.5, 0.8, 1.0

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Figure-20. Ec influence on velocity profiles.

Figure-21. Ec influence on temperature profiles.

The influence of the viscous dissipation

parameter i.e., the Eckert number on the velocity and

temperature are shown in Figures 20 and 21 respectively.

The Eckert number expresses the relationship between the

kinetic energy in the flow and the enthalpy. It embodies

the conversion of kinetic energy into internal energy by

work done against the viscous fluid stresses. Greater

viscous dissipative heat causes a rise in the temperature as

well as the velocity. This behaviour is evident from

Figures 20 and 21. 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

source parameter, Angle of inclination parameter, Casson

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, Diffusion thermo parameter,

Eckert number and Thermal diffusion parameter. The

numerical values of Nusselt and Sherwood numbers are

presented in Table-2. With increase in the Eckert number

and Diffusion thermo parameter, the Nusselt number

increases but for the other parameters such as Prandtl

number and Heat source 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.

0

0.3

0.6

0.9

1.2

0 2 4 6 8 10

u

y

Ec = 0.001, 0.01, 0.1, 1.0

0

0.5

1

0 2 4 6 8 10

θ

y

Ec = 0.001, 0.01, 0.1, 1.0

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1894

Table-11. Numerical values of Skin-friction coefficient (Cf).

Gr Gc M 2

Ko Pr Sc Ec Q Kr Sr Du α γ Cf

3.0 1.0 1.0 1.0 0.71 0.22 0.001 0.1 0.5 0.1 0.5 45o

0.5 2.5681124512

5.0 1.0 1.0 1.0 0.71 0.22 0.001 0.1 0.5 0.1 0.5 45o

0.5 2.6142201589

3.0 3.0 1.0 1.0 0.71 0.22 0.001 0.1 0.5 0.1 0.5 45o

0.5 2.6510029877

3.0 1.0 3.0 1.0 0.71 0.22 0.001 0.1 0.5 0.1 0.5 45o

0.5 2.4508128622

3.0 1.0 1.0 2.0 0.71 0.22 0.001 0.1 0.5 0.1 0.5 45o

0.5 2.6014478231

3.0 1.0 1.0 1.0 7.00 0.22 0.001 0.1 0.5 0.1 0.5 45o

0.5 2.4152213809

3.0 1.0 1.0 1.0 0.71 0.30 0.001 0.1 0.5 0.1 0.5 45o

0.5 2.4013556912

3.0 1.0 1.0 1.0 0.71 0.22 1.000 0.1 0.5 0.1 0.5 45o

0.5 2.5841176939

3.0 1.0 1.0 1.0 0.71 0.22 0.001 0.3 0.5 0.1 0.5 45o

0.5 2.5148999673

3.0 1.0 1.0 1.0 0.71 0.22 0.001 0.1 1.0 0.1 0.5 45o

0.5 2.5014335982

3.0 1.0 1.0 1.0 0.71 0.22 0.001 0.1 0.5 0.3 0.5 45o

0.5 2.5981143336

3.0 1.0 1.0 1.0 0.71 0.22 0.001 0.1 0.5 0.1 1.0 45o

0.5 2.6015329744

3.0 1.0 1.0 1.0 0.71 0.22 0.001 0.1 0.5 0.1 0.5 60o

0.5 2.5231665548

3.0 1.0 1.0 1.0 0.71 0.22 0.001 0.1 0.5 0.1 0.5 45o

1.0 2.5143264782

Table-12. Numerical values of Nusselt (Nu) and Sherwood (Sh) numbers.

Pr Ec Q Du Nu Sc Kr Sr Sh

0.71 0.001 0.1 0.5 - 0.5941258615 0.22 0.5 0.1 - 0.5299854128

7.00 0.001 0.1 0.5 - 0.6841125479 0.30 0.5 0.1 - 0.6212218462

0.71 1.000 0.1 0.5 - 0.5841233548 0.60 0.5 0.1 - 0.6874121154

0.71 0.001 0.5 0.5 - 0.6152344897 0.22 1.0 0.1 - 0.6012154887

0.71 0.001 0.1 1.0 - 0.5012458147 0.22 0.5 0.5 - 0.4811279962

7. CONCLUSIONS

Simultaneous effects of thermal diffusion and

diffusion thermo on unsteady MHD free convection flow

past a vertically inclined plate filled in porous medium in

the presence of viscous dissipation, joule dissipation, heat

absorption, chemical reaction, heat and mass transfer have

been studied numerically. Finite element method is

employed to solve the governing equations of the flow.

From the present investigation, the following conclusions

have been drawn:

a) The velocity profiles increases with an increase in the

Grashof number for heat and mass transfer,

Permeability parameter, Thermal diffusion parameter,

Eckert number and Diffusion thermo parameters.

b) The velocity profiles decreases with an increase in

Magnetic field parameter, Chemical reaction

parameter, Heat source parameter, Prandtl number,

Schmidt number, Angle of inclination parameter and

Casson fluid parameter.

c) There are increases in the temperature profiles with

increases in Eckert number and Diffusion thermo

parameter. When there is increase in Prandtl number

and Heat source parameter there is decrease in

temperature profiles.

d) Concentration decreases with an increase in Schmidt

number and Chemical reaction parameter. When

increasing the value of Thermal diffusion parameter

there is increase in the Concentration boundary layer.

e) Local skin-friction increases with an increase in

Grashof number for heat and mass transfer,

Permeability parameter, Thermal diffusion parameter,

Eckert number and Diffusion thermo parameters

while it decreases for rising values of Magnetic field

parameter, Chemical reaction parameter, Heat source

parameter, Prandtl number, Schmidt number, Angle

of inclination parameter and Casson fluid parameter.

f) Nusselt number decreases with an increase in Prandtl

number and Heat source parameter while it increases

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1895

with an increase in Eckert number and Diffusion

thermo parameter. Sherwood number decreases with

an increase in Schmidt number, Chemical reaction

parameter while it increases with increase of Thermal

diffusion parameter.

g) Finally, the present numerical results coincides with

the published results of Ibrahim and Suneetha [26] in

absence of Casson fluid, Diffusion thermo and Angle

of inclination.

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