effects of radiation and thermal diffusion on mhd heat transfer flow … · 2018-12-08 · flow...

VOL. 13, NO. 22, NOVEMBER 2018 ISSN 1819-6608

ARPN Journal of Engineering and Applied Sciences ©2006-2018 Asian Research Publishing Network (ARPN). All rights reserved.

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8863

EFFECTS OF RADIATION AND THERMAL DIFFUSION ON MHD HEAT

TRANSFER FLOW OF A DUSTY VISCOELASTIC FLUID BETWEEN

TWO MOVING PARALLEL PLATES

B. Mallikarjuna Reddy

1, D. Chenna Kesavaiah

2 and G. V. Ramana Reddy

1

1Koneru Lakshmaiah (Deemed to be University), Vaddeswaram, Guntur, Andhra Pradesh, India 2Department of H & S, K G Reddy College of Engineering and Technology, Chilkur, Moinabad, R R Dist, TS, India

E-Mail: [email protected]

ABSTRACT

An anticipated outcome of present analysis is effects of radiation and thermal diffusion on MHD heat transfer

flow of dusty viscous, incompressible, electrically conducting fluid between two parallel plates with constant suction on

the upper plate and constant injection on the lower plate, first order chemical reaction, variable temperature and uniform

mass diffusion taking into an account. The governing partial differential equations which describe for motion of the

problem change into dimensionless equations and solved by using perturbation technique. The various analytical quantities

for the velocity profiles (for dusty fluid and dust particles), temperature profiles, concentration profiles and skin friction

coefficient are examine and depict graphically in detail.

Keywords: radiation, thermal diffusion, MHD flow, viscoelastic fluid.

INTRODUCTION Dusty flows driven by combined effects of

diffusion and thermal diffusion of chemical taxonomic

group are encountered in many areas like astrophysical,

geophysical and engineering applications. These are

considered in the chemical treatment industries, powder

technology, moisture transport in thermal insulation, food

processing industries, pore water convection hear salt

domes, air craft icing, movement of contaminants in

ground water, waste dissolution and eliminate in

underground nuclear waste disposal, dust entrainment in

clouds in nuclear explosion and manner of functioning of

solid fuels in rocket nozzles, etc. Convective flows

accompanying temperature and concentration differences

at as the transfer of heat and mass carry out place have

been studied widely and distinctly extensions of the

problem have been reported in the literature. In practice,

ammonia, water – vapor, oxygen, hydrogen, helium etc are

main gases which are added in particulate air or implied

liquids embedded in the dust particles flowing though

channels.

In such flows, the buoyancy forces moves

towards within existence due to temperature with the

addition of concentration differences MHD channel flows

are studied due to their important applications in MHD

pumps, MHD flow meters and MHD generators etc. in

view of the above some of the authors studied Ashok

Kumar and Lal [1] considered effect of oscillatory motion

of a viscoelastic dusty fluid passes through a porous

medium under the presence of magnetic field, Attia et al.

[2] Studied MHD Couette flow and heat transfer of a dusty

fluid with exponential decaying pressure gradient, Chenna

Kesavaiah [3] investigated Natural convection heat

transfer oscillatory flow of an elastico viscous fluid form

vertical plate, Das [5] has heat transfer to MHD oscillatory

dusty viscoelastic fluid flow in an inclined channel filled

with a porous medium, Gosh et al. [6] produced the

hydromagnetic flow of a dusty Visco-elastic fluid between

two infinite parallel plates, Ibrahim Saidu et al. [7] studies

MHD effects on convective flow of dusty viscous fluid

with volume fraction of dust particles, Liu [8] focused

flow induced by the impulsive motion of an infinite flat

plate in a dusty gas, Madhura and Kalpana [9] analyzed

thermal effect on unsteady flow of a dusty viscoelastic

fluid between two parallel plates under different pressure

gradients.

The multiphase fluid systems are solicitude with

the motion of a liquid or gas containing immiscible having

only a limited ability to react identical particles. All

multiphase fluid systems found in nature, flows in rocket

tubes, sand dust in gas cooling systems to enhance heat

transfer processes, blood flow in arteries, act of inert solid

particles in atmosphere or other suspended particles in sea

or ocean beaches are the most common examples of

multiphase fluid systems. In this regard some of the

researcher are investigated Madhura and Swetha [10]

Influence of volume fraction of dust particles on dusty

fluid flow through porous rectangular channel, Mohaned

Ismail and Ganesh [11] Unsteady Stokes flow of dusty

fluid between two parallel plates through porous medium,

Mudassar Jalil et al. [12] An exact solution of MHD

boundary layer flow of dusty fluid over a stretching

surface, Singh and Atul Kumar Singh [13] MHD effects

on heat and mass transfer in flow of a dusty viscous fluid

with volume fraction, Om Prakash [14] Effects of thermal

diffusion and chemical reaction on MHD flow of dusty

viscoelastic fluid, Saffman [16] observed on the stability

of laminar flow of a dusty gas, Sandeep and Saleem [17]

MHD flow and heat transfer of a dusty nanofluid over a

stretching surface in a porous medium.

In spite of the above studies, the dynamic model

of a two phase conducting fluid flow in a nonrotating

system has received relatively less attention. The main

intend of this paper investigation of the motion of an

incompressible viscoelastic fluid with embedded small

spherical inert particles bounded by two infinite moving

mailto:[email protected]




8864

parallel plates in the presence of transverse magnetic fluid

under the thermal diffusion and radiation effect. The flow

is produce by time varying pressure gradient. The velocity

fields of the fluid flow and dust particles are determined

by perturbation method.

MATHEMATICAL FORMULATION Consider the unsteady laminar dusty flow of

incompressible, viscous, slightly conducting, viscoelastic

fluid with uniform distribution of the dust particles

between two heated porous infinite moving parallel plates

with the subject to the affect of uniform magnetic field

normal to the flow field with heat source under the effects

of thermal diffusion, radiation and chemical reaction. The

dust particles are uniformly distributed in a porous

medium and adopted in order to deceive to be spherical in

shape and uniform in size. The number of density of the

dust particles is taken as a constant throughout the flow.

Originally, when 0t the temperature and concentration

of the dusty fluids is axis is 0T and 0C respectively. When

0t the temperature profiles and concentration profiles

are evaluated to w

T and w

C respectively. Regarding these

assumptions the flow will be a parallel flow in which the

streamlines are along the x axis as shown below:

2

200 0 0 02

1KNu u

g T T g C C K v u B u ut t y K

(1)

vm K u v

t

(2)

2

002

1T r

p p p

QK qT TT T

t C y C y C

(3)

2 2

02 2T

C T TD Kr C C D

t y y

(4)

The initial and boundary conditions of the problem

0

0 0

0 0

0 0

0 0

0 : 0 , ,

0 : 0 , 1

1

0 , 1

1

at

w

at

w

at

w

at

w

t u v T T y d d

t u v T T T T e

C C C C e for y d

u v T T T T e

C C C C e for y d

(5)

The radiative heat flux of the equation (3) in the

spirit of Cogley et al. [4] 04rq

T T Iy

Where

0

biw

eI K d

T

, where iw

K is the absorption

in the wall and be Planck’s function Introduce the

following non – dimensional quantities

2

2

0 0

, , , , , ,o o

w w

T T C Cy t u v d ay t u v T C a

d d d d T T C C

Introducing these non dimensional quantities,

equations (1) – (4) reduce to

2

2

2

1

1u u u

GrT GmC E v u M ut t y w K

(6)

vW u v

t

(7)

2

2Pr

T TS R T

y t

(8)

2 2

2 20T

C C TSc KrScC S

y t y

(9)

Initial and boundary conditions of equation (5)

according to new system become

0 0

0 : 0 : 0 , 0 1,1

0 : 0 , 1 1

0 , 1 , 1 1

at

w

at at

t T u v T y

t u v T T T T e for y

u v T e C e for y

(10)

where u dusty fluid phase velocity, v dust

particle velocity, t time, density of the fluid,

P pressure of fluid, x coordinate axis in the direction

of the flow, g acceleration die to gravity, volumetric thermal expansion coefficient, mean free

path of diffusing particles, y coordinate axis normal to

the plate, ADarcy number, t time, N Number

density of dust particles, T temperature of fluid, 0T Initial temperature, C concentration of fluid, 0C

initial uniform concentration at 0T , 0mN

(mass

concentration of dust particle),0M B d

(Hartmann




8865

number), 0wg d T T

Gr

(Grashof number),

2

1K dKr

(chemical reaction parameter),

2

4 IR

d

(radiation parameter) 2

mW

K d

(relaxation time

parameter for particles), 0w

g d C CGm

(modified Grashof number), 0

2

KE

d (viscoelastic

parameter), ScD

(Schmidt number),

2

T

QdS

K (Heat

source parameter), Prp

T

C

K

(Prandtl number),

1 2

kK

d ,

0

0

T w

T

w

D T TS

C C

(thermal diffusion

parameter)

SOLUTION OF THE PROBLEM

Solve the system of equations (6) – (9) subject to

the boundary conditions (10) according to Pop [15]

assume that

0 1

0 1

0 1

0 1

, ,

, ,

, ,

, ,

at

at

at

at

u y t u y u y t e

v y t v y v y t e

T y t T y T y t e

C y t C y C y t e

(11)

Substituting equation (11) in to the equations (6)

– (9) and equating harmonic and non – harmonic terms get

the flowing set of equations

0 0 0 0u u GrT GmC (12)

1 1 1 1 1 11 aE u u v GrT GmCw

(13)

0 0 1 1& 1u v u v aw (14)

0 0 0T S R T (15)

1 1Pr 0T S R a T (16)

0 0 0 0T

C Kr ScC S T (17)

1 1 1 0T

C Kr a Sc C S T (18)

where dashes represents differentiation w. r. t. to y

The initial and boundary condition are reduced to

0 0 1 1 0 0 1 1

0 0 1 1 0 0 1 1

10, 0 , 1

10, 0 , 1

u v u v T C T C at y

u v u v T C T C at y

(19)

Solve the equations (12) – (18) under the initial

and boundary conditions (19) we get the solution

13 4 3 4

31 25 6 5 6

1 2 3

,

at

Cosh yCosh y Cosh yu y t A A A A

Cosh Cosh Cosh

Cosh yCosh y Cosh ye A A A A

Cosh Cosh Cosh

13 4 3 4

1

31 23 4 5 6

1 2 3

,

1

1

at

Cosh yCosh y Cosh yv y t A A A A

Cosh Cosh Cosh

Cosh yCosh y Cosh ye A A A A

aw Cosh Cosh Cosh

2

2

, at Cosh yCosh yT y t e

Cosh Cosh




8866

3 21 1 2 2

3 2

, 1 1at Cosh y Cosh yCosh y Cosh yC y t A A e A A

Cosh Cosh Cosh Cosh

Skin Friction

Let f

and p

be the skin friction for dusty fluid

and dust particles respectively then we have

3 4 3 4 1 1

1

5 6 1 1 5 2 2 6 3 3

tanh tanh tanh

tanh tanh tanh

f

y

at

uA A A A

y

e A A A A

3 4 3 4 1 1

1

5 6 1 1 5 2 2 6 3 3

tanh tanh tanh

1tanh tanh tanh

1

f

y

at

uA A A A

y

e A A A Aaw

RESULTS AND DISCUSSIONS

The graphical presentations variations of velocity

profiles (for dusty fluid and dust particles), temperature

profiles, concentration profile and skin friction for dusty

fluid has been analyzed. Figures 2(a) and 2(b) determined

the velocity for both (dusty fluid and dust particles) of

distinct values of the chemical reaction parameter Kr , it

is found that the velocity increases with an increasing

values of chemical reaction parameter. Figures 3(a) and

3(b) depicted velocity profiles (for dusty fluid and dust

particles) of various values of Schmidt number Sc ,

from this figure it is clear that, an increases in the values

of Schmidt number the velocity increases in both dusty

fluid as well as dust particles. The velocity profiles for

both (dusty fluid and dust particles) are observed form

figures 4(a), 4(b) and 5(a), 5(b) of thermal Grashof

number Gr and mass Grashof number Gm . It is

decided that velocity profiles increases where as increases

in the values of thermal Grashof number and mass Grashof

number. This is an account of the buoyancy effects enter

more significant and as consider like. It implies that,

heaviest fluid is entrained out of the free stream due the

having strength buoyancy effects as thermal Grashof

number or mass Grashof number increases. This confirms

the downward flow to a thin region near the plates. Figures

6(a) and 6(b) predicted the velocity for different values of

the porosity parameter 1K . We observed that an

increasing in porosity parameter the velocity also

increases. Effect of Hartman number M evidence to a

resistive type of force similar to drag force, which tends to

resist the retarding flow of viscoelastic fluid flow. From

figures 7(a) to 7(b) clear that an increasing values of

Hartman numbers the velocity decreases for both (dusty

fluid and dust particles). This is because increasing

Hartman number the opposing Lorentz force increases

resulting in the decrease of the fluid velocity. The velocity

profiles for dusty fluid and dusty particles are shown in

figures 8(a) and 8(b) for different values of Prandtl

number Pr . From these figures it is evident that the

velocity profiles decreases with increasing values of

Prandtl number. The effects of heat source parameter S

on velocity profiles (dusty fluid and dust particles) are

found in figures 9(a) and 9(b), we noticed that the velocity

(dusty fluid and dust particles) decreases with increasing

values of heat source parameter. Thermal diffusion

parameter TS effects are shown in figure 10(a) and

10(b) for velocity profiles for both dusty fluid and dust

particles observed, it is being perceived the velocity

decreases in both dusty fluid and dust particles as

increasing values of diffusion parameter. It is known that

the radiation parameter R plays an important role in

flow phenomena because; it is a measure of the relative

magnitude of viscous boundary layer thickness to the

thermal boundary layer thickness. In view of the above the

variation of velocity profiles illustrated in figures 11(a)

and 11(b), it is noticed that the velocity reduces as

increasing values of radiation parameter for both dusty

fluid as well as dust particles. Figures 12(a) and 12 (b)

shown the variation of velocity profiles for viscoelastic

parameter E , it is readily apparent that an increases the

viscoelastic parameter the velocity of dusty fluid and dust

particles also increases. Temperature profiles have been

plotted in figures (13) to (15) indicate the effects of

radiation parameter R , Prandtl number Pr and heat

source parameter S . From these figures found that the

temperature profiles decreases in with increasing values of

radiation parameter and Prandtl number, but the reverse

effect observed in heat source parameter. In the

concentration field the chemical reaction parameter effects

are much influenced. This profile have the common

feature that the concentration profiles decreases in a

monotone fashion form the surface to a zero value far

away in the free stream. Concentration profile for the

different values of chemical reaction parameter Kr are

shown in figure (16), we noticed that an increasing the

chemical reaction parameter the temperature profiles

decreases. The Schmidt number Sc effect on

temperature profiles are observed in Figure-17, where as

Schmidt number increases the temperature profiles also

increases. From figure (18) determined that the

temperature profiles for different values of the heat source

parameter S , it is observed that increases in the heat

source parameter the temperature profiles decreases. For

various values of diffusion parameter TS on

temperature profiles indicated in Figure-19, it is evident




8867

that an increase in diffusion parameter the temperature

profiles decreases. The effects of radiation parameter R

on temperature profiles are indicate in figure (20), form

this figure noticed that an increases in radiation parameter

reduces the temperature profiles. Figure (21) pretending

that the skin friction for Schmidt number Sc versus

Grashof number Gr , it is determined by that skin

friction increases where the Schmidt number increases.

APPENDIX

1 2 3

2 22 2

1 1 22 2 2 2

1 1 2 3 1

1 11 13 4 5 62 2 2 2 2 2 2 2

3 3 1

, , Pr ,

1 1 1, , ,

1

1 1, , ,

T T

S R Kr Sc S R a K a Sc

S SM M a A A

K aw K w

Gm A Gm AGr AGm Gr AGmA A A A

-1 -0.5 0 0.5 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

yFigure 2(a): Velocity Profiles for different values of Kr

Flu

id V

elo

cit

y (

u)

a=0.2;M=2.0;t=1.0;w=0.5;E=1.0;K1=2.0;Gm=2.0

Pr=0.71; ST=1.0; Sc=0.65;Gr=10.0; S=1.0; R=1.0

Kr=5.0,10.0,15.0,20.0

-1 -0.5 0 0.5 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

yFigure 2(b): Velocity Profiles for different values of Kr

Du

st

Ve

loc

ity

(v

)

Kr=5.0,10.0,15.0,20.0

a=0.2;M=2.0;t=1.0;w=0.5;E=1.0;K1=2.0;Gm=2.0

Pr=0.76; ST=1.0; Sc=0.65; Gr=10.0; S=1.0;R=1.0

-1 -0.5 0 0.5 1-0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

yFigure 3(a): Velocity Profiles for different values of Sc

Flu

id V

elo

cit

y (

u)

a=0.2;M=2.0;t=1.0;w=0.5;E=1.0;K-1=2.0;Gm=2.0Kr=10.0; Pr=0.71; S

T=1.0; Gr=2.0; S=1.0; R=1.0

Sc=1.0,2.0,3.0,4.0




8868

s

-1 -0.5 0 0.5 10

0.01

0.02

0.03

0.04

0.05

0.06

0.07

yFigure 3(b): Velocity Profiles for different values of Sc

Du

st

Ve

loc

ity

(v

)

a=0.2;M=2.0;t=1.0;w=0.5;E=1.0;K1=2.0;Gm=2.0

Kr=10.0; Pr=0.76; ST=1.0; Gr=2.0; S=1.0; R=1.0

Sc=1.0,2.0,3.0,4.0

-1 -0.5 0 0.5 1-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

yFigure 4(a): Velocity Profiles for different values of Gm

Flu

id V

elo

cit

y (

u)

Gm=1.0,2.0,3.0,4.0

a=0.2;M=2.0;t=1.0;w=0.5;E=1.0;K1=2.0;Kr=1.0

Pr=0.71; ST=1.0; Sc=0.65; Gr=10.0;S=1.0;R=1.0

-1 -0.5 0 0.5 1-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

yFigure 4 (b): Velocity Profiles for different values of Gm

Du

st

Ve

loc

ity

(v

)

Gm=1.0,2.0,3.0,4.0

a=0.2;M=2.0;t=1.0;w=0.5;E=1.0;K1=2.0;Kr=1.0

Pr=0.76; ST=1.0; Sc=0.65;Gr=10.0;S=1.0;R=1.0

-1 -0.5 0 0.5 1-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

yFigure 5(a): Velocity Profiles for different values of Gr

Flu

id V

elo

cit

y (

u)

Gr=1.0,2.0,3.0,4.0

a=0.2;M=2.0;t=1.0;w=0.5;E=1.0;K1=2.0;Gm=2.0

Kr=10.0; Pr=0.71; ST=1.0; Sc=0.65; S=1.0;R=1.0

-1 -0.5 0 0.5 1-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

yFigure 5(b): Velocity Profiles for different values of Gr

Du

st

Ve

loc

ity

(v

)

a=0.2;M=2.0;t=1.0;w=0.5;E=1.0;K-1=2.0;Gm=2.0Kr=10.0; Pr=0.76; S

T=1.0; Sc=0.65; S=1.0; R=1.0

Gr=1.0,2.0,3.0,4.0

-1 -0.5 0 0.5 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

yFigure 6(a): Velocity Profiles for different values of K

1

Flu

id V

elo

cit

y (

u)

K1=1.0,2.0,3.0,4.0

a=0.2;M=2.0;t=1.0;w=0.5;E=1.0;Gm=2.0;Kr=1.0Pr=0.71; S

T=1.0; Sc=0.65; Gr=10.0; S=1.0;R=1.0

-1 -0.5 0 0.5 1-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

yFigure 6(b): Velocty Profiles for different values of K

1

Du

st V

elo

city

(V)

a=0.2;M=2.0;t=1.0;w=0.5;E=1.0;Gm=2.0;Kr=1.0Pr=0.76; S

T=1.0; Sc=0.65; Gr=10.0; S=1.0; R=1.0

K1=1.0,2.0,3.0,4.0

-1 -0.5 0 0.5 1-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

yFigure 7(a): Velocty Profiles for different values of M

Flu

id V

elo

cit

y (

u)

a=0.2;t=1.0;w=0.5;E=1.0;K1=2.0;Gm=2.0;Kr=1.0

Pr=0.71; ST=1.0; Sc=0.65; Gr=10.0; S=1.0; R=1.0

M=2.0,4.0,6.0,8.0




8869

-1 -0.5 0 0.5 1-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

yFigure 7(b): Velocty Profiles for different values of M

Du

st

Ve

loc

ity

(v

)

a=0.2;t=1.0;w=0.5;E=1.0;K1=2.0;Gm=2.0;Kr=1.0

Pr=0.76; ST=1.0; Sc=0.65; Gr=10.0; S=1.0; R=1.0

M=2.0,4.0,6.0,8.0

-1 -0.5 0 0.5 1-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

yFigure 8(a): Velocty Profiles for different values of Pr

Flu

id V

elo

cit

y (

u)

a=0.2;M=2.0;t=1.0;w=0.5;E=1.0;K-1=2.0;Gm=2.0Kr=1.0; S

T=1.0; Sc=0.65; Gr=10.0; S=1.0; R=1.0

Pr=1.0,2.0,3.0,4.0

-1 -0.5 0 0.5 1-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

yFigure 8(b): Velocty Profiles for different values of Pr

Du

st

Ve

loc

ity

(v

)

a=0.2;M=2.0;t=1.0;w=0.5;E=1.0;K1=2.0;Gm=2.0

Kr=1.0; ST=1.0; Sc=0.65; Gr=10.0; S=1.0;R=1.0

Pr=1.0,2.0,3.0,4.0

-1 -0.5 0 0.5 10

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

yFigure 9(a): Velocty Profiles for different values of S

Flu

id V

elo

cit

y (

u)

S=0.2,0.4,0.6,0.8

a=0.2;M=2.0;t=1.0;w=0.5;E=1.0;K1=2.0;Gm=2.0

Kr=10.0; Pr=0.71; ST=1.0; Sc=0.65; Gr=2.0;R=1.0

-1 -0.5 0 0.5 1-0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

yFigure 9(b): Velocty Profiles for different values of S

Du

st

Ve

lco

ity

(v

)

S=0.2,0.4,0.6,0.8

a=0.2;M=2.0;t=1.0;w=0.5;E=1.0;K1=2.0;Gm=2.0

Kr=10.0; Pr=0.76; ST=1.0; Sc=0.65;Gr=2.0;R=1.0

-1 -0.5 0 0.5 10

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

yFigure 10(a): Velocty Profiles for different values of S

T

Flu

id V

elc

oit

y (

u)

ST=0.1,0.2,0.3,0.4

a=0.2;M=2.0;t=1.0;w=0.5;E=1.0;K1=2.0;Gm=2.0

Kr=10.0; Pr=0.71; Sc=0.65; Gr=2.0;S=1.0;R=1.0

-1 -0.5 0 0.5 1-0.01

-0.005

0

0.005

0.01

0.015

0.02

yFigure 10(b): Velocty Profiles for different values of S

T

Du

st

Ve

loc

ity

(v

)

ST=0.1,0.2,0.3,0.4

a=0.2;M=2.0;t=1.0;w=0.5;E=1.0;K1=2.0;Gm=2.0

Kr=10.0; Pr=0.76; Sc=0.65; Gr=2.0; S=1.0;R=1.0

-1 -0.5 0 0.5 10

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

yFigure 11(a): Velocty Profiles for different values of R

Flu

id V

elo

cit

y (

u)

R=0.2,0.4,0.6,0.8

a=0.2;M=2.0;t=1.0;w=0.5;E=1.0;K1=2.0;Gm=2.0

Kr=10.0; Pr=0.71; ST=1.0; Sc=0.65; Gr=2.0;S=1.0




8870

-1 -0.5 0 0.5 1-0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

yFigure 11 (b): Velocity Profiles for different values of R

Du

st

Ve

loc

ity

(v

)

R=0.2,0.4,0.6,0.8

a=0.2;M=2.0;t=1.0;w=0.5;E=1.0;K1=2.0;Gm=2.0

Kr=10.0; Pr=0.76;ST=1.0; Sc=0.65; Gr=2.0; S=1.0

-1 -0.5 0 0.5 10

0.01

0.02

0.03

0.04

0.05

0.06

0.07

yFigure 12 (a): Velocity profiles for dusty fluid different values of E

Flu

id V

elo

cit

y (

U)

E=1.0,2.0,3.0,4.0

a=0.2,M=2.0,t=1.0,w=0.5,R=1.0,K1=2.0,Gm=2.0

Kr=10.0,Pr=0.76,ST=1.0, Sc=0.65,Gr=2.0, S=1.0

-1 -0.5 0 0.5 10

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

yFigure 12(b): Velocity profies for dust particles different values f E

Du

st V

elo

cit

y (

V)

E=1.0,2.0,3.0,4.0

a=0.2,M=2.0,t=1.0,w=0.5,R=1.0,K1=2.0,Gm=2.0

Kr=10.0,Pr=0.76,ST=1.0, Sc=0.65,Gr=2.0, S=1.0

-1 -0.5 0 0.5 1-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

yFigure (13).: Temperature Profiles for different values of R

Tem

pera

ture

a=0.2,S=1.0,t=1.0,Pr=0.76

R=1.0,2.0,3.0,4.0

-1 -0.5 0 0.5 1-0.05

0

0.05

0.1

0.15

0.2

yFigure (14).: Temperature Profiles for different values of Pr

Te

mp

era

ture

a=0.2,S=1.0,t=1.0,R=1.0

Pr=0.71,0.76,0.8,0.9

-1 -0.5 0 0.5 1-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

yFigure (15).: Temperatur Profiles for different Values of S

Tem

pera

ture

S=2.0,4.0,6.0,8.0

a=0.2,t=1.0,R=1.0,Pr=0.76

-1 -0.5 0 0.5 10

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

yFigure (16): Concentration Profiles for different values of Kr

Co

ncen

tra

tio

n

Kr=1.0,2.0,3.0,4.0

a=0.2;Sc=0.65;t=1.0;ST=2.0;Pr=0.71;R=1.0;S=1.0

-1 -0.5 0 0.5 1-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

yFigure (17): Concentration Profiles for different values of Sc

Co

nc

en

tra

tio

n

Sc=0.2,0.4,0.6,0.8

a=0.2;t=1.0;ST=2.0;Kr=0.5; Pr=0.71;R=1.0;S=1.0




8871

REFERENCES

[1] Ashok Kumar and M Lal. 2017. Effect of oscillatory

motion of a viscoelastic dusty fluid passes through a

porous medium under the presence of magnetic field,

International Journal of Engineering and Technology.

9(4): 3402-3407.

[2] Attia H A, Al-kaisy A M A and Ewis K M. 2011.

MHD Couette flow and heat transfer of a dusty fluid

with exponential decaying pressure gradient,

Tamkang Journal of Science and Engineering. 14(2):

91-96.

[3] Chenna Kesavaiah D, Satyanarayana P V,

Sudhakariah A and Venkataramana S. 2013. Natural

convection heat transfer oscillatory flow of an elastico

viscous fluid form vertical plate. International Journal

of Research in Engineering and Technology. 2(6):

959-966

[4] Cogly A C, Vincentry W C and Gilles S E. 1968. A

Differential approximation for radiative transfer in a

non-gray gas near equilibrium. AIAA Journal. 6: 551-

555

[5] Das U J. 2017. Heat transfer to MHD oscillatory

dusty viscoelastic fluid flow in an inclined channel

filled with a porous medium. Latin American Applied

Research. 47: 145-150.

[6] Gosh N C, Gosh B C and Debnath L 2000. The

hydromagnetic flow of a dusty Visco-elastic fluid

between two infinite parallel plates, Computers and

Mathematics with Applications. 39: 103-116.

[7] Ibrahim Saidu, Waziri M Y, Abubakar Roko and

Hamisu Musa. 2010. MHD effects on convective flow

of dusty viscous fluid with volume fraction of dust

particles, ARPN Journal of Engineering and Applied

Sciences. 5(10): 86-91.

[8] Liu J T C. 1967. Flow induced by the impulsive

motion of an infinite flat plate in a dusty gas,

Austronautica Acta. 13: 369-377.

[9] Madhura K R and Kalpana G. 2013. Thermal effect

on unsteady flow of a dusty viscoelastic fluid between

two parallel plates under different pressure gradients.

International Journal of Engineering and Technology.

2(2): 88-99.

[10] Madhura K R and Swetha D S. 2017. Influence of

volume fraction of dust particles on dusty fluid flow

-1 -0.5 0 0.5 1-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

yFigure (18): Concentration Profiles for different values of S

Co

nc

en

tra

tio

n

S=1.0,2.0,3.0,4.0

a=0.2;Sc=0.65;t=1.0;ST=2.0;Kr=0.5;Pr=0.71;R=1.0

-1 -0.5 0 0.5 1-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

yFigure (19): Concentration Profiles for different values of S

T

Co

nc

en

tra

tio

n

ST=0.1,0.2,0.3,0.4,0.5

a=0.2;Sc=0.65;t=1.0;Kr=0.5;Pr=0.71;R=1.0;S=1.0

-1 -0.5 0 0.5 1-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

yFigure (20): Concentration Profiles for different values of R

Co

nc

en

tra

tio

n

a=0.2;Sc=0.65;t=1.0;ST=2.0;Kr=0.5;Pr=0.71;S=1.0

R=1.0,2.0,3.0,4.0

-1 -0.5 0 0.5 14.2

4.25

4.3

4.35

4.4

4.45

4.5

Gr

Figure (21).: Skin friction for different values of S versus Gr

Sk

in f

rict

ion

S=2.0,4.0,6.0,8.0

a=0.2, M=5.0, t=1.0, w=0.5, Sc=2.0

Kr=1.0,Gm=5.0,K=10,Pr=0.71,E=1.0

dust Particles (p)

dusty Fluid (f)




8872

through porous rectangular channel. International

Journal of Mathematics Trend and Technology. 50(5):

261-275.

[11] Mohaned Ismail A and Ganesh S. 2014. Unsteady

Stokes flow of dusty fluid between two parallel plates

through porous medium. Applied Mathematical

Sciences. 8(5): 241-249

[12] Mudassar Jalil, Saleem Asghar and Shagufta

Yasmeen. 2017. An exact solution of MHD boundary

layer flow of dusty fluid over a stretching surface,

Hindawi Mathematical Problems in Engineering,

Article ID 2307469, 5 pages.

[13] N P Singh and Atul Kumar Singh. 2001. MHD effects

on heat and mass transfer in flow of a dusty viscous

fluid with volume fraction, Indian Journal of Pure and

Applied Physics. 39: 496-509.

[14] Om Prakash, Devendra Kumar and Dwivedi Y K.

2010. Effects of thermal diffusion and chemical

reaction on MHD flow of dusty viscoelastic (Walter’s

liquid model – B) fluid, J. Electromagnetic Analysis

and Applications. 2: 581-587.

[15] Pop I. 1968. Effects of thermal diffusion and

chemical reaction on MHD flow of dusty visco-elastic

(Walter’s liquid model-B) fluid, Rev. Roum.

Physics.13: 41.

[16] Saffman P G. 1962. On the stability of laminar flow

of a dusty gas. Journal of Fluid Mechanics. 13(1):

120-128.

[17] Sandeep N and Saleem S. 2017. MHD flow and heat

transfer of a dusty nanofluid over a stretching surface

in a porous medium, Jordan Journal of Civil

Engineering. 11(1)

effects of radiation and thermal diffusion on mhd heat transfer flow … · 2018-12-08 · flow...

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