2-45-1445916906-1. mathematics - ijmcar - unsteady hydromagnitic free convective

7/21/2019 2-45-1445916906-1. Mathematics - Ijmcar - Unsteady Hydromagnitic Free Convective

http://slidepdf.com/reader/full/2-45-1445916906-1-mathematics-ijmcar-unsteady-hydromagnitic-free-convective 1/16

www.tjprc.org [email protected]

UNSTEADY HYDROMAGNITIC FREE CONVECTIVE HEAT TRANSFER

FLOW OF VISCO-ELASTI C FLUID THROUGH POROUS MEDIUM WITH

HEAT SOURCE AND VISCOUS DISSIPATION

D. PRAVEENA1, S. V. K.VARMA

2, C. VEERESH

3 & M. C. RAJU

4

1,2,3Department of Mathematics, S. V. University, Tirupati, Andhra Pradesh, India

4Department of Humanities and Sciences, Annamacharya Institute of Technology and

Sciences Rajampet (Autonomous), Rajampet, Andhra Pradesh, India

ABSTRACT

An investigation has been made to study MHD boundary layer flow of visco-elastic fluid of combined heat source

and viscous dissipation over a moving inclined plate through the porous media with constant heat flux. The governing

equations are solved by Perturbation method. The effect of various physical parameters on the velocity and temperature

distributions of the flow filed are discussed and shown through graphs. The results are analysed for various physical

parameters such as magnetic field, Grashof number, Schmidt number, Prandtl number, Eckert number, porosity and heat

absorption. we observed that visco-elastic parameter increases the velocity slightly significant and also thermal boundary

layer thickness slightly increases as the visco-elastic parameter is considerably significant.

KEYWORDS: MHD, Porous Medium, Heat Source, Viscous Dissipation, Non-Newtonian Fluid

Received: Oct 07, 2015; Accepted: Oct 19, 2015; Published: Oct 26; Paper Id.: IJMCARDEC20151

1. INTRODUCTION

Natural convection flow over vertical surfaces immersed in porous media has paramount importance

because of its potential applications in soil physics, geohydrology, and filtration of solids from liquids, chemical

engineering and biological systems. Study of fluid flow in porous medium is based upon the empirically

determined Darcy’s law which accounts for the drag exerted by the porous medium analysed by joseph and neild

[5]. Ingham and Pop [19] have investigated the problem of transport phenomena in porous media. Such flows are

considered to be useful in diminishing the free convection, which would otherwise occur intensely on a vertical

heated surface. Gomaa and Taweel [21] have examined the effect of oscillatory motion on heat transfer about

vertical flat surfaces. The heat transfer enhancement of oscillatory flow in channel with periodically upper and

lower walls mounted obstacles has been analyzed by Abdelkader and Lounes [22]. Singh [7] analyzed the MHD

free convective flow past an accelerated vertical porous plate by finite difference method. The study of non-

Newtonian fluids offers many interesting and exciting challenges due to their technical relevance in modelling

fluids with complex rheological properties(such as polymer melts, synovial fluids, paints, etc). Visco-elastic fluids

also present some highly peculiar characteristics and mathematical features such as the non-unidirectional nature

of the flow of such fluids and the increase in the order of the differential equations characterizing such flowsanalized by Ariel [20]. A lot of work on the flow and heat transfer characteristics of non-Newtonian fluids has also

Or i gi n al Ar t i c l e

International Journal of Mathematics and

Computer Applications Research (IJMCAR)

ISSN(P): 2249-6955; ISSN(E): 2249- 8060

Vol. 5, Issue 6, Dec 2015, 1-16

© TJPRC Pvt. Ltd



2 D. Praveena, S. V. K. Varma, C. Veeresh & M. C. Raju

Impact Factor (JCC): 4.6257 NAAS Rating: 3.80

been done in order to control the quality of the end product in many manufacturing and processing industries, see for

instance by Fakhar [ 24]. Previous studies deals with the studies concerning non-Newtonian flows and heat transfer in the

absence of magnetic fields, but presently we find several industrial applications such as polymer technology and

metallurgy have been analyzed by Chakrabarti and Gupta [2], where the magnetic field is applied in the viscoelastic fluidflow. Sarpakaya [1] was mostly first researcher to investigate MHD flows of non-Newtonianfluids. The study of

magnetohydrodynamic boundary layer flows of electrically conducting fluids finds applications in several industrial and

technological fields such as meteorology, electrical power generation, solar power technology, nuclear engineering, and

geophysics. Georantopoulos et al.[3] studied the problems of hydromagnetic free convection effects on the stokes problem

for an infinite vertical platel. Singh and Dikshit [12] investigated the study of hydromagnetic flow past a continuously

moving semi-infinite plateat large suction. Tokis [11] has examined the exact solutions of the unsteady

magnetohydrodynamic free-convection flows.MHD free-convection flow near a vertical oscillating plate were studied by

Nanousis and Tokis [10]. Raptis and Singh [8] has studied the MHD free convection flow past an accelerated vertical plate.

Sharma and Singh [25] investigated unsteady MHD free convective flow and heat transfer along a vertical porous plate

with variable suction and internal heat generation. In recent years hydromagnetic flows and heat transfer have become

more important because of numerous applications, for example, metallurgical processes in cooling of continuous strips

through a quiescent fluid, thermonuclear fusion, aerodynamics, among others. Sacheti et al. [15] solved the exact solution

for unsteady magnetohydrodynamic free convection flow with constant heat flux Jha[14] has studied the MHD free and

forced convection flow past an infinite vertical plate with heat source. The study of flow and heat transfer for an

electrically conducting fluid past a porous plate under the influence of a magnetic field has attracted the interest of many

investigators in view of its applications in many engineering problems such as magnetohydrodynamic(MHD) generator,

plasma studies, nuclear reactors, geothermal energy extractions and However, it seems less attention was paid on

hydromagnetic free convection flows near a vertical plate subjected to a constant heat flux boundary condition even though

this situation involves in many engineering applications. Chandra et al. [16] analyzed the effects of magnetic field and

buoyancy force on the unsteady free convection flow of an electrically conducting fluid when the flow was generated by

uniformly accelerated motion of an infinite vertical plate subjected to constant heat flux. They obtained an exact solution

with the help of Laplace transform technique and the numerical results were computed with the approximated error

functions appeared in the solution. Pop and Soundalgekar [4] investigated free convection flow past on accelerated infinite

plate. Singh [9] analyzed the MHD free convective flow past an accelerated vertical porous plate by finite difference

method. Singh and Dikshit [13] studied hydromagnetic flow past a continuously moving semi-infinite plateat large suction.

Sharma and Singh [26] investigated unsteady MHD free convective flow and heat transfer along a vertical porous plate

with variable suction and internal heat generation. Toki and Tokis[23] studed the exact solutions for the unsteady free

convection flows on a porous plate with time-dependent heating. Later on,MHD two-fluid convective flow and heat

transfer in composite porous medium was analyzed by Malashetty et al. [18]. All of the above studies pertain to steady

flow. However, a significant number of problems of practical interest are unsteady. The flow unsteadiness may be caused

by achange in the free stream velocity or in the surface temperature or in both. When there is an impulsive change in the

velocity field, the inviscid flow is developed instantaneously, but the flow in the viscous layer near the wall is developed

slowly which becomes fully developed steady flow after sometime. Raptis and Kafousias [6] studied the influence of a

magnetic field on the steady free convection flow through a porous medium bounded by an infinite vertical isothermal

plate with a constant suction velocity. Umavathi [17] studied oscillatory flow of unsteady convective fluid in a infinite

vertical stratum. Recently, MHD convective heat and mass transfer flow of a Newtonian fluid past a vertical porous plate



Unsteady Hydromagnitic Free Convective Heat Transfer Flow of Visco-Elasti C Fluid 3

through Porous Medium with Heat Source and Viscous Dissipation


with chemical reaction, radiation absorption and thermal diffusion was studied by Umamaheswar et al. [28 & 29].

The aim of the paper is to investigate steady MHD boundary layer flow of visco-elastic fluid of combined heat

source and viscous dissipation over a moving inclined plate through the porous media with constant heat flux.

The governing equations are solved by perturbation method. The effect of the parameters on the velocity and temperature

distributions of the flow filed are discussed and shown through graphs. The results are analysed for various physical

parameters such as magnetic field, Grashof number, Schmidt number, Prandtl number, Eckert number, porosity and heat

absorption.

2. MATHEMATICAL FORMULATION

Consider an unsteady hydromagnetic free convective heat transfer, oscillatory flow of electrically conductive

visco-elastic fluid through porous medium, past an impulsively started infinite porous plate with constant heat flux in the

presence of viscous dissipation and heat source effects. Choosing a rectangular Cartesian coordinate system( , )O x y′ ′

, the

x′ -axis is take along the plate in the vertically upward and parallel to the direction of the fluid flow and the y′

-axis is

normal to it. A uniform magnetic field is applied normal to the flow direction. Let,u v′ ′

be the velocity component along

x′ and y′

-axis respectively.

We made the following assumptions:

• Induced magnetic field and electric field are neglected.

• Vertical plate is of infinite length so all the flow variable are functions lf normal distance y′

and time t ′ only.

Initially, the plate and the fluid are at the same temperatureT

∞′ in a stationary condition. Under these assumptions

the governing boundary layer equations are

0v

y

′∂=

′∂ (1)

( )220( )

2

3 3

2 3

Bu u dU uv g T T U u

t y dt K y

K u uv

y t y

σ υ β υ

ρ

ρ

′ ′ ′ ′∂ ∂ ∂′ ′ ′ ′ ′+ = + − + + + −∞

′ ′ ′∂ ∂ ′∂

′ ′ ′∂ ∂′− +

′ ′ ′∂ ∂ ∂

(2)

( )22

02

QK T T T uT v T T t y C C C y y p p p

υ

ρ ρ

′ ′ ′ ′∂ ∂ ∂ ∂′ ′ ′+ = − − +∞

′ ′∂ ∂ ∂′∂

(3)

The boundary conditions are

, at 00 T qu U y y K T

′ ′∂ −′ ′= = =

′∂ and

( ) , asu U t T T y′ ′ ′ ′ ′ ′→ → → ∞∞ (4)





The continuity equation(1) gives

0v V ′ = (5)

Where 0 ( 0)V > is the suction velocity of the fluid introducing the following non-dimensional quantities

2( )

0 0 0, , , , ,

0 0

2 2 220 0 0, , , ,

13 2 20 0 0

y V tV T T K V u U T y t u U

U U q

B KV K V g qGr M K K p

K V U V T

θ υ υ υ

σ υ β υ

υ ρ ρυ

′ ′ ′−′ ′ ∞= = = = =

′

′′= = = =

2 2 20 0Pr , , .

20

C Q V U K p o T S E

K q C K V p pT

µ υ

υ

= = =

′ (6)

Using the above non-dimensional quantities and(5), equations(2) and(3) reduce to

( )2 3 3

1 12 2 3

u u dU u u uK Gr M U u

t y dt y y t yθ

∂ ∂ ∂ ∂ ∂− = + − − + + −

∂ ∂ ∂ ∂ ∂ ∂

(7)

22

Pr Pr2

uS E

t y y y

θ θ θ θ

∂ ∂ ∂ ∂− = + +

∂ ∂ ∂∂

(8)

1

1 M M

K p

= +

Where

The corresponding boundary conditions in dimensionless form are

1u = ,

1 y

θ ∂= −

∂ at

0 y =

( )u U t →, 0θ → as

y → ∞ (9)

3. SOLUTION OF THE PROBLEM

In order to solve the coupled non-linear partial differential equations(7) and(8), assuming that the expressions for

velocity( , )u y t

and temperature( , ) y t θ

,in neighbourhood of the plate(i.e in the boundary layer) as

int

0 1( , ) ( ) ( )u y t u y u y eε = + (10)

int

0 1( , ) ( ) ( ) y t y y eθ θ ε θ = + (11)






And for free stream

int( ) 1U t eε = +

(12)

Where ( 1)ε << is a small parameter substituting the equations(10),(11) and(12) into equation(7) and(8) and

equating the coefficients of like powers of ε , neglecting the terms of2ε , we obtain

1 0 0 0 1 0 0 1K u u u M u Gr M θ ′′′ ′′ ′+ + − = − − (13)

1 1 1 1 1 2 1 1 2(1 )K u inK u u M u Gr M θ ′′′ ′′ ′+ − + − = − − (14)

2

0 0 0 0Pr PrS Euθ θ θ ′′ ′ ′+ − = − (15)

1 1 1 1 0 1Pr 2PrS Eu uθ θ θ ′′ ′ ′ ′+ − = − (16)

1

1

p

M M K

= +

, 2 1 M M in= +, 1 PrS S in= +

Where

The corresponding conditions are

0 1u = , 1 0u = , 0 1θ ′ = − , 1 0θ ′ = at 0 y =

0 1u =, 1 1u =

, 0 0θ →, 1 0θ →

as y → ∞

(17)

Where prime denotes the differentiation with respect to y

. The equations(13) and(14) are third order differential

equations when 1 0K ≠and for 1 0K =

they reduced to equations governing the Newtonian fluid. Hence, the presence of

the elasticity of the fluid invreases the order of the governing equations from two to three and therefore, they need three

boundary conditions in(17), therefore for a unique solution,we follow Beard and Walter() and assume that the solutions for

0u , 1u , 0θ and 1θ in the form

( ) ( ) ( ) ( )2

0 01 1 02 1u y u y K u y o K = + +

( ) ( ) ( ) ( )2

1 11 1 12 1u y u y K u y o K = + +

( ) ( ) ( ) ( )2

0 01 1 02 1 y y K y o K θ θ θ = + +

( ) ( ) ( ) ( )2

1 11 1 12 1 y y K y o K θ θ θ = + + (18)



Unsteady Hydromagnitic Free Convective Heat Transfer Flow of Visco-Elasti C Fluid 7 through Porous Medium with Heat Source and Viscous Dissipation


021 021 1 021 021 011u u M u Gr uθ ′′ ′ ′′′+ − = − − (31)

022 022 1 022 022 012u u M u Gr uθ ′′ ′ ′′′+ − = − − (32)

111 111 2 111 111 2u u M u Gr M θ ′′ ′+ − = − − (33)

112 112 2 112 112u u M u Gr θ ′′ ′+ − = − (34)

121 121 2 121 121 111 111u u M u Gr u inuθ ′′ ′ ′′′ ′′+ − = − − + (35)

122 122 2 122 122 112 112u u M u Gr u inuθ ′′ ′ ′′′ ′′+ − = − − + (36)

011 011 011Pr 0S θ θ θ ′′ ′+ − = (37)

2

012 012 012 011Pr PrS uθ θ θ ′′ ′ ′+ − = −

(38)

021 021 021Pr 0S θ θ θ ′′ ′+ − = (39)

022 022 022 011 021Pr 2 PrS u uθ θ θ ′′ ′ ′ ′+ − = − (40)

111 111 1 111Pr 0S θ θ θ ′′ ′+ − = (41)

112 112 1 112 011 111Pr 2 PrS u uθ θ θ ′′ ′ ′ ′+ − = − (42)

121 121 1 121Pr 0S θ θ θ ′′ ′+ − = (43)

122 122 1 122 011 121 021 111Pr 2 Pr( )S u u u uθ θ θ ′′ ′ ′ ′ ′ ′+ − = − + (44)

The corresponding boundary conditions are

At0 y =

; 011 1u =, 021 0u =

, 111 0u =, 121 0u =

012 0u = , 022 0u = , 112 0u = , 122 0u =

011 1θ ′ = −, 021 0θ ′ =

, 111 0θ ′ =, 121 0θ ′ =

012 0θ ′ =, 022 0θ ′ =

, 112 0θ ′ =, 122 0θ ′ =

As y → ∞

; 011 1u →, 021 0u →

, 111 1u →, 121 0u →

012 0u →, 022 0u →

, 112 0u →, 122 0u →

011 0θ →, 021 0θ →

, 111 0θ →, 121 0θ →

012 0θ →, 022 0θ →

, 112 0θ →, 122 0θ →

(45)





Solving equations(29)-(44) under the boundary conditions(45) and by using(180 and(280, we obtain the mean and

fluctuating parts of velocity and temperature distributions in the boundary layers as

( ) ( )1 2

0 11a y a y

u y G e e− −

= + −

+

( )32 1 1 22 2

2 4 5 6 7

a ya y a y a y a y E A e P e Pe P e P e

−− − − −+ + + +

+31 2 1 2 1 22 2

1 8 14 15 19 16 17 18{ ( ) [ ( ) ]}a ya y a y a y a y a y a y

K P e e E P e P P e P e P e P e−− − − − − −

− + + + + + + (46)

( ) 1

11 b yu y e−

= −+

( )32 1 1

23 24 25 26

b yb y m y b y E P e P e P e P e

−− − −+ + +

+

31 2 1

1 30 31 32 33 34{ [ ( ) ]}b ym y b y b yK E P e P e P e P P e−− − −

+ + + + (47)

( )1

0

1

a ye

ya

θ −

=

+ ( )31 1 22 2

1 1 2 3

a ya y a y a y E Ae Pe P e Pe

−− − −+ + +

+

31 1 22 2

1 13 10 11 12[ ]a ya y a y a y

K E P e P e P e P e−− − −

+ + + (48)

( )1 yθ = ( )32 1

20 21 22

b yb y m y E P e P e P e

−− −+ +

+

31 2

1 27 28 29( )b ym y b y

K E P e P e P e−− −

+ + (49)

Skin-Friction, Nusselt Number

We now study Skin-friction(

τ ) from velocity field. It Is given by

τ = 0=

∂

∂

y y

u

1 1 2 2 2 4 1 5 1 6 21 7 3

1 8 1 8 2 14 1 15 19 2 16 3 17 1 18 2

int

1 23 2 24 3 25 1 26 1

1 30 1 31 2 32 3 33 34 1

( ) ( 2 2 )

{ [ ( ) 2 2 ]}

[ ( )

[ ( ) ]

G a a E A a p a p a p a p a

K p a p a E p a p p a p a p a p a

e b E p b p b p m p b

K E p m p b p b p p b

τ

ε

= − + + − − − − −

+ − + + − − + − − −

+ + − − − −

+ − − − − + (50)

Nusselt number: In non-dimensional form, the rate of heat transfer is given by

1 1 1 1 2 2 3 3 1 13 1 10 1 11 2 12 3

0

int

20 2 21 3 22 1 1 27 1 28 2 29 3

1 ( 2 2 ) ( 2 2 )

[ ( ) ( )

y

E A a p a p a p a K E p a p a p a p a y

e E p b p b p m K E p m p b p b

θ

ε

=

∂= − + − − − − + − − − −

∂

+ − − − + − − − (51)

4. RESULTS AND DISCUSSIONS

The formulation of the problem that accounts for the effects of elastic- viscous parameter and viscous dissipation

on the flow of a viscous incompressible fluid past an impulsively started infinite vertical plate. From figure 1 it is observed

that an increase in Grash of number(Gr) leads to rise in the values of velocity due to enhancement in boundary force. In






addition the curve show that the peak values of the velocity increases rapidly near the plate and there after it decreases as

distance(y) increases. It is observed from figure 2 &3 that the mean velocity decreases when Prandtl number(Pr) &

Schmidt number(S) increases. It is noticed from figure 4 that Eckert number decreases velocity profile not considerably

significant. The maximum velocity occurs near the plate and thereafter it decreases as distance(y) increases. From figure 5we noticed that velocity decreases when n increases. From figure 6 It is seen that the effect of porosity parameter(Kp) on

the velocity slightly significant. From the figure 7 we observed that visco-elastic parameter(K1) increases the velocity

slightly significant. In figure 8 the effect of Temperature in the boundary layer increases with the increases of Gr values. It

is observed that the thermal boundary layer thickness decreases as Pr increases in figure 9. From figure 10 as heat

absorption parameter(S) increases the temperature decreases. From the figure 11 it is observed that the thermal boundary

layer thickness increases with the Eckert number increases. From figure 12 it seen that the temperature not considerably

significant when n increases from figure 13 it is noticed that the effect of the porosity parameter on the temperature is

considerably significant. We noticed that the thermal boundary layer thickness slightly increases as the visco-elastic

parameter is considerably significant in figure 14.

0 1 2 3 4 5 6 7 81

1.02

1.04

1.06

1.08

1.1

1.12

1.14

1.16

y

u ( y

, t )

Gr = 1.0

Gr = 2.0

Gr = 3.0

Gr = 4.0

Pr = 0.71 ; S = 0.22 ; Kp = 1.0 ; M = 15.0 ; ε =0.01;

E = 1.0 ; K1 = 0.01 ; t = 1.0 ; n = 1.0

Figure 1: Velocity Profiles against y for Different Values of Gr

0 1 2 3 4 5 6 7 80.995

1

1.005

1.01

1.015

1.02

1.025

1.03

1.035

1.04

y

u ( y ,

t )

Pr = 0.71

Pr = 1.0

Pr = 3.0

Pr = 7.0

Gr = 1.0 ; S = 0.22 ; Kp = 1.0 ; M = 15.0 ; ε = 0.01 ;

E = 1.0 ; K1 = 0.01 ; t = 1.0 ; n = 1.0

Figure 2: Velocity Profiles against y for Different Values of Pr





0 1 2 3 4 5 6 7 80.995

1

1.005

1.01

1.015

1.02

1.025

1.03

1.035

1.04

y

u ( y ,

t )

S = 0.22

S = 0.6

S = 0.78

S = 0.96

Gr = 1.0 ; Pr = 0.71 ; Kp = 1.0 ; M = 15.0 ; ε = 0.01 ;E = 1.0 ; K1 = 0.01 ; t = 1.0 ; n = 1.0

Figure 3: Velocity Profiles against y for Different Values of S

0 1 2 3 4 5 6 7 81

1.005

1.01

1.015

1.02

1.025

1.03

1.035

1.04

1.045

y

u ( y ,

t )

E = 1.0

E = 0.75

E = 0.05

E = 0.25

Gr = 1.0 ; Pr = 0.71 ; S = 0.22 ; Kp = 1.0 ; M = 15.0 ;

ε = 0.01 ; K1 = 0.01 ; t = 1.0 ; n = 1.0

Figure 4: Velocity Profiles against y for Different Values of E

0 1 2 3 4 5 6 7 80.99

1

1.01

1.02

1.03

1.04

1.05

y

u ( y , t )

n = 1.0

n = 2.0

n = 3.0

n = 4.0

Gr = 1.0 ; Pr = 0.71 ; S = 0.22 ; Kp = 1.0 ; M = 15.0 ;

ε = 0.01 ; E = 1.0 ; K1 = 0.01 ; t=1.0

Figure 5: Velocity Profiles Against y for Different Values of n

0 1 2 3 4 5 6 7 80.995

1

1.005

1.01

1.015

1.02

1.025

1.03

1.035

1.04

1.045

y

u ( y , t )

Kp = 1.0

Kp = 10.0

Kp = 100.0

Kp = 200.0

Gr = 1.0 ; Pr = 0.71 ; S = 0.22 ; M = 15.0 ; ε = 0.01 ;

E = 1.0 ; K1 =0.0 1 ; t = 1.0 ; n = 1.0

Figure 6: Velocity Profiles against y for Different Values of KP






0 1 2 3 4 5 6 7 81

1.005

1.01

1.015

1.02

1.025

1.03

1.035

1.04

1.045

y

u ( y , t )

K1 = 0.01

K1 = 0.1

K1 = 0.5

K1 = 1.0

Gr = 1.0 ; Pr = 0.71 ; S = 0.22 ; Kp = 1.0 ; M = 15.0 ;

ε = 0.01 ; E = 1.0 ; t = 1.0 ; n = 1.0

Figure 7: Velocity Profiles against y for Different Values of K1

0 1 2 3 4 5 6 7 80

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

y

θ

( y

, t )

Gr = 1.0

Gr = 2.0

Gr = 3.0

Gr = 4.0

Pr= 0.71 ; S = 0.22 ; Kp = 1.0; M = 1.0 ;

ε = 0.01; E = 1.0 ; K1 = 0.01 ; t = 1.0; n= 1.0

Figure 8: Temperature Profiles against y for Different Values of Gr

0 1 2 3 4 5 6 7 8-0.2

0

0.2

0.4

0.6

0.8

1

1.2

y

θ ( y

, t )

Pr = 0.71

Pr = 1.0

Pr = 3.0

Pr = 7.0

Gr = 1.0 ; S = 0.22 ; Kp = 1.0 ; M = 1.0 ;

ε = 0.01 ; E = 1.0 ; K1 = 0.01 ; t = 1.0 ; n =1.0

Figure 9: Temperature Profiles against for Different Values of Pr

0 1 2 3 4 5 6 7 80

0.2

0.4

0.6

0.8

1

1.2

1.4

y

θ ( y

, t )

S = 0.22

S = 0.60

S = 0.78

S = 0.98

Gr = 1.0 ; Pr = 0.71 ; Kp = 1.0 ; M = 1.0 ;

ε = 0.01 ; E = 1.0 ; K1 = 0.01 ; t = 1.0 ; n = 1.0

Figure 10: Temperature Profiles Against for Different Values of S





0 1 2 3 4 5 6 7 80

0.2

0.4

0.6

0.8

1

1.2

1.4

y

θ ( y ,

t )

E = 1.0

E = 0.8

E = 0.6

E = 0.4

Gr = 1.0 ; Pr = 0.71 ; S = 0.22 ; Kp = 1.0 ; M = 1.0 ;

ε = 0.01 ; K1 = 0.01 ; t = 1.0 ; n = 1.0

Figure 11: Temperature Profiles Against for Different Values of E

0 1 2 3 4 5 6 7 80

0.2

0.4

0.6

0.8

1

1.2

1.4

y

θ ( y

, t )

n = 1.0

n = 2.0

n = 3.0

n = 4.0

Gr = 1.0 ; Pr = 0.71 ; S = 0.22 ; Kp = 1.0 ; M = 1.0 ;

ε = 0.01 ; E = 1.0 ; K1 = 0.01 ; t = 1.0

Figure 12: Temperature Profiles Against for Different Values of n

0 1 2 3 4 5 6 7 80

0.2

0.4

0.6

0.8

1

1.2

1.4

y

θ ( y ,

t )

Kp = 1.0

Kp = 5.0

Kp = 10.0

Kp = 15.0

Gr = 1.0 ; Pr = 0.71 ; S = 0.22 ; M = 1.0 ;ε = 0.01 ; E = 1.0 ; K1 = 0.01 ; t = 1.0 ; n = 1.0

Figure 13: Temperature Profiles Against for Different Values of KP

0 1 2 3 4 5 6 7 80

0.2

0.4

0.6

0.8

1

1.2

1.4

θ ( y ,

t )

K1 = 1.0

K1 = 0.8

K1 = 0.6

K1 = 0.4

Gr= 1.0 ; Pr = 0.71 ; S = 0.22 ;Kp = 1.0 ; M = 1.0 ;

ε = 0.01 ; E = 1.0 ; t = 1.0 ; n = 1.0

Figure 14: Temperature Profiles Against for Different Values of K1






2. CONCLUSIONS

MHD boundary layer flow of visco-elastic fluid of combined heat source and viscous dissipation over a moving

inclined plate through the porous media with constant heat flux is studied. The governing equations are solved by

Perturbation method. The effect of the parameters on the velocity and temperature distributions of the flow filed are

discussed and shown through graphs. Some concluding observations from the investigation are given below.

• Velocity increases with increasing values of Grashof number, permeability of the porous medium and visco

elasticity parameter where as it shows reverse tendency in the case of Prandtl number and Schmidt number.

• Thermal boundary layer thickness reduces when the values of Prandtl number and heat absorption parameter but

Eckert number has a different scenario on temperature distribution.

REFERENCES

1.

Sarpakaya, T.(1961). Flow of Non-Newtonian Fluids in Magnetic Field. AICHE Journal, Vol. 7, pp. 324-328.

2. Chakrabarti and Gupta, A.S.(1979). Hydromagnetic Flow and Heat Transfer over a Stretching Sheet. Quarterly Journal of

Mechanics and Applied Mathematics. Vol. 37, pp. 73-78.

3. Georgantopoulos, G. A.; Douskos, C. N.; Kafousias, N. G.: and Goudas, C.L.(1979). Hydromagnetic free convection effects on

the Stokes problem for an infinite vertical plate, Lett. Heat Mass Transf. 6, 397–404.

4. Pop, I. and Soundalgekar, V. M.(1980). Free convection flow past an accelerated infinite place. Z. Angrew, Math. Mech.60

167-168.

5. Joseph, D. D.; Nield, D. A. and Papanicolaou, G.(1982). Water Resources Research, Vol. 18, p. 1049.

6.

Raptis, A. A; Kafousias, N.(1982). Heat transfer in flow through a porous medium bounded by an infinite vertical plate under

the action of a magnetic field. Int. J. Energ. Res., 6, pp. 241–245.

7. Singh, A. K.(1983). Finite difference analysis of MHD free convective flow past an accelerated vertical porous plate. Astrophys

Space Sci., 94,395-400.

8. Raptis, A. A.; and Singh, A.K.(1983). MHD free convection flow past an accelerated vertical plate. Int. Commun. Heat Mass.

10, 313–321.

9.

Singh, A. K..(1983). Finite difference analysis of MHD free convective flow past an accelerated vertical porous plate.

Astrophys Space Sci., 94, 395-400.

10.

Nanousis, N. D; and Tokis, J.N.(1984). MHD free-convection flow near a vertical oscillating plate, Astrophys. Space Sci. 98,

397–403.

11.

Tokis, J. N.(1985). A class of exact solutions of the unsteady magnetohydrodynamic free-convection flows. Astrophys. Space

Sci. 112, 413–422.

12.

Singh, A.K. and Dikshit, C.K..(1988). Hydromagnetic flow past a continuously moving semi-infinite plate at large suction.

Astrophys, Space Sci., 248, 249-256.

13.

Jha, B.K.(1991). MHD free and forced convection flow past an infinite vertical plate with heat source. Astrophys. Space Sci.

183,169–175.

14.

Sacheti, N.C.; Chandran, P.; and Singh, A.K.(1994). An exact solution for unsteady magnetohydrodynamic free convection

flow with constant heat flux. Int. Commun. Heat Mass. 21, 131–142.





15.

Chandran, P.; Sacheti, N. C.; and Singh, A.K.(1998). Unsteady hydromagnetic free convection flow with heat flux and

accelerated boundary motion. J. Phys. Soc. Jpn. 67, 124–129.

16. Umavathi, J.C.; Palaniappan, D.(2000). Oscillatory flow of unsteady Oberbeck convection fluid in an infinite vertical porous

stratum. J. Appl. Mech., 69(2), pp. 35–60.

17. Malashetty,M. S.; Umavathi, J.C.; Kumar, J. P.(2001). Two fluid magnetoconvection flow in an inclined channel, Int. J.

Transport Phenomena, 3, pp. 73–84, 2001.

18. Ingham, D. B. and Pop, I.(2002). Transport Phenomena in Porous Media. Pergamon, Oxford.

19.

Ariel, P.D.(2003). Flow of a third grade fluid through a porous flat channel. Int. J. Engg.Sci., Vol. 41, pp. 1267 – 1285.

20.

Gomaa, H.; and Al Taweel, A.M.(2005). Effect of oscillatory motion on heat transfer at vertical flat surfaces. Int. J. Heat and

Mass Transfer, vol. 48, pp. 1494–1504.

21.

Abdelkader, K.; and Lounes, O.(2007). Heat transfer enhancement in oscillatory flow in channel with periodically upper and

lower walls mounted obstacles. Int. J. Heat and Fluid Flow, vol. 28, 1003–1012.

22.

Toki, C. J.; and Tokis, J.N.(2007). Exact solutions for the unsteady free convection flows on a porous plate with time-dependent

heating. Z. Angew. Math. Mech. 87, 4–13.

23. Fakhar, K.(2008). Exact solutions for nonlinear partial differential equation arising in third grade fluid flows. Southeast Asian

Bulletin of Mathematics, Vol. 32, pp. 65 – 70.

24. Sharma, P.R. and Singh, G.(2008). Unsteady MHD free convective flow and heat transfer along a vertical porous plate with

variable suction and internal heat generation. Int. J. Appl. Maths& Mechanics, 4,1-8.

25. Raju,K.V.S.; Reddy, T.S.; Raju, M.C.; Satyanarayana, P.V. and Venkataramana, S.(2014). MHD convective flow through porous

medium in a horizontal channel with insulated and impermeable bottom wall in the presence of viscous dissipation and Joule’s

heating. Ain Sham’s engineering Journal, 5(2), 543-551.DOI: 10.1016/j.asej.2013.10.007.

26.

Ravikumar, V.; Raju, M.C.; Raju, G.S.S.(2014). Combined effects of heat absorption and MHD on convective Rivlin-Ericksen

flow past a semi-infinite vertical porous plate. Ain Shams Engineering Journal, 5(3), 867–875, DOI:

10.1016/j.asej.2013.12.014

27.

Raju, M.C.; Varma, S. V. K.; Seshaiah, B.(2015). Heat transfer effects on a viscous dissipative fluid flow past a vertical plate in

the presence of induced magnetic field. Ain Shams Engineering Journal, Vol.6, No.1, 2015, 333-339.

doi:10.1016/j.asej.2014.07.009.

28. Umamaheswar, M.; Raju, M.C.; and Varma, S.V.K.(2016). MHD convective heat and mass transfer flow of a Newtonian fluid

past a vertical porous plate with chemical reaction, radiation absorption and thermal diffusion, International Journal of

Engineering Research in Africa Vol. 19 37-56, doi:10.4028/www.scientific.net/JERA.19.37.

29.

Umamaheswar, M; Varma, S.V.K; Raju, M.C.(2013). “Unsteady MHD Free Convective Visco-Elastic Fluid Flow Bounded By

an Infinite Inclined Porous Plate In The Presence Of Heat Source, Viscous Dissipation And Ohmic Heating”, International

journal of Advanced Science and Technology, Vol. 61, pp. 39-52. http://dx.doi.org/10.14257/ijast.2013.61.05

APPENDICES]






Table 1

1

1

p

M M K

= +

1 2

1 1 1 1( )

G r G

a a a M

−=

− −

2 1 M M in= +

12 (1 1 4 )

2 M a

+ +=

1 P rS S in= +

3 1 2a a a= +

2

1

P r 4

2

P r S a

+ +=

2 2

1 11 2

1 1

Pr

4 2 Pr

G aP

a a S

−=

− −

2 2

1 22 2

2 2

P r

4 2 P r

G aP

a a S

−=

− −

3

2 215 2

2 2 1

a AP

a a M =

− −

2 2

1 13 2

3 3

2 Pr

Pr

G a aP

a a S

−=

− −

3

12 3 716 2

3 3 1

( )GrP a PP

a a M

− −=

− −

1 1 2 2 3 31

1

(2 2 )a P a P a P A

a

+ +=

3

10 1 517 2

1 1 1

( 8 )

4 2

G rP a PP

a a M

− −=

− −

14 2

1 1 1

rG AP

a a M

−=

− −

15 2

1 1 1

r

4 2

G PP

a a M

−=

− −

26 2

2 2 1

r

4 2

G PP

a a M

−=

− −

37 2

3 3 1

rG PP

a a M

−=

− −

31 18 2

1 1 1

G aPa a M

=− −

9 1 82 PrP G P= −

21 910 2

1 14 2 Pra PP

a a S =

− −

2

2 911 2

2 24 2 Pr

a PP

a a S =

− −

1 2 912 2

3 3

2

Pr

a a PP

a a S

−=

− −

1 10 2 11 3 1213

1

(2 2 )aP a P a PP

a

− + +=

( )3

13 1 4

14 2

1 2 1

GrP a PP

a a M

− −=

− −

2

1

1 1 4

2

M b

+ +=

2

1

1

Pr Pr 4

2

S m

+ +=

112 bab +=

3 2 1b a b= +

1 1 1 12 Pr N a b G=

2 2 1 12 Pr N a b G= −

19 14 15 16 17 18( )P P P P P P= − + + + +

2 4 5 6 7( ) A P P P P= − + + +

120 2

2 2 1P r

N P

b b S

=

− −

22 1 2

3 3 1P r

N P

b b S

=

− −

2 2 0 3 2 122

1

( )b P b PP

m

− +=

2023 2

2 2 2

G rPP

b b M

−=

− −

2 124 2

3 3 2

G rPP

b b M

−=

− −

2225 2

1 1 2

GrPP

m m M

−=

− −

2 6 2 3 2 4 2 4( )P P P P= − + +

2 2 8 3 2927

1

( )b P b PP

m

− +=

1

820

28G

PPP =

1

821

29G

PPP =

33 0 2

1 1 2

N P

m m M =

− −

43 1 2

2 2 2

N P

b b M =

− −

53 2 2

3 3 2

N P

b b M =

− −

633 2

1 1 2

N P

b b M =

− −

3 4 3 0 3 1 3 2 3 3( )P P P P P= − + + +

2 3

3 2 7 1 1 25( ) N G rP in m m P= − + +

2 3

4 28 2 2 23( ) N GrP inb b P= − + +

2 3

5 2 9 3 3 24( ) N G rP in b b P= − + +

2 3

6 1 1 26( ) N inb b P= +

Table 2: Variations in Skin Friction, NUSSELT

Number for Different Values of Gr, Pr, S, KP, M, E, K1

Gr Pr S KP M E K1 τ Nu

1.0 0.71 0.22 1.0 15 1.0 0.01 0.2614 -1.8851

2.0 0.71 0.22 1.0 15 1.0 0.01 0.5051 -1.8712

3.0 0.71 0.22 1.0 15 1.0 0.01 0.7614 -1.8480

1.0 1.0 0.22 1.0 15 1.0 0.01 0.2027 -2.40171.0 3.0 0.22 1.0 15 1.0 0.01 0.0712 -0.4337

1.0 7.0 0.22 1.0 15 1.0 0.01 0.0388 -50.4384





Table 2: Contd.,

1.0 0.71 0.6 1.0 15 1.0 0.01 0.1987 -2.4546

1.0 0.71 0.78 1.0 15 1.0 0.01 0.1817 -2.7059

1.0 0.71 0.96 1.0 15 1.0 0.01 0.1686 -2.9503

1.0 0.71 0.22 5 15 1.0 0.01 0.2678 -1.8847

1.0 0.71 0.22 10 15 1.0 0.01 0.2671 -1.8847

1.0 0.71 0.22 12 15 1.0 0.01 0.2658 -1.8847

1.0 0.71 0.22 1.0 60 1.0 0.01 0.1728 -1.8890

1.0 0.71 0.22 1.0 90 1.0 0.01 0.1597 -1.8893

1.0 0.71 0.22 1.0 110 1.0 0.01 0.1562 -1.8894

1.0 0.71 0.22 1.0 15 0.75 0.01 0.2611 -1.6638

1.0 0.71 0.22 1.0 15 0.50 0.01 0.2608 -1.4425

1.0 0.71 0.22 1.0 15 0.25 0.01 0.2605 -1.2213

1.0 0.71 0.22 1.0 15 1.0 0.1 0.2601 -1.8851

1.0 0.71 0.22 1.0 15 1.0 0.5 0.2546 -1.8851

1.0 0.71 0.22 1.0 15 1.0 1.0 0.2478 -1.8851

2-45-1445916906-1. mathematics - ijmcar - unsteady hydromagnitic free convective

Documents