The research paper published by IJSER journal is about Effects of Variable Fluid Properties and Viscous Dissipation on Mixed Convection Fluid Flow past a Vertical Plate in Porous Medium 1

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Effects of Variable Fluid Properties and Viscous Dissipation on Mixed

Convection Fluid Flow past a Vertical Plate in Porous Medium

P.K.Singh
Department of Mathematics, University of Allahabad, Allahabad-211002, INDIA

Abstract-In the present work, free and forced convection boundary layer flow of an incompressible and viscous dissipative fluid with variable thermal conductivity and temperature dependent viscosity past a an isothermal vertical plate is investigated. The c onvective flow is taking place in such a porous medium whose permeability is assumed to be spatially variable. The convective flow is due to two factors which influence the flow simultaneously- (1) free stream along the plate and (2) the buoyancy force caused by the variations in density due to temperature difference. The governing equations for the the boundary layer flow are converted into to a system of cou pled ordinary differential equations by using suitable similarity transformations. These equations are solved numerically and effects of Grashof number, Eckert number and permeability parameter on the velocity and temperature are discussed and presented graphically

Key Words- Porous medium, variable permeability, thermal conductivity, viscosity, viscous dissipation,vertical plate.

—————————— ——————————

1 INTRODUCTION

The combined free and forced convection flow and heat transfer problems in a fluid saturated porous medium have been the subject matter of extensive investigations due to their immense applications in a number of engineering and indus- trial applications in industry, science and technology- for ex- ample in petroleum industry, chemical engineering, geothermal resources and cooling processes of nuclear reactors etc., to name a few. The investigations on free convection flows which were put on firm theoretical foundations by Polhausen[1], were extended by Merkin[2] and [3] and Cheng and Minko- wycz [4] etc.. Ranganathann and Viskanta [5] and Chen et al. [6] have studied the combined free and forced convection from vertical plates in porous media.

Here it is mentionworthy that all these studies have been car- ried out for the fluid flows which have constant properties. The physical properties of the fluids, mainly viscosity and thermal conductivity may change significantly with temperature Schlichting [7]. Kays and Crawferd [8] have described in de- tails various relations between the physical properties of fluids and temperature. It is to be noted that that different fluids be- have differently with temperature. Choi [9] studied the effects of variable properties on the boundary layer flow. Lai and Ku- laki [10], Pop et al. [11] and Eswara and Nath [12] have shown that temperature dependent viscosity has quite significant ef- fect on momentum and thermal transport in the boundary layer flow. Elbashbeshy [13], Seddeek [14] and Seddeek and Almu- shigeh [15] considered the hydromagnetic flow and heat trans- fer past a continuously moving porous boundary with simulta- neous effects of radiation and variable viscosity.
Most of the studies involving variable viscosity have consi- dered the thermal conductivity as constants. In order to have a clearer and more insightful picture of the thermal transport, it would be helpful to study the effects of thermal conductivity variation on heat transfer boundary layer phenomena. Elbash- beshy and Ibrahim [16] and Khound and Hazarika [17] have considered the variation of viscosity parameter along with thermal conductivity/ diffusivity parameter and their findings show a significant influence on the velocity and temperature
distribution.
Also, the studies of Tierney et al. [18] and Benenati and Bro- silow [19] have shown that the porosity of the medium may change which in turn may cause a change in the permeability of the medium. Chandrasekhar et al.[20], took into account the variation of permeability in their investigations on mixed con- vection flows in the presence of horizontal impermeable sur- faces in saturated porous media and have shown that the va- riability in porosity has significant effect on the velocity distribu- tion and heat transfer. Chandrasekhar and Namboodiri [21] studied the mixed convection flows about inclined surfaces in a saturated porous media incorporating the variation of per- meability and thermal conductivity due to packing of particles. Massour and El-Shaer [22] have considered the effects of va- riable permeability and thermal conductivity while Reddy and Reddy [23] have taken into account the effects of variable vis- cosity and thermal conductivity on an electrically conducting fluid flow past a moving vertical plate. Hassanien [24] consi- dered the influence of variable permeability and thermal con- ductivity on the mixed convection flow from an impermeable vertical wedge wherein he obtained a non similarity solution for the case of variable surface heat flux.
In view of the increasing technological applications of mixed convective flows, it would be interesting to study this type of flow with variable fluid properties. To my knowledge, simulta- neous effects of variations in viscosity, thermal conductivity and medium permeability have not been studied so far. Thus the aim of the present paper is to investigate the effects of variable permeability and thermal conductivity on the mixed convection boundary layer dissipative flow with temperature dependent viscosity past a vertical plate in porous medium.
a drop cap.

2 MATHEMATICAL FORMULATION

Let us consider a steady two dimensional boundary layer flow of an incompressible and viscous dissipative fluid along a ver- tical plate in a porous medium . x direction is taken along the plate and y is normal to it ,i.e., the plate starts at x=0 and ex-

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The research paper published by IJSER journal is about Effects of Variable Fluid Properties and Viscous Dissipation on Mixed Convection Fluid Flow past a Vertical Plate in Porous Medium 2

ISSN 2229-5518

tends parallel to the x axis and is of semi infinite length. The

a (8)

0

plate temperature is uniformly maintained at

Tw and the tem-

where a is the viscosity variation parameter and it’s value
perature Tw

is higher than the temperature T of the fluid far
depends on the nature of the fluid and

0 is viscosity of the

away from the plate. A steady flow parallel to the plate with

free stream velocity U is assumed to take place. The mixed convective flow is assumed to take place due to the simulta-
neous effects of (1) the free stream along the plate and (2) the
buoyancy force caused by the variations in density due to
temperature difference. Let u and v be the velocity compo- nents in the boundary layer region along the x and y- axes
fluid at the edge of the boundary layer.

Thus, with these assumptions on the physical parameters giv- en by (7) and (8), the equations (1), (2) and (3) with the help of equations (5) and (6), reduce to the following ordinary differen- tial equations:

1 a Gr a f

respectively. The viscosity and the thermal conductivity of the f

e ff a f e

2 R 2

1

R (1

d *e )

0 (9)

fluid are assumed to be variable.Also, the permeability of the e e
porous medium is supposed to be avariable. Then under the

1 [ (1

de )

b{1 (1

de )}]

d (b

1)e 1 f

usual boundary layer approximations, the governing equations

0 0 0

r


for the present Darcy type flow, following Nield and Bejan [18] 2 2

a 2

(10)

and Schlichting [19], are given by:

Ec e f R [ (1

] 0

d *e

u v 0 (1)

x y

Gr g Tw

T x3

u u v u

( u )

g T T U u

(2) 2 Pr

x y y y

0 k0 0 k ( y) 0

U 2

c

0

R U x

u T v T

2

( y) T u u 2

(3)

cp (Tw T ) 0

x y y y

Boundary conditions

0cp y

0 k ( y)


(112) x k0

The transformed boundary conditions are given by-

'

0 f 0 1

at 0

1 0 at

(12)

We introduce the following non-dimensional variables:

1

The physical problem is now mathematically represented by


U 2 the equatioGnrs (6) 2and (7) and these equations involve five pa-

y rameters

,

R 2 R

, Pr , Ec

and . Gr being a non dimen-
0 x sional buoyae ncy pe arameter gives the free convection while

R , the Reynolds number gives the forced convection and,

( 0U x ) f , such that u , v

y x

(5)

e Gr


therefore,

R 2

gives the relative importance of forced and

free convectioe n in determi2ning the over all flow. In the similar
Now, we have u U f , v

1 U 0 f f

2 x

fashion, the parameter

R

, being a ratio of non dimensional
permeability parameter aned the Reynolds number represents

the relative importance of Darcy and general viscous drag.
And theTse vTalues of the velocity components satisfy the equa

The Eckert number Ec
appears in the viscous dissipation

Tw T

(T6h)e variations in the permeability and thermal conductivity have been assumed, following Chandrasekhara et al.[*], as
given below -
term. The parameter appears in the Darcy term.

3 RESULTS AND DISSCUSION

k ( )

k0 (1

d *e )

(7)
Figure 1(a & b) shows the effects of

Gr

2 on the profiles of

( ) 0 [ 0 (1

de )

b{1

0 (1

de )}]

Re


the non dimensional velocity f and temperature . As
where d * and d are constants,

0 , k0 and

0 are the values

reported in Chandrasekhar et al.[], the flows for the values of
of the diffusivity, permeability and porosity respectively at the edge of the boundary layer, b being the ratio of the thermal conductivity of the solid to that of the fluid.
The variation of viscosity with dimensionless temperature, fol- lowing Slattery (1978), is assumed to be of the form-

Gr

2 = 0.25 and 0.35, represent the mixed flow with some

e

margin of error. We see that for these values , the velocity (graph a) initially increases from the no slip condition near the plate acquiring a peak value and then gradually decreases

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The research paper published by IJSER journal is about Effects of Variable Fluid Properties and Viscous Dissipation on Mixed Convection Fluid Flow past a Vertical Plate in Porous Medium 3

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and attains the free stream velocity. Same is true for tempera- ture (graph b) also. We have also plotted the boundary layer behaviour for opposing flow also, i.e., for the negative values

Gr


of , here we have taken it as -1. For velocity and tempera-

2

e

ture both, the profiles steadily approach respectively the val- ues 1 and 0, the far end boundary conditions in the respective cases. While solving the problem, we noticed that the velocity and temperature profiles become oscillatory for the values of

Gr


> 0.75 and to show this kind of behaviour, we have plot-

2

e

has the effect to lower the velocity profile. The effects of uni- form i.e., no variations, in the permeability are shown by the thick lines in both the graphs and, the effects are quite visible in both the profiles. While in the temperature profile, the effect is to increase the temperature, on the other hand, it affects the velocity profile adversely. The effects of constant viscosity are shown by dotted lines and it is observed that velocity field re- mains almost unaffected but, it has the effects of increasing the temperature profile.
Figure 4 shows the effects of viscosity variations on the veloci-
ty and temperature profiles. We know that for the values of a >
0 , the viscosity decreases and from the graph it is noticed
that velocity profiles show decreasing patterns corresponding
ted it for

Gr


=3 . Also, we have investigated the velocity and

2

e

to any increase in the values of a. In both the figures, the ef-
fects of clear medium i.e., the absence of prous medium are shown by thick lines showing a decreasing trend in both the
temperature distribution taking a=0 which means no variation in the viscosity. In this case, the patterns are like those for

Gr


=3, but with a much greater peak. These unusual beha-

2

e

viours can be attributed to the simultaneous exponential varia- tions of the thermal conductivity, viscosity and medium per- meability.

2

Figure 2 shows the influence of permeability parameter

Re

on the flow field. As the values of this parameter are in- creased, we observe that velocity and temperature both show increasing trends and then they gradually acquire their respec-
profiles. Also, we have investigated the effects of uniform
thermal conductivity and medium permeability which are
shown by dotted lines and the effects are to lower the velocity and temperature values and this result, perhaps, provides an explanation as to why we got a somewhat unusual pattern with exponential variation of viscosity, thermal conductivity and medium permeability.

4 CONCLUSION

we observe that the velocity initially increases from no slip condition and it acquires a peak value and then gradually de- creases and attains the free stream velocity. It is also noticed

Gr

tive boundary values. Also, we have plotted these profiles for
two values of Pr, 7 & 0, which are shown respectively by
that for some values of

2 , the flow behaviour is somewhat

e

dashed and dotted lines.
The effects of Eckert number Ec
are shown in the figures 3

unusual. This may be attributed to the simultaneous variations

2

and it is observed from both the figures that an increase in Ec
in the fluid properties. The effects of the parameter
, vis-

Re

has the effect of increasing both velocity and temperature. The
effects of Nield modification in the energy equation are shown by the dashed lines and the effect is quite prominent in that the temperature profile has significantly decreased which, in turn,
cosity variation and the permeability variation on the velocity and temperature profiles are clearly visible.

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The research paper published by IJSER journal is about Effects of Variable Fluid Properties and Viscous Dissipation on Mixed Convection Fluid Flow past a Vertical Plate in Porous Medium 4


ISSN 2229-5518

(a)

(b)

(a) (b)

Figure 1

Figure 2


(a) (b (a (b

Figure 3

Figure 4

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The research paper published by IJSER journal is about Effects of Variable Fluid Properties and Viscous Dissipation on Mixed Convection Fluid Flow past a Vertical Plate in Porous Medium 5

ISSN 2229-5518

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