mixed convection over an isothermal vertical flat plate embedded in a porous medium with magnetic...

MIXED CONVECTION OVER AN ISOTHERMAL VERTICAL FLAT

PLATE EMBEDDED IN A POROUS MEDIUM WITH MAGNETIC

FIELD, RADIATION AND VARIABLE VISCOSITY WITH HEAT

GENERATION.

T.RajaRani1 , C.N.B.Rao2

Higher College of Technology, Muscat, Oman1

S.R.K.R. Engineering College, Bhimavaram, India2

[email protected], [email protected]

ABSTRACT

This paper focuses on the numerical solutions of

the effects of magnetic field, radiation, variable

viscosity and heat generation on similarity solutions

of mixed convection adjacent to an isothermal

vertical plate which is embedded in a porous

medium. A similarity transformation is used to

reduce the partial differential equations governing

the problem into ordinary differential equations and

the equations are solved numerically subject to

appropriate boundary conditions by the use of

Runge-Kutta-Gill method together with a shooting

technique. The flow and heat transfer quantities of

similarity equations are found to be the functions of

QRdC ,,, and RP where C is the magnetic

interaction parameter, Rd is the radiation parameter,

is viscosity variation coefficient, Q is the heat

generation/absorption parameter and RP is the

mixed convection parameter which is the ratio of

Rayleigh to cletep numbers. In the present work

the cases of assisting and opposing flows are

discussed. It has been found that in opposing flow

case, dual solutions exist for negative values of RP

and boundary layer separation occurs. It is observed

that depending on the values of RP there exists no

solution, a unique solution or dual solutions and

also the temperature decreases significantly with

increase in Q and C. Skin friction, heat transfer

coefficient, velocity and temperature fields are

studied and discussed with the help of a table and

graphs.

KEY WORDS Mixed Convection , Porous medium, Darcy model,

Heat generation, Variable viscosity.

1. 1.INTRODUCTION 2. 3. During last five decades much insightful

work has been done on mixed convection

boundary-layer flows in porous media. The

analogous problems have important

applications in the fields such as geothermal

systems, food processing and grain storage,

solar power collectors, oil reservoir modeling

and the dispersion of chemical contaminants in

different industrial processes in the

environment. References [10] and [14] stand

evident to the fact that convection flows in

porous media are of vital importance to such

processes.

Reference [12] discussed the Internal heat

generation and Radiation effects on a Certain

Free convection Flow and reference [1]

discussed the Numerical study of the combined

free-forced convection and mass transfer flow

past a vertical porous plate in a porous medium

with heat generation and thermal diffusion.

Reference [8] discussed the effect of variable

viscosity on convective heat transfer in the

three different cases of natural convection,

mixed convection and forced convection taking

fluid viscosity to vary inversely with

temperature. The authors have discussed the

effect of the appropriate parameters on the flow

and heat transfer quantities. The authors

however did not discuss hot plate and cold

plate cases separately in free convection and

did not discuss opposing flow case in mixed

convection.

International Journal of Digital Information and Wireless Communications (IJDIWC) 2(1): 7-17 The Society of Digital Information and Wireless Communications, 2012(ISSN 2225-658X)

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Reference [11] discussed laminar natural

convection flow (nonporous) and heat transfer

of fluids with and without heat sources in

channels with constant wall temperatures.

Reference [5] discussed mixed convection

boundary layer flow over a vertical surface for

the Darcy model when viscosity varies

inversely as a linear variation of temperature.

Results of both assisting flow and opposing

flow were discussed as function of mixed

convection parameter and variable viscosity parameter c . In the opposing flow case, the

existence of dual solutions and boundary layer

separation were noticed. Mixed convection

boundary layer flow on a vertical surface in a

saturated porous medium is studied in reference

[9]. In that paper the flow of a uniform stream

past an impermeable vertical surface embedded

in a saturated porous medium and which is

supplying heat to porous medium at a constant

rate is considered. Reference [6] discussed

radiation effects on natural convection in an

inclined porous surface with internal heat

generation.

The aim of the present paper is to

investigate the effects of variable viscosity or

temperature dependent viscosity, magnetic

field, radiation and heat generation/absorption

on mixed convection boundary layer flow over

a vertical surface embedded in a porous

medium. It is assumed that viscosity of the

fluid varies as a linear function of temperature

as in [4] & [13]. Both cases of assisting and

opposing flows are considered. The physical

coordinate system of the problem is presented

in Fig-1. The governing partial differential

equations are transformed into ordinary

differential equations by similarity variable and

the equations are solved numerically using

Runge-Kutta-Gill method with a shooting

technique for some values of the governing

parameters. Variations in flow and heat transfer

characteristics are presented in a table and in

some figures and discussed. Quantitative

comparisons with the existing results for the

case of constant viscosity, without magnetic

field and radiation effects as reported by

Merkin [9], Chin and Nazar [5] are presented.

2. FORMULATION AND SOLUTION

Let an isothermal flat plate be embedded

vertically in a porous medium saturated with

a viscous incompressible electrically

conducting, gray, emitting, absorbing and

non-scattering fluid. It is also assumed that

internal heat generation/absorption is

present. The plate is maintained at a constant

temperature, x-axis is taken vertically along

the plate and y-axis perpendicular to it so

that the plate can be described by y = 0.

Orientation of the plate can be seen in Fig-1.

Using the boundary layer and Boussinesq

approximations the equations governing the

problem, i.e., the Continuity equation, the

Darcys law and the energy equations are presented as:

0

y

v

x

u (2.1)

020

UuBUuK

gx

pp

(2.2)

0

v

Ky

p (2.3)


8

TT

c

Q

y

q

y

T

c

k

y

Tv

x

Tu

p

r

p

m

0

2

2

(2.4)

Here u, v are fluid velocity components, T is

fluid temperature, K is permeability, mk is

effective thermal conductivity of the fluid

saturated porous medium, 0B is the magnetic

flux, is the electric conductivity and rq is

the radiative heat flux transverse to the

vertical plate. The Rosseland approximation

is used in the energy equation to describe the

thermal radiative heat transfer. It may be

noted that by the use of Rosseland

approximation, the applicability of the

present analysis is limited to optically thick

fluids only and 0Q is the heat

generation/absorption constant.

The appropriate boundary conditions are:

0,,,

0,0,0,0

xyasUuTT

xyatvATT

(2.5)

where U is free stream velocity. Taking

U as BU where B is a constant,

introducing a stream function , cleteP

number xPe , non-dimensional functions

,f ; a similarity variable and radiative

flux rq through the relations

x

UcQQ

y

T

k

Tq

Pex

y

TT

TT

Pe

f

xUPe

p

c

sr

x

xm

m

x

0

3

2

1

0

2

1

3

16

)(

)(

(2.6)

where s is the Stefan-Boltzmanns

constant and ck is the mean absorption

coefficient and eliminating fluid pressure

from (2.2), (2.3) the governing equations are

obtained as:

)(1

2

11 RPCfCfC

(2.7)

fQ

Rd2

3

41

(2.8)

Here 2*

*

MK

KC

is the magnetic field

parameter, K

LK

LBM

f

2*

22

02 &

e

s

kk

TRd

34

is the radiation parameter, and

x

x

Pe

RaRP is Mixed convection parameter,

f

x

xTTgKRa

0 is the Rayleigh

number,

2

11 f ( Ref. [4] &

[13]) where viscosity variation

coefficient.

The boundary conditions (2.5) in terms of f

and are

0,0,

,0,1,0

fas

fat

(2.9)

Equation (2.7) can be integrated once using

the condition on f at to get

( )1

11

2

C RPf

C

(2.10)

which gives the slip velocity.


9

3. SOLUTION OF THE PROBLEM

3.1 Parameters of the Problem and Their Effect on the Flow and Heat Transfer

The flow and heat transfer depend on the

parameter C, , RP, Rd and Q where C is

the magnetic field parameter, is the

viscosity variation coefficient, RP is the

mixed convection parameter which is the

ratio of the Rayleigh number to cleteP

number, Rd is the radiation parameter and

Q is the heat generation parameter. Positive

and negative values of A correspond to

TT0 , TT0 and in turn to assisting

flow and opposing flow respectively. The

parameter takes positive as well as

negative values, the limiting values being

-2 and +2. Irrespective of the values of

0T and T , zero value of corresponds to

constant viscosity. In this paper solutions are

found for the values of -1, 0, 0.5 and 1 of .

The mixed convection parameter RP

takes positive values for assisting flow and

negative values for opposing flow. When RP

is zero, the results correspond to the forced

convection case. Enhanced flow can

correspond to an increase in the positive

value of RP, as an increase in its value can

be due to an increase in the temperature

difference )( 0 TT . Calculations are done

for a wide range of positive and negative

values of RP.

When there is no magnetic field, the

parameter C takes the value unity and for

increasing intensity of the magnetic field,

the parameter takes values smaller than

unity. In the present study, solutions are

found for the values 0.5 and 1 of C.

Reduced flow can be expected for smaller

values of C or for increased intensity of the

magnetic field as the Lorentz force (due to

the magnetic field) obstructs the flow.

When transfer of heat energy through

radiation is neglected, the parameter Rd

takes zero value and for increasing intensity

of thermal radiation, the parameter takes

larger values. Solutions are found for the

values 0, 0.5, 10 of the parameter Rd.

Thermal radiation causes thickening of the

thermal boundary layer and hence increasing

values of the parameter Rd can increase

thermal boundary layer thickness.

When transfer of heat energy through

generation of heat is neglected the parameter

Q takes zero value and for heat generation,

the parameter Q takes positive values like

0.5 and 0.7. However for heat absorption,

the parameter Q takes negative values, say,

-0.2. When heat absorption occurs (Q is negative) the fluid exerts a dragging force

on the surface and opposite is the case for

heat generation (Q is positive). Effects of

Simultaneous variation of the values of the

parameters on the flow and heat transfer are

presented in the discussion.

3.2 NUMERICAL SOLUTION

The equations for f and , i.e.,

equations (2.7), (2.8) are integrated

numerically subject to boundary conditions

(2.9) by Ruge-Kutta-Gill method together

with a shooting technique. The accuracy of

the method is tested by comparing

appropriate results of the present analysis

with available results. Present work agreed

well with Merkin [9], Chin etal.[5], which is

shown in Fig-2 for skin friction )0(f and in


10

Table-1 for heat transfer coefficient ' (0) '

respectively for no magnetic field, no

radiation, constant viscosity and no heat

generation i.e., 1, 0, 0 & 0C Rd Q

Table -1

Variation of heat transfer )'0(' with RP

)'0(' for Upper Solution RP Chin[5] Present work

0.0 0.79790 0.7978845

-0.1 --------- 0.7720349

-0.3 0.71730 0.7172853

-0.5 0.65750 0.6574822

-0.8 0.55396 0.5539434

-1.0 0.46962 0.4695999

-1.1 0.41915 0.4191351

-1.2 0.35848 0.3584588

-1.3 0.27448 0.2743369

-1.35 0.19388 0.1908006

RPc=-1.354 0.15589 0.1556460

)'0(' for Lower Solution -1.1 0.00176 0.0017597

-1.2 0.01849 0.0184873

-1.3 0.06528 0.0652853

-1.35 0.13223 0.1322655

RPc=-1.354 0.15589 0.1561500

3.3 Discussion of the Results

In the following, discussion of both

assisting and opposing flows are presented.

However more attention is paid to the

solutions of the opposing flow case. When

0T T the opposing flow case arises in

which RP takes negative values and dual

solutions exist. The solution corresponding

to relatively larger values of (0)f and

' (0) ' is referred to as the upper solution

and the one corresponding to smaller values

of (0)f and ' (0) ' as the lower solution.

It can be understood that at the critical

values of RP ( )cRP , the first solution (upper

solution) and the second solution (lower

solution) meet. When 0T T the assisting

flow case arises in which RP takes positive

values and single solution exist. The local

drag coefficient is proportional to the skin

friction, (0)f and the local Nusselt number

is directly proportional to the heat transfer

coefficient or wall transfer rate, ' (0) ' .

Qualitatively interesting results related to

the skin friction, heat transfer coefficient,

velocity and temperature profiles are

presented; some of them are in the form of

table-1 and others in the form of figures 2 to

16. The changes in skin friction with

negative values of RP (opposing flow) are

shown in the Fig-2, 3, 4 and with positive

values of RP(assisting flow) in the Fig-5.

The corresponding changes in heat transfer

rate are shown in Fig-6, 7 & 8.

From figures 3 and 4, it can be observed

that when Q is positive (heat generation) the

local drag coefficient (the values of (0)f )

takes negative values and dual solution

exist. However for Q negative (heat

absorption) the local drag coefficient takes

positive values and single solution exists.

Physically, positive sign of (0)f implies

that the fluid exerts a dragging force on the

surface and negative sign implies the

opposite. It is also observed as Q increases

from 0.5 to 0.7 the range of admissible

absolute values of RP also decrease. From

Fig-3 it is observed that local drag

coefficient takes larger values in the

presence of magnetic field (C=0.5) than in

its absence (C=1). The coefficient of drag is

also observed to be larger in the absence of

radiation than in its presence. The range

(critical value of RP) over which solution

exist can be seen to be considerably larger

for negative variation of viscosity

( 0.5) than for constant variation of


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viscosity ( 0 ). However for positive

variation of viscosity ( 0.5) the range is

less than for constant variation of viscosity

( 0) .

From Fig-5 (assisting flow), it is

observed that as RP increases the magnitude

of (0)f increases for all values of

, , &C Rd Q . This is because the fluid

velocity increases when the buoyancy force

increases and hence increases the skin

friction.

It is observed from Fig-6 that ' (0) '

decreases when RP becomes more negative.

This is because the buoyancy force works

against the fluid flow and therefore heat

transfer process is retarded. Like skin

friction ( (0))f , heat transfer coefficient

(' (0) ') is negative for heat generation and

is positive for heat absorption. From Fig-6

and 7 it is observed that, as the intensity of

magnetic field increases i.e., (C decreases

from 1 to 0.5) heat transfer coefficient

' (0) ' increases in case of opposing flow

and opposite is the behavior in case of

assisting flow. Also it can be observed that

for a fixed value of RP, the value of

' (0) ' for negative variation of viscosity

)5.0( is always higher than the value

of ' (0) ' for constant viscosity ( 0)

and the value of ' (0) ' for positive

variation of viscosity ( 0.5) is always

less than for constant viscosity ( 0) . It is

observed that with variable viscosity, the

separation of boundary layer is delayed for

positive variation of viscosity than for

negative variation of viscosity. In case of

assisting flow, from Fig-8, it can be viewed

that heat transfer coefficient ' (0) '

increases as RP increases positively. This is

because the buoyancy force works along

with the fluid flow, therefore heat transfer

process is accelerated.


12

Variations in temperature are shown in

Fig-9 to Fig-13. From Fig-9, it may be

observed that thermal boundary layer

thickness is more in case of heat generation

(Q is positive) than in case of heat absorption

(Q is negative). Boundary layer thickness is

more for lower solution than for upper

solution. As magnitude of RP increases

thermal boundary layer thickness increases.

From Fig-10 it may be viewed that

boundary layer thickness is more in presence

of radiation than in its absence. Also it may

be observed that increase in thermal radiation

parameter (Rd) produces significant increase

in the thickness of the thermal boundary layer


13

of the fluid and so the temperature profile

( ) increases and tends to zero at the edge of

the boundary layer. This is due to the fact that

the presence of thermal radiation causes

thickening of the thermal boundary layer. As

C decreases (magnetic effect increases) there

is no significant change in upper solution but

there is an increase in the lower solution. This

may be due to the resistance offered by the

Lorentz force due to the flow, and as a result

an increase in temperature.

In Fig-11 are shown variations of

temperature with variations of viscosity for

fixed values of C, Rd, Q and RP. It may be

noticed that the nature of the lower solution is

just opposite to the nature of the upper

solution.

From Fig-12 it may be observed that

temperature and thermal boundary layer

thickness are more for opposing flow than for

assisting flow. It can be seen from Fig-13 that

in case of assisting flow, thermal boundary

thickness for positive variation of viscosity

)5.0( is more than for constant viscosity

)0( . But, for negative variation viscosity

)5.0( it is less than for constant viscosity.

As RP (mixed convection parameter) increases

thermal boundary layer thickness decreases and

also it is observed for RP=50, after certain

stage, temperature is becoming negative.


14

From Fig-14, slip velocity 9.0)0( f is

same in case of upper and lower solutions for

heat generation (Q= 0.5, 0.7) as well as for heat

absorption )2.0( Q when 0,1.0 RP .

It can be observed that as RP becomes more

negative slip velocity decreases for both heat

generation and absorption. The velocity of the

fluid is more for heat absorption than for heat

generation. Hydrodynamic boundary layer

thickness is more for lower solution than for

upper solution.

From Fig-15, it can be observed that slip

velocity and fluid velocity for negative

variation of viscosity ( 5.0 ) as well as for

positive variation of viscosity )5.0( is

more than for constant viscosity ( 0

).Velocities of aiding and opposing flows are

shown in Fig-16. As RP increases, velocity and

slip velocity increase and slip velocity of the

aiding flow is found to be more than unity

whereas for opposing flow they are less than

unity. It may be noted that slip velocity can be

directly calculated from the equation 2.10.

From Fig-9 and Fig-14 it can be observed

that thermal boundary layer thickness is more

than hydrodynamic boundary layer thickness.


15

4. Conclusions

This paper analyzes momentum and heat

transfer as affected by magnetic field,

radiation, variable viscosity and heat

generation in similarity solutions of mixed

convection adjacent to an isothermal vertical

plate embedded in a porous medium.

Numerical solutions for momentum and heat

transfer are obtained by employing Runge-

Kutta Gill method together with a shooting

technique. The following conclusions are

drawn from the numerical results.

1. In case of opposing flow in the presence of heat generation (Q is positive), the local

drag coefficient (skin friction) takes

negative values and dual solutions exist.

However in case of heat absorption (Q is

negative), skin friction takes positive values

and single solution exists. Physically

positive sign of skin friction implies that the

fluid exerts a dragging force on the surface

and negative sign implies opposite.

2. Local drag coefficient takes larger values in the presence of magnetic field than in its

absence. The coefficient of drag is also

observed to be larger in the absence of

radiation than in its presence.

3. The separation of boundary layer is delayed for positive variation of viscosity than for

negative variation of viscosity.

4. As mixed convection parameter (RP) is more negative (opposing flow), heat transfer

coefficient decreases because the buoyancy

force works against the fluid flow, therefore

heat transfer process is retarded. In case of

assisting flow heat transfer coefficient

increases as RP increases positively. This is

because the buoyancy force works along

with the fluid flow, therefore heat transfer

process is accelerated.

5. Thermal boundary layer thickness is more than hydrodynamic boundary layer

thickness. The velocity of the fluid is more

for heat absorption than for heat generation.

Slip velocity in aiding flow is more than in

opposing flow.

6. Thermal boundary layer increases as Rd(radiation) increases, due to the fact that

the presence of thermal radiation causes

thickening of the thermal boundary layer.

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