MIXED CONVECTION FLOW IN HEAT ABSORBING DARCIAN MEDIA WITH VISCOUS DISSIPATION AND THERMAL RADIATION...

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e-ISSN: 24085162; p-ISSN: 20485170; April, 2017: Vol. 2 No. 1B pp. 330 – 333

320

MIXED CONVECTION FLOW IN HEAT ABSORBING DARCIAN MEDIA

WITH VISCOUS DISSIPATION AND THERMAL RADIATION

Sanjib Sengupta*and Bishnu Ram Das

Department of Mathematics, Assam University, Silchar, India

*Corresponding author: [email protected]

Received: December 21, 2016 Accepted: March 16, 2017

Abstract: The present paper deals with the study of a mixed convective flow of Newtonian, incompressible, viscous fluid

past a porous vertical plate embedded in heat absorbing Darcian porous media in presence of viscous dissipation

and thermal radiation. A system of non-linear, coupled partial differential equations has been developed based on

the basic conservation principles of physics such as conservation of mass, conservation of momentum and

conservation of energy. Multi – parameter regular perturbation technique has been used to solve the governed

system of equations subject to a set of favourable boundary conditions. The exact closed form of solutions for the

velocity, temperature as well as skin – friction and Nusselt number have been obtained in terms of some governed

physical parameters. Numerical simulations in terms of graphs and table have been made to investigate the effects

of pertinent parameters. It is observed that, an increase in Reynolds number decreases the temperature as well as

the velocity while the presence of Reynolds number increases the Nusselt number as well as skin- friction. Again

the fluid velocity and temperature are found to increase due to increase in Eckert number but a reversed effect has

observed in case of Nusselt number and skin – friction.

Keywords: Newtonian fluid, mixed convection, heat transfer, heat absorption, Darcian

Introduction

The phenomena of mixed or combined convection arise when

both free and forced convection occur simultaneously in a

flow domain. Several researchers investigated the theory of

mixed convection due to its applications in many industrial

and mechanical devices. The pioneering work in mixed

convection flow was made by Ostrich (1954), where he

investigated the combined natural and forced convection

laminar flow and heat transfer of fluid with and without heat

source in channels with linearly varying wall temperature. The

case of mixed convection cooling of a heated, continuously

stretching surfaces was considered by Chen (1998). Hadhrami

et al. (2001) justified the study of combined free and forced

convection in vertical channels of porous media. Mahmud and

Fraser (2003) studied the mixed convection – radiation

interaction in a vertical porous channel with entropy

generation. Gireesha et al. (2013) enunciated the study of

mixed convective flow of a dusty fluid over a vertical

stretching sheet with non-uniform heat source/sink and

radiation. Recently, Dawood et al. (2015) reviewed the

process of forced, natural and mixed-convection heat transfer

and fluid flow in annulus.

On the other hand, radiation heat transfer plays an important

role in manufacturing industries for the design of reliable

equipment. If the temperature of the surrounding fluid is

rather high, radiation effects play an important role. The

theory of thermal radiation has been considered by many

researchers in various flow situations. Babu et al. (2011),

analyzed radiation and chemical reaction effects on unsteady

MHD convection flow past a vertical moving porous plate

embedded in a porous medium with viscous dissipation. Shit

and Halder (2011), Ahmed et al. (2011), investigated the

effects of thermal radiation on MHD viscous fluid flow over a

shrinking sheet. The effect of thermal radiation on boundary

layer flow and heat transfer of dusty fluid over an unsteady

stretching sheet was put forward by Manjunatha and Gireesha

(2012), Christophand Johannes (2014), investigated

numerically the heat transfer with thermal radiation in an

enclosure in case of buoyancy driven flow. Sengupta (2015)

investigated the thermal radiation effect with chemical

reaction and radiation absorption along with the conditions of

variable wall temperature and concentrations. Very recently,

Sahooet al. (2016) analyzed thermal radiation heat transfer

model and its application for automobile exhaust components.

The problem of thermal radiation in unsteady mixed

convection flow is attracted attention of many researchers due

to some practical applications in engineering, technology and

in cooling of nuclear reactors. Researchers like, Elsayed et al.

(2012), Reddy et al. (2013) and Khan et al. (2014) developed

the theory of thermal radiation on various flow situations. Off

late, Devi et al. (2016) investigated the effect of radiation on

an unsteady MHD mixed convective flow past an accelerated

vertical porous plate with suction and chemical reaction.

It is interesting to observe that, if a fluid is highly viscous or

fluid shear rate is of high in magnitude, the viscous

dissipation is predominant. Viscous dissipation is a

mechanical process defined as the heat that produces due to

work done by fluid particles on adjacent layers to counter the

shearing forces. In case of mixed convection flow, the effect

of viscous dissipation is significant and cannot be ignored.

The significant contribution in viscous dissipation was made

by Gebhart (1962), who had investigated the effect of viscous

dissipation in natural convection. Considering the importance

of viscous dissipation, Fand and Brucker (1983), Mahajan and

Gebhart (1989) investigated viscous dissipation in natural

convection and buoyancy induced flows respectively. Barletta

(1998), studied laminar mixed convection with viscous

dissipation in a vertical channel, Nield et al. (2003),

considered the viscous dissipation effect in forced convection

thermally developing flow through parallel porous channel

with walls at uniform temperature. Pentokratoras (2005)

investigated the effect of viscous dissipation in natural

convection along a heated vertical plate.

Again, the study of viscous dissipation with mixed convection

is significant in industrial and technological point of view.

Hung and Tao (2009), considered the effects of viscous

dissipation on fully developed forced convection flow in

porous media. Das (2014) investigated the influence of

chemical reaction as well as viscous dissipation on MHD

mixed convection flow. Abdollahzadeh and Hyun (2014)

undertook the problem of viscous dissipation with thermal

radiation and Joule heating for the study on MHD forced

convection flow. Pal and Samad (2015) used similarity

transformations and obtained numerical solutions for

analyzing the combined effects of viscous dissipation and

thermal radiation on non-Newtonian fluid along a surface with

heat generation and uniform surface heat flux.

Following the importance of the aforesaid phenomena, the

purpose of the present paper is to study the flow behavior of a

mixed convection flow of a Newtonian, incompressible,

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Mixed Convection Heat Absorbing Flow with Viscous Dissipation and Thermal Radiation



321

viscous fluid through a vertical porous plate immersed in heat

absorbing Darcian porous media in presence of thermal

radiation and viscous dissipation effects.

Basic equations and assumptions undertaken

Basic equations

The vector forms of equations that describe the flow situation are formulated as:

. 0 (Continuity Equation)q

2( . ) ( Modified Navier-stokes equation)q

q q p g q qt K

2 *( ( . ) ) (Energy equation)p R

TC q T k T q Q T

t

( ) (Boussinesq approximation)T T T

Where, q = ( u , v , 0) : the fluid velocity, p : the hydrostatic pressure, : the fluid density, :g the acceleration due to

gravity, : the dynamic viscosity, k: the thermal conductivity of the medium,( , ) , ,0x y

x y

: 2-D vector gradient

operator, Rq : the radiative heat flux, : the densities of fluid particles in free stream, :T coefficient of volumetric

thermal expansion, , :T T Fluid temperature near the plate and in free stream, * :Q First order heat source parameter, :pC

specific heat at constant pressure.

Basic assumptions The fundamental assumptions considered for the study are as

follows:

a) All the fluid properties except possibly the pressure are

independent of variations of x*-scale.

b) All the fluid properties are considered constant except

the physical influence of the density term with

temperature in the thermal buoyancy force.

c) The empirical relation between the pressure gradient

and the flow velocity is assume as linear as such, the

Brinkman– Darcy porous term is considered; while

Forchheimer’s non-linear porous effect is neglected

therein.

d) The flow domain is considered as homogenous and

isotropic.

e) Due to moderate viscous fluid, the viscous dissipation of

energy is considered in the energy equation.

f) The temperature of fluid particles near the plate surface

is supposed to be more than their respective components

at the free stream region.

g) The radiation heat fluxes are considered prominent

along normal to the plate i.e. towards the fluid domain

and are thus taken negligible along the radial direction.

Mathematical formulation of the problem

The mixed convective flow of an unsteady, laminar, semi-two

dimensional incompressible viscous fluid over an infinite

vertical porous flat plate embedded in Darcian porous medium

is considered for the study. The positive -coordinate is

measured along the plate in the vertically upward direction

and the positive - coordinate is taken normal to the plate in

the outward direction towards the fluid region.

Fig. 1: A schemetic representation of the flows coordinate

system

Under the Boussinesq and boundary layer approximations, the basic boundary-layer equations thus governed as:

Continuity equation

0 v

y

(1)

momentum equationx

2

2

1T

u u p uv u g T T

t y x y K

(2)

Energy equation

22

2

1 ry

s

p p p

qT T k T uv Q T T

t y C y C y C y

(3)

x

y





322

With use of the relevant boundary condition as:

0, 0, for all y 0 and t 0u v T T (4.1)

, 1 exp , expp o mu u v V i t T T T T i t at y 0 when t > 0 (4.2)

1 exp , for y when t >0 ou U U i t T T (4.3)

Where, ( , ), , , , , , , , ,p o ou v t K u V U U are respectively the ( x , y ) component velocities, time variable, kinematic

co-efficient of viscosity, permeability parameter, slip velocity of the plate, fluctuation parameter ( << 1), frequency of

oscillation, mean plate suction velocity, free stream velocity, mean free stream velocity.

The Bernoullie’s pressure equation gives,

1exp( )o o

pU i U i t

y k k

(5)

Now using (5) in (2) we get,

2

2exp( )o o T

u u uv U u i U i t g T T

t y y k k

(6)

We now use a set of non-dimensional quantities as:

, , , , , , , , , , Re ,p o o o

p o L

o o o o m o o

u tV U V Lx y u v L T T Ux y u v u t U U

L L V V V L V T T V V

3

1

2

0 1

Q 4 4, , , Re , Q = , = , =1+

3

T m po o sr r L s

o p

g T T L cKV V L L T NK G P N R

L V k V C k

Where, ,Re , , , and L r r SL G P Q N represents some reference length along the normal scale, local Reynolds

number, thermal Grashof number, Prandtl number, heat source/generation parameter, thermal radiation parameter, respectively.

The non - dimensional set of equations on using Rossolandradiative approximation model for optically thick medium is obtained

as:

0v

y

(7)

2

2

1 1 1exp( )

Reo o r

L

u u uv U u i i t U G

t y y K K

(8) 22

2Rec s

r L

N uv E Q

t y P y y

(9)

With non-dimensional boundary condition as:

0, 0, 0, for y 0 when t 0u v

(10.1)

, 1 exp , 1 exp at y=0 when t>0pu u v i t i t (10.2)

1 exp , 0 for y when t>0ou U U i t (10.3)

Method of solution

We use a first set of perturbation solutions as:

1 , exp( )of y t f y f y i t

Where, f stands for and u etc.

On using the form, we obtain a set of mean and perturbed equations for temperature as well as velocity fields as: 22

2 Re Re Reo o o

r o s r o cL L Lr

d d duN P v Q P E P

dy dy dy

(11)

2

1 1 11 12

Re Re Re Re2 L Lo o

r o s r r c rL L

d dud d duN P v Q i P P v E P

dy dy dy dy dy

(12)

2

2

Re ReRe Reo o L L

o L o o r L o

d u duv u U G

dy dy k k

(13)





323

2

1 11 1 1 1 12

Re Re Re Re ReoL o L L L o r L

dud u duv M u v U M G

dy dy dy

(14)

1

1where M sayi

k

With the boundary condition as:

1 1 1, 1, 1, 0, 1, 1 at y =0,t > 0o p o ou u v u v (15.1)

1, 0, , 0 for y , t > 0o o o o ou U u U (15.2)

We now use a second set of perturbation as:

2

01o oo c cf y f y E f y o E

2

1 1 11o c cf y f y E f y o E

Using the above trial forms, we obtain a set of unperturbed as well as perturbation parts of equations as: 2

2Re Re 0 (16)oo oo

r L oo s r L oo

d dN P v Q P

dy dy

2

201 0101 012

Re Re Re Re ( ) oo oor L oo s r L r L r L

d d d duN P v Q P v P P

dy dy dy dy

(17)

2

2

Re ReRe Re (18)oo oo L L

L oo oo o L r oo

d u duv u U G

dy dy k k

2

01 01 0001 01 012

ReRe Re Re (19)L

L oo L r L

d u du duv u v G

dy dy k dy

2

10 1010 102

Re ( ) Re Re (20)oor L oo s r L r L

d d dN P v Q i P P v

dy dy dy

2

01 1011 1111 102

P Re ( )P Re P Re 2P Re (21)oor L oo s r L r L r L

d du dud dN v Q i v

dy dy dy dy dy

2

10 101 10 10 1 102

Re Re Re Re Re (22)ooL oo L L L o r L

d u du duv M u v M U G

dy dy dy

2

10 0111 1101 1 11 11 112

Re Re Re Re ( ) Re ( ) Re (23)ooL oo L L L L r L

du du dud u duv v M u v G

dy dy dy dy dy

With the following sets of zeroth-order and first order boundary conditions as:

01 10 11 10 11, 0, 1, 0, 0, 1, 0,oo p oou u u v u u v v

01 01 10 110, 1, 0, 1, 0 at y =0, when t >0oov (24.1)

01 10 11 10 11, 0, , 0, 1, 0,oo o ou U u u U u v v

10 11 010, 0, 0, 0 for y , t>0oo (24.2)

Finally, the mean as well as the perturbed parts of the solutions for temperature and velocity fields are calculated as:

1exp( )oo m y

2 1 3 2exp( ) exp( )oo ou U b m y b m y

01 4 1 1 2 2 1 3 1exp( ) exp( 2 ) exp( 2 ) exp( )a m y a m y a m y a y

01 4 1 5 2 6 1 7 1 8 2exp( ) exp( 2 ) exp( 2 ) exp( ) exp( )u b m y b m y b m y b y b m y

10 6 1 1 6 1 7 1 1(1 )exp( )cos exp( ) exp( )sinR a y y a m y a y y

10 12 15 19 2 2 13 16 20 2 2

12 2 15 1 19 1 1 20 1 1

( )exp( )cos ( )exp( )sin

exp( ) exp( ) exp( )cos exp( )sin

R o ou U U b b b y y b b b y y

b m y b m y b y y b y y





324

11 9 1 12 2 15 1 18 1 22 1 1 1 23 1 1 1

26 1 2 2 27 1 2 2 30 2 1 1 31 2 1 1

exp( ) exp( 2 ) exp( 2 ) exp( ) exp( ( ) )cos exp( ( ) )sin

exp( ( ) )cos exp( ( ) )sin exp( ( ) )cos exp( ( ) )sin

R a m y a m y a m y a y a m y y a m y y

a m y y a m y y a m y y a m y

34 2 2 2 35 2 2 2 36 3 3 37 3 3exp( ( ) )cos exp( ( ) )sin exp( )cos exp( )sin

y

a m y y a m y y a y y a y y

11 36 2 39 1 42 2 45 1 48 1

52 1 1 1 53 1 1 1 56 2 1 1

57 2 1 1 60 2 2 2 61

exp( ) exp( ) exp( 2 ) exp( 2 ) exp( )

exp( ( ) )cos exp( ( ) )sin exp( ( ) )cos

exp( ( ) )sin exp( ( ) )cos exp(

Ru b m y b m y b m y b m y b y

b m y y b m y y b m y y

b m y y b m y y b

2 2 2

64 1 2 2 56 2 1 1 57 2 1 1

60 2 2 2 61 2 2 2 64 1 2 2

65 1 2 2 68 3

( ) )sin

exp( ( ) )cos exp( ( ) )cos exp( ( ) )sin

exp( ( ) )cos exp( ( ) )sin exp( ( ) )cos

exp( ( ) )sin exp( )cos

m y y



b m y y b y

3 69 3 3

70 4 4 71 4 4

exp( )sin

exp( )cos exp( )sin

y b y y

b y y b y y

Discussion on some quantities of engineering importance

Non-dimensional skin-friction1 2 3 4 5 6

1( ( ) (( ( )) cos ( ( ))sin ))

ReR

L

Ec Ec t Ec t

1 2 3 1 2where, ( )m b mb , 2 1 4 2 5 1 6 1 7 2 8( 2 2 )mb m b mb b m b

3 2 12 15 19 2 13 16 20 2 12 1 15 1 19 1 20( ) ( )oU b b b b b b m b mb b b

4 2 36 1 39 2 42 1 45 1 48 1 1 52 1 53 2 1 56 1 57 2 2 60{ 2 2 ( ) ( ) ( )m b mb m b mb b m b b m b b m b

2 61 1 2 64 2 65 3 68 3 69 4 70 4 71( ) }b m b b b b b b

5 2 12 15 19 2 13 16 20 2 13 1 16 1 19 1 20( ) ( )oU b b b b b b m b mb b b

6 2 37 1 40 2 43 1 46 1 49 1 52 1 1 53 1 56 2 1 57 2 60 2 2 61{ 2 2 ( ) ( ) ( )m b mb m b mb b b m b b m b b m b

2 64 1 2 65 3 68 3 69 4 70 4 71( ) }b m b b b b b

Non-dimensional heat transfer rate (Nusselt number)

1 2 3 4 5

0

1 1[ (( )cos ( )sin )]

Re Re

RR

r L r Ly

Nu Nu Nu EcNu t Nu EcNu tP y P

1 1 2 1 1 2 1 3 1 4 2 1 1 6 1 6 1 7where, 2 2 , Nu m Ec m a Ec m a Ec a Ecm a Nu a m a a ,

3 1 9 2 12 1 15 1 18 1 1 22 1 23 1 2 26 2 27 2 1 30( 2 2 ( ) ( ) ( )Nu Ec m a m a m a a m a a m a a m a

1 31 2 2 34 2 35 3 36 3 37 4 1 1 6 1 7 1 7( ) ), a m a a a a Nu a a m a

5 1 10 2 13 1 16 1 19 1 22 1 1 23 2 26 1 2 27 1 30 2 1 31( 2 2 ( ) ( ) ( )Nu Ec m a m a m a a a m a a m a a m a

2 34 2 2 35 3 36 3 37( ) )a m a a a

Results & Discussion

To get some physical insight into the problem, numerical

simulation has been made to justify the influence of varies

physical parameters that governed by the system due to

inclusion of various physical situations, on the fluid variables,

such as velocity, temperature and concentration as well as on

some quantity of engineering interest like, skin-friction and

Nusselt number.

In figures (1) to (5) the effect of pertinent parameters such as,

Prandtl number ( )rP , Permeability parameter ( )K , heat sink

parameter ( )SQ , thermal Grashof number ( )rG and thermal

radiation parameter ( )N on the temperature profile ( , )R y

is depicted graphically. It is observed from these figures that

the fluid temperature increases due to increase in parametric

values of ( )K , ( )rG , ( )N while the increase of ( )rP , ( )SQ

found to be decreased the temperature within the thermal

Fig. 1: Temperature against normal distances for arbitrary

change in values of Prandtl number for Pr=7.0, ReL=0.5,

QS=0.01, N=2.5, Gr=5.0, U0=0.5, up=0.5, 𝜔=7.85714,

𝜔𝑡=1.57142857, K=0.5, 𝜖=0.001, Ec=0.001.

0 5 10 15 20 250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Normal distance(y)

Tem

pera

ture

Pr=0.71

Pr=1.00

Pr=4.00

Pr=7.00





325


change in values of Permeability parameter for Pr=7.0,

ReL=0.5, QS=0.1, N=3.6, Gr=10.0, U0=0.5, up=0.5,

𝜔=7.85714, 𝜔𝑡=1.57142857, K=0.5, 𝜖=0.001, Ec=0.01


change in values of Thermal Grashof number for Pr=7.0,

ReL=0.05, QS=0.1, N=1.2, Gr=10.0, U0=0.5, up=0.5,

𝜔=7.85714, 𝜔𝑡=1.57142857, K=0.5, 𝜖=0.001, Ec=0.01


change in values of heat sink parameter for Pr=7.0, ReL=0.05,

QS=0.1, N=1.2, Gr=5.0, U0=0.5, up=0.5, ω=7.85714,

ωt=1.57142857, K=0.5, 𝜖 =0.001, Ec=0.001

Fig.5: Temperature against normal distances for arbitrary

change in values of Thermal radiation for Pr=7.0, ReL=0.1,

QS=0.01, N=1.2, Gr=1.0, U0=0.5, up=0.5, 𝜔=7.85714,

𝜔𝑡=1.57142857, K=0.5, 𝜖=0.001, Ec=0.001

In Fig. 6, the influence of Eckert number ( )Ec on the

temperature profile ( , )R y is shown graphically. Due to

increase in values of ( )Ec , the fluid experiences frictional

heating in the intermediate layers; this thus, contributed in the

thickening of thermal boundary layer and thus increases the

temperature within the boundary layer significantly.


change in values of Eckert number for Pr=7.0, ReL=0.1,

QS=0.01, N=2.6, Gr=10.0, U0=0.5, up=0.5, 𝜔=7.85714,

𝜔𝑡=1.57142857, K=0.5, 𝜖=0.001, Ec=0.001

The influence of Reynolds number (Re )L on the

temperature profile ( , )R y is demonstrated graphically in

figure (7). The increase in values of (Re )L increases the

strength of the suction velocity on the plate which minimizes

the growth of thermal boundary layer. This thus diminishes

the temperature R near to the plate surface.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Normal distance(y)

Tem

pera

ture

K=0.1

K=1.0

K=2.0

K=3.0

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Normal distance(y)

Tem

pera

ture

Gr=3.0

Gr=5.0

Gr=7.0

Gr=9.0

0 5 10 150

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Normal distance(y)

Tem

pera

ture

Qs=0.0

Qs=0.5

Qs=1.0

Qs=1.5

0 2 4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Normal distance(y)

Tem

pera

ture

N=1.0

N=2.3

N=3.6

N=5.0





326

Fig. 7: Temperature against normal distances for arbitrary change in

values of Local Reynolds number for Pr=7.0, ReL=0.05, QS=0.1,

N=1.2, Gr=5.0, U0=0.5, up=0.5, 𝜔=7.85714, 𝜔𝑡=1.57142857, K=0.5,

𝜖=0.001, Ec=0.001

The parametric effect of the pertinent parameters such as

Prandtl number ( )rP thermal Grashof number ( )rG , heat

absorption parameter ( )SQ , permeability parameter ( )K and

thermal radiation parameter ( )N on the velocity profiles

( , )Ru y is demonstrated graphically interns of figures (8) to

(12). The prime velocity is seen to increase due to increase in

parametric values of ( )rG ( )K and ( )N while a reverse

phenomenon as observed due to presence in values of ( )rP as

well as ( )SQ . The presence of ( )rG increases the thermal

buoyancy force, which in turn accelerates the flow rate and

thus increases the value of Ru . Due to rise in values of the

permeability parameter, the resistance of the porous media

increases, This results in increasing the velocity fluxes inside

the momentum boundary layer, which accelerates the flow

rate and increases the value of .Ru As the presence of thermal

radiation parameter increases the temperature of the fluid

particles near the plate surface, this results in effecting an

increment in the flow rate by increasing the kinetic energy of

the fluid particles as a results the value of Ru is found

increasing.

Fig. 8: Velocity against normal distances for arbitrary change in values of Prandtl number for Pr=7.0, ReL=0.3, QS=0.5, N=2.6, Gr=1.0,

U0=0.5, up=0.5, 𝜔=7.85714, 𝜔𝑡=1.57142857, K=0.5, 𝜖=0.001, Ec=0.001

Fig. 9: Velocity against normal distances for arbitrary change in values of Thermal Grashof number for Pr=7.0, ReL=0.05, QS=0.1,

N=1.2, Gr=5.0, U0=0.5, up=0.5, 𝜔=7.85714, 𝜔𝑡=1.57142857, K=0.5,

𝜖=0.001, Ec=0.001

Fig. 10: Velocity against normal distances for arbitrary change in values of heat sink parameter for Pr=7.0, ReL=0.05, QS=0.1, N=1.2,

Gr=5.0, U0=1.0, up=1.0, 𝜔=7.85714, 𝜔𝑡=1.57142857, K=0.5,

𝜖=0.001, Ec=0.001

Fig. 11: Velocity against normal distances for arbitrary change in values of Permeability parameter for Pr=0.7, ReL=0.1, QS=0.1, N=1.2,

Gr=10.0, U0=0.5, up=0.5, 𝜔=7.85714, 𝜔𝑡=1.57142857, K=0.5,

𝜖=0.001, Ec=0.01

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Normal distance(y)

Tem

pera

ture

ReL=0.2

ReL=0.5

ReL=0.7

ReL=1.0

0 1 2 3 4 5 6 70.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

normal distance(y)

Velo

city(u

R)

Pr=0.71

Pr=1.50

Pr=4.00

Pr=7.00

0 2 4 6 8 10 12 14 16 180.5

0.52

0.54

0.56

0.58

0.6

0.62

0.64

0.66

0.68

normal distance(y)

Velo

city(u

R)

Gr=3.0

Gr=5.0

Gr=7.0

Gr=9.0

0 2 4 6 8 10 12 14 16 18 201

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

1.45

1.5

normal distance(y)

Velo

city(u

R)

Qs=0.0

Qs=0.1

Qs=0.5

Qs=0.9

0 2 4 6 8 10 12 14 16 18 200.4

0.6

0.8

1

1.2

1.4

1.6

1.8

normal distance(y)

Velo

city(u

R)

K=0.5

K=1.0

K=2.0

K=3.0





327

Fig. 12: Velocity against normal distances for arbitrary

change in values of Thermal radiation for Pr=7.0, ReL=0.05,

QS=0.1, N=1.2, Gr=5.0, U0=0.5, up=0.5, 𝜔=7.85714,

𝜔𝑡=1.57142857, K=0.5, 𝜖=0.001, Ec=0.001

The influence of physical parameters like Eckert number

( )Ec and Reynolds number (Re )L on the velocity profile

( , )Ru y is shown graphically in figures (13) and (14)

respectively. It is observed that, the velocity increases due to

increase in values of ( )Ec while a reverse phenomenon is

seen due to presence of ReL . As due to increase in values of

Ec, frictional heating in the thermal boundary layer increases.

This in turn increases the kinetic energy of the fluid particle in

the intermediate layers as a result accelerates the flow rate and

thus increases the velocity. Again due to raise in values of

ReL , the plate suction velocity increases. This in turn

decreases the flow rate within the boundary layer and thus

diminishes the velocity.


change in values of Eckert number for Pr=7.0, ReL=0.2,

QS=0.5, N=2.6, Gr=1.0, U0=0.5, up=0.5, 𝜔=7.85714,

𝜔𝑡=1.57142857, K=0.5, 𝜖=0.001, Ec=0.001


change in values of Local Reynolds number for Pr=7.0,

ReL=0.05, QS=0.5, N=1.2, Gr=5.0, U0=0.5, up=0.5,

𝜔=7.85714, 𝜔𝑡=1.57142857, K=0.5, 𝜖=0.001, Ec=0.001

In figures (15) to (17), the influence of permeability parameter

(K), heat absorption parameter ( )sQ and thermal radiation

parameter (N) on the Nusselt number profiles ( , )RNu t are

demonstrated graphically. The Nusselt number is found

decreasing due to increase in values of (K) and (N)

respectively while, a reverse phenomenon is observed due to

presence of ( )sQ . Due to an increment in values of (K) and

(N), the thickness of the thermal boundary layer increases, this

restricts the heat transfer process to transmit plate region to

free stream region results of which declines the heat transfer

rate. This in turn decreases the value ofRNu . Again due to

thinning of thermal boundary layer affected by presence of

heat absorption parameter, the heat fluxes transmit

spontaneously towards the free stream region. This

transmission of heat transfer effects in increasing the values of

RNu .

Fig. 15:Nusselt number against time for arbitrary change in

values of Permeability parameter for Pr=7.0, ReL=0.05,

QS=0.01, N=3.6, Gr=10.0, U0=1.0, up=1.0, 𝜔=7.85714,

𝜔𝑡=1.57142857, K=0.5, 𝜖=0.001, Ec=0.01

0 5 10 15 20 25 300.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

normal distance(y)

Velo

city(u

R)

N=1.0

N=2.3

N=3.6

N=5.0

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.5

0.55

0.6

0.65

0.7

0.75

normal distance(y)

Velo

city(u

R)

Ec=0.0

Ec=0.2

Ec=0.5

Ec=1.0

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.5

0.52

0.54

0.56

0.58

0.6

0.62

0.64

normal distance(y)

Velo

city(u

R)

ReL=0.3

ReL=0.5

ReL=0.7

ReL=1.0

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.265

0.27

0.275

0.28

0.285

0.29

0.295

0.3

0.305

time

Nusselt n

um

ber

K=1.0

K=2.0

K=3.0

K=4.0





328

Fig. 16:Nusselt number against time for arbitrary change in

values of heat sink parameter for Pr=7.0, ReL=0.05, QS=0.1,

N=2.6, Gr=8.0, U0=0.5, up=0.5, 𝜔=7.85714, 𝜔𝑡=1.57142857,

K=0.5, 𝜖=0.001, Ec=0.001

Fig. 17:Nusselt number against time for arbitrary change in values of

Thermal radiation parameter for Pr=7.0, ReL=0.05, QS=0.1, N=3.6,

Gr=8.0, U0=0.5, up=0.5, 𝜔=7.85714, 𝜔𝑡=1.57142857, K=0.5,

𝜖=0.001, Ec=0.001

Figures 18 to 20 show how the skin-friction R against time t

is being affected due to influence of parameters like heat

absorption parameter ( ),SQ permeability parameter (K) and

thermal radiation parameter (N). It is clearly seen from these

figures that, the skin-friction increases due to increase in

parametric values of (K) and (N) but a reversed phenomenon

are being observed due to presence of ( )SQ . Due to an increase

in value of (K), though the skin-friction shows an increasing

trend, but it is clearly shows from figure 18 that, the skin-

frictional values attain a steady state due to presence of (K).

The increase of skin-friction values in presence of (K)and (N)

is due to the fact that, the flow rate in both the cases

accelerates, as a result the plate surface experiences a drag

force opposite to the motion. This force is responsible for the

enhancement of the skin-frictional values on the plate.

Fig. 18: Skin-friction against time for arbitrary change in values of

heat sink parameter for Pr=7.0, ReL=0.05, QS=0.01, N=2.6, Gr=5.0,

U0=0.5, up=0.5, 𝜔=7.85714, 𝜔𝑡=1.57142857, K=0.5, 𝜖=0.001, Ec=0.001

Fig. 19:Nusselt number against time for arbitrary change in values of

Permeability parameter for Pr=7.0, ReL=0.05, QS=0.1, N=2.3, Gr=8.0,

U0=0.5, up=0.5, 𝜔=7.85714, 𝜔𝑡=1.57142857, K=0.5, 𝜖=0.001, Ec=0.001

Fig. 20: Skin-friction against time for arbitrary change in values of Thermal radiation parameter for Pr=7.0, ReL=0.05, QS=0.01, N=3.2,

Gr=5.0, U0=1.0, up=1.0, 𝜔=7.85714, 𝜔𝑡=1.57142857, K=0.5,

𝜖=0.001, Ec=0.001

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

time

Nusselt n

um

ber

Qs=0.0

Qs=1.0

Qs=1.5

Qs=2.0

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

time

Nusselt n

um

ber

N=1.0

N=1.5

N=2.0

N=2.5

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 54

4.5

5

5.5

6

6.5

7

time

skin

friction

Qs=1.0

Qs=2.3

Qs=3.6

Qs=5.0

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 58

10

12

14

16

18

20

time

skin

friction

K=0.1

K=0.5

K=0.7

K=1.0

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 57

8

9

10

11

12

13

14

time

skin

friction

N=1.0

N=2.5

N=4.0

N=5.5





329

In Table 1, a comparison of the present work with the work of

Babuet al. (2011) is made to validate the effects of the Eckert

number on the Local Nusselt number. It is observed that in

both of these works, the Nusselt number decreases gradually

with the increase of Eckert number against the fixed time

t=1.0. The table shows a good agreement in the present work

with that of Babuet al. (2011).Table 2 shows numerically the

variations in values of skin-friction as well as the Nusselt

number for arbitrary change in values of local Reynolds

number against change in time. It is clearly shows that, the

Skin-friction and the Nusselt number decreases steadily by

increase in values of Reynolds number. It is also seen that,

due to passage of time, both the skin-frictional values and that

of the Nusselt number show a fluctuating trends.

Table 1: A comparison in the values of Nusselt number against

Eckert number at t=1.0 for Pr=7.1, ReL=1.0, QS=0.0, N=1.2,

Gr=2.0, U0=1.0, up=0.5, 𝝎=0.1, K=0.5, 𝝐=0.001, Ec=0.001

Ec Babuet al. (2011) Present paper

0.00

0.01

0.02

0.03

1.1376

0.8652

0.5429

0.2546

0.8317

0.8286

0.8255

0.8225

Table 2: Table displaying the variation in values of skin-friction and Nusselt number due to influence of local Reynolds

number against time

Time

( )RSkin friction

number(Nu )RNusselt

Re 0.5L Re 0.7L

Re 1.0L Re 1.5L

Re 0.5L Re 0.7L

Re 1.0L Re 1.5L

0.0000

0.5000 1.0000

1.5000

2.0000 2.5000

3.0000

3.5000 4.0000

4.5000

5.0000

1.3293

1.3244 1.3279

1.3278

1.3244 1.3293

1.3258

1.3258 1.3293

1.3244

1.3279

1.0055

1.0012 1.0043

1.0042

1.0013 1.0055

1.0025

1.0026 1.0055

1.0012

1.0043

0.7543

0.7507 0.7534

0.7532

0.7508 0.7544

0.7518

0.7519 0.7543

0.7507

0.7534

0.5696

0.5666 0.5688

0.5686

0.5666 0.5696

0.5674

0.5675 0.5696

0.5666

0.5688

0.4574

0.4597 0.4580

0.4582

0.4596 0.4574

0.4591

0.4589 0.4574

0.4597

0.4580

0.4505

0.4525 0.4510

0.4511

0.4525 0.4505

0.4519

0.4519 0.4505

0.4525

0.4510

0.4453

0.4471 0.4458

0.4458

0.4471 0.4453

0.4466

0.4466 0.4453

0.4471

0.4459

0.4413

0.4429 0.4419

0.4417

0.4430 0.4413

0.4424

0.4426 0.4413

0.4429

0.4419

Table 3: Table displaying the variation in values of skin-friction and Nusselt number due to influence of Eckert number

against time

Time ( )RSkin friction

number(Nu )RNusselt

0.0Ec 0.1Ec 0.2Ec 0.3Ec 0.0Ec 0.1Ec 0.2Ec 0.3Ec 0.0000

0.5000

1.0000 1.5000

2.0000

2.5000 3.000

3.5000

4.0000 4.5000

5.0000

12.0870

12.0595

12.0788 12.0790

12.0594

12.0869 12.0677

12.0673

12.0871 12.0596

12.0787

11.5166

11.5000

11.5197 11.5084

11.5046

11.5213 11.5016

11.5127

11.5167 11.4999

11.5196

10.9461

10.9404

10.9605 10.9378

10.9498

10.9556 10.9354

10.9581

10.9463 10.9403

10.9605

10.3757

10.3809

10.4013 10.3672

10.3949

10.3899 10.3693

10.4034

10.3759 10.3807

10.4015

0.6290

0.6348

0.6294 0.6312

0.6340

0.6282 0.6336

0.6318

0.6290 0.6348

0.6294

0.6207

0.6265

0.6211 0.6230

0.6258

0.6199 0.6254

0.6235

0.6207 0.6265

0.6211

0.6124

0.6183

0.6128 0.6147

0.6175

0.6116 0.6171

0.6153

0.6124 0.6183

0.6128

0.6042

0.6101

0.6045 0.6064

0.6093

0.6034 0.6089

0.6070

0.6041 0.6101

0.6046

In Table 3, the influence of Eckert number on the skin-friction

and on the Nusselt number is depicted for arbitrary change in

values of time. It is observed that, both the skin-friction and

the Nusselt number are also showing a decreasing trend due to

increase in Eckert number. As shown earlier, the skin-friction

and the Nusselt number both exhibit a fluctuating trend due to

increment of time variable in presence of Eckert number.

Conclusions

A mixed convective flow problem of Newtonian,

incompressible, viscous fluid past a porous plate through

Darcian porous media and under the influence of thermal

buoyancy forces is studied in presence of viscous dissipation

and thermal radiation. A multi - parameter perturbation

scheme is developed to solve the governed equations and the

influence of various physical parameters on the flow variables

are numerically simulated and interpreted through graphs and

tables. The significant outcomes of the investigation are as

follows:

The fluid temperature decreases due to increase in

values of Reynolds number, Prandtl number and heat

absorption parameters, while an increase in values of

Eckert number, Grashof number, permeability

parameter and thermal radiation parameter increases

the fluid temperature.

The increase in values of thermal Grashof number,

permeability parameter, thermal radiation parameter

and Eckert number increases the fluid velocity but a

reversed phenomenon has observed on the velocity

due to increase in parametric values of Prandtl

number, heat source parameter and Reynolds number.





330

The Nusselt number is found decreasing due to

increase in values of permeability parameter, thermal

radiation parameter, Reynolds number and Eckert

number, while the Nusselt number is found fluctuating

with time.

The Skin-friction on the plate increases as thermal

radiation and permeability parameters increase, but

the skin-friction is found decreasing due to increase in

heat sink parameter, Reynolds number and Eckert

number. The skin-friction is also found in fluctuating

mode against time.

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331

Appendix

2

1

Re Re 4 Re

2

r L r L r L SP P P Q Nm

N

,

2

2

4ReRe Re

2

LL L

km

, 2

1 1 1

ReRe L

Lb m mk

2

1

Rer LGb

b , 3 2p ob u U b ,

2

3

Re Re 4 Re

2

r L r L r L SP P P Q Nm

N

2 2

2 31 2

2 2

Re

4 2 Re Re

r L

r L r L s

P m ba

Nm m P P Q

,

2 2

1 22 2

1 1

Re

4 2 Re Re

r L

r L r L s

P m ba

Nm m P P Q

1 2 2 33 2

1 1

Re 2

Re Re

r L

r L r L s

P m m b ba

N P P Q

4 1 2 3 1 1 2( ), a a a a m m ,

2

4

4ReRe Re

2

LL L

km

, 44

2

1 1

Re

ReRe

r L

LL

G ab

m mk

15

2

2 2

Re

Re4 2 Re

r L

LL

G ab

m mK

26

2

1 1

Re

Re4 2 Re

r L

LL

G ab

m mk

, 37

2

1 1

Re

ReRe

r L

LL

G ab

k

, 8 4 5 6 7( )b b b b b

2

2 2 22 2

1

1 1 [ Re Re 4 Re Re 4 Re 16 Re ]

2 2r L r L S r L r L S r L r LP P Q NP P Q NP N P

N

2

2 2 22 2

1

1 1[ Re 4 Re 16 Re Re 4 Re ]

2 2r L S r L r L r L S r LP Q NP N P P Q NP

N ,

2

5 1 1 Re Rer L r L sa Nm m P P Q

1 56 2 2 2

5

Re

( Re )

r L

r L

m a Pa

a P

,

2

17 2 2 2

5

( Re )

( Re )

r L

r L

m Pa

a P

,

2

2 2 2 2

2

4Re 4Re1 1[Re Re Re 16 Re ]

2 2

L LL L L L

k k

2 2 2 2 2

2

1 1 4Re 4Re[ (Re ) 16 Re Re ]

2 2

L LL L L

K K

, 9 1 2 6Re ReL r Lb m b G a

10 6Re Rer L r Lb G G a , 2

11 2 2

ReRe L

Lb m mk

, 2 3 11

12 2 2 2

11

Re

Re

L

L

m b bb

b

,

2

3 213 2 2 2

11

Re

Re

L

L

b mb

b

2

14 1 1

ReRe L

Lb m mk

,

2

9 14 715 2 2 2 2 2 2

14 14

Re( )

Re Re

r L

L L

b b G ab

b b

,

9 7 1416 2 2 2 2 2 2

14 14

Re Re( )

Re Re

L r L

L L

b G a bb

b b

2 2

17 1 1 1

ReRe L

Lbk

, 18 1 1 12 Re ReL Lb ,10 17 7 18

19 2 2 2 2

17 18 17 18

Re( )r Lb b G a b

bb b b b

10 18 7 1720 2 2 2 2

17 18 17 18

Rer Lb b G a bb

b b b b

,

2

2 2 22 2

3

1 1[ Re Re 4 Re Re 4 Re 16 Re ]

2 2r L r L S r L r L S r L r LP P Q NP P Q NP N P

N

2

2 22 2 2

3

1[ Re 4 Re 16 ( Re ) Re 4 Re ]

2r L S r L r L r L S r LP Q NP N P P Q NP

N

21 12 15 19( )b b b b ,22 13 16 20( )b b b b ,

23 2 2 21 2 22ob U b b

24 2 2 22 2 21ob U b b , 25 1 19 1 20b b b , 26 1 20 1 19b b b

27 1 2 3 15 1 2 2 12b m m b b m m b b , 28 1 2 3 16 1 2 2 13b m m b b m m b b

2 1 1m , 3 1 2m ,

4 2 1m ,5 2 2m

6 2 1i ,7 4 1i ,

8 5 2i ,9 3 2i

2

29 2 1 2 3 122 P Re 2 Rer L r Lb m a P m b b , 2

30 2 3 132 Rer Lb P m b b , 2

31 1 2 1 2 152 Re 2 Rer L r Lb m a P P m b b

2

32 1 2 162 Rer Lb P m b b ,

33 1 3 27Re 2 Rer L r Lb a P P b , 34 282 Rer Lb P b ,, 2

8 1 1 Re Rer L S r La Nm m P Q P





332

1 4 89 2 2 2

8

Re

( Re )

r L

r L

P m a aa

a P

,

2

1 410 22 2

8

( Re )

Re

r L

r L

P m aa

a P

, 2

11 2 24 2 Re Rer L S r La N m m P Q P

3011 29

12 2 22 2 2 2

11 11

Re

Re Re

r L

r L r L

P ba ba

a P a P

,

29 30 11

13 2 22 2 2 2

11 11

Re( )

Re Re

r L

r L r L

P b b aa

a P a P

2

14 1 14 2 Re Rer L S r La N m m P Q P ,

3214 31

15 2 22 2 2 2

14 14

Re( )

Re Re

r L

r L r L

P ba ba

a P a P

31 32 14

16 2 22 2 2 2

14 14

Re( )

Re Re

r L

r L r L

P b b aa

a P a P

, 2

17 1 1 Re Rer L r L Sa N P P Q

3417 33

18 2 22 2 2 2

17 17

Re(

Re Re

r L

r L r L

P ba ba

a P a P

,

34 17

19 2 22 2 2 2

17 17

33 Re[ ]

Re Re

r L

r L r L

b ab Pa

a P a P

2 2

20 1 1 1 1 1{( ) } Re ( ) Rer L r La N m P m P , 21 1 1 1 12 ( ) Re Rer L r La N m P P

1 2 25 20 1 2 26 2122 2 2 2 2

20 21 20 21

2 Re 2 Rer L r LP m b b a P m b b aa

a a a a

,

1 2 25 21 1 2 26 2023 2 2 2 2

20 21 20 21


a a a a

2 2

24 1 2 2 1 2{( ) } Re ( ) Rer L r L sa N m P m P Q , 25 2 1 2 22 ( ) Re Rer L r La N m P P

1 2 23 24 1 2 24 2526 2 2 2 2

24 25 24 25

2 Re 2 ReL r LP m b b a P m b b aa

a a a a

,

1 2 23 25 1 2 24 2427 2 2 2 2

24 25 24 25

2 Re 2 Rer L r LP m b b a P m b b a

aa a a a

2 2

28 2 1 1 2 1{( ) } Re ( ) Rer L r L sa N m P m P Q , 29 1 2 1 12 ( ) Re Rer L r La N m P P

2 3 25 28 2 3 26 2930 2 2 2 2

28 29 28 29


a a a a

,

2 3 25 29 2 3 26 2831 2 2 2 2

28 29 28 29


a a a a

2 2

32 2 2 2 2 2{( ) } ( ) Re Rer L r L sa N m m P P Q ,

2 2

33 2 2 2 2 2{( ) } Re ( ) Rer L r L sa N m P m P Q

2 3 23 32 2 3 24 3334 2 2 2 2

32 33 32 33


a a a a

,

2 3 23 33 2 3 24 3235 2 2 2 2

32 33 32 33


a a a a

36 9 12 15 18 22 26 30 34( )a a a a a a a a a , 37 10 13 16 19 23 27 31 35( )a a a a a a a a a

2

2 2 2 2

4

4Re 4Re1 1[Re Re Re 16 Re ]

2 2

L LL L L L

k k

2

2 2 2 2

4

4Re 4Re1 1[ Re 16 Re Re ]

2 2

L LL L L

k k

, 38 2 5 12Re 2 ReL La m b Gr a

39 13ReLa Gr a , 40 1 6 15Re 2 ReL La mb Gr a

, 41 16ReLa Gr a , 42 1 7 18Re ReL La b Gr a

43 19ReLa Gr a , 2

35 2 2

ReRe L

Lb m mk

,

2 8 3536 2 2 2

35

Re

Re

L

L

m b bb

b

,

2

2 837 2 2 2

35

Re

Re

L

L

m bb

b

2

38 1 1

ReRe L

Lb m mk

,

1 4 3839 2 2 2

38

Re

Re

L

L

m b bb

b

,

2

1 440 2 2 2

38

Re

Re

L

L

m bb

b

,

2

41 2 2

Re4 2 Re L

Lb m mk

38 41 3942 1 2 2 1 2 2

41 41

Re

Re Re

L

L L

a b ab

b b

,

38 39 4143 2 2 2 2 2 2

41 41

Re

Re Re

L

L L

a a bb

b b

2

44 1 1

Re4 2 Re L

Lb m mK

,





333

40 44 4145 2 2 2 2 2 2

44 44

Re

Re Re

L

L L

a b ab

b b

,

40 41 4446 2 2 2 2 2 2

44 44

Re

Re Re

La a bb

b L b L

2

47 1 1

ReRe L

LbK

,

42 47 4348 2 2 2 2 2 2

47 47

Re

Re Re

L

L L

a b ab

b b

,

43 474249 2 2 2 2 2 2

47 47

Re

Re Re

L

L L

a bab

b b

2 2

50 1 1 1 1 1

Re{( ) } Re ( ) L

Lb m mk

, 51 1 1 1 12 ( ) Re ReL Lb m

22 50 23 5152 2 2 2 2

50 51 50 51

a b a bb

b b b b

,

22 51 23 5053 2 2 2 2

50 51 50 51

a b a bb

b b b b

,

2 2

54 1 2 2 1 2

Re{( ) } Re ( ) L

Lb m mk

55 2 1 2 22 ( ) Re ReL Lb m ,

26 54 27 5556 2 2 2 2

54 55 54 55

a b a bb

b b b b

,

26 55 27 5457 2 2 2 2

54 55 54 55

a b a bb

b b b b

2 2

58 2 1 1 2 1

Re{( ) } Re ( ) L

Lb m mk

, 59 1 2 1 12 ( ) Re ReL Lb m

30 58 31 5960 2 2 2 2

58 59 58 59

a b a bb

b b b b

,

30 59 31 5861 2 2 2 2

58 59 58 59

a b a bb

b b b b

,

2 2

62 2 2 2 2 2

Re{( ) } Re ( ) L

Lb m mk

63 2 2 2 22 ( ) Re ReL Lb m ,

34 62 35 6364 2 2 2 2

62 63 62 63

a b a bb

b b b b

,

34 63 35 6265 2 2 2 2

62 63 62 63

a b a bb

b b b b

2 2

66 3 3 3

ReRe L

Lbk

, 67 3 3 32 Re ReL Lb

,

36 66 37 6768 2 2 2 2

66 67 66 67

a b a bb

b b b b

36 67 37 6669 2 2 2 2

66 67 66 67

a b a bb

b b b b

, 70 36 39 42 45 48 52 56 60 64 68( )b b b b b b b b b b b

71 37 40 43 46 49 53 57 61 65 69( )b b b b b b b b b b b .


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MIXED CONVECTION FLOW IN HEAT ABSORBING DARCIAN MEDIA WITH VISCOUS DISSIPATION AND THERMAL RADIATION...

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