Free convective flow through a porous medium bounded by an infinite vertical plate with oscillating temperature

MECHANICS RESEARCH OCPMJNICATIONS Voi.15(2), 131-137, 1988. Printed in the USA. 0093-6413/88 $3.00 + .00 Cc~5~ight (c) 1988 Pergamon Press plc

FREE CONVECTIVE FLOW THROUGH A POROUS MEDIUM BOUNDED BY AN INFINITE VERTICAL PLATE WITH OSCILLATING TEMPERATURE

F. Khan and K.A. Narayan Department of Chemical Engineering, Indian Institute of Technology, Kanpur - 208016, India.

(

(Received 29 October 1987; accepted for print 9 March 1988)

Introduction

Unsteady two-dimensional free convective flow through a porous medium bounded by an infinite vertical plate is considered when the temperature of the plate is oscillating with time about a constant nonzero mean value. The governing equations are solved by a finite difference numerical scheme for all values of the frequency parameter, ~ , with constant as well as variable permeability parameter K and the constant amplitude parameter e. The effects of these parameters on resulting velocity and temperature J~elds are discussed. In recent years considerable interest has been shown in unsteady flow through a porous medium. Numerous studies have dealt with the systems heated ~om below. Some attention has also been given to investigations of free convection caused by a temperature gradient normal to the gravitational field. Combarnous and Bories i l l and Cheng [2] have provided extensive reviews of most studies concerning the free convection in porous media. Recently Raptis [3] has also investigated unsteady free convection flow through a porous medium. His analysis, however, is valid only for small values of frequency parameter and is based on the constant values o f permeability throughout the entire flow domains-. This assumption is not in accordance with experimental observations. The inhomogeneities which can be present in a porous medium are widespread and well known. For example, in oil recovery from underground reservoirs, drilling fluid may cause well damage and the permeability around the wellbore is a function of reservoir distance and thickness. Hever et al. [4] reported that the permeability was maximum near the well due to damage and decreased as one moved away from the wall. Among the works on convection in porous media with continuous permeability variations we mention Gheorghitza [5], Ribando and Torrance [6]. They considered linear variation in permeability with distance whereas Gjerde and Tyvand [7] analysed the stability in a horizontal porous layer heated from below by taking the periodic permeability variation. In this communication, results are presented of a finite difference numerical solution to the pertinent equations covering the entire range of values of frequency parameter. Over the limited region, the results are compared with those of Raptis [3]. The paper is concluded by discussing the implications of permeability variation on the flow and temperature fields. 131

132

F. KHAN and K.A. NARAYAN

G o v e r n i n g Equations and M e t h o d o f S o l u t i o n

The

equations

for

unsteady

flow

two-dimensional

of

a

viscous

fluid

through

a

porous m e d i u m bounded by an i n f i n i t e v e r t i c a l porous p l a t e are

(~V

=

~y

Here u time,

6u ~t'

+

v

6T ~t'

+

v

and v T the

are

the

6u 6y

0

-

~T 6y

g6 (T-T

- ~

velocities

temperature,

g

(J)

the

)

62u

+y

T u k

6y2

(2)

~2T 2 6y

(3)

in x and

y

acceleration

directions due

to

respectively,

g r a v i t y , B the

due to v o l u m e expansion, K the p e r m e a b i l i t y o f the porous m e d i u m ,

t'

is the

coefficient

c~ the t h e r m a l

di f f u s i v i t y and y is the k i n e m a t i c viscosity.

F o r a c o n s t a n t suction, eqn. (1) gives on i n t e g r a t i o n

V

=

-

(4)

V 0

where v

o the p l a t e .

is a c o n s t a n t and the n e g a t i v e sign i n d i c a t e s t h a t the suction is t o w a r d s

The boundary c o n d i t i o n s are

u :

O,

T : T w + e ( T w - T oo) e i

u + 0 , T-~ T

where

¢

as y *

is a c o n s t a n t (0

w't'

at y = 0,

(5)

oo ,

< e < I),

T

W

is the

mean t e m p e r a t u r e

of

the

plate

OONVECrIVE FLOW IN PORf~dS MEDIUM

and Too is the temperature

the

of

fluid away

from

133

the plate, Introducing the

dimensionless variables, t , r l , to) U and 0 defined by

t = to' t')

U

rl

=

VoY

-

'

Y

V

u

Vo(l

0

eiti

+e

to'y

to =

2

'

O

T-

T

T®)

(I

=

oo

'

(r w-

+ c e it)

equations (2) and (3) become

6U 6q

+

2

I K

U

+ G0

6u

-to(-TT

+

U )=o

6n

where

F = |

(6)

!

~ 20

#

2

an

+ Ee

it

a9

+

=

an

'YgB (T w ,

G

= V

T,x))

3 O

kv p

= .IF_

and

K

(7)

0

2

0 2,,

-

The corresponding boundary conditions become

q

=

O,

U

q

÷

oo,

U "* 0 ,

To incorporate

the

=

spatial

O) 0 = 0

1)

(8)

"* 0

variation

of

permeability,

the

following

functional

form was used. -C2q K where Ko) C ]

:

K°

(I

+ CI e

)

(9)

and C 2 are constants, and have been determined using the data

o f Hever et al. [4] as t

K°

--

3.5 ;

C I =

0.286;

C2

--

0.35

134

F. KHAN and K.A. NARAYAN

It is seen that Furthermore,

equations (6) and (7) are coupled partial

the

d i f f e r e n t i a l equation~.

inclusion o f the spatial variation o f permeability adds to the

c o m p l e x i t y o f the equations. Numerical solutions using i m p l i c i t

finite difference

schemes were sought, and no d i f f i c u l t y

was

regarding the s t a b i l i t y

experienced

throughout the whole range o f conditions entailed in this study.

Results and Discussion

The v e l o c i t y and temperature

fields in the porous medium have been computed

as functions o f distance for a range o f the relevant parameters, namely~ frequency parameters, permeability o f the porous medium (Ko, C I and C 2) and amplitude parameter ¢. All results relate, however, to the constant values o f Grashof number (G = 4), Prandtl number (P = 0.71) and for a single value o f dimensionless time t = 3 ~ /z~; though the numerical scheme is found to yield useful results for any other values of these parameters.

In the

range o f low

in Figures

l

parameter (to = 0.3), typical

results are shown

3 where

velocity and temperature are plotted as functions of

the spatial coordinate ( r l )

for a combination o f to, ¢ and K values as indicated

on the each

to

frequency

figure.

study o f Raptis [3]

Included in each for

low

figures are the results o f the analytical

frequency range.

Good agreement

is seen to exist

between, the present results and those o f Raptis [3]. Further examination o f these figures reveals that the value o f maximum velocity decreases with both decreasing permeability as well as the amplitude o f oscillation. However, the decrease in v e l o c i t y is much more sensitive to the drop in permeability than that in the amplitude

of

oscilllations.

On

the

otherhand,

the

temperature-distance

not appreciably influenced by the variation Jn ¢ (see Figure the v e l o c i t y profiles

for low

behaviour

is

3). Figure ~ shows

( c0 -- 0.6), intermediate ( to = 3) and large ( m

10)

values o f the frequency parameter and for d i f f e r e n t values o f permeability w h e r e as the amplitude o f oscillation was kept constant at ¢ = 0.2. It is seen that in the case o f to= 0.67 the v e l o c i t y reaches a maximum value o f about is less than that

observed

l.g which

D r to = 0.3 with everything else being constant.

As

the frequency o f oscillation increases (tom 3 and 10), the maximum value o f velocity becomes

progressively smaller, and the e f f e c t o f permeability also vanishes for

a given value o f

to. In all the above cases the temperature profiles (not shown

here) have the same trend as that seen in Figure 1. However, the drop in temperature is quite steep as to increases.

OONVECTIVE FLOW IN POROUS MEDILM

135

Until now the discussion has been restricted to the case where the permeability is assumed to remain at a constant value. As remarked at the outset, this assumption is not in accordance with the experimental results of Hever et al. [4], and hence it is desirable to relax this condition. This however) precludes the possibility of even an approximate analytical solution whereas the numerical approach such as one adopted in this work poses no difficulty. Typical results of velocity profiles for two

different

in Figure more

values of

6. Intuitively

frequency parameters (co = 0.5 and 6) are plotted

one would expect the variations in permeability

to be

prominent near the plate, and this is also borne out by the results shown

in Figure 6. For example, comparing the values of velocity (Figure

I and 6),

for co = 0.3, K ° = 3.5 at q = 0.5, U = 1.5 whereas when the permeability varlation is taken into account the corresponding value is ].g. But for large values of q _> 3) no difference between the behaviours shown in Figures I and 6 is observed. Thus the present study can successfully cope with the spatial variation o f permeability. Conclusion The

free convective

flow in a porous medium bounded by an infinite

plate with oscillating mean temperature has been numerically results are in agreement with

Raptis [3] in the low

vertical

investivated. The

frecluency region. Unlike

the earlier study [3] the present study is applicable over the complete of

range

frequency of oscillations. Furthermore, the influence of the spatial variation

o f permeability has also been examined and it results in higher values of velocity than that expected when a constant value is assigned to permeability.

Re ferences

I.

M.A. Combarnous and S.A. Bories, Adv. Hydroscience, I0, 232 (1975).

2.

P. Cheng) Adv. Heat Transfer, 14, I (1978).

3.

A.A. Raptis, Int. 3. Engng. Sci., 21(4), 345 (1983).

4.

G.3. Hever, 3r.) G.C. Clark and 3.N. Dew) Trans. A.I.M.E.) 222, 469 (1961).

5.

St. I. Gheorghitza, Proc. Camb. Phil. Soc. 57, 871 (1961).

6.

R. Ribando and K.E. Torrance) Trans. A.S.M.E, Series C, 3. Heat Transfer,

98, ~2 (1976). 7.

K.M. Gjerde and P.A. Tyvand, Int. 3. Heat Mass Transfer, 27, 2289 (1984).

136

F. FJ4AN and K.A. NTLRAYAN

| 25~-

----

!

this work Reference 3

20~-- ~

!

~ll/z~\

>~

,.~= 03

h'~--.

1sb-t,'/,I

",\

~ :o2

k\

/-K=35

" \ \ ',~

/F .z//_

:2o :o.5

.?5'

0

20

40

6.0

8.0 ,

10C

\

/

...... This woix .....

\

Fiq 1 - Velocity vS distance for d}fferent values of

Reference 3

'\

permeability at low frequency

b

!

K :(35 ~ - - E =0.8 .-=~f15 :02

( 75F

\ 050

\ \,

0.2~

\\ \

--

This work

- - -

Reference 3

\ \ \. ::,, X -.<-

0 L ¢

Fig

_

~

J 2.0

L 40

...±. _

_ L 6.0

(

(

0

20

40

6-0

8.0

q Fig 3 - T~rnperalure vs distGnce for different volueS Of Qmplitude rolio at low frequency

~ 8.0

2 -Velocity vs dit, tonce for differer~t vo!ues of amplitude ratio at low frequency

~=03 .6 =0.8 =0.5

_

CONVECFIVE FLOW IN POROUS MEDIUM

~N'~K~.~K/~.

~ =0.6

P

£ =0.2

~

137

_; 1.0

0.5 0

1.O

20

3-O

1.0

Tt

4,0

K= 3,5 2.0

?

5.0

60

~: 3.O E =0.2

1.0i

~=10.0 £ =02

O.?5

i

0

1.O

~ 0,4

2-0

K= - - 3"520 ~ 1 1-0 2.0

0 O

Fig

4 -

3.0

tt

I

3.0

4,0

I

4.0

i

5.0 ~E

Velocity vs. distance at low,intermediate and high frequency.

0.50

0-25

0

0

0-2

O.&

0.6

0.8

1.0

1.2

1.4

Fig 5 -Temperature vs distance tar different voiues of frequency 2.4

2.01.6 ~

~

=0.30

1.2 /

~

K=3.5. e_0.35~ E=0.20

0.8 0.4

6-0

Oo

1.0

2,0

3.0

4.0

5.0

6.0

7.O

8,0

Fig. 6 -Effect of spatial variation in permeability on velocity tield for different values of frequency