ACTA

Aeta Meehanica 35, 223--229 (1980)

_MECHANICA

| by Springer-Verlag 1980

MHD Flow Through a Porous Straight Channel By D. Bathaiah, Tirupati, India With 4 Figures

(Received April 3, 1978, revised October 4, 1978) Summary -

Zusammenfassung

MHD Flow Through a Porous Straight Channel. The unsteady two-dimensional incompressible viscous flow through a straight channel with porous flat walls distant h apart in the presence of a uniform transverse magnetic field is studied. The velocity distribution both in the unsteady and steady cases is obtained. The coefficients of skin friction at both the walls which are subjected to injection and suction are evaluated and the effect of magnetic field on velocity distribution is investigated. MHD-Striimung dureh einen porSsen~ geraden Kanal. Es wird die unstetige, zweidimensionale, inkompressible, z~he Str6mung durch einen geraden Kanal mit por6sen, ebenen W~nden im Abstand hunter Einwirkung eines homogenen ~agnetfeldes quer zur Str6mungsrichtung nntersucht. Die LSsung liefert die Geschwindigkeitsverteilung sowohl fiir den station~iren, als auch fiir den instation~iren Fall. Die Koeffizienten der geibung an den beiden Wgnden, an denen eingespritzt and abgesaugt wird, werden berechnet und der EinfluB des Magnetfelds auf die Geschwindigkeitsverteilung untersucht. 1. Introduction Fluid flow through porous media is of fundamental importance to a wide range of disciplines in various branches of natural science and technology. Civil engineers, Mining engineers, Pet~roleum engineers and hydrogeologists are interested in seepage problems in rock mass, sand beds and subterranean acquifers. Civil and agricultural engineers are interested in the same phenomenon for efficient layout of drainage system for irrigation and recovery of swampy area. The nuclear engineer is interested in fluid flow through reactors to maintain a uniform temperature throughout the bed. The textile technologist is interested in fluid flow through fibres. Biologists are interested in water movement through plant roots and through and out of the cells of living systems. Verma and Mathur [5] have studied magnetohydrodynamic flow between two parallel plates, one in uniform motion and the other at rest with uniform suction at the stationary plate. They have observed that the coefficient of skin friction decreases with the increase of Hartmann number. Satyaprakash [3] has obtained the exact solution of the problem of unsteady viscous flow through a porous straight channel. He has obtained the result that the velocity increases with time and tends ultimately towards the steady state at both the points, as should have been the ease in the presence of a pressure gradient which remains constant for all times.

0001-5970/80/0035/0223/$01.40

224

D. Bathaiah:

I n vhis paper the flow of a viscous incompressible slightly conducting fluid through a porous straight channel u n d e r a uniform transverse magnetic field is considered. The pressure gradient is t a k e n as constant q u a n t i t y as a special case. U n d e r t.he constant pressure gradient the case of s t e a d y flow is obtained b y taking the time since the start of the motion to be infinite. 2. F o r m u l a t i o n and Solution of the P r o b l e m

Unsteady two-dimensional incompressible viscous flow t h r o u g h a straight channel with porous flat walls distant h apart in the presence of a uniform transverse magnetic field is considered. The lower plate is t a k e n as x-axis and a straight line perpendicular to t h a t as y-axis. I t is assumed t h a t the fluid injected into the channel t h r o u g h the wall at y = 0 and sucked t h r o u g h the wall at y ---- h. L e t u and ~, be the velocity components of the fluid at a point (x, y) in the direction of axes of coordinates respectively. I t is assumed t h a t the fluid is of smM1 electrical conductivity with magnetic R e y n o l d s n u m b e r m u c h less t h a n u n i t y so t h a t the induced magnetic field can be neglected in comparison with the applied magnetic field (Sparrow and Cess [4], BathMah et al. [1]). The equations of motion of an incompressible, viscous slightly conducting fluid, in the absence of input electric field, are .

.

.

.

.

.

~v ~_ u - - + v -~t ~x 8y

~o /~y

. -~ v

7~ -~ ~ x ~" 8y ~ ]

~u 32v = 0 0x @ ey

(2.1) (2.2) (2.3)

where o is the density of the fluid, ~, the coefficient of kinematic viscosity, t the time measured since the start of the motion, p the pressure at a point (x, y), a the electrical conductivity, /~e the magnetic permeability and H 0 the uniform applied magnetic field. The initial a n d b o u n d a r y conditions are: when t ~ 0, u = 0 and v = 0 for 0 =< y=< h (2.4) when t > 0, u = 0 and v ---- vo = constant > 0 for y ---- 0, h. F r o m the initial and b o u n d a r y conditions (2.4) we m a y say t h a t the velocity distribution is independent of x. Hence Ou ~v -- = 0 and - - = O. (2.5) Ox ax On substituting ~

= 0 and using (2.4) the Eq. (2.3) yields v - v0.

Substituting v = vo in Eqs. (2.1) and (2.2) we obtain - 8u -

3t

~_

Vo

~y

o

~x

O~u

~t~tt~

~y~

~o

(2.6)

MHD Flow Through a Porous Straigh~ Channel and

0-

@ ~y

225 (2.7)

Let v0 be the characteristic velocity, h the characteristic length, ~v02 the characteristic pressure and h/vo the characteristic time. We define the dimensionless qunatities u', x', y', p ' and t' as follows: U~

u

X~

x

vo

y~

h '

p'--

P

(2.8)

t

t'=

~Vo2 ~

y

h (h/Vo)

In view of Eq. (2.8), Eqs. (2.5) to (2.7) reduce to Ox"

and

Ou'

~u' __

= 0

(2.9)

3/3' 4- 1 32u'

~t-~- 4- Oy'

O~

R @,2

]cu'

0 = @' Oy' where

k

--

al~e2H~

--

(2.10) (2.11)

Magnetic p a r a m e t e r

~OVo

R - - v~ - - Reynolds n u m b e r .

The initial and boundary conditions (2.4) in view of (2.8) lead to when

t' ~ O,

u'=0

for

O <= y' ~ 1

when

t'>0,

u'=0

for

y'=-0,1.

(2.12)

I t is observed that u' is independent of x' from Eq. (2.9). Hence u' is a function of y' and t' only. From Eq. (2.11) we m a y say that p' is independent of y'. Therefore it follows from the Eq. (2.10) that @' is a function of t' only. We

assume

that Op' _

(2.13)

/(t') .

Then Eq. (2.10) becomes 0u'

1 ~2u'

~t--= + ~

=/(t') + ~ or

ku'.

(2.14)

We define Laplace Transform as oo -~' = f u ' e -~t"

dt'

(2.15)

with the inversion ~+icr

u' = - - 1 ~ ~' e ~t" d2 2,hi .~! y-- i~z

(2.~6)

226

D. Bathaiah:

where R,(2):> 0 and y is greater t h a n the real p a r t of all the singularities of ~'. W e denote o~

]()~) = f /(t') e -;r dt'.

(2.17)

o

In view of Eqs. (2.15) and (2.17) the Eq. (2.14) is t r a n s f o r m e d into d ~ 7' _ 1~ dg' _ R(t~" q- ;t) ~' --~ - - R [ ( ~ ) .

dy"2

(2.18)

dy'

The t r a n s f o r m e d b o u n d a r y conditions are ~' = 0

for

y'--~ 0,1.

(2.t9)

I n view of the conditions (2.19), the solution of Eq. (2.18) is ](;9

e -(1-y}Y Sin h I/R2

(lc+)OSinh( -~+'4R(1~''+~)2 )

2

Y'

(2.20) + e-?-- Sin h

R~ + 4R(/c + ,~) (1 -- y') 2

4- (k + Z) '

"

A Special Case L e t us assume t h a t the pressure gradient is a constant quantity. H e n c e let --

-fit') = - - P

(2.21)

where P is a positive constant. p [()~) = f Pe -xt" dt' = -Z"

(2.22)

o

B y taking the inversion of Eq. (2.20) (Carslaw and J a e g e r [2]) we obtain velocity distribution U' =

P

e - ( 1 - y ' 1 2 S i n ]~ ~1

{(R 2 + 4Rk)l/2y'}

K Sin h --~ (R e + 4Rlc)l/~ Ry__~'

]

1

+ e ~ S i n h ~- {(R 2 + 4Rk) 1/2 (l -- y ' ) j +

Pe-~t I e + --k--

[

-(~-v')~

l

--

]

s i . t~ • 2 e"~- Sin ( ~ y ' ) { e - R ~ 2 ( - - 1 ) n - - 1} exp

+ 32PR~ ,=o .~'

1

Sinh~-(Ry') 4- e 2 Sinh ~- {R(1 -- y')}

- - ~ - ~ (R e + 4~2,~e + 4 R K )

(R ~ + 4n2Jr~ + 4Rk) (R 2 + 4~2~ ~)

P

(2.23)

MHD Flow Through

a

Porous Stra.ight Channel

227

Steady State The flow becomes s t e a d y after a lapse of infinite time since the start of the motion. U n d e r a constant pressur e gradient - - P , the s t e a d y flow m a y be deduced from t h e Eq. (2.23). I n the case of s t e a d y state the velocity distribution is i _P e -(1-y )7 Sin h 1 {(R2 + 4Rk)l/2 y,} /c Sin h 1 (R 2 -k 4RIr 1/2 2

u' = --

Ry'

~- e- y Sin h 1 {(R2 _}_ 4Rk)~/2 (1 - - y')}

]

(2.24)

p

-}- ~--.

N o w let us find the s t e a d y state solution directlY from the equation of motion. After substituting P for ](t'), for s t e a d y state the Eq. (2.14) becomes

d2u---~' -- R du' _ k R u ' = - - P R . dy "~ dy'

(2.25)

The b o u n d a r y conditions are same as those given in (2.19). I n view of the b o u n d a r y conditions (2.19), the solution of Eq. (2.25) is P

[ -(~-u'~= R 1 e ~ Sin h - - {(R 2 + 4Rk) 1/2 y'} ]~Sin h -~ 1 (R ~ -k 4Rio) a12 [ 2

u' =

(2.26)

-}- e-K~u'Sin h 12 {(R2 q- 4Rk)a/2 (1 -- y')} -k --'K The results (2.24) and (2.26) are identical.

S k i n Friction The shearing stress at the wall y' = 0 is

a n d the coefficient of skin friction is given b y 270

_

o~vo:

2

.

(2.27)

R \ Oy' ]y'=o

The coefficient of skin friction at the wall y' = 0 is

CI = --

[ ( R 2 + 4~R k ) l / 2 { e _ . / 2 _ ~ _ C o s h 1 P k Sin h - - (R ~ -k 4Rk) ~]2 2 2+4Rk)1/2

~-Sinhl(R

+

hl k Sin

R 2

oo

Acta

Mech.

e R/~_ coshlR-kSinh 2

!]

~{e-~/2(--1) n -- 1} exp --~-~ (2~~-+ 4~2~~ + 4Rio)

n=O

15

1 (R2 -~- 4Rk)l/e } -~-

35/3--4

/~ (2.28)

228

D. Bathaiah:

The coefficient of skin friction at the wall y' : I is C/ :

R

1 (R~ + 4R~)~/~ k sin h ~I

q-Sinh 2

~_

( R 2 + 4Rk)1/2

e~/2 + cos h

1 kSinh--R 2

2

+ s i n h ~ R - e~/2 2 (2.29)

~ n2( 1)n en/~{e m~(--1) ~ -- l} exp --~-~ (~'-' + 4n2~~ -F 4/~k) + 64P:~ 2 ~ n=o (R 2 + 4n2~ 2 4- 4Rk) (R 2 4- 4n2~~) 3. Conclusions W e obtain the velocity distribution in u n s t e a d y and s t e a d y cases. W h e n the magnetic p a r a m e t e r K tends to zero, our results coincide with those of Satyaprakash [3]. W e plot a graph taking magnetic p a r a m e t e r K against velocity distribution u' (Fig. 1). I t is observed t h a t the velocity increases with the increase of magnetic parameter. W e draw another graph (Fig. 2) taking velocity distribution against time. W e find t h a t the velocity increases with time and tends ultimately towards the s t e a d y state at b o t h the points, as should have been the case in the presence of a pressure gradient which remains constant for all times. I t is also observed t h a t the s t e a d y state is obtained earlier t h a n in the nonmagnetic c a s e . From Fig. 2 we conclude that the velocity at the point near the wall which is subjected to injection is less than that at the point near the wall which is subjected to suction. Verma and Mathur [5] have observed that the coefficient of skin friction decreases as the magnetic field strength increases at the stationary wall which is subjected to suction. But we observe from the Figs. 3 and 4 that the coefficients of skin friction increase as the magnetic parameter 0'08

0.05

O-O&

0"065 I

i u ~ 0.02 y' =0.9 LIJ 0

O' 0~-5

-0,02

9,035

2

3

&

K

Fig. I. 31a.gnetic loa.rameger K vs. velocity u"

f f

yf =0.1

0!G

1.&

~cI _

21,2

Fig. 2. Time t' vs. velocity ~l"

3~-0

MHD Flow Through a Porous Straight Channel

229

-2

C~

Cf._~

f

,

I

1

-6

-8 I 2

{ 3

l 4-

I 5

-8

K

I I

l 2

I 3

( 4-

I 5

K-

Fig. 3. Magnetic parameter K vs. Coefficient of skin friction C/' at the wall y' = i

Fig. 4. Magnetic parameter K vs. Coefficient of skin friction Cf at the wall y' = 0

increases b o t h a t t h e walls which are s u b j e c t e d to injection a n d suction. F r o m Figs. 3 a n d 4 it is also observed t h a t t h e coefficient of skin friction a t t h e p o i n t n e a r t h e wall which is s u b j e c t e d to suction is less t h a n t h a t a t t h e p o i n t n e a r t h e wall which is s u b j e c t e d to injection. I t h a n k Prof. K. S i t a r a m for his c o n s t a n t e n c o u r a g e m e n t . References

[1] Bathaiah, D., Krishna, D. V., Scetharamaswamy, ~.: Stratified rotating viscous flow between two disks under transverse magnetic field. Proc. Indian Acad. Sci. 82A, 17--30 (1975). [2] Carslaw, H. S., Jaeger, J. C. : Operational methods in applied mathematics. New York: Dover Publications, Inc. 1963. [3] Satyaprakash: An exact solution of the problem of unsteady viscous flow through a porous straight channel. Proc. Natn. Inst. Sci. (India) 35A, 123 (1969). [4] Sparrow, E. M., Cess, R. D.: Magnetohydrodynamic flow and heat transfer about a rotating disc. Trans. ASME, J. Appl. Mech. 29, t81--187 (1962). [5] Verma, P. D., Mathur, A. K. : Magnetohydrodynamic flow between two parallel plates, one in uniform motion and the other at rest, with uniform suction at the stationary plate. Proc. Natn. Inst. Sci. (India) 35A, 507--517 (1969). D. Bathaiah Sri Venkateswara University Tirupati - 517502 (A.P.) India

15"