INT. CGMM. HEAT MASS TRANSFER 0735-1933/84 $3.00 + .00 Vol. Ii, pp. 399-406 g~ergamon Press Ltd. Printed in the United States

HYDROMAGNETIC FREE-CON'gECTION FLOW PAST AN IMPULSIVELY STARTED

VERTICAL PLATE IN A ROTATING FLUID

A. ~. Singh Banaras Hindu University

Department of Mathematics Varanasi-221005 India

(~icated by D.B. Spalding)

ABSTRACT The three dimensional flow of a viscous incompressible and electrically conducting flui8 past an infinite vertical plate started impulsively in its own plane is discussed in a rotating system. The magnetic Reynolds number is taken small so that the induced magnetic field can be neglected. An exact solution is obtained for the axial and the transverse components of the velocity and the skin-friction by defining a complex velocity. It is observed that both velocity components of water (primary and secondary) decrease with rotation parameter in both cases, cooling and heating of the plate.

Introduction

The flow of an incompressible viscous fluid past an

impulsively started infinite plate was first studied by Stokes[7].

This is also kno~ as Rayleigh's problem in the literature. This

problem is of fundamental importsnce as it provicles one of the exact

solution of the Navier-Stokes equation and reveals how the velocity

profiles varies with time. Due to importance of hydromagnetic

rotating flow in astrophysical, geophysical and engineering problems,

the effects of cor~olis force as well as magnetic force have been

presented by many researchers for different aspects (cf. Soundalgekar

and Pop [5], Debnath [I], Purl and Ku]shrestha [3] etc.).

in many industrial applications~ the flow past an infinite

vertical plate, started impulsively from rest, plays an important

399

400 A.K. Singh Vol. ii, No. 4

role. The flow past a vertical plate moving impulsively in its

o~¢n plane was studied by Sotundal~(ekar [6], where the effects of

natural convection currents due to the cooling or heating< of the

plate were discussed. ']'he extenslon of this proble~ to hydro-

magnetic was presente ~] by Oeor~ntor)ou]os et al [2~ when the

magnetic field Js fixed re]Ftive to tbe f]~li~]. ~urt, her, R~{~)tis and

Singh [4] studied the effect of magnetic rj~],i on this ~rob].em for

another phys~ca] situation when the m~£netic lines oC force "~re

fixed re]ative to the elate.

In nature~ the phenomenon of rotation is alwa~:s eneounterea

and is very often observed in meterorolo~<y. On Lhe other han d~ the

free-convection effect on flow problems is very important in heg~t

transfer studies. The present paper is undertaken in order to st,~,gy

the effect of rotation on the hydromaF[netic free-convection flow of,

an ].ncompress]ble viscous ~md eIect~dcai]y eond~et]n,! f]~,] p~st ~n

]mpu]_sively started infinite vertic<~l ~)!ate. The tem~ e~mtu~'e of the

plate differs from the temperature of the f]_ui0, c~1~sin~ the f]ow of

free-convection in the boundary layer. The Laplace tr,nsfo<'m

technique is used to obtain the solution of the ~;rob]em.

M~'at h e mat _ic al An <mlj{.si__s

We consiHer the unsteady flow ~enerate£ in ~n electrically

conductin~ viscous fluid boundeS hi," an j_nfinite non-eonductin~

vertical p]ste (~al]) at z'--O. T~,e flui<:l ~n'] plate rotate as ~,

r,i~irl befiT; with uniform an~ular veIocJt,~T~ about the z~-nxj, ,~ in

the presence of "~,n £mpo*er] ~nJ Corm ma~,netie fJ e]H ~o norm~l to th~

plate. The p]ate initiril]y at rest~ sur~i~nIy starts ~mDuls:ivelv

with a ve]ocJ ty U in its o~rn p]sme ana its temperature ]nstL~n-

taneous]y rise <~ or Cn~l~( to T ~ wh'Lch is; thereafter ma~nt~,ineJ

eonst:tnt. The x'-~xis is chosen a]onc the ]nCJnJi:e [~:t~ in L~e

upwarS d~rectJnns~ and th(~ y'-sxJs norm~] to x'-n×J.& Jn the n]ane

of the p].ste. Since th~ T~]s.tn ~', inCi!dte i.~ ]~n,tb~ :~11 tb~

physfico~] quantities are functions of z' a n d t' only. (In,let t, he

usual Bouss~nesq ~s approximation the b~:~sio equations in % rot, atin ,~

fluid in non-dimensional form are

"~u ~L2 u ~-~- 2Jtv = O~ + ~z~- I~iu, .. I)

Vol. Ii, No. 4 H Y D ~ C FREE--ION PAST A PLATE 401

%v "~2u -- - Mv~ ~t + 2~u = ~z 2

~t - P ~z 2 ~

with the initial and boundary conditions

U(~,t) = O, V(Z,t) = O, O(z,t) = 0

U(O,t) = I, v(O,t) = O, O(O,t) = I

for t~O;

for t~O;

. . ( 2 )

• . ( 7 )

u(~,t)-- O, v(~,t)-- O, O(~,t)= 0 for t~O. .. (k)

The non-dimensional quantities introduced in the above equations are

as follows

o v')/U,P= ~Cp/k,Ja=-~/U 2, Z=Z'U/'~ ,t:t'U'-/~ ,(u,v):(u',

. 2 y 2 . M:~e,!ol U2,G=~g~ (T'-T~)/U3,0-(T '-~'_-._ ~)/(T~-r~),' ' .. (5')

where all the physical qu~.tities have their usual meanings.

The hvdroma~net~c rotstln~, free-convection £]ow past the

mnvic~ ........... ~.rl]? is dosC~d by the syste~n o£ equations (I)-(~.,~ under

the hound~.ry eonJJti,~n,<: (h]. The solution of equg.tion (3) has been

obt;,ine,i by eounaa]~ekar [6]. Hence we now have to solve equations

(I) ~:md (2) only. Introducin~ the complex velocity q:u+iv~

equations (I) and (2) c~.n he combined into a sin{{le equation

C-O m-~2q ~t -- + ~z ~ - mq, .. (6)

subject to initial and bound0ry conditions

q(z~t) : 0 for t~O,

q(o~t) = I and q(~,t) -- 0 for t~O, .. g)

wh e T" e

m = H + i2J~. .. 8)

The so]ution o f equation (6)~ under the OoiJndcr 2 conditions

(7) by t h e Lsplace transform technique~ is

q= 9- - ( 1 - m' e - e r f c ( ~ z ) - 4 ~ ) + e z J ~ e r f e ( z - - + J'6"t) ?_r~ 2JZ

O e~ t e -Z ~ - ~ . Z - ~ 7 " t ' z ~ . z

- e 2~-t erFet--~2~t +~ + ~ er~ ct---'~94t '

402 A.K. Singh Vol. ii, No. 4

1.0

0.8 - !

0 . 6 -

0.4-

0.2- -

0.0

CURVE G ,.,3_ M t

1 10.O 0.5 O.0 O,L

2 10.0 1.0 0.0 0.~

3 5.0 0.5 0.0 0.~

&. 5.0 0.5 1.0 oJ,

~ , 5 5.0 1,0 1.0 oJ. 1

2 3 6 5.0 0,5 1.0 0.2

/,. 7 5.o 1.o o.o 0.2

5 8 5.0 1.0 1.0 0.2

1.0 2.0 3.0

Z

FIG. 1 Primary Velocity Profiles of Water (P = 7.0) for Cooling the Plate

1.0

0 .8

I 0 .6

0. / ,

0 .2

k CURVE G ~ M t

~ 1 -5.0 0.5 0.0 0.,

%~L"~ - - ' ' ' - 1 2 - 5 . 0 1.0 o.o o.i,

, - , o o , , o o ,

, ' % " % k - , , - l o o ° 5 l o o , \ \ \ , - 5 o , o o o o ,

',k~A , - 5 . o 0.5 ~.o o.~

~ ~ s . o ~.o ~.o 0.2

- , i " ~ ~ ' ~ ' ~ ' ~ l - , " ' - - ' ~ " - - . . . . . . . . . J J

0.0 1.0 2.0 3.0

Z ~

FIG. 2 Primary Velocity Profiles of Water (P = 7.0) for Heating the Plate

Vol, ii, No. 4 H Y D ~ I C FREE--ON PAST A PLATE 403

where

~ = m / ( p - 1 ) . . . ( lc')

If" ~x and ~y are the axia3 and the transverse components of sk~n-friction~ we have

"Cx + i ~y = - ~zl z=o

G I -mt = (I - ~)~ err (~'~) + ~ e

+G ~ ~-(~_--~ P e~{t [erf (~-~t) - erf (~'~. .. (11)

In equations (9) and (11), the argument of the complementary

error function and error function is complex. Therefore, to obtain

h ]ocitv and skin- the axial and the transverse components of t..e ve

friction~ it is necessary to introduce some properties of' the

complementary error function with complex arquments due to Strend [8]

For any complex number c = a + ib, e-~c'(c) = erfc (~) and

erfc(c) = e -2iab f(a,b)~ a>O, b>.O~ .. (12)

where f(a,b) is a complex function given by

f(a,b) = £ (ab) 2n~gn(a)-i(n+1)gn+1(a) ~ , .. (13) n=o

which tends to zero as a---~ and

gn+1 (a)= ~n--~+ ( i~I/2(n+1): a~?;7 n;7--h ; n=O,1,2,..... (lh)

where

go (a) = erfc(a). .. (15)

Discussions and Results

In view of these equations~ exDresslons for u~ v :~qd~x~'[y

are obtained but they are omitted here to save the space. In

order to have a physical view of the oroblem, these expressions are

used to obts.in the numerical values of u, v and~x~ "[y for different

values of various parameters. The Prandtl number P is taken equa7

to 7.0 correspondinK to water.

The primary velocity profiles of water are shown Jn

figures 1 and 2 for cooling and heating of the plate respectively.

It is seen from these figures that the primary velocity profile of

water decreases with increase in either magnetic parameter M or

404 A.K. Singh Vol. ii, No. 4

rotation parameter ~ in both cases, cooling and heating of the

plate. The secondary velocity profiles of water for cooling and

heating of the plate are plotted in figures 3 and 4 respectively.

The influence of the magnetic field is to increase v for both case~

(G>0 and GgO). But the effect of rotation on v is just reverse

to that of magnetic parameter. Further, we can see that with more

cooling of the plate, due to free-convection currents, v increases

while reverse phenomenon takes place for greater heating of the

plate.

The skin-friction components at the wall for different

values of G, t, ~- and M have been shown in Table 1. It follows

Table I. Values of skin-friction components in water (P=7.0)

t ~ M ~x 1"y

-5.0 0.2 0.5 0.0 3.4151 -1.6346 I .0 2. 5906 -0. 8104

1.0 0.0 3.2022 -I .3126 1.0 2.8453 -0.8207

0.4 0.5 0.0 3.6486 -2.4561 1.0 2.6321 -I .0202

1.0 0.0 3- 1139 -2.0~71 1.0 2. 7889 -1.0720

5.0 0.2 O.5 O.C -1.1295 2.3311 1.0 0.k283 1. 5091

1.0 0.0 -0.9116 2.8882 1.0 O. 2551 2. 2600

0.4 O. 5 0.0 -2. 1693 3.4855 1.0 0.1118 1.9038

1.0 0.0 -1. 2853 ~+. 2586 1.0 -0. C009 2. 833~

that Tx decreases for G~0 and increases for G>0 as M increases

while ~y increases for G~O and decreases for G>0. The effect of

rotation on skin-friction components is very interesting. It

decreases Tx for G40 and increases for G>0 in hydrodynamic flow

(M=0.0) while reverse effect occurs in hydromagnetic flow (M=I.0).

Further, the effect of rotation on ~ is to increase it in both

cases, G~0 and G,O in hydrodynamic flow. In hydromagnetic flow,

~y decreases for G~0 and increases for G~O as Jtincreases.

Vol. II, No. 4 HYD}~X~%~WETIC FREE--ON PAST A PLATE 405

Z

0.0 1.0 2.0 3.0

-0.02 3

/h M t - 0 . 0 0.5 1.o 0.2

0.5 0.0 0.2

l, 5.0 0.5 1.0 0.1,

5 5.0 0.5 0,0 0,4

oo,f/ / .o,oooo , ~.o 1.o o .o o.~

~ ~.o , . 0 1 .o o .~

-°'1° f K_J -0.12 L-

FIG. 3 Secondary Velocity Profiles of Water (P = 7.0) for Cooling the Plate

0.0

- 0.04

- 0 . 0 8

t > - 0 . 1 2

- 0 . 1 6

- 0 . 2 0

1.0 z 2 . 0 i I "

,~C_~ /// //// cuRvE o ~ . , \~ //// I _~oo o, 1o o~

~ \ % - . " ' / - - , . //I I ~ - s . o o.s ~.o o.,.

~~//// 3 -~0.0 1.0 ~.0 0.~ ! : o , o o 6 S -10.0 0.5 0.0 0./,

5.0 1.o 1.o 0.~.

\~/ / 7 -10.0 1.0 1.0 0.&

8 -10 0 1.0 0.0 OJ.

FIG. 4 Secondary Velocity Profiles of Water (P = 7.0) for Heating the Plate

406 A.K. Singh Vol. ii, No. 4

References

[I~ Debnath L.: ZAMM 52, 623 (1972).

[2] Georgantopolous G.A., Douskos C.N., Kafousias N.G. and Goudas C.L.: Lett. in Heat and Mass Transfer 5, 379 (1978).

[3] Puri P. and Kulshrestha P.K.: J.Appl.Mech. 28, 205 (1976) .

~] Raptis A. and Singh A.K.: Int. Comm. Heat Mass Transfer 10, 313 (1983).

t~] Soundalgekar V.M. and Pop I.: Bull.Math. de la Soc. Math. de la R.S. de Roumanie Tome 14, 375 (1970).

[6] Soundalgekar V.M.: J.Heat Transfer (Trans ASME) 99C, ~99 (1977)

~] Stokes G.G.: Camb. Phil. Trams. 19, 127 (1851).

~I Strand O.N.: Math. Comput. 19, 127 (196~).