Convection driven by non-uniform surface tension

L-N HEAT A~DMASS TRANSFER 0094-4548/79/0901-0375502.00/0 Vol. 6, pp. 375-383, 1 9 7 9 ©Pergamon Press Ltd. Printed inGreat BrieRin

CONVECTION DRIVEN BY NON-UNIFORM SURFACE TENSION Luciano M. de Socio Istituto di Meccanica Applicata alle Macchine Politecnico di Torino Torino, Italy

(Cc,,u~nicated by U. Grigull)

ABSTRACT Creeping motions induced by thermocapillarity in a f l u i d f i l l i n g a rectangular enclosure are considered at low Marangoni numbers. The problem is successfully solved in analytic form by means of biorthogonal functions. As an application thermal gradients at the free surface are assumed to originate from uniformly distributed h e a t sources in the f l u i d and/or temperature differences between the vertical walls. The solution for the stream function is evaluated in terms of the dimensionless products which govern the problem. Introduction

Thermocapillarity, diffusocapillarity, and thermoelectric flows are mduced by sizable surface tension variations due to temperature, composition, and electrical potential gradients along the interface between two fluid pha~es. A comprehensive and updated review of the existing literature can be found in [ l ] . Relatively l i t t l e work has been done on this subject as gravity

usually

tends to suppress the effects of a non uniform surface tension distribution. Other phenomena, where surface tension variations play an important were described in the pioneer paper [3].

role

Research has been mostly devoted to

the so-called interfacial s t a b i l i t y problems [2] related to a varying

surface

tension at the free surface of a fluid layer. More recently technological processes have been proposed, concerning the handling of liquids in reduced gravity environments. In this circumstance, tension variations at the free surfaces can determine significant effects, as shown, for example, in the experiments of [4] . Creeping, or h~ghly viscous convection flows are expected to take place in very viscous fluids as oils and 375

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L.M. de Socio

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glass in the presence of micro-gravitational fields. Therefore, i t seems interesting to show how the corresponding flow

field

can be easily evaluated by an analytic solution of the pertinent equations. On this purpose use w i l l be made of the biorthogonal series which received

con-

siderable mathematical attention in [5,6] with noticeable applications in

some

free convection problems [7,8,9]. As a significant example, the case of a f l u i d f i l l e d rectangular

cavity

w i l l be considered here. Steady convection is driven by thermocapillarity to a non uniform temperature at the free surface.

The temperature f i e l d

responds to a uniform heat source distribution in the f l u i d and/or to perature difference between the isothermal vertical walls.

The wall

bottom and the free surface are assumed to be adiabatic whereas the

due cor-

a temat

the

Marangoni

number Ma is small so that heat transfer w i l l be solely due to conduction. Finally, the free surface w i l l be supposed to remain f l a t since i t

pres-

ents negligible displacements. Analysis

Following the analysis of the order of magnitude of 10 one obtains the basic set of dimensionless equations for the creeping flow in a rectangular vity of aspect ratio

Ar = l

at small

ca-

Ma :

Energy equation = 0

(1)

V ~ = m0 x

(2)

These equations are to be solved in the domain

{ ( x , y ) l - I ~ x ~ l ; -2Ar~Y~<0}.

V20 + ~

- Momentum equation

The x-axis corresponds to the free surface of the f l u i d at rest and they-axis points positively upward along the centerline of the cavity, F i g . l . In Eq.(2) the temperature difference with respect to a reference value tributed heat source

AT , and

AT = T2 - TI

y is the ratio between the

~ and the reference value

walls are kept at the same temperature erwise

T - T I has been made dimensionless

TI , then

Q . y =l

In

case the

and AT = ~ d 2 / l , o t h -

and ~ = },AT/d2 . The dimensionless stream function

referred to the product

Ud , with

dis-

vertical is

U a reference velocity.

The characteristic number ~ = Gr/Re is indicative of the relative portance of the buoyancy effects with respect to the viscous forces.

im-

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CDNVECFICN D ~

BY SURFACE T~qSICN

377

Equations (1) and (2) are subject to the boundary conditions O(-l,y) : O; 4(+l,y)

O(l,y) : O* ; Oy(x,O) : 8y ( x , - 2 A r ) : 0

= ~x(+l,Y)

~(x,O) = 0 ;

=0 ;

~(x,-2Ar) = ~y(X,-2Ar) = 0

(3)

Cyy ( x , O ) = - c 2 0 x

The last condition expresses the shear stress balance at the free surface, when the viscosity of the external f l u i d phase is negligible in comparison with . The parameter

c2 corresponds to the ratio

(@o/@T)AT/~U. For

c2 = 0

,

thermocapillarity effects are neglected and the problem reduces to the usual natural convection in the enclosure. take

U = (gBATd) ~ and ~

In this particular case, i t is customary to

becomes equal to

surface tension variations are considered,

Gr½ .

When motions

driven

by

c2 is put equal to unity and, as

consequence, the reference velocity is given by

a

U = (@o/@T)AT/~ , and the ratio

is represented by the modified Bond number pSgd2/(@o/BT) which provides

a

qualitative estimate of the relative importance of the buoyancy and surface~tension forces. and 09". = l

Furthermore O* = 0

for isothermal walls at the same temperature,

when TI f T2 .

As already said in the Introduction, the free surface has been assumed to be given by the equation presented by the line

y=O.

Actually, i t s trace in the plane (x,y) is

y = h ( x ) , with

h(x)

re-

obeying to the equation of balance

between the normal component of the stress jump and the surface tension.

How-

ever, as i t was already shown in a similar situation [7], the height rise is extremely small with respect to h(x)

Ar

and can be neglected.

The evaluation of

is beyond the goal of this paper, but i t can be carried out, as in [7],via

a lenghty but straightforward procedure. The temperature distribution is immediately evaluated from (1) and (3)

in

the form O(x)

=

½0*(l+x)+½y(l-x

2)

which corresponds to the regime of pure conduction which takes place Ma .

(4) at

small

Let now ~(x,y) = ~p(x)+~H(x,y)

(5)

~p(x) = ~(x 2 - I ) 2 ( 0 * / 4 8 - yx/120)

(6)

where is a particular solution of (2) and ~H is the solution of the homogeneous equation VW~H = 0

(7)

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subject to the following boundary conditions ~ H ( ± l , y ) = ~H,x(± l ,y) = 0 ;

~H(X,O)=~H(X,-2Ar) = - ~p(x)

~H,y(X,-2Ar) = 0 ;

~H,yy(X,O) = -c 2 0x

(8)

The system (7-8) represents an edge problem where the inhomogeneous condition is assigned along the free surface. The solution can be sought in the form of a series of even and odd eiigen-functions, #!n)(x) and @!n)(x),respectively ¢!n)(x) = snsins ncossnx-snxcoss nsins nx ¢! n)(x) = PnC°SPnsinPnx-pn xsinpncOspnx associated to the eigenvalues sn and Pn which are the complex root~ in f i r s t and in the fourth quadrant of the equations sin2s = -2s

;

the

sin2p = 2p

In fact, i f one considers the functions @!n) = _(Snsins n+2cossn)coss nx + SnXCOSSnsinsnx $!n) = "(PnC°SPn-2sinPn)sinPnX + pnxsinpncospn x ~I n) = (s n sin sn- 2 cos Sn)COSSn~X - SnXCOSsn sin sn x ~I n) = (PnC°SPn +2sinpn)sinpnx - pnxsinpncospnx ~n)=

@In)

then the vectors [¢I n), ¢~n)]

and [~i(n),~2(n)] satisfy a biorthogonal%ty condition and. likewise, the vectors [$~h). $i,)] and [ ~ n ) , ~!n)] ,[5]. Solut ion

The solution to the differential problem (7-8) can be formally written oo

as

~H(x,y) = ~-~ I(cneSnY +dne-Sn(Y+2Ar))¢In)(x)/s~ + n_--oo

n#o

+(~n ePnY+ (J e-PR(Y+2Ar))}¢In)(x)/P~

(9)

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(IICvIITIONDIEV]~ BY ~

T~NSION

where the coefficients are to be determined from the (8).

379

The mathematical de-

t a i l s leading to the evaluation of the unknown Cn , an , dn , dn are here omitted for the sake of compactness as they represent a st rai ght application

of

well established technique (see, for i n s t a n c e , [ 7 , 8 , 9 ] ) . More important

a

is

to

pay some attention to the convergence of the series at the right side of (9). According to Joseph's theorem 16], for edge data as those given in (8), al though the series does not converge at the endpoints Leibniz c r i t e r i o n for conditional convergence.

x=± l , i t satisfies

Furthermore, numerical

tions prove that the series (9) converges to the data for

-l < x< l

the

computaand that

this convergence is usually fast. The results of actual computations of the low f i e l d were compared withtthose obtained by numerical integrating the basic equations in finite-difference form. In this last case, the procedure of I l l ] was adopted.

The proposed analytic

so-

lution shows an excellet agreement with the numerical solution of the flow f i e l d in a wide range of parametric conditions. From the boundary and the biorthogonality conditions one has the

following

system of linear algebraic equations for the coefficients which appear in (9): cm = Wm/km - exp(-2smAr ) d m {1 -

exp(-4smAr)} kmdm- E Anm{exp(-4sn Ar) + (I + Sn)/(l-sn)}dn = Vm-Wmexp(-2smAr)+

n¢o - ~gnm Wn exp(-2Sn Ar ) / kn n¢o am : Wm/ km - exp(-2PmAr) dm Co

{l

-

exp(-4pmAr)} kmdm- E

Anm{exp(-4Pn Ar) +(l+Pn ) / ( I -Pn )}dn : Vm- WmeXp(-2PmAr)+

n:-oo

nfo -

}~ Anm Wn exp(-2PnAr ) / kn nfo

where km = -4 cos" sm ;

km : -4 sin" Pm ;

Vm = ( 4 ~ y / p ~ ) ( l - s i n 2 pm/p~);

Vm = 2 ae*/s~ wm = vn - 2 e * . c 2 ;

;

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L.M. de Socio

Vol. 6, No. 5

Wm = Vm-4yc2( l "sin2Pm / P~n) and 16(sn cos~Sm - s~cos2sn)(Sn - l ) /

n#m

(Sm- s~)2Sn

Anm = cos2sn(~S~ + cos2sn l ( s n - l ) / s n

n =m

16(p~ sin 2 Pm - PmSin2Pn)(Pn-l)/ (p~-p~)2pn

n # m

sin2Pn(~ Pn + s i n 2 P n ) ( P n - l ) / P n

n : m

Anm =

Inspection of the d i f f e r e n t i a l problem indicates that i t s solution linearly depends upon the dimensionless parameters y, ~ , c 2 and 8" . Therefore the lution at given

Ar

in a general case can be immediately found a f t e r the

lowing four basic situations are considered: (1) (y=O, c 2 = l , e : ( I I ) (y=O, c2 =0, e= I , c 2 = 0,~ = I , 0 " :

so-

8'= l ) ; ( I l l )

fol-

0,8"= l )

;

(y = l , c 2 = l , ~ = O,@*=:O);(IV)(y=l,

0).

While a f u l l parametric investigation is the aim of a furthen'research,Figs. 2 and 3 show typical flow patterns for the cases (1) and ( I l l ) The n~ximum absolute value of ~ ,l~Imax meter

relative toA r = l .

is usually taken as indicative

para-

of the flow f i e l d in the cavity. The values of this parameter as a

ction of

Ar

fun-

were evaluated in a few cases. The results are given in Table l ,

which shows that a decreasing tance of the effects of

Ar

corresponds to an increasing loss of

O* with respect to those of y • This can

impor-

be easily

understood on physical grounds. Table

Ar

Case

I

[ ¢[max

Ar

Case

[~[max

i

I

0.255

iO -I

0.5

III

0.746

10 -2

1

III

0.771

10 -2

0.25

I

0.462

10 -2

0.5

I

0.158

10 -I

0.25

III

0.434

10 -2

I t is worth mentioning that in the case Ar=O.25; ¥ = 0 , c 2 = I , ~ = I , 0 " = I , one finds I.

I~DIn~x = 0.478 lO-2 in comparison with the value 0.462 lO-2 o f Table

At increasing

~ , all the other parameters being kept constant,

I~Imax = 0.544 lO -2 at ~= 5 and 0.622 lO -2 at

~ = lO.

one has

Recalling theph,ysical

meaning of ~ , these results show the effectiveness of the buoyancy forces

in

VOI. 6, No. 5

~IONDRIVI~

BY ~JIRFACE TI~ISION

381

activating the recirculating flow in the enclosure. Acknowledgement

This work was partially supported by the Italian National Research Coun~l, G.N.F.M. Thanks to Prof.S.Ostrach for is kind bibliographic assistance. Nomenclature

Ar

~spect ratio of the cavity, D/2d

C2

(aoI~T)Z~TIIJU semi-width of the cavity depth of the cavity

d D g Ma Pr

gravitational acceleration Marangoni number, Pr Re Prandtl number, ~/~ distributed heat source

Q Gr

reference heat source Reynolds number, pUd/p Grashof number, gSATd3/v2

T

temperature

U

reference velocity non-dimensional Cartesian co-ordinates, (x,y) = (X,Y)/d Cartesian co-ordinates

Re

x,y X,Y

Gr/Re B Y AT 8 X lJ,'0 P o

qJ

coefficient of thermal expansion non-dimensional heat source Q/Q reference temperature difference thermal d i f f u s i v i t y non-dimensional temperature difference, (T - TI)/AT non-dimensional temperature difference at the hot wall thermal conductivity viscosity coefficient and kinematic viscosity,respectively density surface tension non-dimensional stream function

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Vol. 6, No. 5

Subscripts 1

temperatureof the cold wall

2

temperatureof the hot wall

x,y

derivative along x and y , respectively References

I. 2. 3.

S.Ostrach, Motion Induced by Capillarity, in Physicochemical Hydrodyno3nics, (B.D.Spalding ed. ), Adv.Publ.Ltd. ,London 1979 G.Z,Gershuni and Zhukhovitskii, Convective Stability of Incompressible~ids, Keter Publ. House, 8erusalem, 1976

4.

V.G. Levich and V.S. Krylov, Surface Tension Driven Phenomena, Ann.Rev.Fluid Mech., l , 293, Annual Reviews, Palo Alto, 1969 S. Ostrach and A.Pradhan, AIAA Journal, I_66,419, 1978

5.

D.D.Joseph, SIAMJ.Appl .Math. ,33, 337, 1977

6. 7.

D.D.Joseph and L.Sturges, SIAMJ.Appl.Math., 34, I , 1978 D.D.Joseph and L.Sturges, J.Fluid Mech., 69, 565, 1975

8.

L.M.de Socio,L.Misici and A.Polzonetti, Natural Convection in Heat Generating Fluids in Cavities, ASMEPaper, 18th Nat. Heat Transfer Conf.,S.Diego 1979

9. L.M.de Socio, Mech.Res.Comm., 6_,33, 1979 lO. S.Ostrach, Convection due to Surface Tension Gradients, Proc.COSPAR Meeting, Space Res.VIII, Insbruck, 1978 I I . L.M.de Socio, E.M.Sparrow and E.R.G.Eckert, Comp.and Fluids, l_, 273, 1973

X

2d

FIG.I A sketch of the cavity

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CONVECTICND~BY

SURFACE T~gSION

Y--O

383

Y--O l

I,

F

J

S J FIG. 2 Typical flow patterns for case (1), Ar = l

FIG. 3 Typical flow patterns for case ( I l l ) , Ar= l