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Formation of patterns induced by thermocapillarity and gravity
Alexander Oron, Philip Rosenau
To cite this version:
Alexander Oron, Philip Rosenau. Formation of patterns induced by thermocapillarity and gravity.
Journal de Physique II, EDP Sciences, 1992, 2 (2), pp.131-146. �10.1051/jp2:1992119�. �jpa-00247619�
Classification Physics Abstracts
47.20 47.35
Formation of patterns induced by thermocapillarity and
gravity
Alexander Oron and Philip Rosenau
Department of Mechanical Engineering, Technion Israel Institute of Technology, Haifa 32000, Israel
Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, NM 87545, U-S-A-
(Received 24 June 199I, accepted 8 November I991)
Abstract. Consider a liquid film on a slightly inclined plane driven by gravity and
thermocapillarity. We derive an equation describing the nonlinear evolution of the interface between the liquid and the atmosphere. In line with an approach introduced by us elsewhere we preserve the full impact of the curvature that contributes toward formation of bubbles and
together with convection induced by the inclined plane contributes toward either breaking of
waves or prevention of rupture that otherwise will always occur. Travelling waves are also
possible. A variety of possible equilibrium states is also discussed.
1. Introduction.
A liquid film at rest bounded on one side by a heated flat plane and on the other open to the
atmosphere exhibits under certain conditions an unstable behavior. One of the sources of that
instability is the temperature variation of the interfac1al tension which leads to surface tractions. This so-called Marangoni, or the thermocapillary effect II, 2], is important when the liquid film is thin or when the gravity is very low, such as in the extra-terrestrial
conditions.
Experimental studies of the thernlocapillary effect are well documented, references [2-4]
and the references therein. The linear analysis of the Marangoni effect was presented in references [1, 5, 6] and the conditions for the onset of the instability were derived. Weakly
nonlinear aspects of the phenomenon based on amplitude expansion were studied in
references [7-12]. Those studies differ each other in asymptotic representations of the independent and the dependent variables lead to different evolution equations describing the behavior of the interface in different parametric regimes. However, no numerical studies of those equations were reported except for the overly simplified problem in reference [7].
The three-dimensional problem of a thermocapillary flow was treated in reference [10]
using what might be best described as a boundary layer approach. An equation describing the
nonlinear evolution of the interface of the film supported by the cool plane was derived there in the form
h~ -V.
~~~~~
~+ ~~V(Gh-sv~h)j = 0 (1)
2(1+Bh)
where h is the location of the interface, B is the Biot number, G is the Bond number
characterizing the relation between the effects of gravity and the thernlocapillarity and s is the surface tension number measuring the relative importance of the surface tension and the
thernlocapillarity. The derivation of equation (I) given in reference [10] implicitly uses the
assumption that in addition to the amplitude its spatial gradients are also small. However, it tums out that the solutions of equation (I) do not necessarily obey that assumption.
While in this work we follow reference [10] with respect to the asymptotic scaling we extend it in two ways. First, we allow for a slight plane inclination. This adds a convective part into
the dynamics. Second, we preserve the full effect of the interfacial curvature on surface tension to derive an evolution equation
h~ +
~~°
(h~)~ V ~'~~
~ ~~
~ +
~~
V(Gh sN V~h
)j = 0 (2)
3 2(1 + Bh ) 3
being N
= [I + e~(Vh)~]~~/~ and m
= ± I. The parameter G is positive when the film is
supported from below and negative when it is bounded by a rigid plane from above and by the
ambient air flom below (negative gravity). These modifications will be shown to be
important : each of them can contribute toward prevention of rupture which otherwise will
always occur.
We also carry out the numerical studies of equations (1), (2) to study the impact of different
physical mechanisms on the behavior of the interface.
Note the presence oi ~, the perturbation scale, in N. Though fornlally of higher order, (h~
=
o(e~'),
see below), it enters into a sensitive balance between backward diffusion induced by the destabilizing effect of the gravity (for negative gravity) and the stabilization due to the surface tension. As shown later, preserving the full effect of curvature is under
certain conditions crucial and results in a dramatic change in the fornling patterns. It will be also shown that equation (2) exhibits a rich variety of possible pattems. That includes the film rupture and formation of dry areas, travelling waves and the waves propelled toward breaking
similar to the shallow water waves. More about the asymptotic approach used here is said in section 4.
A similar asymptotic approach to the problem of the thermocapillary flow albeit using a
fornlally different scaling of the independent and the dependent variables was presented in references [9, 13]. The point of bringing it up here is that their scaling leads to an equation
identical to ours though they assumed the fluid layer to be of thickness O(I). This is
summarized in appendix where using scale invariance it is also shown that the two approaches
are identical. The main difference worth noting is that due to a different scale length for nornlalization the resulting dimensionless quantities are of different order. This effectively
renders equation (2) of interest for a wide parametric range.
It is of interest to note that the problem of the Marangoni flow was usually investigated for the case of positive gravity (a film supported from below). In the numerical studies we stress the case of negative gravity where the surface tension is essential to balance the destabilizing
effect of gravity.
The plan of the paper is as follows. In section 2, we fornlulate the problem and derive the evolution equation using the regularization approach [14-19]. In section 3 we present the
essence of our numerical studies and the emergence of a rich variety of pattems. This is followed by summary and concluding remarks in section 4. In the appendix, we present an
evolution equation for a film of a finite thickness for the combined case of Benard-Marangoni
flow.
2. Derivation of the interfacial equation.
Consider a thin liquid film on a slightly inclined plane. We assume the surface tension being a function of the temperature (the Marangoni effect). The thernlal conductivity, the viscosity and the density of the fluid are assumed to be temperature-independent and the phenomenon
of the buoyancy is neglected.
We start with the goveming equations of the incompressible flow :
v~+ (v.V)v= ~Vp+vv~v+F (3)
P div v
= 0 (4)
T~+v.VT= ~ V~T (5)
wherein v
= (u, v, w), p are the fluid flow velocities and the pressure, respectively, T is the temperature, v and ~ are the kinematic viscosity and the thernlal conductivity of the fluid, respectively, and F is the body force with components (g sin p, 0, -gcosfl) in the coordinate system shown in figure I, fl being the inclination angle of the plane and g the
gravitational acceleration.
~ x
U&
Fig. I. The basic geometry of the problem ; a film flowing on a floor. If the film flows on a ceiling the direction of g should be reversed.
The boundary conditions are :
I) at the plane x~ = 0, we use the non-slip conditions for the velocities and the specified temperature Tj
v=0, T=Tj. (6)
2) at the free surface x~ =
H (xi, x~, t) we employ :
(al the stresses balance :
~p-p~)n+2~D.n=2«Kn+V~« (7)
(b) the heat transfer condition :
~~
+ q(T T~)
= 0 (8)
(c) the kinematic condition :
fl+v.VH=0 (9)
where T~ and p~ are the given, unifornl temperature and the pressure of the ambient air.
D is stress tensor, ~ is the fluid viscosity, « is the temperature-dependent surface tension,
« = ma « (T To), To being a reference temperature, K is the mean interfacial curvature,
q is the ratio between the heat transfer rates by convection and conduction, n and
V~ are the unit nornlal vector and the surface gradient at the interface, respectively.
Introducing the longitudinal (transverse) length scale I (a) and a characteristic velocity U
we define the following dimensionless quantities :
x =
xiii
, y = x~li
, z = xja, r = utli
, e =
all
T_T~ ~2 ~
@~~ ~ >
P
~ ~P~palj, h"j, (lo)
u = ui/u, u = u~/u, w = u~ il(ua)
wherein T~, TI~ are the low and the high temperatures at the boundaries.
Projecting equation (7) on the interface r and comparing the order of magnitude of various
tennis one obtains
~~ ( II)
U
= ~
~ ~~~ ~~~
~~~~~ ~
~~
Using e equations (10) and (iii read xi =
xi
,
x~ =
yi
,
x~ =
ezi, t
= e- i riluo,
vi = euUo, u~
= euUo
, u~ =
s~wUo
,
(12)
T-T~ u~~l
~"~n _~n' P~Pa~~~P, H=6hi
where the reference velocity Uo is given by
« (TI~ T~)
Uo =
lL
Introducing equations (12) into the equations of motion one obtains
u~~ P~ + GE ' sin p
=
e~Re (u~ + uu~ + uu~ + wu~) e~(u~ + u~ ) (13)
v~~ P~ = s~Re (v~ + uv~ + vv~ + wv~) e~(u~ + v~~) (14) P~ G cos p = e~Re (w~ + uw~
+ vw~ + ww~ s~(w~ + w~) s~w~~ (15)
u~+v~+w~=0 (16)
9~~ =
e~ Re Pr
(@~ + u@~ + v@~ + w@~) e~(@~ + @~~) (17)
Here, Re=Uoa/v is the Reynolds number, Pr=v/~ is the Prandtl number and
G
=
ga~/(Uo v) is the Bond number characterizing the relative importance of gravitational vs.
surface tension effects.
We now rescale the boundary conditions to obtain
u=v=w=0@z=0 (18)
= @j @ z
= 0 (19)
@~ e~(
@~h~ +
@~h~) + aq( @~) N '/~
= 0 @ z
=
h (x, y, r) (20)
P
=
2[e~(u~ hj +
u~h) w~ w~h~) + e~h~h~(u~ + v~) + s~(w~ uzh~ v~h~)] N ~/~
~ [s~ h~(I + s~h)) 2 s~h~ h~ h~ + e~ h~(I + e~hj) N @
z = h (x, y, vi (21)
Uo lL
le~ u~h~ s~h~w~ + 2 s~h~(u~ + v~) + 2 s~h~h~(v~ + e~w~) +
+ (uz + 6~Wxl (6~h] 1lj = (@x + hx @zl N -1/3@ z
=
h (x, y, vi (22j
~~~Y
~Y ~~~Y ~z ~ ~~~x(Uy ~ ~x) ~ ~~~x ~y(Uz ~ ~~~x) ~
~ ~~Z ~ ~~~Y~ ~~~~~ ~j j
~~Y ~ ~Y ~zl'~ ~~~@ ~ ~(X> Y' ~) (~~) h~ + uh~ + vh~ w = 0 @ z
= h (x, y, r ) (24)
where N
= [I + s~(Vh)~]~~/~ is the metrics that measures the interfacial distortion.
Next, take SW1, and assume the angle p to be small, fl
= spa. Altematively to keep p
= O(I) one has to assume G
= O(s) which is proper for microgravity conditions. Dropping higher order tennis in equations (13)-(24) leads to the following, simplified set of equations
and boundary conditions
u~~ P~ + Gpo = 0 (25)
vzz Py = 0 (26)
P~
= G (27)
u~+v~+w~=0 (28)
@zz = 0 (29)
at z = 0
u = v = w = 0 (30)
= @j (31)
at z = h :
~~ ~ ~~ ~~ ~21'~ '~~
~
° ~~~j
P
= sN V~h (33)
u~ = (b~ + h~ @~) N ~/~ (34)
v~ = (@~ + h~ @~) N ~/~ (35)
h~+uh~+vh~-w=0 (36)
wherein s
= «o s~/~Uo («/~Uo
=
O(s~~)) and Bi
= aq is the Biot number. Here we have
also assumed that the values e~Re and s~ Ma
are of higher order of magnitude, Ma
= RePr
being the Marangoni number.
In carrying out the asymptotic expansion we have allowed h~, the derivative of h in the direction of the flow along the plate to be as large as o(e~ ~). While it allows us to account for
possible large gradients [14-19], it also ensures that all the higher order tennis are rightfully neglected.
Using the boundary conditions (31), (32) integration of equation (29) yields the temperature profile
=
'~~~
~ z
+ @j m
bz + @j (37)
1+
W~~~h
where m
= I, @i
=
I in the case of a cool ambient air and a hot plane. In the opposite case
of the hot ambient air and the cool plane m
=
I and @j = 0. The surface temperature
@r is therefore given via
Note that as [h~[ - oJ, @r - @~.
Integrating equation (27) and using the boundary condition (33) one finds that the pressure is distributed according to
P =G(h-z)-sNV~h. (39)
The first ternl corresponds to the hydrostatic part. The second one accounts for the effects of the distorted interface.
Substituting equation (39) into equations (25), (26) and using the boundary conditions (30), (34), (35) yields the velocity field components u, v :
u = [Gh_, s(N V~h)~ Gp al (z~/2 hz) zN ~/~ @r (40)
v = [Gh~ s(N V~h)~] (z~/2 hz) zN ~/~ @r (41)
The component w of the velocity is found using equations (28), (40), (41) and the boundary
condition (30) :
w =
~
Vh V [Gh sN V~h (z~/6 hz~/2)V~[Gh sN V~h] +
2 2
~ ~~O j ~x ~ j [16'~ ~~~~r~)x ~ l'~ ~~~~r,)yl (~~)
Introducing equations (40)-(42) and (38) into the kinematic condition (36), leads to an
equation describing the evolution of the liquid-air interface
h I
(h3j~ ~ v
I B'~h ~ hj
+ fl V [h~ V(N V~h )]
=
0 (43)
~ 3 3 2(1 + Bh)~ 3
Here y= -G and we have also introduced B as an effective Biot number. As before
N
= 11 + e2(vhj2j-3/2
The second ternl of equation (43) represents the convective effect due to the plane
inclination. The next two tennis are of diffusive nature. The first is due to the gravity while the
second one is due to the Marangoni phenomenon. The gravity stabilize; the evolution ol'the interiace when the film I; ;upported from below, y~ 0 (the film flowing on a floor) and
destabilizes it when the film is ;upporied l'row above, y~ 0 (tlow on ±t ceiling). It'the free ,urface I; exposed to the air cooler then the rigid plate (m the Marangoni et'iect ha; a
de;tabilizing impact. In the contrary case of the air hotter than the plate (ni
= the
Marangoni effect tends to stabilize the forming pattern. The last term in equation (43) I; due to the surface tension. It always ha; a stabilizing effect.
In u;ing an effective Biot number, B, we can safely omit it; dependence on the metric term N due to the very mild variation of N. However, as past experience ha; taught u, [14-19[, this
variation may have a crucial impact on the balance between the ;uriace ten;ion and the
driving instability becau;e it affects the width oi the band of un;table mode;.
In the one-dimensional case (3,
=
0)
fi- ~
(fi~), + ~
~
~'~~ ~
~
i,j
+ (fi'(fi,, N ), (, =1). (44) -( + Bfi )-
,
flow N
# [1 + ~~/l)(~~~.
Equation; (43) (>I (44) ;ire oui'kcy re;ult. Equation (43) ie(iuce; to equation (?.l~) of' ret'erence I)( ii'the p±ti'aineter, ~ and flu are ;et to be leio.
Unto;; one deal; in condition; ol'iiiici'ogi'aviiy ii I, clear that the gravity I; the trio,t
important niechani;ni of';tabilizaiion or de;tabilization. The impact of'tfic licit tran;t'cr at the intcrl~=tcc, de;cribed by Biot number B I; minor l'or large ;in(f po;itivc y. carte,pending to
thicker t'ilni, or,mailer temperature dilTei-once, between the bound;irie,. Coiiipari;on of'the
two eli'ect; ;how; that therinocapillarity (ioiiiinale; gravity only it'the average height
fi,j ;ati;l'ie,
/i~~ II l'~=<i
which I; the equivalent to the condition a
~ 3 U<j t,/~ g an(I il'ihe Biot nuniber range, in
;onto interval encoiiipa;;ing ihc valuc B
= , ~ y/3_
A; an illu;tration, a,;vine a water layer ol' iiini average thicLnc,, at ?11 °C at the terra;trial value ol'gravity Jn(I the teiiipcrJturc (lili'eience between the hour(lane, ~T being °C. For thi, ,ituation y161 an(I fi~ =I),I)~. Tfieielkire. the heat Iran,f'er at the interfbce can
,igniflcantly afi'ect only the,e part, of'the interlhce fk>r which fi
~ ().l)~ nun. Unto,; thi; I, (he
ca;e heat tram;l'er at the Ii-cc ,urlbcc can bc (li,icgar(fed. Equation (44) then beconic, fi_ ~'(fi~), + ~'( yfi +.ifi,, N ),j = () (45)
,
which eerie,pond, to the evolution of an cf'f'ectively thermally in,ul;ite(I iitcrliicc.
Note that equation (43), in(I it, de,con(rant; a, well. equation, (44) ;ii(I (45), pie;erve the total nia,,, i-e-
lfi iii)
fi<, = con,(. (46)
~,i
where ii I, either inllniic or periodic doiiiain. For equation (45) (the I-D ca;e) another con,ervation law I, available, naiiiely
~ ~'
=
~ d; [yfi) .iNh), w1 (7 (47)
d7
jj
fi 3
jj
Equation (47) has been derived from equation (45) by multiplying the latter by h~~ and integrating that over il. (For B # 0 in equation (47) y
- y Bm/2 h (I + Bh)~.) A
similar conservation law is also valid for the 2-D case under substitution h~-Vh,
h~-V~h.
For the case of a horizontal plane, pa
=
0, equation (44) can be rewritten in the conservative fornl
h~ = (h~(Q(h) + fl h~Nj (48)
where
~~~~~
3 2 ~~~l +Bh~(I+Bh)~
6 where A is an integration constant.
Multiplying equation (48) by Q(h) + (s/3) h~N and integrating it over il by parts using periodic boundary conditions one has
~m) j (P(h)+(( I+s~hj-I)j
dx=- h~[(Q(h)+£h~N) j~dx (49)
dT T
n 3s
n 3
x
where
F
= [P (h) + s( I + e~hj 1)/3 e~] dx
n
and
P(h)=-jQ(h)dh=-~+~)hln ~ +~h.
I+Bh 6
The right-hand side of equation (49) is always non-negative, therefore
$
« o (50)
Note also that equation (48) can be written as
~~~~~~3%~~4~~'3F (51)
This fornl is reminiscent of the Cahn-Hilliard equation with F playing the role of free energy and h~ that of mobility coefficient.
Relations (46)-(51) provide a start for any meaning full analysis of equation (44).
3. Numerically aided studies.
Our numerical investigation was limited to one-dimensional study, equation (44) with
periodic boundary conditions. Nevertheless, we have found equation (44) to predict a rich
variety of pattems which are discussed next.
We consider first equation (45) (I.e. B
= 0) in the interval 0 « x « 2 ar amended with the
periodic boundary conditions and initial conditions of the fornl
h(x, 0)
m ho(x)
= h~ + r sin kx (52)
