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VOLUME
86, NUMBER 7 PHYSICAL REVIEW LETTERS 12F
EBRUARY
2001
Suppression of Dripping from a Ceiling
John M. Burgess,* Anne Juel,
†
W. D. McCormick, J. B. Swift, and Harry L. Swinney
‡
Center for Nonlinear Dynamics and Department of Physics, The University of Texas at Austin, Austin, Texas 78712
(Received 18 July 2000)
An isothermal layer suspended from a surface is gravitationally (Rayleigh-Taylor) unstable. We find
that, when a vertical temperature difference DT above a critical value 共DT 兲
c
is imposed across the
liquid-gas layer system (heated from below), the restoring force provided by the temperature-dependent
surface tension (thermocapillarity) can stabilize the layer. Our measurements of the most unstable wave
number for DT , 共DT兲
c
agree well with our linear stability analysis. The instability occurs at long
wavelengths: the most unstable wavelength at 共DT兲
c
is infinite.
DOI: 10.1103/PhysRevLett.86.1203 PACS numbers: 47.20.Dr, 47.20.Ma, 68.15. +e
A layer of water suspended from a ceiling will drip,
as anyone with a leaky roof can attest. In general, any
thin liquid coating applied to the underside of a surface
will drip. That is, the interface between a liquid layer
above a gas layer is unstable to infinitesimal deformations,
as shown in the classic works of Rayleigh, Taylor, and
Lewis [1]. Suppression of this Rayleigh-Taylor instability
was demonstrated by applying vertical oscillations [2] and
was predicted for applied electric fields [3] and tempera-
ture gradients [4]. In our paper, we demonstrate that the
Rayleigh-Taylor instability can be suppressed with a ver-
tical temperature gradient.
A perturbation of the depth of the liquid layer produces
thicker regions where the interface becomes warmer and
thinner regions where it becomes colder. Since the surface
tension decreases with temperature, fluid is pulled along
the interface from the warmer regions of lower surface
tension toward the colder regions of higher surface tension,
as indicated in Fig. 1(a). Thus thermocapillary stresses act
to stabilize the liquid layer [4].
Our experiments examine the stability of a silicone oil
layer of thickness d (0.0125 cm) suspended above a gas
layer of thickness d
g
(0.0275 cm) with an imposed tem-
perature on the lower boundary hotter than the temperature
of the upper boundary [see Fig. 1(a)]. Properties of the flu-
ids are given in [5]. We will first describe our analysis and
experimental methods and then will present a comparison
of experiment and theory.
Our analysis uses a modified form of an evolution
equation obtained by VanHook et al. [6], who studied
the stably stratified problem of a gas layer above a liquid
layer; in that case thermocapillarity is destabilizing and
gravity is stabilizing, while in our case thermocapillarity
is stabilizing and gravity is destabilizing. With a change
of sign of the temperature gradient and gravitational
acceleration terms, the evolution equation for the liquid
depth h from [6] applies to our thin-film Rayleigh-Taylor
instability:
≠h
≠t
苷 =?
Ω
M
2
Qh
2
=h
共Q 2 Fh兲
2
2
G
3
h
3
=h 2
S
3
h
3
=
2
=h
æ
,
(1)
where the three terms in the brackets describe ther-
mocapillary, gravitational, and surface tension effects,
respectively. The Marangoni number for the liquid layer
is M 苷 共ds
T
DT兾rnk兲共1 1 d
g
k兾dk
g
兲
21
, the Gali-
leo number is G 苷 gd
3
兾nk, the inverse crispation num-
ber is S 苷 sd兾rnk, the two-layer Biot number is F 苷
共k兾k
g
2 1兲共1 1 d
g
k兾dk
g
兲
21
, Q 苷 1 1 F. The thick-
ness and thermal conductivity of the liquid (gas) layer,
respectively, are d 共d
g
兲 and k 共k
g
兲, and the temperature
difference imposed across the liquid-gas system is DT [7].
The temperature coefficient of surface tension is s
T
⬅
2ds兾dT ; r, n, and k are the liquid density, kinematic
viscosity, and thermal diffusivity, respectively. Time has
been nondimensionalized by d
2
兾k and length by d.
A linear stability analysis of (1) about a flat interface
with periodic boundary conditions yields a stable layer for
DT . 共DT兲
c
, where
FIG. 1. (a) Suspended liquid layer geometry with an imposed
temperature difference DT 苷 T
b
2 T
t
. 0. (b) Schematic of
the apparatus.
0031-9007兾01兾86(7)兾1203(4)$15.00 © 2001 The American Physical Society 1203
VOLUME
86, NUMBER 7 PHYSICAL REVIEW LETTERS 12F
EBRUARY
2001
共DT兲
c
苷
2
3
rgd
2
s
T
共1 1 d
g
k兾dk
g
兲
2
共k兾k
g
兲共1 1 d
g
兾d兲
. (2)
The (dimensionless) growth rate is
g共q兲 苷
G
3
q
2
∑
e2
µ
q
q
cap
∂
2
∏
, (3)
where e 苷 关共DT兲
c
2DT兴兾共DT兲
c
is the reduced tempera-
ture, q is the wave number of the perturbation, and q
cap
苷
p
rgd
2
兾s is the (dimensionless) capillary wave number.
The predicted growth rate dependence on wave number
and temperature is shown in Fig. 2 for the conditions of
our experiment [5].
For DT , 共DT兲
c
, the analysis predicts that the wave
number of the mode with the maximum growth rate is
q
ⴱ
苷 q
cap
q
e兾2, (4)
and that modes are unstable up to a maximum wave
number, q
max
苷
p
2 q
ⴱ
; perturbations to the height of the
interface with q . q
max
will have negative growth rates.
This is a long wavelength instability: The wave number q
ⴱ
of the most unstable mode goes to zero as 共DT兲
c
is
approached.
Our oil-air layer system is sandwiched between two
sapphire windows (5.0 cm in diameter), each of which is
in contact with a temperature-controlled circulating water
bath [see Fig. 1(b)]. DT is controlled to 60.05 K, which
is small compared to the typical DT , 10 K. The win-
dows are adjusted interferometrically, parallel and level to
共1.2 6 0.1兲 3 10
24
rad. The thickness of the oil layer,
d 苷 0.0125 6 0.0003 cm, is set by the height of a brass
sidewall ring that encircles the oil layer; the horizontal sur-
face of the ring is treated with a nonwetting agent.
The oil layer is prepared by first inverting the window
encircled by the brass sidewall and then applying the oil
using a microliter glass syringe. The viscous oil (200 cS)
spreads slowly over the window and pins at the corner of
the sidewall. After several hours, this window is reinverted
to obtain a suspended oil layer, as shown in Fig. 1(b).
0 0.02 0.04 0.06 0.08 0.10 0.12
−0.010
−0.005
0
0.005
0.010
q
γ (s
−1
)
∆T = 20 K
10 K
0 K
−10 K
14.9 K
FIG. 2. Predicted growth rate g [(3) dimensionalized by d
2
兾k]
as a function of the dimensionless wave number q for several
applied temperature differences DT.ForDT . 14.9 K, the
layer is predicted to be stable for all wave numbers.
The oil-air layer interface position is visualized using
deflectometry, in which the oil layer serves as a vari-
able-thickness lens [8]. A fine grid is imaged through
the oil layer using a CCD camera interfaced to a com-
puter. The grid, a thin transparent plastic sheet with square
transparent spaces (0.018 cm wide) between the grid lines
共0.036 cm兲, is placed directly on the upper bath top win-
dow. Light passing through the oil layer is refracted at the
oil-air interface, distorting the image of the grid as the in-
terface deforms. The displacement of an imaged grid point
(in units of d)isd 苷 b共n
air
2 n
oil
兲共2d
s
兾n
s
1 d
w
兾n
w
兲,
where jbjø1 is the local slope of the interface; n
air
苷 1,
n
oil
苷 1.41, n
s
苷 1.77, and n
w
苷 1.33 are the indices of
refraction for air, oil, sapphire, and water, respectively, and
d
s
苷 0.3175 cm and d
w
苷 2.54 cm are the thickness of
each sapphire window and the water between the windows
in each bath, respectively. Then, with d in units of pix-
els in the image, we have b 艐 0.008d. The minimum
slope we can detect is about 0.3
±
; thus a mode with a half-
wavelength equal to the width of the layer must have an
amplitude comparable to the layer depth to be detectable.
The formation of a drop is illustrated in Fig. 3. The
cumulative displacement vectors for the grid points in the
images [Fig. 3(a)] are shown in Fig. 3(b); the vector slope
of the interface is calculated at each grid point; numerical
integration then yields the surface height [Fig. 3(c)].
Measurements of the most unstable wave number as
a function of DT are compared with our linear stability
analysis in Fig. 4. Theory and experiment agree well. To
obtain each data point, we set DT , 共DT兲
c
, inverted the
oil layer, and observed the emergence of deformations us-
ing the technique described above. We determine the most
unstable wave number by scaling the average width of a
reconstructed drop at half its maximum height by 3, since
the width at half the maximum height of a sinusoidal defor-
mation is one-third of the full wavelength. We determine
the wave number by this method since Fourier analysis
is inappropriate for a long wavelength instability, where
the wavelength is comparable to the width of the system.
Far below the onset of instability the wave number would
be large enough so that many drops would form, produc-
ing more regular patterns if the air layer were sufficiently
thick, as has been observed for an isothermal suspended
layer [8].
The growth rate of the most unstable wave number pre-
dicted by linear analysis assumes exponential time de-
pendence of the most unstable mode in the perturbation.
We examine this point by setting DT , 共DT 兲
c
and then
inverting the oil layer. We then observe an initial ex-
ponential increase in the depth of a drop, as Fig. 5(a)
illustrates. At longer times there is a transition to a con-
stant growth rate; this change in behavior is likely due to
an increase in nonlinear effects as the drop grows larger.
Comparison of theory and experiment for the regime with
exponential growth is shown in Fig. 5(b). The data ex-
hibit a decrease in growth rate with increasing DT as
1204
VOLUME
86, NUMBER 7 PHYSICAL REVIEW LETTERS 12F
EBRUARY
2001
FIG. 3. Drop formation: (a) A sequence of grid images (at
t 苷 0, 720, 760, and 800 s) illustrating the formation of an
isothermal drop. The width of the region shown is 1.2 cm.
(b) The accumulated vector displacement field of the grid points.
(c) A cross section through the center of the reconstructed drop
(the ordinate zero is at the original position of the interface);
the surface ripples arise from sparse sampling of the image,
while the asymmetry is due to the formation of a nearby drop.
In (c) the vertical scale is expanded 25 times compared to the
horizontal scale.
predicted, but the uncertainty in the data is large because
the exponential regime is observable over only a small frac-
tion of an e-folding time.
The observed decrease in the drop growth rate toward
zero with increasing DT [Fig. 5(b)] suggests that it should
be possible to stabilize the gravitationally unstable oil layer
with a sufficiently large DT. Indeed, we find that sus-
pended layers can be stabilized and, once stabilized, they
can be maintained indefinitely — some layers have been
maintained stable for more than a week. However, most of
our attempts to obtain a stable inverted layer were not suc-
cessful because the process of producing a suspended layer
by inverting an upward-facing layer inevitably produced a
large amplitude perturbation at the longest possible wave-
length, which is the most unstable mode: If the window is
inverted slowly, the layer is thicker at its lower end and a
drop forms there, while, if the window is inverted rapidly,
the layer is thicker at the upper end and a drop forms there.
0 0.05 0.10 0.15
−20
−10
0
10
20
q
∆ T (K)
q*
q
max
2πd/L
FIG. 4. Measurements 共≤兲 of the most unstable wave number
q
ⴱ
compared with the prediction of a linear stability analysis
(dashed curve). The solid curve corresponds to the maximum
unstable wave number (see text). The measurements are made
immediately before a drop touches the bottom boundary; the
maximum deformation of a drop is typically only 1% of the
measured wavelength. The vertical dotted line shows the value
of q corresponding to the diameter of the experimental system.
The uncertainties in q are approximately equal to the size of
a data point in the plot. Negative DT corresponds to the top
boundary hotter than the bottom boundary.
Nevertheless, with DT set a few degrees above critical, we
were able to produce stable layers about 10% of the time.
The lower limit of stability of stabilized layers was de-
termined by decreasing DT in small steps, starting with
−10 0 10
0
0.005
0.010
0.015
∆ T (K)
γ (s
−1
)
700 800 900
0
0.01
0.02
Time (s)
Drop amplitude (cm)
0 10 20 30 40 50 60
8
10
12
14
16
18
20
Time (hours)
∆ T (K)
(a)
(c)
(b)
FIG. 5. Drop growth: (a) The drop size grows exponentially
(dashed curve) at early times and linearly (solid line) at later
times. (b) Predicted g共q
ⴱ
兲 (solid curve) and measured drop
growth rate 共≤兲. In (a) and (b), as in Fig. 4, we set DT , 共DT 兲
c
before inversion of the layer. (c) Two experimental determi-
nations of the loss of stability of a liquid layer initially sta-
bilized at DT 苷 19.3 K 关DT . 共DT兲
c
兴. The temperature was
decreased at different rates, but in both cases the drop formed
at 9.7 6 0.2 K.
1205
VOLUME
86, NUMBER 7 PHYSICAL REVIEW LETTERS 12F
EBRUARY
2001
a layer that had been stabilized for approximately 4 h.
The results were reproducible for a wide range of tem-
perature histories, as Fig. 5(c) illustrates for two cases.
The DT at which the layer became unstable, 9.7 K, was
lower than that given by (2), 14.9 K [9], perhaps because,
after an inverted layer had relaxed to form a flat layer,
the pinned edge of the layer suppressed the most unstable
(long wavelength) mode. We measured the instability for
three air layer depths [including d
g
苷 0.0275 cm corre-
sponding to Figs. 4 and 5(c)] and found that the difference
between the observed DT at onset and the predicted value
changes sign at F 艐 1兾2.Ford
g
苷 0.0275 cm 共F 苷
0.35兲, 0.0206 cm 共F 苷 0.45兲, and 0.0138 cm 共F 苷 0.65兲,
the observed DT at onset was 30% below, 10% below, and
30% above the predicted value, respectively. In a study
of the stably stratified case [10], a dependence of onset
on initial interface profile was found to be minimal for
small jF 2 1兾2j and stronger for large jF 2 1兾2j. The
pinning condition was found to be crucial in determining
the non-uniform base interface shape in [10]. Therefore,
we conjecture that pinning of the liquid layer at the side-
wall is a source of the deviation from the predictions of
linear analysis we observe. We emphasize that the data in
Figs. 4 and 5(c) were obtained under different protocols:
for Fig. 4 and Figs. 5(a) and 5(b) we set DT , 共DT 兲
c
and
then the oil layer was inverted, while for Fig. 5(c) we set
DT . 共DT兲
c
before inversion. This difference, together
with the pinning, may be the source of the difference be-
tween the 共DT兲
c
measured in Fig. 5(c) and the data in
Fig. 4.
In summary, we have demonstrated thermocapillary sta-
bilization of a Rayleigh-Taylor unstable interface. Future
experiments could use interferometry to detect the growth
of the most unstable (long wavelength) mode, which is
not observable in our deflectometry measurements. Future
experiments should also examine the effect of boundary
conditions by varying the wetting properties and geome-
try of the sidewalls and by examining larger width layers
and various liquid layer depths. Such measurements to-
gether with linear and nonlinear analyses of the evolution
equation (1) including realistic boundary conditions with
sidewall pinning in a finite (rather than infinite) width layer
should provide a better understanding of thermocapillary
stabilization of Rayleigh-Taylor unstable layers.
We thank S. VanHook, D. Goldman, M. Shattuck,
P. Matthews, and E. Knobloch for helpful discussions.
This work was supported by the Office of Naval Re-
search and by NASA Grants No. NAG3-1839 and
No. NCCS5-154.
*Electronic address: jburgess@chaos.ph.utexas.edu
†
Present address: Department of Physics and Astronomy,
University of Manchester, Manchester M13 9PL, United
Kingdom.
‡
Electronic address: swinney@chaos.ph.utexas.edu
[1] Lord Rayleigh, Scientific Papers (Cambridge University
Press, Cambridge, England, 1900), Vol. II
; G. I. Taylor,
Proc. R. Soc. London A 201
, 192 (1950); D. J. Lewis, Proc.
R. Soc. London A 202
, 81 (1950).
[2] G. H. Wolf, Phys. Rev. Lett. 24
, 444 (1970).
[3] N. A. Bezdenezhnykh, V. A. Briskman, A. A. Cherepanov,
and M. T. Sharov, Fluid Mech. Sov. Res. 15
, 11 (1986).
[4] B. K. Kopbosynov and V. V. Pukhnachev [Fluid Mech. Sov.
Res. 15
,
95 (1986)] predicted thermocapillary stabilization
of a Rayleigh-Taylor unstable layer; they incorrectly at-
tributed the first prediction to K. A. Smith [J. Fluid Mech.
24
, 401 (1966)]. Stable localized deformations have been
predicted by R. J. Deissler and A. Oron [Phys. Rev. Lett.
68
, 2948 (1992)].
[5] For silicone oil (Dow Corning 200) and air, respec-
tively, at 25
±
C: r 苷 0.969
a
, 0.00118
b
g兾cm
3
; n 苷
2.0
a
, 0.1568
b
cm
2
兾s a 苷 0.00096
a
, 0.00339
b
K
21
; k 苷
15500
a
, 2650
b
erg兾共cm s K兲; C
P
苷 1.469
a
, 1.005
c
3
关10
7
erg兾共gK兲兴; k ⬅
k
rC
P
苷 0.001088, 0.2235 cm
2
兾s.
The surface tension is s 苷 21.0
a
dyn兾cm and s
T
苷
0.068
d
dyn兾共cm K兲. These values are obtained from (a)
Dow Corning 200 Silicone Oils data sheet; (b) Properties
of Materials at Low Temperature, Natl. Bur. Stand. (U.S.)
(Pergamon Press, New York, 1959); (c) H. J. Palmer
and J. C. Berg, J. Fluid Mech. 47
, 779 (1971); (d) D. J.
Poferl and R. Svehla, NASA Technical Note D-7488
(unpublished).
[6] S. J. VanHook, M. F. Schatz, J. B. Swift, W. D. McCormick,
and H. L. Swinney, J. Fluid Mech. 345
, 45 (1997).
[7] M involves the temperature difference DT controlled in the
experiment rather than the temperature difference across
the liquid layer, which is given by (assuming conductive
heat transfer) DT
liq
苷 DT共1 1 d
g
k兾dk
g
兲
21
.
[8] M. Fermigier, L. Limat, J. E. Wesfreid, P. Boudinet, and
C. Quilliet, J. Fluid Mech. 236
, 349 (1992).
[9] A correction for finite cell size, 1 2 关2pd兾共q
cap
L兲兴
2
,gives
only a 3% reduction from (2).
[10] R. Becerril, S. J. VanHook, and J. B. Swift, Phys. Fluids
10
, 3230 (1998).
1206
