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Suppression of Dripping from a Ceiling

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An isothermal layer suspended from a surface is gravitationally (Rayleigh-Taylor) unstable. We find that, when a vertical temperature difference DeltaT above a critical value (DeltaT)(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 DeltaT<(DeltaT)(c) agree well with our linear stability analysis. The instability occurs at long wavelengths: the most unstable wavelength at (DeltaT)(c) is infinite.
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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 DT
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 , DT
c
agree well with our linear stability analysis. The instability occurs at long
wavelengths: the most unstable wavelength at DT
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
=?
M
2
Qh
2
=h
Q 2 Fh
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 ds
T
DTrnk兲共1 1 d
g
kdk
g
21
, the Gali-
leo number is G gd
3
nk, the inverse crispation num-
ber is S sdrnk, the two-layer Biot number is F
kk
g
2 1兲共1 1 d
g
kdk
g
21
, Q 1 1 F. The thick-
ness and thermal conductivity of the liquid (gas) layer,
respectively, are d d
g
and k k
g
, and the temperature
difference imposed across the liquid-gas system is DT [7].
The temperature coefficient of surface tension is s
T
2dsdT ; r, n, and k are the liquid density, kinematic
viscosity, and thermal diffusivity, respectively. Time has
been nondimensionalized by d
2
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 . DT
c
, where
FIG. 1. (a) Suspended liquid layer geometry with an imposed
temperature difference DT T
b
2 T
t
. 0. (b) Schematic of
the apparatus.
0031-90070186(7)1203(4)$15.00 © 2001 The American Physical Society 1203
VOLUME
86, NUMBER 7 PHYSICAL REVIEW LETTERS 12F
EBRUARY
2001
DT
c
2
3
rgd
2
s
T
1 1 d
g
kdk
g
2
kk
g
兲共1 1 d
g
d
. (2)
The (dimensionless) growth rate is
gq
G
3
q
2
e2
µ
q
q
cap
2
, (3)
where e 关共DT
c
2DT兴兾共DT
c
is the reduced tempera-
ture, q is the wave number of the perturbation, and q
cap
p
rgd
2
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 , DT
c
, the analysis predicts that the wave
number of the mode with the maximum growth rate is
q
q
cap
q
e2, (4)
and that modes are unstable up to a maximum wave
number, q
max
p
2 q
; 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
of the most unstable mode goes to zero as DT
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
1.2 6 0.1 3 10
24
rad. The thickness of the oil layer,
d 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 rst 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
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
deectometry, in which the oil layer serves as a vari-
able-thickness lens [8]. A ne 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
0.036 cm, 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 bn
air
2 n
oil
兲共2d
s
n
s
1 d
w
n
w
,
where jb1 is the local slope of the interface; n
air
1,
n
oil
1.41, n
s
1.77, and n
w
1.33 are the indices of
refraction for air, oil, sapphire, and water, respectively, and
d
s
0.3175 cm and d
w
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 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 , DT
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 sufciently
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 , DT
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
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VOLUME
86, NUMBER 7 PHYSICAL REVIEW LETTERS 12F
EBRUARY
2001
FIG. 3. Drop formation: (a) A sequence of grid images (at
t 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 eld 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 sufciently large DT. Indeed, we nd that sus-
pended layers can be stabilized and, once stabilized, they
can be maintained indenitely 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 of the most unstable wave number
q
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 gq
(solid curve) and measured drop
growth rate . In (a) and (b), as in Fig. 4, we set DT , DT
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 19.3 K DT . DT
c
. The temperature was
decreased at different rates, but in both cases the drop formed
at 9.7 6 0.2 K.
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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 at 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
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 12.Ford
g
0.0275 cm F
0.35, 0.0206 cm F 0.45, and 0.0138 cm F 0.65,
the observed DT at onset was 30% below, 10% below, and
30% above the predicted value, respectively. In a study
of the stably stratied case [10], a dependence of onset
on initial interface prole was found to be minimal for
small jF 2 12j and stronger for large jF 2 12j. 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 , DT
c
and
then the oil layer was inverted, while for Fig. 5(c) we set
DT . DT
c
before inversion. This difference, together
with the pinning, may be the source of the difference be-
tween the DT
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 deectometry 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 nite (rather than innite) 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 Ofce 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 rst 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 0.969
a
, 0.00118
b
gcm
3
; n
2.0
a
, 0.1568
b
cm
2
s a 0.00096
a
, 0.00339
b
K
21
; k
15500
a
, 2650
b
erg兾共cm s K; C
P
1.469
a
, 1.005
c
3
10
7
erg兾共gK兲兴; k
k
rC
P
0.001088, 0.2235 cm
2
s.
The surface tension is s 21.0
a
dyncm and s
T
0.068
d
dyn兾共cm K. 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
DT1 1 d
g
kdk
g
21
.
[8] M. Fermigier, L. Limat, J. E. Wesfreid, P. Boudinet, and
C. Quilliet, J. Fluid Mech. 236
, 349 (1992).
[9] A correction for nite cell size, 1 2 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
... For instance, Deissler and Oron theoretically found that cooling the substrate could produce a steady, nonruptured suspending film, which was explained by minimizing an appropriate Lyapunov functional of the system [26]. Afterwards, Burgess et al. experimentally demonstrated that heating the liquid-gas interface could also prevent the liquid droplets dripping from the ceiling when the produced temperature gradient was above a critical value [27]. More recently, the experimental finding of Burgess et al. [27] was theoretically explained by Alexeev and Oron [28], where the time-dependent Navier-Stokes (NS) and long-wave evolution equations in both two-dimensional and three-dimensional situations were solved and analyzed. ...
... Afterwards, Burgess et al. experimentally demonstrated that heating the liquid-gas interface could also prevent the liquid droplets dripping from the ceiling when the produced temperature gradient was above a critical value [27]. More recently, the experimental finding of Burgess et al. [27] was theoretically explained by Alexeev and Oron [28], where the time-dependent Navier-Stokes (NS) and long-wave evolution equations in both two-dimensional and three-dimensional situations were solved and analyzed. In the above-mentioned studies [26][27][28], the substrate was horizontally placed. ...
... More recently, the experimental finding of Burgess et al. [27] was theoretically explained by Alexeev and Oron [28], where the time-dependent Navier-Stokes (NS) and long-wave evolution equations in both two-dimensional and three-dimensional situations were solved and analyzed. In the above-mentioned studies [26][27][28], the substrate was horizontally placed. However, how the thermocapillary stress controls or modifies the RT instability for the situation of an inclined substrate remains unclear. ...
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... When placed over a less dense medium, a liquid layer will typically collapse downwards if it exceeds a certain size, as gravity acting on the lower liquid interface triggers a destabilizing effect called a Rayleigh-Taylor instability 1,2 . Of the many methods that have been developed to prevent the liquid from falling [3][4][5][6] , vertical shaking has proved to be efficient and has therefore been studied in detail [7][8][9][10][11][12][13] . Stabilization is the result of the dynamical averaging effect of the oscillating effective gravity. ...
Preprint
When placed upside down a liquid surface is known to destabilize above a certain size. However, vertical shaking can have a dynamical stabilizing effect. These oscillations can also make air bubbles sink in the liquid when created below a given depth. Here, we use these effects to levitate large volumes of liquid above an air layer. The loaded air layer acts as a spring-mass oscillator which resonantly amplifies the shaking amplitude of the bath. We achieve stabilization of half a liter of liquid with up to 20 cm width. We further show that the dynamic stabilization creates a symmetric Archimedes' principle on the lower interface as if gravity was inverted. Hence, immersed bodies can float upside down under the levitated liquid.
Article
We discuss the effect of odd viscosity on Rayleigh–Taylor instability of a thin Newtonian liquid film with broken time-reversal symmetry as it flows down a uniformly heated, inclined substrate. Although considerable experimental and theoretical studies have been performed regarding Rayleigh–Taylor instability, there is still a need to understand the instability mechanism in the presence of odd viscosity, which creates nondissipative effects. Odd viscosity represents broken time reversal and parity symmetries in the two-dimensional active chiral fluid and characterizes deviation of the system from one that contains a passive fluid. Adopting the long-wave approach allows a nonlinear free surface evolution equation of the thin film that considers the influence of odd viscosity to be derived. New, interesting linear stability analysis results illustrate that larger odd viscosity leads to a lower perturbation growth rate ω r and cutoff wave number k c . In other words, odd viscosity has a stabilizing effect on the Rayleigh–Taylor instability. Numerical simulations are conducted using the method of lines to solve the nonlinear evolution equation. The numerical results show that enhancing the odd viscosity effect suppresses the disturbance amplitude and wave frequency. In addition, the numerical results show that the inclination angle and the Weber number have stabilizing effects on the Rayleigh–Taylor instability. However, the Biot number has the opposite effect when the thin liquid film conductivity is poor. Also, the oscillation tends to accumulate downstream of the inclined substrate if the evolution time is sufficiently long.
Article
The Rayleigh-Taylor instability in viscosity varying fluid layers is studied. A heavier fluid layer is superimposed on a lighter fluid layer. Both the fluid layers are bounded between two parallel isothermal horizontal walls. Out of the two walls, one wall is isothermally heated, while the other wall is isothermally cooled. The Rayleigh-Taylor instability is studied for axisymmetric configuration. Viscosities of both the fluids are considered to be varying with temperature. The effect of viscosity ratio, temperature ratio, heating location, Prandtl number and Weber number on the instability is studied. When the viscosity of fluid layers vary with temperature, the configuration becomes more unstable compared to that for constant viscosity fluid layers. For varying viscosity fluid layers the spike undergoes large deformations. At lower viscosity ratios, mushroom shaped spike is formed with elongated skirt structure. Whereas, at higher viscosity ratios, spike with fluid column structure is formed. It is found that increasing viscosity ratio shows stabilizing effect. Increasing temperature ratio is found to show destabilizing effect with complex spike structure formation at high temperature ratios. In top wall heating configuration, formation of mushroom shaped spike and skirt of the spike occurs earlier in a thinner and elongated form compared to that of bottom wall heating configuration. Prandtl number showed insignificant effect on the instability. For the parameter values of the study, when Weber is lower than 55 the configuration is stable and does not undergo instability. Overall, surface tension has shown stabilizing effect on the instability.
Article
Pendant drops suspended on the underside of a wet substrate are known to accumulate fluid from the surrounding thin liquid film, a process that often results in dripping. The growth of such drops is hastened by their ability to translate over an otherwise uniform horizontal film. Here we show that this scenario is surprisingly reversed when the substrate is slightly tilted (≈2°); drops become too fast to grow and shrink over the course of their motion. Combining experiments and numerical simulations, we rationalize the transition between the conventional growth regime and the previously unknown decay regime we report. Using an analytical treatment of the Landau-Levich meniscus that connects the drop to the film, we quantitatively predict the drop dynamics in the two flow regimes and the value of the critical inclination angle where the transition between them occurs.
Article
Instability of a thin viscous film flowing under an inclined substrate: steady patterns - Volume 898 - Gaétan Lerisson, Pier Giuseppe Ledda, Gioele Balestra, François Gallaire
Article
Full-text available
The Rayleigh-Taylor instability of a viscous liquid superimposed upon air can be dynamically stabilized by oscillating the liquid perpendicularly to its horizontal equilibrium surface, thus maintaining the position of the liquid for an arbitrary time. The viscosity of the liquid was found to have a strong influence on the stability of the short-wavelength modes. In the parameter regime investigated the effect of the compressibility of the gas was negligible.
Article
Full-text available
We study experimentally and theoretically the evolution of two-dimensional patterns in the Rayleigh—Taylor instability of a thin layer of viscous fluid spread on a solid surface. Various kinds of patterns of different symmetries are observed, with possible transition between patterns, the preferred symmetries being the axial and hexagonal ones. Starting from the lubrication hypothesis, we derive the nonlinear evolution equation of the interface, and the amplitude equation of its Fourier components. The evolution laws of the different patterns are calculated at order two or three, the preferred symmetries being related to the non-invariance of the system by amplitude reflection. We also discuss qualitatively the dripping at final stage of the instability.
Article
This paper discusses mechanisms for stabilizing the surfaces of liquids by application of variable force fields including parametric stabilization. The literature is surveyed and new data of the present authors are presented. The various stability situations attainable by application of various physical forces are described in graphical form and by means of photographs.
Article
Surface-tension-driven Bénard (Marangoni) convection in liquid layers heated from below can exhibit a long-wavelength primary instability that differs from the more familiar hexagonal instability associated with Bénard. This long-wavelength instability is predicted to be significant in microgravity and for thin liquid layers. The instability is studied experimentally in terrestrial gravity for silicone oil layers 0.007 to 0.027 cm thick on a conducting plate. For shallow liquid depths (<.017 cm for 0.102 cm2 s[minus sign]1 viscosity liquid), the system evolves to a strongly deformed long-wavelength state which can take the form of a localized depression () or a localized elevation (), depending on the thickness and thermal conductivity of the gas layer above the liquid. For slightly thicker liquid depths (0.017–0.024 cm), the formation of a dry spot induces the formation of hexagons. For even thicker liquid depths (>0.024 cm), the system forms only the hexagonal convection cells. A two-layer nonlinear theory is developed to account properly for the effect of deformation on the interface temperature profile. Experimental results for the long-wavelength instability are compared to our two-layer theory and to a one-layer theory that accounts for the upper gas layer solely with a heat transfer coefficient. The two-layer model better describes the onset of instability and also predicts the formation of localized elevations, which the one-layer model does not predict. A weakly nonlinear analysis shows that the bifurcation is subcritical. Solving for steady states of the system shows that the subcritical pitchfork bifurcation curve never turns over to a stable branch. Numerical simulations also predict a subcritical instability and yield long-wavelength states that qualitatively agree with the experiments. The observations agree with the onset prediction of the two-layer model, except for very thin liquid layers; this deviation from theory may arise from small non-uniformities in the experiment. Theoretical analysis shows that a small non-uniformity in heating produces a large steady-state deformation (seen in the experiment) that becomes more pronounced with increasing temperature difference across the liquid. This steady-state deformation becomes unstable to the long-wavelength instability at a smaller temperature difference than that at which the undeformed state becomes unstable in the absence of non-uniformity.
Article
An apparatus for accelerating small quantities of various liquids vertically downwards at accelerations of the order of 50g ( g being 32.2 ft./sec. ² ) is described, and the behaviour of small wave-like corrugations initially imposed on the upper liquid surface has been observed by means of high-speed shadow photography. The instability observed under a wide variety of experimental conditions has been analyzed, and the initial phases have been found to agree well with the first-order theory given in part I. When the disturbance has attained a considerable amplitude the first-order equations cease to apply and it changes from a wave into a form which has the appearance of large round-ended columns of air extending into the liquid and separated by narrow sheets of liquid. The air columns attain a steady velocity relative to the accelerating liquid and continue to penetrate into the liquid until the lower surface of the liquid is reached. In spite of these very large surface disturbances, the main body of liquid below them is accelerated as though they did not exist.
Article
It is shown that, when two superposed fluids of different densities are accelerated in a direction perpendicular to their interface, this surface is stable or unstable according to whether the acceleration is directed from the heavier to the lighter fluid or vice versa. The relationship between the rate of development of the instability and the length of wave-like disturbances, the acceleration and the densities is found, and similar calculations are made for the case when a sheet of liquid of uniform depth is accelerated.
Article
Recent experimental results [J. Fluid Mech. 345, 45 (1997)] for long-wavelength surface-tension-driven rupture of thin liquid layers (∼0.01 cm) found the onset for significantly smaller imposed temperature gradients than predicted by linear stability analyses that assume an initially flat interface with periodic boundary conditions. The presence of sidewalls and other aspects of the experiment, however, led to deformed interfaces even with no imposed temperature gradient. These sidewall effects were not due to a small system size since experiments with aspect ratios as large as 450 were significantly affected. The stability analysis presented here takes into account the effects of the deformed interface profile and shows that these effects account for some of the disagreement between experiment and theory. In addition, deviations from standard linear stability theory caused by these effects have the same qualitative behavior as the deviations seen in the experiments.
Article
The linear hydrodynamic stability analysis of liquid pools heated from below combining surface tension and buoyancy effects as presented by Nield (1964) is confirmed by experiment for a series of silicone oils. The experimental method used is an adaptation of the Schmidt–Milverton technique, in which the stability limit is located by the change of slope in the plot of heat flux versus temperature drop across the liquid pool.
Article
The linearized stability problem for steady, cellular convection resulting from gradients in surface tension is examined in some detail. Earlier work by Pearson (1958) and Sternling & Scriven (1959, 1964) has been extended by considering the effect of gravity waves. In order to avoid the use of an assumed coupling mechanism at the interface, the relevant dynamical equations were retained for both phases. It is shown that the existence of a critical Marangoni number is assured, and that for many situations this critical value is essentially that which is appropriate to the case of a non-deformable interface. Usually, surface waves are important only at very small wave-numbers, but they are dominant for unusually thin layers of very viscous liquids.