PHYSICS OF FLUIDS
VOLUME 16, NUMBER 8
AUGUST 2004
Wave patterns of a rivulet of surfactant solution in a Hele-Shaw cell
W. Drenckhan,a) S. Gatz, and D. Weaire
Department of Physics, Trinity College Dublin, Dublin 2, Ireland
共Received 4 December 2003; accepted 23 April 2004; published online 6 July 2004兲
When a stream of surfactant solution is injected at a constant rate between two narrowly spaced,
vertical glass plates, the straight rivulet becomes unstable at a critical flow rate. Various regimes are
encountered, in which upward and downward traveling waves of remarkable regularity are
observed. We present extensive data for these waves and some tentative theory to account for some
of their properties. © 2004 American Institute of Physics. 关DOI: 10.1063/1.1766211兴
I. INTRODUCTION
work, so far as we are aware. For a more detailed comparison of both systems refer to Sec. V.
As part of an investigation into various instabilities that
are encountered in the physics of foams,1,2 we have explored
the behavior of a single rivulet of surfactant solution, descending under gravity in a vertical Hele-Shaw cell. A surprisingly rich variety of phenomena is observed for flow
rates such that the straight downward motion of the rivulet
becomes unstable, causing it to meander. The meandering
ranges from sinusoidal waves of small amplitude to serpentine waves of larger amplitude. The waves can travel either
upward or downward. More complex waves and disordered
motion are observed in some regimes. The ensemble of these
results presents a complicated scenario and a considerable
challenge to theory, of which we have only some elements at
this stage.
Extensive measurements in one regime are found to conform to a remarkably simple relation between wavelength,
amplitude, and the separation of the two plates.
The experimental arrangement is very elementary 共see
Fig. 1兲. Surfactant solution is injected with a glass nozzle at
various flow rates at the top of two narrowly spaced, vertical
glass plates of variable separation D. The solution used here
is 0.4% commercial ‘‘Fairy’’ dishwashing liquid in deionized
water. Variable flow rates are obtained and monitored by
attaching a flow meter with a high precision needle valve to
a large tank, supplying a constant pressure head.
As the liquid descends under gravity, a vertical rivulet is
formed between the plates. Figure 2共a兲 indicates its horizontal cross section, consisting of two Plateau borders attached
to the plates and connected by a thin film. At low flow rates,
for which the vertical rivulet is stable, the size of the Plateau
borders is small compared with the plate separation D. In the
regime in which instabilities occur, the Plateau border width
may be comparable to D, and the two borders merge for the
highest flow rates considered here 关Figs. 2共b兲 and 2共c兲兴.
Analogous, but much less regular patterns are familiar
from investigations of the meandering motion of a rivulet on
an inclined hydrophobic plane.3– 6 However, the present system seems to be quite distinct in its behavior and produces
elegant wave patterns that have no counterpart in earlier
II. OBSERVATIONS AND ANALYSIS
A. Main observations
For low flow rate Q, a straight rivulet is observed and
found upon closer observation to have the expected structure
关Fig. 2共a兲兴. Measurement of the width W of the Plateau border as a function of flow rate gives results in good agreement
with
冉 冊 冉 冊
W⫽ 2⫺
⫺ 1/2
f␩
2␳g
1/4
Q 1/4,
共1兲
which is predicted by foam drainage theory.1 Here g is gravity, and ␳ and ␩ are the density and viscosity of the solution,
respectively. The bulk parameters of the solution are effectively unchanged by the presence of surfactants and hence
equal to that of pure water. f is a numerical factor,1 which
depends on the boundary conditions, in particular the surface
mobility of the Plateau border. Our computer simulations
show that this value should be between 10 and 51 for surface
Plateau borders, corresponding to full slip 共low surface viscosity兲 and no slip 共high surface viscosity兲 of the gas/liquid
interface. A typical set of data, see Fig. 3, gives an exponent
of 0.22⫾0.01 and a constant of proportionality of 0.039
⫾0.002 (ms) 1/4. This would imply f ⬇9 and hence extremely mobile interfaces, which we know is not the case, as
Fairy dishwashing liquid produces very rigid interfaces.
Questions posed by this deviation, which is well outside possible measurement errors, are very much at the core of current research activities in foam physics.2,7,8
Figure 4 encapsulates much of the content of this paper.
It indicates the various regimes that are encountered as the
flow rate is increased, starting with regime I, in which the
rivulet remains straight. It becomes unstable at a critical flow
rate Q 1 , above which an upward traveling, sinusoidal wave
is observed. The velocity is finite at the point of onset Q 1 ,
and decreases steadily in regime II up to a second critical
flow rate Q 2 , at which it vanishes. At this point, a stationary
wave is observed. Beyond Q 2 , we observe a motion similar
to the garden hose instability just below the inlet, whose
frequency increases with flow rate. This results in downward
a兲
Electronic mail: [email protected]
1070-6631/2004/16(8)/3115/7/$22.00
␲
2
3115
© 2004 American Institute of Physics
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3116
Phys. Fluids, Vol. 16, No. 8, August 2004
Drenckhan, Gatz, and Weaire
FIG. 3. Variation of Plateau border width W with flow rate. It is fitted to a
power law, conforming well to Eq. 共1兲.
B. Regime II
FIG. 1. Sketch of the experimental setup, incorporating a photograph of a
meandering rivulet. Surfactant solution is injected at various flow rates between two narrowly spaced, vertical glass plates. The rivulet formed by the
liquid becomes unstable at a critical flow rate, producing extraordinarily
regular and stable wave patterns traveling upward or downward.
In regime II the variation of the wave velocity V 共upward and hence negative in our convention兲 with flow rate Q
may be explained or rationalized as follows. In a simple
model we consider the forces acting horizontally on a segment of the slender rivulet at the point of maximum amplitude. We believe the two key ingredients to be the destabilizing centrifugal force F c , caused by the liquid being forced
around the bend of local curvature ␬,
F c ⫽2A Pb ␳ 共 v ⫺V 兲 2 ␬ ,
traveling waves—but of irregular form—in regime III. Only
when a much higher flow rate Q 3 is reached does the pattern
restabilize 共beginning at the inlet兲 such that the pendulum
motion below the inlet generates essentially perfect, downward traveling waves of large-amplitude, serpentine form in
regime IV. In this regime the amplitude increases linearly
close to the inlet, but eventually tends to a limiting value
further down. This value decreases with flow rate, leading to
increasingly sinusoidal wave forms at the end of regime IV.
For even higher flow rates, the rivulet breaks into subrivulets
which, due to the surfactants, do not detach from the main
rivulet and hence generate a foam between the plates. Of the
three regimes which exhibit traveling waves, II and IV have
proven to provide the most reproducible behavior. We shall
concentrate on these for the analysis in Secs. II B and II C.
共2兲
and the stabilizing surface tension force F ␥ related to the
same curvature,
F ␥ ⫽2D ␥␬ .
共3兲
Here v is the average fluid velocity in the Plateau borders, ␳ the fluid density, D the plate separation, A Pb the cross
section of one Plateau border, and ␥ the surface tension.
Liquid flow through the thin film connecting the Plateau borders is neglected in the derivation of F c . The surface tension
force is roughly estimated by ignoring the Plateau borders,
so that a film of width D spans the plate. Note that both
forces are proportional to ␬.
The flow rate Q is related to v and A Pb by the drainage
theory1 in a single Plateau border:
A Pb ⫽
v⫽
冉 冊
冉 冊
f␩
2␳g
1/2
共4兲
Q 1/2 ,
␳g
Q
⫽
2A Pb
2 f␩
1/2
共5兲
Q 1/2.
For a description of the variables see Sec. II A. Equating
共2兲 and 共3兲, and using 共4兲 and 共5兲, we obtain for the wave
velocity
V⫽⫺aQ ⫺ 1/4⫹bQ 1/2,
FIG. 2. The different possible geometries of the rivulet cross section, depending on the flow rate. 共a兲 Two separate Plateau borders connected by a
thin film. 共b兲 Borderline case of the two Plateau borders touching. 共c兲
Merged Plateau borders forming a rivulet whose surfaces have curvature
2/D.
with
a⫽ 共 ␥ D 兲 1/2
冉 冊
2g
␳f␩
共6兲
1/4
and b⫽
冉 冊
␳g
2 f␩
1/2
.
共7兲
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Phys. Fluids, Vol. 16, No. 8, August 2004
Wave patterns of a rivulet of surfactant solution
3117
FIG. 4. Overview of the physical characteristics of the
wave patterns in the four distinct regimes found for
different flow rates. These data are taken for a plate
separation of D⫽2.1 mm. A detailed description of the
phenomena can be found in Sec. II A. 共a兲 Regime. 共b兲
Wave velocity. 共c兲 Wavelength and amplitude. 共d兲 Photograph of the pattern.
This is negative for low Q and becomes positive beyond a
critical value which we identify with Q 2 in Fig. 4. Downward traveling waves, roughly consistent with this trend, are
indeed observed in regime III. But we have refrained from
trying to characterize them here on account of their disordered form. Note furthermore that since both forces scale
with ␬, this model is not elaborate enough to reproduce the
observed wavelengths.
For the steady wave at Q 2 we can establish a relationship between the flow rate and the plate separation by setting
V⫽0 in Eq. 共6兲,
Q 2⫽
冉 冊
2 f ␩ ␥ 2 2/3
D .
␳
g
C. Regime IV
Figure 7 shows the variation of amplitude A and wavelength ␭ with distance from the inlet in regime IV. In some
cases a fitting procedure is needed to infer the limiting
values.24 Unlike in regime II, these values depend on the
nozzle size, with smaller nozzles resulting in higher undulation frequency, higher wave velocity, and smaller
amplitudes/wavelengths. We found, however, that amplitudes
共8兲
Unfortunately, Q 2 is difficult to determine experimentally, as the wave velocity varies gradually over a fairly large
range of flow rates. However, measurements of the flow rate
Q 1 at the very well-defined onset of the instability as a function of plate separation 共Fig. 5兲 display the proportionality
predicted in Eq. 共8兲 very well. We have as yet no theoretical
explanation of the onset of the instability, but the above argument leads us to believe that it occurs at a certain large
fraction of Q 2 . A more elaborate model is necessary, in particular, to account for the measured wavelengths at the onset
of the instability 共see Fig. 6兲.
FIG. 5. Flow rate Q 1 at the onset of the instability as a function of plate
separation D. The data are fitted with the power law Q 1 ⬃D 2/3 关compare Eq.
共8兲兴.
-
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3118
Phys. Fluids, Vol. 16, No. 8, August 2004
Drenckhan, Gatz, and Weaire
FIG. 6. Wavelength ␭ at the onset of the instability as a function of plate
separation. We have no model yet to describe these data.
FIG. 8. All data collapse on the same graph when ␭D ⫺1/2 is plotted
against A.
and wave lengths for all nozzle sizes can be related to the
plate separation D in a very simple way, as all of the acquired data conform well to
because the trend to larger amplitudes, and hence larger
maximum curvature, eventually encounters an upper bound
on the possible curvature of the rivulet. To see this, consider
the same type of argument as advanced above, but applied
only to the thin film which spans the plates between the
Plateau borders. This has negligible inertia and hence negligible centrifugal force: hence the surface tension force on an
element of film must be approximately zero. This means that
the two principal radii of curvature—one parallel to the plane
of the plates and the other perpendicular—must be equal and
opposite. Any increase in the curvature of the rivulet must be
matched by an equal and opposite increase of the transverse
curvature. So long as the Plateau borders are small, this
transverse curvature cannot exceed 2/D, where D is the plate
␭ 2 ⬃AD.
共9兲
This is demonstrated in Fig. 8, which comprises data for
different plate separations and nozzle sizes. The fitted curve
is given by
␭D ⫺ 1/2⫽CA 0.454,
共10兲
where C⫽4.55⫾0.01.
It is clear that the large-amplitude serpentine waves of
regime IV are not a simple continuation of the trends discussed in Sec. II B for lower flow rates. We believe this is
FIG. 7. Wave pattern in regime IV 共here D⫽2.1 mm): 共a兲 Photographs of the rivulet for different flow rates. The length of the pattern is determined
by the instability of large amplitudes. 共b兲 Variation and saturation of amplitude and 共c兲 wavelength with distance from inlet, including the data from the images
of 共a兲.
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Phys. Fluids, Vol. 16, No. 8, August 2004
Wave patterns of a rivulet of surfactant solution
3119
FIG. 9. Observation of the wetting film using monochromatic light interference. Images 1– 6 display successive time steps of a downward sliding wave
in regime IV.
separation. We therefore attribute to the remarkable relation
共9兲 the significance of maintenance of approximate constant
curvature, close to its maximum allowed value. This general
idea may be mathematically expressed in an admittedly simplistic model: The local curvature ␬ (x) of a sinusoidal wave
pattern y(x)⫽Ae ikx with k⫽2 ␲ ␭ ⫺1 can be derived to be
␬共 x 兲⫽
d2y共 x 兲
4␲2
⫽⫺
y 共 x 兲.
dx 2
␭2
共11兲
This, assuming constant curvature ␬ 0 for the points of maximum amplitude, would give us the relationship
␭ 2⫽
4␲2
A.
␬0
共12兲
Assuming that ␬ 0 ⫽2/D, as reasoned above, we obtain
␭D ⫺ 1/2⫽& ␲ A 1/2,
共13兲
identifying the constant of proportionality as C⫽& ␲
⬇4.44, which agrees extraordinarily well with the fitting results from Eq. 共10兲.
However, while this argument may have identified the
correct physical basis for the relation, in detail it relies on
assumptions, namely, small Plateau borders and sinusoidal
waves, which do not apply well in regime IV. Clearly a detailed analysis is called for and should be possible using such
tools as the Surface Evolver9 in combination with more detailed fluid dynamic simulations.
III. THE ROLE OF THE WETTING FILM
Due to the vanishing contact angle of surfactant solution
on glass, the surfaces of the plates are coated with a wetting
film in the vicinity of the moving rivulet. The precise role of
this surrounding film is not clear, but it may well be responsible for some of the patterns and trends observed. In particular, it could have a significant contribution to the stabilization of the wave patterns and hence play a similar role to
that of contact angle hysteresis in the traditional meandering
experiments on hydrophobic surfaces. Stabilizing forces
could, for instance, be related to the Gibbs elasticity of the
wetting film. This could be caused by local depletion of surfactant concentration in the gas/liquid interface as a result of
the rivulet dynamics.
Figure 9 gives some indication of the existence and motion of the wetting film. It shows monochromatic light interference patterns of a downward sliding wave pattern in a
sequence of successive time steps 共0.2 s intervals兲. The interference pattern in the background of the black rivulet dis-
FIG. 10. Some oddities observed in regime IV: 共a兲 Example of an experimentally observed and computationally generated ‘‘beating’’ pattern of two
sinusoidal waves with a wavelength ratio of 5:6. The component with
smaller wavelength travels down faster within the pattern. 共b兲 For large
amplitudes a thickening instability develops on the horizontal sections of the
rivulet. 共c兲 For very high flow rates a localized thickening might develop
and travel downward in the rivulet without spreading out. The image sequence shows this process with the images being 0.2 s apart.
plays the thickness variations of the wetting film as the rivulet is sliding across it (V⫽20 mm/s). Surprisingly, this
pattern seems to be more or less static with respect to the
glass plates, rather than to the rivulet.
IV. FURTHER OBSERVATIONS
In view of the wide extent of the data presented here, we
will not elaborate it with any detailed consideration of various further effects that we have observed. These include the
occasional observation of waves which appears to contain
two Fourier components, ‘‘beating’’ together 关Fig. 10共a兲兴.
Also, for wave patterns with large amplitudes or at very high
flow rates, there is a local thickening of the rivulet, leading to
‘‘varicose’’ instabilities 关Figs. 10共b兲 and 10共c兲兴.
The rivulet width varies significantly along the wave pattern for high flow rates, with the horizontal parts of the rivulet being distinctively thicker than the vertical parts. There
does not seem to be a general rule for the respective rivulet
cross sections 共according to Fig. 2兲 required to produce a
stable wave pattern, as these depend on the plate separation
D. We can state, however, that if D is chosen such that the
vertical parts of the rivulet have a cross-section of type a
共Fig. 2兲, the waves are much better behaved. We believe that
this can be attributed to the stabilizing effect of the thin film
formed between the Plateau borders.
We occasionally observed that the wavelength decreased
with flow rate in regime II, leading to a dispersion relation
with two branches sharing the same wavelength and wave
velocity at the onset of the instability. There is a possibility
that this might be related to temperature effects, as the experiment seems to be quite sensitive to temperature changes.
Unlike what is observed in the meandering on nonwetting planes,10 hysteresis does not seem to play a role in this
experiment as long as the plates are wetted evenly. This
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3120
Phys. Fluids, Vol. 16, No. 8, August 2004
means that a pattern obtained for a specific flow rate Q does
not depend on whether Q has been increased or decreased up
to that point.
Attempts to reproduce the patterns with chemically more
pure and better characterized surfactants than Fairy liquid
have not proven successful in generating such regular and
stable patterns. Using, furthermore, pure liquids with very
low surface tension 共e.g., silicone oil兲 resulted in irregular
patterns. Experiments with pure deionized water between
glass or Plexiglas plates produce patterns very similar to
those encountered in the meandering of water on hydrophobic surfaces 共refer to Sec. V兲, displaying very clearly the
effects of pinning and contact angle hysteresis. But despite
the lack of regularity of the wave patterns, all systems mentioned in this paragraph display the well-defined instability at
a critical flow rate.
V. COMPARISON WITH RIVULET MEANDERING ON
INCLINED PLANES
Despite its significance for industrial applications and
fundamental science, research on stream meandering has
been surprisingly limited. Most of the experimental works up
to date investigate the properties of rivulets of various liquids
on inclined, nonwetting planes.3–5,10–13 As in our case, these
systems display several regimes with fairly well-defined
transition regions. With increasing flow rate these regimes
have been identified to be 共1兲 individual droplets, 共2兲 straight
rivulet, 共3兲 stable meandering rivulet, 共4兲 unstable meandering rivulet, and 共5兲 restable rivulet. Due to the irregularity of
the patterns, quantitative analysis has been restricted to measurements of the sinuosity of the meanders 共ratio of total
length of the rivulet to the distance between inlet and bottom
of rivulet兲. Several attempts have been made to map out
phase diagrams for the several regimes, with one of the key
parameters being the inclination of the surface. Theoretical
work has been similarly scarce, being limited to rather specific aspects of the problem or suffering from fairly strong
simplifications.14 –20 Some of the models predict aspects of
the experimental results, but fail to describe a more complete
picture. In general, the gap between theory and experiment
seems to be rather large, seemingly dominated by the
struggle to fully understand and implement the role of contact angle hysteresis.
Our system resembles that described above, the key
similarities being the stabilizing effect of surface tension
forces and the destabilizing effect of centrifugal forces of the
liquid being forced around bends in the rivulet. However, as
a result of the different set-up and wetting properties, certain
aspects differ significantly and might help to overcome some
of the difficulties encountered in the traditional work on meandering phenomena. The advantage of our system lies in its
accessibility and the remarkable regularity of the wave patterns observed. Due to the rivulet being contained between
two vertical surfaces, its cross section is well defined and
measurable. Furthermore, the existence of a wetting film 共see
Sec. III兲 eliminates the problems of contact angle hysteresis
and pinning effects and hence leads to nearly perfect, traveling wave patterns, which can easily be analyzed and com-
Drenckhan, Gatz, and Weaire
pared to theoretical models in terms wavelength, amplitude,
local curvature, or rivulet width.
Detailed comparison between experimental results for
stream meanders is difficult as a result of the different geometries and physical properties of the investigated systems and
the different parameters which researchers have focused on.
However, a few general comparisons can be made: We do
not observe rivulet breakup into separate droplets 共low flow
rate兲 or subrivulets 共high flow rate兲 as a result of the surfactant loaded interfaces forming highly stable films. The overall characteristics of the various regimes, however, seem to
be similar and we are tempted to draw parallels between the
‘‘stable meandering’’ and our regime II, and between the
‘‘unstable meandering’’ and our regime IV. Nakagawa4 reports an increase of sinuosity with flow rate in the first and a
decrease in the latter, which is what we see in terms of increasing 共II兲 and decreasing 共IV兲 wavelengths and amplitudes. The ‘‘unstable meander’’ was termed ‘‘pendulum rivulet’’ by Schmuki.5 Its properties, especially its relationship
between decay frequency and flow rate, are strikingly similar
to our observations in regime IV. Our rivulet, however, does
not break up into subrivulets 共thanks to the surfactants兲, but
stabilizes into very regular, traveling waves.
We have not observed any surface waves, which seem to
precede the onset of meandering on nonwetting surfaces.4,5
VI. CONCLUSIONS AND OUTLOOK
We have introduced investigations of a simple system
which we believe will greatly enhance our understanding of
the science of meandering. Our system poses many advantages over those employed for the traditional work on meandering on hydrophobic surfaces 共see Sec. V兲. It produces
extremely regular and well controlled wave patterns, which
will permit straightforward analysis and study of a great variety of physical and chemical parameters. It is very rich in
interesting effects, and a successful and comprehensive
theory may well be just as rich in interest. We therefore
expect that this work will trigger a wave of theoretical and
experimental studies of meandering problems.
It remains to be seen how the underlying physics relates
to seemingly similar problems such as that of Sec. V and also
to fluid-structure interactions,21 such as the ‘‘garden hose instability’’ or viscous fluid buckling.22,23
In addition, we hope to employ our investigations to
improve our understanding of foam drainage and dynamic
effects in high velocity flows in Plateau borders and soap
films.
ACKNOWLEDGMENTS
The authors would like to thank Simon Cox and Florence Elias for motivating this research, and Norbert Kern,
Adrian Daerr, Laurent Limat and Emanuel de Langre for
interesting discussions. This research was supported by the
Prodex and ELIPS programmes of ESA, and is a contribution to ESA Contract No. C14308/AO-075-99. W.D. received
additional funding from the ULYSSES France-Ireland exchange scheme and the German National Merit Foundation.
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Phys. Fluids, Vol. 16, No. 8, August 2004
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The data were fitted to F(x)⫽A(1⫹Bx ⫺C ) ⫺1 , with A, B, and C being
the fitting parameters. There is no physical reason for using this formula,
apart from the fact that it describes the data very well.
14
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