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arXiv:nlin/0511044v1 [nlin.CD] 22 Nov 2005
Clustering of floaters by waves
P. Denissenko1, G. Falkovich†and S. Lukaschuk1
1Fluid Dynamics Laboratory, University of Hull, HU6 7RX, UK
†Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100, Israel
We study experimentally how waves affect distribution of particles that float on a water surface.
We show that clustering of small particles in a standing wave is a nonlinear effect with the clustering
time decreasing as the square of the wave amplitude. In a set of random waves, we show that small
floaters concentrate on a multi-fractal set.
PACS numbers: 47.27.Qb, 05.40.-a
Even for incompressible liquids, surface flows are gen-
erally compressible and can concentrate pollutants and
floaters. Spatially smooth random flows can be char-
acterized by the Lyapunov exponents whose sum is the
asymptotic in time rate of volume change in the La-
grangian frame (co-moving with the fluid element). Since
contracting regions contain more fluid particles and thus
have more statistical weight than expanding ones, the
rate is generally negative in a smooth flow (for volume
in the phase space, this is a particular case of the second
law of thermodynamics) [1, 2, 3, 4]. As a result, density
concentrates on a fractal (Sinai-Ruelle-Bowen) measure
in a random compressible flow [5, 6, 7, 8]. Indeed, it has
been observed experimentally that random currents con-
centrate surface density on a fractal set [9, 10, 11, 12, 13].
Moreover, recent theory predicts that the measure must
actually be multi-fractal i.e. the scaling exponents of the
density moments do not grow linearly with the order of
the moment [2, 14, 15].
Here we study the effect of clustering by waves on the
water surface. In a single-mode standing wave, fluid sur-
face expands and contracts periodically. Only in a set
of random waves, one may find regions where contrac-
tions accumulate and lead to the growth of concentration.
This is true, yet for potential waves the respective rate
of clustering of the points on the water surface appears
only in the sixth order in wave amplitudes [17, 18]. Since
wave amplitudes are typically much less than the wave-
lengths (otherwise, waves break), such a rate is usually
so small as to be unobservable. For example, for waves
with periods in seconds and the (pretty large) ratio of
the amplitude to wavelength 0.1, the clustering time is
in weeks. However, even small particles can move rel-
ative to the fluid. The physical mechanism that causes
drift of floaters relative to water surface is the capillarity
which breaks Archimedes’ law and makes a floater iner-
tial (i.e. lighter or heavier than the displaced liquid). As
a result, the floaters cluster already in a standing wave
(either in the nodes or in the antinodes depending on the
sign of the capillary force) — brief report of the discov-
ery of this phenomenon has been published in [16]. The
theory of particle motion in a standing surface wave, pre-
sented in the Supplement to [16], predicts that the drift
must appear in the second order in wave amplitudes. The
first part of our experimental results described here shows
FIG. 1: Experimental setup.
that this is indeed the case.
We measured clustering time for small hydrophilic hol-
low spheres with the average size 30 µm and density 0.6
g/cm3. Particles were sift from the dry powder of glass
bubbles (S60HS, 3M), separated by flotation in acetone
(density 0.78 g/cm3) and washed in clean water. The
particles spread readily over flat water surface and do
not form stable clusters. This is possibly caused by dou-
ble layer repulsion, which compensates an attraction due
to the surface tension.
The experimental set-up is shown in Figure 1. Surface
waves are generated in a rectangular cell (C) (horizon-
tal size 9.6 x 58.3 mm, and depth 10 mm) through the
parametric instability [19]. The cell is filled with puri-
fied water (resistivity 18 MOhm·cm) up to the edge of
the lateral walls to eliminate the meniscus effect - ”brim-
full” boundary conditions. The cell is sealed and me-
chanically coupled to the electromagnetic shaker V (V20,
Gearing and Watson Electronics Ltd), whose vertical os-
cillation amplitude and frequency is controlled by digital
synthesizer (Wavetek 81). The oscillation amplitude is
measured by iMEMS accelerometer ADXL150 (Analog
Device) attached to the moving frame. The cell is illu-
minated from below by the expanded collimated beam
(pin-hole P1, lenses L1-L2) from the continuous wave
laser (CWL). A spatial filter, the lens L3 with 0.1 mm
pin-hole P2 at focal distance F=250 mm, rejects all re-
fracted light and forms an image of the anti-nodes on the
screen S. Two high-resolution cameras CCD1 and CCD2
(2048 x 2048 pixels) are controlled through the Dantec
2
PIV system. The shutters of both cameras are synchro-
nized in phase with the shaker oscillation. The camera
CCD1 collects the images of anti-nodes. Its shatter is
open for a time equal to the one period of the parametric
wave. The camera CCD2 collects the light scattered by
the particles on the surface. It is positioned off axis to
avoid straight laser light and its shutter is opened for a
shorter time (∼1 ms) to prevent smearing of particle in
the images. The CCD2 shutter is opening at the phase
when the liquid surface is nearly flat. This allows to
keep CCD2 at a minimal angle to the system’s optical
axis. The optical axis of CCD2 is perpendicular to the
cell long axis.
The measurements of the clustering time as a function
of the wave amplitude were performed as follows. For a
given cell geometry and chosen frequency, we determined
the parametric instability threshold, an oscillation ampli-
tude Ac. This procedure was similar to that described
in [20]. Each experimental run has been started from
mixing: the shaker amplitude was kept at A∼5Acfor
a couple minutes and then lowered to A∼0.9Ac. Next,
a desired amplitude of vibration Ai> Acis set and the
acquisition of the images from both CCDs started. The
unstable parametric wave appears after a time delay with
an amplitude growing up to a stationary value propor-
tional (A−Ac)2. A set of collected images always starts
from the moment when there are neither waves nor parti-
cle motion and ends when a new stationary state reached
with the developed wave and clustered particles. After
the parametric wave appears, the homogeneous area in
CCD1 images is replaced by a network of lines corre-
sponding to the wave anti-nodes (since the water surface
curved by the wave serves as a lens). The local line width
decreases as the wave amplitude increases, and the max-
imum of intensity is constant across the line. So the
variance of the light intensity averaged over an image
area can be chosen as a characteristic of the wave ampli-
tude. The variance of intensity amplitude measured from
the particle images (CCD2) represents the amplitude of
the growing inhomogeneity in particle concentration. A
number of frames collected for each camera is 100 and
the frame rate is adjusted using preliminary test runs.
Figure 2a shows the original images of anti-nodes and
particle clusters. Figures 2 b,c,d present the results of
image processing - the variances of light intensities from
CCD1 and CCD2 versus time showing how the particle
clustering and the wave develop. The bottom curve 2d
shows the wave amplitude taken from CCD1 while the
upper two curves, 2b and 2c, present respectively the
longitudinal and lateral variances of particle concentra-
tion taken from the variances of light intensity measured
by CCD2. This allows to observe clustering along each
wave vector of the standing wave separately.
The time delay between the stabilization of the wave
amplitude and the saturation of concentration inhomo-
geneity is used as a characteristic time of clustering. The
inverse clustering time is plotted versus squared wave am-
plitude in Figure 3. The averaged surface wave amplitude
FIG. 2: Visualization of the standing wave and particles clus-
tering at the nodes (a), the variance of light intensities as a
function of time showing the growth of lateral (b) and longi-
tudinal (c) inhomogeneities in the concentration of particles
and the growth of wave amplitude (d)
FIG. 3: The inverse characteristic time of (lateral) clustering
as a function of the squared wave amplitude.
< A2>was determined from the sizes Sx, Syof the im-
age producing by refracted light in the focal plane P2 of
the lens L3. For the refractive index of water 1.33 and
small wave amplitudes, the angles Sy/F and Sx/F are
equal to one third of the maximum surface inclination.
It follows from Fig. 3 that within 10% the inverse clus-
tering time is proportional to the square of amplitude, as
predicted by the theory [16].
Quasi-linear standing waves exist only at small am-
plitudes of shaker vibrations. Increasing the amplitude
one observes more and more complicated patterns, from
spatio-temporal chaos to developed wave turbulence, see
e.g. [20, 21, 22, 23, 24]. Random compressible flows
are generally expected to mix and disperse at the scales
larger than the correlation scale of the velocity gradi-
3
ents and to produce very inhomogeneous distribution at
smaller scales see [2, 4, 14, 17, 25, 26, 27, 28] for the-
ory and [9, 10, 11, 12, 29, 30] for experiments. Here
we show experimentally that this is also true for a set
of random surface waves (for what follows, it is actually
enough if particle motion corresponds to so-called La-
grangian chaos). Let us briefly describe relevant math-
ematical quantities of interest. Consider the number of
particles inside the circle of the radius raround the point
x:nr(x) = R|r′−x|<r n(r′)dr′. One asks how the statis-
tics of the random field nr(x) changes with the scale of
resolution r. That can be characterized by the scaling
exponents, ζm, of the moments: hnm
ri ∝ rζm. Note that
ζ0= 0 and ζ1= 2. When the distribution is uniform on a
surface, one expects ζm= 2m. When this equality breaks
for some m, one usually calls the distribution fractal.
First, the fractal (information) dimension for a random
surface flow has been measured by Sommerer and Ott,
who found non-integer d=dζm/dm|m=0 [9]. Then, the
scaling of the second moment has been found and related
to the correlation dimension (again non-integer) [10, 11].
Therefore, fractality of the distribution has been estab-
lished in [9, 10, 11]. To the best of our knowledge, dif-
ferent dimensions have not been compared for the same
flow (if found different, that would give a direct proof of
multifractality).
On the other hand, a theory recently developed for a
short-correlated compressible flow gives the set of expo-
nents [2, 14] ζmwhich depend nonlinearly on m(for com-
parison, note that those theoretical formulas give the La-
grangian exponents which in our notations are ζn+1 −2).
Such nonlinear dependence corresponds to a multifrac-
tal distribution. Multifractality of the measure predicted
in [2, 14] means that the statistics is not scale-invariant:
strong fluctuations of particle concentration are getting
more probable as one goes to smaller scales (increases
resolution).
In the second part of the experiment we measured con-
centration moments and scaling exponents for the sus-
pension of small hydrophilic particles mixed by a surface
wave turbulence (at the driving amplitude ∼2Ac). Note
that at such an amplitude, it is not yet developed tur-
bulence but rather few modes that interact nonlinearly
and provide for the Lagrangian chaos. The experiment
has been done for a set of oscillation frequencies from 30
to 220 Hz and amplitudes 1.8−2.5Ac. We reproduce
here a typical result for the parametric wave with the
frequency 32 Hz, wavelength about 7 mm at the oscilla-
tion amplitude 198µm ≃2Ac. A snapshot of the floaters
distribution for this set is shown in Figure 4.
We checked several approaches to quantify the particle
concentration and found that the most reliable algorithm
should base on recognition and counting the individual
particles. We developed such an algorithm and imple-
mented it for 95 µm fluorescent microspheres (No. Duke
Scientific Corp.). The particle density is 1.05 g/cm3.
To make them floating we used 20% salt (NaCl) solu-
FIG. 4: The image of particle distribution in random waves
(20 x 20 mm). The number of particles 920.
tion. Fluorescence method greatly improves the image
contrast, eliminates the problem of spurious refractions,
and allows positioning the CCD2 camera with an optical
filter on the system optical axis. In this part of the ex-
periment we used the cell with the horizontal span 50x50
mm and the depth 10 mm. The size of observation area is
about 30x30 mm, the pixel size 15 µm and mean particle
diameter corresponds to 6-7 pixels.
Images from the experiment with chaotic clustering
were preprocessed. A background noise was subtracted
using individual threshold for each frame equal to the
mean intensity plus 3 standard deviations. The resulted
images were smoothed by low-passed 5x5 pixels filter.
The particle coordinates were determined maximizing a
correlation of 3x3 matrix (75 ×75 µm). The method was
validated by comparison the number of particles with
that estimated on the stage of emulsion preparation. The
particle detection in the dense clusters was verified by di-
rect visual inspection of images.
Up to 1000 images with particle distributions were
recorded at sampling rate 4 sec. The first six moments of
the coarse-grained concentration, Nm=hnm
rir−2m, are
shown in Figure 5 versus the scale of averaging (bin size
r). We see that indeed the moments with m > 1 grow
when rdecreases below the wavelength of the parametri-
cally excited mode. We see that this growth slows down
when rdecreases below r= 50 pixels (lg10(50) ≃1.7).
This is possibly due to the dense clusters where the finite
particle size, short range repulsion, and the particle back
reaction on the flow are important. An additional reason
may be an insufficient representation of dense regions by
the finite number of particles. On this log-log plot the
straight lines correspond to the power laws. The scaling
exponents for the interval 50 < r < 300 pixels are shown
4
FIG. 5: Moments of concentration (2,3,4,5 and 6th) versus
the scale of coarse-graining. Inset: scaling exponents of the
moments of particle number versus moment number.
in the inset. The nonlinearity of the dependence of ζon
mis the first experimental demonstration of multifrac-
tality in the distribution of particles.
Another interesting aspect of particle distribution is
related to the inertia which may cause particle paths to
intersect. This phenomenon was predicted in [31] and
called sling effect, it must lead to appearance of caus-
tics in particle distribution [32]. At weak inertia (like in
our system), caustics are (exponentially) rare [31, 32] yet
we likely see one at the center of Fig. 4. Indeed, as ar-
gued in [31], breakdowns of particle flow are mostly one-
dimensional so that caustics must look locally as two par-
allel straight lines. At higher inertia, breakdowns provide
for extra mixing that makes the sum of Lyapunov expo-
nents positive and measure smooth (rather than multi-
fractal) [33, 34]. Statistical signatures of co-existence of
caustics and multi-fractal distribution needs further stud-
ies.
We believe that the new effect of clustering by waves
is of fundamental interest in physics and may be of prac-
tical use for particle separation, cleaning of liquid sur-
faces, better understanding of environmental phenomena
associated with the wave transport. Chaotic motion of
floaters produces the multi-fractal distribution and caus-
tics and is a good model to study behavior of inertial
particles in random flows important for water droplets
in clouds, fuel droplets in internal combustion engines,
formation of planetesimals etc.
The work is supported by the grants of Royal Society,
Israel Science Foundation and European network. We
thank V. Steinberg for useful discussions. GF is grateful
to V. Vladimirov for hospitality and support.
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