Floaters on Faraday waves for Final Motimo Meeting - Toulouse Sep. 22-23, 2015

Properties of active suspensions Modelling Approach Results Individual Modelling Validations

Large populations of swimmingmicro-organisms and individual modelling

Blaise Delmotte12, Eric Climent12, Franck Plourabou12, Pierre Degond3

1Institut de Mcanique des Fluides, University of Toulouse INPT-UPS, Toulouse, France2IMFT CNRS, UMR 5502 1, Alle du Professeur Camille Soula, 31 400 Toulouse, France

3Institut de Mathmatiques de Toulouse,UMR 5219, Toulouse, France

Workshop Nice 09/25/2013 B. Delmotte

5 mm

Inset: The displacement field demonstrates local heterogeneities in the flow.

A typical snapshot of an experiment: The white spots indicate the positions of the beads floating on surface waves.

[email protected]

http://fcxn.wordpress.com ! Month 6 General Meeting

http://xn.unamur.be

Fluctuations drive viral memes in online social media: Integrating criticality into network science

Ceyda Sanl, Vsevolod Salnikov, Lionel Tabourier, and Renaud Lambiotte

CompleXity Networks, naXys, University of Namur, Belgium.

To spread our posts throughout online social network such as Twitter: When do we need to post? How often should a #hashtag be posted? These questions emphasize the time features of our twitting activity. They would be controlled much more easily compared to the followings: What we post and how many number of followers we have.

To be mobile in dense granular media such as highly packed beads on surface waves: Do single beads move independently or form a group? Is the trajectory of each bead regular in time? The quantification of the bead dynamics shows that the beads perform heterogenous motion with a distinct time scale to characterize this heterogeneity.

restricted amount of attention restricted amount of space

Restricted amount of sources forces social and physical systems to present emergence of order.

hypothesis

Twitter users want to spread their messages and beads under driving want to be mobile. As a result, the twitter users collectively advertise and the beads form groups to move together. Both systems self-organize and create dynamic heterogeneity.

The origin of the fluctuations in dynamics would be the same origins:

Therefore, the interpretation of the dynamic heterogeneity of the beads in a critical limit would help to characterize viral memes (#hashtags) in twitter.

Refs: 1 C. Sanl et al. (arXiv - 2013). 2 L. Berthier (2011).

Refs: 1 H. Simon (1971). 2 L. Weng et al. (2012). 3 J. P. Gleeson et al. (2014).

0 12 24 36 48 60 72 840

10

20

30

40

time (hours)

num

ber o

f tw

eets

/uni

t tim

e daily tweet cycles

propogations of #hashtags 0 10 20 30 40 50 60 70 80 90

0

5

10

15

20

25

4(l=

2R,

)

(s)

=0.652=0.725=0.741=0.749=0.753=0.755=0.760=0.761=0.762=0.766=0.770=0.771

(a)

4(l, )

h

4(l, )

mobility of beads

spatiotemporal granular flow

time

0.2

0.6

1

1.4

1.8

*

P(r

=2R

,t)

MM

Twitter #hashtag analysis

single beads: groups:

perim

eter

quantifying dynamic heterogeneity:

time (s) 0 0.5 1 1.5 2

[Ref:2]

[Ref:2]

#nice: #pepsi:

observation: artificial representation:

0 10 20 30 40 50 60

time (hours) 0 1 2 3 4 5 6 7 8

time (hours)

analysis:

time (hours)

cum

ulat

ive

time (hours)

cum

ulat

ive

Floaters on Faraday waves: Clustering and heterogeneous flow

CompleXity Networks

Final MOTIMO workshop, Collective dynamics of active particles, swimmers, motile cells, September 22-23, 2015, Institut de Mecanique des Fluides de Toulouse (IMFT).

September 23, 2015, Toulouse.

@CeydaSanli

Physics of Fluids

UNamur

Greetings,

C. Sanli, CompleXity Networks, UNamur

[email protected]

[email protected]

[email protected]

Floaters on Faraday waves 1

[email protected]

Samenstelling promotiecommissie:

Prof. dr. G. van der Steenhoven (voorzitter) Universiteit TwenteProf. dr. Detlef Lohse (promotor) Universiteit TwenteDr. Devaraj van der Meer (assistent-promotor) Universiteit TwenteProf. dr. Karen E. Daniels North Carolina State UniversityProf. dr. J. Frits Dijksman Universiteit TwenteDr. Dirk van den Ende Universiteit TwenteProf. dr. Stefan Luding Universiteit TwenteDr. Peter Schall Universiteit van Amsterdam

The work in this thesis was carried out at the Physics of Fluids group of the Facultyof Science and Technology of the University of Twente. It is part of the research pro-gramme of the Foundation for Fundamental Research on Matter (FOM), which is fi-nancially supported by the Netherlands Organisation for Scientific Research (NWO).

Nederlandse titel:Drijvende deeltjes op Faraday golven: over clustering en heterogene dynamica.

Publisher:Ceyda Sanl, Physics of Fluids, University of Twente,P.O. Box 217, 7500 AE Enschede, The [email protected] & ceydasanli.com (WordPress)

Cover Illustration: Ceyda Sanl & Deniz Cakr

c Ceyda Sanl, Enschede, The Netherlands 2012No part of this work may be reproduced by printphotocopy or any other means without the permissionin writing from the publisher.ISBN: 978-90-365-3359-1

mailto:[email protected]


Two Directions of my PhD Thesis:

Clustering on standing Faraday waves

Heterogeneous flow on capillary Faraday waves

C. Sanl et al, Phys. Rev. E 89, 053011 (2014).

C. Sanl et al, Phys. Rev. E 90, 033018 (2014).


Observation

Floaters on Faraday waves 3C. Sanli, CompleXity Networks, UNamur

5 mm

a ~ 0.8 mm f = 19 Hz

5 mm

a ~ 1.2 mm f = 20 Hz

low :

high :

!

!

Experimental set-up


SANLI, LOHSE, AND VAN DER MEER PHYSICAL REVIEW E 89, 053011 (2014)

water

air floaters

(a)

(b)

(d)

(c)

(e)

(f)

(g)

(h)

l

drift

attraction

drift

NN

A

A

FIG. 1. (Color online) Experimental setup: (a) shaker, (b) glasscontainer, 814510 mm3, (c) Schott fiber light source, (d) PhotronFastcam SA.1, (e) an illustration of a camera image, (f) pinnedbrim-full boundary condition, (g) the surface deformation aroundour hydrophilic heavy floaters causes an attractive force, (h) thedirection of the period-averaged drift of a single floater, where Aand N represent the antinode and the node, respectively.

taken to be large enough to capture the maximum verticaldisplacement (2.5 0.1 mm) of the floaters.

In the period-averaged context, there are two mechanismsthat drive the floaters on the standing Faraday wave. The firstone is the attractive capillary interaction [6,7] due to the surfacedeformation around the floaters [Fig. 1(g)], which is significantwhen the distance between the floaters l is smaller than thecapillary length lc = (/g)1/2. Here, is the surface tensioncoefficient of the interface, the liquid density, and g theacceleration of gravity. (For an air-water interface at 20 C,lc = 2.7 mm.) The second is due to the standing Faraday wave,which causes a time-averaged drift of the floaters towardsthe antinodes [Fig. 1(h)], which is observed and described inRefs. [1,3]. This drift, which is discussed in greater detail inAppendix A2, is reminiscent of the famous Stokes drift of anobject on a traveling wave.

The control parameter of the experiment is the floaterconcentration . We simply measure by dividing the areacovered with floaters by that of the total horizontal field ofview. In Fig. 2 we show a top view of the distribution of theparticles in two distinct limits, namely for low and high .The remarkable difference between the two states is clear: Forlow [Fig. 2(a)] small clusters float around the antinodes,whereas for high [Fig. 2(b)] there is one large cluster aroundthe nodal lines. This completely inverts the pattern and theparticles now seen to avoid the antinodal regions.

To inspect this concentration-dependent clustering weintroduce the correlation factor c, which quantifies to whatextent the position of the clusters is correlated with the waveantinodes

c (r,t)a(r)r,t(r,t)r,t

, (1)

where the brackets r,t indicate that the average is taken withrespect to both space r = (x,y) and time t [13]. Here, (r,t)is the floater concentration and the wave distribution a(r) is atest function that is positive at the antinodes and negative at

(a) (b)

(c) (d)

xy

FIG. 2. (Color online) (a), (b) Clustering of floaters on a rectan-gular standing wave in experiment. The snapshots show the stationarystate when the surface wave elevation is nearly zero. The small yellowrectangles mark the location of the antinodes and the yellow lines thatof the nodal lines. Clearly, for = 0.08 (a) particles cluster at theantinodes, whereas for = 0.61 (b) the pattern is spontaneouslyinverted into a large cluster around the nodal lines. Note that in (b) allparticles touch whereas the average distance between particles in (a)is somewhat larger. This is due to the breathing effect explained inthe text. (c), (d) Artificial antinode clusters at = 0.10 (c) and nodeclusters at = 0.44 (d) as used in the potential energy calculation.The white bars indicate a length scale of 5 mm.

the nodes. More specifically, a(r) is defined as

a(r) ={acos(r) when acos(r) > 0 (antinodes),acos(r) when acos(r) < 0 (nodes).

(2)

Here, acos(r) = 2 cos2 kxx cos2 kyy 1, with kx,ky the wavenumbers in the x,y direction. Since with the above definitionthe nodal regions are three times as small as the nodal ones, aconstant = 3 is introduced such that c = 0 when the floatersare equally distributed over the two-dimensional wave surface[14]. To check the robustness of c regarding the precise formof a(r), we also use a step function astep(r), which equals 1 atthe antinodes and 1 at the nodes.

In Fig. 3(a) we present the correlation factor c plottedagainst for both acos and astep. We observe three distinctregions: For low (0.5) we find node clusters for which c < 0(region III). Finally, there is a broad intermediate regionII, in which we observe morphologically rich self-organizedfloater patterns, some of which are steadier than others.These quasisteady patterns cause the large scatter in c in theregion between = 0.2 and 0.35. Between = 0.35 and 0.5,patterns are quite dynamic leading to an even spreading ofparticles over the waves (c 0).

In addition to the position, another remarkable differencebetween the antinode and the node clusters is hidden in theirdynamics during a single wave period: Experimentally we

053011-2

Control parameters

: contact angle of spheres R: radius of spheres a: shaking amplitude f: shaking frequency h: depth of water layer : area fraction

=bead area/total area

!a, f

h


Part-1: Why the antinode clusters at low ?

Why the node clusters at high ?



Floaters on a static surface


Chapter 1. General introduction

Figure 1.1: Champagne bubbles, aggregated at the wall of a wine glass, and accumulatedleaves floating on a wavy sea surface [3].

sion. The surface profile around each floater is not axisymmetric in the presence ofan other floater so that the floaters are driven to each other. This attractive inter-action among floaters is called gravity-induced capillary interaction. (A repulsiveinteraction is also possible, but only if the floaters have different wetting propertiesor densities.) Similarly, the wall disturbs the surface profile, and so it drives an at-traction (or repulsion) between the floaters and the wall. Both floater-to-floater andfloater-to-wall interactions can be well formulated when the meniscus deformationis small by minimizing the total potential energy (both gravity and surface tension)of the floaters [1, 2]. In the present Thesis, floater refers to an object floating ona liquid, which is exposed to not only gravity and Archimedes buoyancy, but alsosurface tension.

Without applying any calculation, one can conceptually understand [1] why abubble floating on champagne is attracted by the wall of a wine glass as shown inFig. 1.1. Consider a bubble floating on champagne. In the static case, there are threevertical forces acting on the bubble, namely the weight, the buoyancy, and the sur-face tension. Since the bubble is much lighter than the champagne, the buoyancy ismuch larger than its weight. To create a balance, the surface tension acts downwardswhich creates a convex-shaped surface profile around the bubble. However, thereis a physical limit such that the surface tension cannot bend the meniscus beyond acertain angle. Therefore, there is an excess upwards force due to lack of downwardsforces. On the other side, the glass wall creates a meniscus with also a convex-shape

2

Bubble in an equilibrium


bubbleinterface

Bubble in a curved interface


glass

Heavy particle in a curved interface


glass

Phenomenological conclusion


bubbles (light particles) drift to a local surface maximum

heavy particles drift to a local surface minimum


!!!


Floaters on a dynamic surfaceChapter 1. General introduction

Figure 1.1: Champagne bubbles, aggregated at the wall of a wine glass, and accumulatedleaves floating on a wavy sea surface [3].

sion. The surface profile around each floater is not axisymmetric in the presence ofan other floater so that the floaters are driven to each other. This attractive inter-action among floaters is called gravity-induced capillary interaction. (A repulsiveinteraction is also possible, but only if the floaters have different wetting propertiesor densities.) Similarly, the wall disturbs the surface profile, and so it drives an at-traction (or repulsion) between the floaters and the wall. Both floater-to-floater andfloater-to-wall interactions can be well formulated when the meniscus deformationis small by minimizing the total potential energy (both gravity and surface tension)of the floaters [1, 2]. In the present Thesis, floater refers to an object floating ona liquid, which is exposed to not only gravity and Archimedes buoyancy, but alsosurface tension.

Without applying any calculation, one can conceptually understand [1] why abubble floating on champagne is attracted by the wall of a wine glass as shown inFig. 1.1. Consider a bubble floating on champagne. In the static case, there are threevertical forces acting on the bubble, namely the weight, the buoyancy, and the sur-face tension. Since the bubble is much lighter than the champagne, the buoyancy ismuch larger than its weight. To create a balance, the surface tension acts downwardswhich creates a convex-shaped surface profile around the bubble. However, thereis a physical limit such that the surface tension cannot bend the meniscus beyond acertain angle. Therefore, there is an excess upwards force due to lack of downwardsforces. On the other side, the glass wall creates a meniscus with also a convex-shape

2


Heavy particle on a standing wave


2.4. The wave elevator Chapter 2. Playing with surface tension ...

The vertical wave acceleration z oscillates with respect to t. (Here, we use = 2/ t2 for simplicity.) The vertical acceleration which the floater experiences isg+ when t < T/2 and g when t > T/2. Since the floater acceleration is larger fort < T/2 the contribution of this part of the wave cycle to the drift is larger. Therefore,in the time averaged situation, the bubble (B< 0) drifts towards N, and so the sphere(B> 0) towards A. This is consistent with the result suggested by the derived driftforce given in Eq. 2.33. The time dependent drift within T is sketched in Fig. 2.9(a) forthe sphere and in Fig. 2.9(b) for the bubble. The mechanism discussed here resemblesan accelerating elevator, and so can be called a wave elevator.

..

A A

A A

(a)

g

(b)

zx

B >0 B T/2g

..-

drift

g +

drift

g..-

g +

N

N..

N

N..

Figure 2.9: The wave elevator: The asymmetry in the vertical floater acceleration and thecorresponding drift are illustrated for a sphere with B> 0 (a), and a sphere or a bubble withB < 0 (b). = 2/ t2 is the vertical surface wave acceleration and T is the wave period.Since the contribution of the drift in the lower cycle (t < T/2) is larger than the one shownabove (t > T/2), on average, the floater in (a) drifts towards the antinode (A), whereas thefloater in (b) drifts towards the node (N).

35

T: the wave period



..

A A

A A

(a)

g

(b)

zx

B >0 B T/2g

..-

drift

g +

drift

g..-

g +

N

N..

N

N..


35



..

A A

A A

(a)

g

(b)

zx

B >0 B T/2g

..-

drift

g +

drift

g..-

g +

N

N..

N

N..


35



..

A A

A A

(a)

g

(b)

zx

B >0 B T/2g

..-

drift

g +

drift

g..-

g +

N

N..

N

N..


35


wave elevator!

Antinode clusters at low



water

air floaters

(a)

(b)

(d)

(c)

(e)

(f)

(g)

(h)

l

drift

attraction

drift

NN

A

A







, (1)


(a) (b)

(c) (d)

xy




(2)




053011-2

The drift force is always towards the antinodes for our floaters.

The drift force is a single floater force.

References References

tional particles due to capillary attraction, Phys. Rev. E 83, 051403-110(2011).

[7] M. R. Maxey and J. J. Riley, Equation of motion for a small rigid spherein a nonuniform flow, Phys. Fluids 26, 883889 (1983).

[8] G. Falkovich, A. Weinberg, P. Denissenko, and S. Lukaschuk, Floaterclustering in a standing wave, Nature (London) 435, 10451046 (2005).

[9] P. Denissenko, G. Falkovich, and S. Lukaschuk, How waves affect thedistribution of particles that float on a liquid surface, Phys. Rev. Lett. 97,244501-14 (2006).

[10] S. Lukaschuk, P. Denissenko, and G. Falkovich, Nodal patterns offloaters in surface waves, Eur. Phys. J. Special Topics 145, 125136 (2007).

[11] J. B. Keller, Surface tension force on a partly submerged body, Phys.Fluids 10, 30093010 (1998).

[12] C. Duez, C. Ybert, C. Clanet, L. Bocquet, Making a splash with waterrepellency, Nat. Phys. 3, 180183 (2007).

[13] P. K. Kundu and I. M. Cohen, Fluid Mechanics, (Elsevier Inc., USA) (2008).

[14] H. J. van Gerner, M. A. van der Hoef, D. van der Meer, and K. van derWeele, Inversion of Chladni patterns by tuning the vibrational accelera-tion, Phys. Rev. E 82, 012301-14 (2010).

[15] H. J. van Gerner, K. van der Weele, M. A. van der Hoef, and D. van derMeer, Air-induced inverse Chladni patterns, J. Fluid Mech. 689, 203220 (2011).

37

! Correlation number c:

antinodes

nodes

Experiment


Full story at low


2.2. Floaters at a static interface Chapter 2. Playing with surface tension ...

face of the liquid (with density l) is deformed to satisfy the vertical force balance.This means that the weight of the sphere, the buoyancy force and the surface tensionforce (with surface tension coefficient ) must sum up to zero.

The surface deformation due to a single sphere in a static equilibrium is shownschematically in Fig. 2.1 for four distinct spheres, namely (a) hydrophilic [wetting,i.e., < /2] and heavy [s > l], (b) hydrophobic [non-wetting, i.e., > /2] andheavy, (c) hydrophilic and light [s < l], and finally (d) hydrophobic and light.

arctan (z )z

l

(a) (b)

g

hydrophilic ( < /2) hydrophobic ( > /2)

(d)(c)

s

l

>

Fc

s

l

0 (antinodes),acos(r) when acos(r) < 0 (nodes).

(2)




053011-2


water

air floaters

(a)

(b)

(d)

(c)

(e)

(f)

(g)

(h)

l

drift

attraction

drift

NN

A

A







, (1)


(a) (b)

(c) (d)

xy




(2)




053011-2


antinodes

nodes

Experiment


antinodes

nodes

Experiment ! Correlation number c:

antinodes

nodes

Experiment


antinodes

nodes

Experiment


antinodes

nodes

Experiment

t is time


antinodes

nodes

Experiment

is the wavelength


antinodes

nodes

Experiment antinodes

nodes

C. Sanli et al. PRE 89, 053011 (2014).Floaters on Faraday waves 17

Correlation number c


FROM ANTINODE CLUSTERS TO NODE CLUSTERS: THE . . . PHYSICAL REVIEW E 89, 053011 (2014)

0.4

0.2

0

0.2

0.4

0.6 aa

I II

III

cosstep

5

4

3

2

1

0x 105

(E/

l )/N

E, antinode(b)

(a)

c( )

t ~~ 0.36

c

210

123

4

( E/l )/ N

c

E, nodeE

(E/

l )/N

c

(c)

0 0.1 0.3 0.4 0.5 0.65

4

3

2

1

0x 105

0.7

E cE d

E cE d

config. of the node clusters

, config. of the clusters w/o breathing

E d

config. of the antinode clusters

0.2

22

2

x 105

FIG. 3. (Color online) Experimental (a) and calculated (b), (c)transition from antinode to node clusters. (a) The correlation factor cis plotted versus the floater concentration for both acos (red circles)and astep (blue squares), where the error bars indicate the standarddeviation of a single experiment. (b) The total potential energy E/Nper floater particle for the artificial patterns [see Figs. 2(c), 2(d)],nondimensionalized by l2c , is plotted versus for both the antinode(black circles) and node (black squares) configurations. #E/N (redstars) represents the energy difference between the antinode and nodeconfigurations. (c) Constituents of E/N versus . Circles indicate thecapillary energy Ec/N (orange) and the drift energy Ed/N (purple)for the antinode configurations, whereas squares indicate the samequantities for the node clusters. For comparison, the purple dashedlines show the drift energy Ed without incorporating the breathingeffect.

observe that in the antinode clusters the floaters periodicallymove away from and towards the antinode [Fig. 4(a)]. Thishappens because when the wave reaches its maximum the(downward moving) floaters move away from the antinode,whereas in the minimum they move towards it. We call thisperiodic motion at the antinode clusters breathing. In contrast,nodal clusters do not breathe; instead the clusters as a wholeoscillate back and forth around the nodal lines [Fig. 4(b)].As a result, the floaters in the node clusters stay closelytogether without changing their relative distance (which isapproximately equal to the particle diameter 2R), whereas theperiod-averaged distance between the particles in the antinode

A N

(c) (d)

xy

A N

A(a) (b)

xy

FIG. 4. (Color online) The breathing effect: When we comparean experimental antinode (a) with a node cluster (b), we clearly seethat particles in the first are much farther apart due to breathing (seetext). Again, the antinodes (A) are marked by small yellow rectanglesand the nodes (N) with yellow lines. The bars indicate a length scaleof 5 mm. We artificially design hexagonal clusters to incorporate thisbreathing effect: An antinode cluster (c) is grown by adding hexagonalrings at decreasing increments rnn starting from a large initial value,whereas a node cluster (d) is grown from a close-packed hexagonalstructure with increasing increments rnn. The color coding identifiesconsecutive rings.

cluster is significantly larger than 2R [15]. The breathingphenomenon is discussed in more detail in Appendix A2.

III. POTENTIAL ENERGY ESTIMATE

Now, what is the reason for the observed pattern inversion?To answer this question we estimate the energy in artificiallycreated node and antinode clusters, which are inspired by ourexperimental observations [Figs. 4(a) and 4(b)]: The antinodecluster is modeled as a two-dimensional static hexagonallypacked cluster where the distance between the neighboringfloaters increases towards the antinode point (A) [Fig. 4(c)]to implement the observed breathing effect. The distancehere can be considered as the period-averaged experimentaldistance between the floaters. The node cluster, in contrast,is designed as a two-dimensional hexagonal cluster where thedistance between the neighboring floaters sitting exactly atthe crossing of two nodal lines (N) is equal to an averagefloater diameter 2R [Fig. 4(d)]. Furthermore, the distanceslightly increases away from N. Further details on the artificialantinode and node cluster configurations can be found inAppendix B.

During the motion of the floaters on the wave there is anintricate exchange of wave energy (input), potential energy,kinetic energy, and dissipation (output). However, in a steadystate the input and output must balance and since the particlesreturn to (approximately) the same positions after each periodof the wave it is sufficient to compare the potential energy

053011-3


0.4

0.2

0

0.2

0.4

0.6 aa

I II

III

cosstep

5

4

3

2

1

0x 105

(E/

l )/N

E, antinode(b)

(a)

c(

)t ~~ 0.36

c

210

123

4( E

/l )/ Nc

E, nodeE

(E/

l )/N

c(c)

0 0.1 0.3 0.4 0.5 0.65

4

3

2

1

0x 105

0.7

E cE d

E cE d



E d


0.2

22

2

x 105



A N

(c) (d)

xy

A N

A(a) (b)

xy






053011-3

Correlation number c



0.4

0.2

0

0.2

0.4

0.6 aa

I II

III

cosstep

5

4

3

2

1

0x 105

(E/

l )/N

E, antinode(b)

(a)

c( )

t ~~ 0.36

c

210

123

4

( E/l )/ N

c

E, nodeE

(E/

l )/N

c

(c)

0 0.1 0.3 0.4 0.5 0.65

4

3

2

1

0x 105

0.7

E cE d

E cE d



E d


0.2

22

2

x 105



A N

(c) (d)

xy

A N

A(a) (b)

xy






053011-3


0.4

0.2

0

0.2

0.4

0.6 aa

I II

III

cosstep

5

4

3

2

1

0x 105

(E/

l )/N

E, antinode(b)

(a)

c(

)t ~~ 0.36

c

210

123

4( E

/l )/ Nc

E, nodeE

(E/

l )/N

c(c)

0 0.1 0.3 0.4 0.5 0.65

4

3

2

1

0x 105

0.7

E cE d

E cE d



E d


0.2

22

2

x 105



A N

(c) (d)

xy

A N

A(a) (b)

xy






053011-3

Experiment

''

I

''

III

''

II

Experiment

''

I

''

III

''

II


antinodes

nodes

Experiment Experiment

''

I

''

III

''

II


antinodes

nodes

Experiment


Why the node clusters at high ? Lets look at the experiment more carefully!

10 m

m10 m

m

antinode clusters node clusters

breathing non-breathing


Potential energy approach


0.4

0.2

0

0.2

0.4

0.6 aa

I II

III

cosstep

5

4

3

2

1

0x 105

(E/

l )/N

E, antinode(b)

(a)

c(

)t ~~ 0.36

c

210

123

4

( E/l )/ N

c

E, nodeE

(E/

l )/N

c

(c)

0 0.1 0.3 0.4 0.5 0.65

4

3

2

1

0x 105

0.7

E cE d

E cE d



E d


0.2

22

2

x 105



A N

(c) (d)

xy

A N

A(a) (b)

xy






053011-3

Energy approach in detail:

! : surface tension

! l : capillary length c ! N : number of floaters

! E : capillary energy

! E : drift energy d

c

drift to the antinodes + capillary attraction + breathingEnergy approach in detail:

! : surface tension

! l : capillary length c ! N : number of floaters

! E : capillary energy

! E : drift energy d

c


0.4

0.2

0

0.2

0.4

0.6 aa

I II

III

cosstep

5

4

3

2

1

0x 105

(E/

l )/N

E, antinode(b)

(a)

c( )

t ~~ 0.36

c

210

123

4

( E/l )/ N

c

E, nodeE

(E/

l )/N

c

(c)

0 0.1 0.3 0.4 0.5 0.65

4

3

2

1

0x 105

0.7

E cE d

E cE d



E d


0.2

22

2

x 105



A N

(c) (d)

xy

A N

A(a) (b)

xy






053011-3


0.4

0.2

0

0.2

0.4

0.6 aa

I II

III

cosstep

5

4

3

2

1

0x 105

(E/

l )/N

E, antinode(b)

(a)

c( )

t ~~ 0.36

c

210

123

4( E

/l )/ Nc

E, nodeE

(E/

l )/N

c

(c)

0 0.1 0.3 0.4 0.5 0.65

4

3

2

1

0x 105

0.7

E cE d

E cE d



E d


0.2

22

2

x 105



A N

(c) (d)

xy

A N

A(a) (b)

xy






053011-3


antinodes

nodes

Experiment


antinodes

nodes

Experiment ! Correlation number c:

antinodes

nodes

Experiment


antinodes

nodes

Experiment


antinodes

nodes

Experiment


Comparison with experiment experiment

! Energy approach

! Experiment

Comparison:

! Energy approach

! Experiment

Comparison:

energy approach0


Floaters on a Faraday wave:


capillary Heterogeneous flows and group formations

Part- 2:

Observation


2 mm

a = 1 mm f = 250 Hz

= 0.63

4 times slower than real time

!

Experimental set-up


= bead area/total area!

SANLI, SAITOH, LUDING, AND VAN DER MEER PHYSICAL REVIEW E 90, 033018 (2014)

air floaters

(a)

(c)

(e)

(b)

(f)

(g)

far field light source

wavelength ~ 2R ~

a =0.1mmf =250 Hz

water

0

0 r 18 M" cm) such that the water level isperfectly matched with the container edge as shown in Fig. 1(g)to create the brim-full boundary condition [44]. Sphericalhydrophilic polystyrene floaters [Fig. 1(d)], contact angle 74,and density 1050 kg/m3, with an average radius R of 0.31 mmare carefully distributed over the water surface to make amonolayer of floaters [45]. The polydispersity of the floatersis approximately 14% and assumed to be just wide enough toavoid crystallization [46]. To avoid any surfactant effects, boththe container and the floaters are cleaned by performing thecleaning protocol as described in Ref. [47].

A continuous white fiber light source (Schott) is used toilluminate the floaters from far away as shown in Fig. 1(b).The positions of the floaters are recorded with a high-speedcamera (Photron Fastcam SA.1) at 60500 frames per second.The lens (Carl Zeiss 60 mm) is adjusted such that it focuseson the floaters at the nondeformed water surface. Here weuse the random capillary Faraday waves to just agitate thedensely packed floaters so that there is no macroscopicapparent amplitude observed. The wave amplitude is alwaysconsiderably smaller than the floater radius (0.31 mm).

The resultant capillary ripples on the water surface inthe container, made from transparent hydrophilic glass with10 mm height and a 81 45 mm2 rectangular cross section,are shown in Fig. 1(h). To eliminate the boundary effects dueto the sharp corners of the container, an elliptic rim madefrom plastic is used to contain the particles. Each image takenwith the high speed camera is 512 640 px2 (36 28 mm2),where px means pixel, as shown by the yellow rectangle (sizeratios are preserved). The horizontal field of view is 35%of the total area of the ellipse. Due to the asymmetric surfacedeformation around each hydrophilic heavy sphere, there isan attraction [23,24] between the spheres so that the floatersare cohesive [48]. For moderate , the monolayer can beconsidered two dimensional [Fig. 1(i)]. In the densely packedregime however [Fig. 1(j)], particles are so close that the layermay (locally) buckle and have three-dimensional aspects [49].

The control parameter of the experiment is the floater (pack-ing) concentration , which (ignoring buckling) is measuredby determining the area fraction covered by the floaters in thearea of interest [Fig. 1(h)]. In this study, is increased frommoderate to large concentrations, = 0.650.77.

Under the influence of the erratic capillary waves and theattractive capillary interaction, a large scale convective motion

033018-2

Driving interactions and forces


attractive capillary interactions

erratic capillary (Faraday) waves

amplitude (a)=0.1mm frequency (f)=250Hz

5 mm

Attractive capillary interaction:

a, f area fraction

=0.63

image sequence

Questions that we address

Capillary Faraday wave:

How can we quantify the heterogeneous dynamics of the grains both in space and in time?

How does the dynamics of the grains vary as a function of the area fraction ?

Four-point dynamic susceptibility*

morphological parameter

(total area of the connected region)

(# of connected regions)

1 2 Methods

by m

easu

ring

the

dist

ance

of e

ach

float

er

by v

isua

lizin

g th

e flo

ater

mot

ion

loca

lly

Illustrations of the break-up of an initial subgroup into pieces

c

average grain radius R

all subgroups

=

When increases, the subgroups are morphologically deformed less.

successfully identifies the slowing down motion at moderate .

The morphological delay time characterizes the time scale of the deformation in the subgroups.

*

low : (=0.63)

high : (=0.67)

t=0 (s) t=2.74 (s) t=5.47 (s)

5 mm

The self-overlap order parameter defined below is calculated for each area fraction .

A.R. Abate and D.J. Durian, PRE (2007).

grain index

number of grains

=

The susceptibility :

time

From this, the four-point dynamic susceptibility is determined.

Gaussian cutoff In the limits:

1, if the grain travels > within .

For the cohesive grains floating on a capillary Faraday wave, the morphological analysis is more sensitive in explaining the slowing down dynamics at moderate .

By applying the morphological analysis, we can visualize the heterogeneous dynamics. We can learn more details about the flow by inventing more advanced morphological parameters.

Inset: The displacement field presents local heterogeneities in the flow.

A typical snapshot of an experiment is shown: The white spots indicate the positions of the floaters.

Morphological analysis**

high speed camera

d


Four-point dynamic susceptibility:


To quantify the heterogeneous dynamics

amplitude (a)=0.1mm frequency (f)=250Hz

5 mm

Attractive capillary interaction:

a, f area fraction

=0.63

image sequence

Questions that we address

Capillary Faraday wave:

How can we quantify the heterogeneous dynamics of the grains both in space and in time?

How does the dynamics of the grains vary as a function of the area fraction ?

Four-point dynamic susceptibility*

morphological parameter

(total area of the connected region)

(# of connected regions)

1 2 Methods

by m

easu

ring

the

dist

ance

of e

ach

float

er

by v

isua

lizin

g th

e flo

ater

mot

ion

loca

lly

Illustrations of the break-up of an initial subgroup into pieces

c

average grain radius R

all subgroups

=

When increases, the subgroups are morphologically deformed less.

successfully identifies the slowing down motion at moderate .

The morphological delay time characterizes the time scale of the deformation in the subgroups.

*

low : (=0.63)

high : (=0.67)

t=0 (s) t=2.74 (s) t=5.47 (s)

5 mm

The self-overlap order parameter defined below is calculated for each area fraction .

A.R. Abate and D.J. Durian, PRE (2007).

grain index

number of grains

=

The susceptibility :

time

From this, the four-point dynamic susceptibility is determined.

Gaussian cutoff In the limits:

1, if the grain travels > within .

For the cohesive grains floating on a capillary Faraday wave, the morphological analysis is more sensitive in explaining the slowing down dynamics at moderate .

By applying the morphological analysis, we can visualize the heterogeneous dynamics. We can learn more details about the flow by inventing more advanced morphological parameters.

Inset: The displacement field presents local heterogeneities in the flow.

A typical snapshot of an experiment is shown: The white spots indicate the positions of the floaters.

Morphological analysis**

high speed camera

d

Self-overlap order parameter


Qt(l, ) =1

N

NX

i=1

!l(ri) (1)

ri = |ri(t+ ) ri(t)| (2)

!l(ri) = eri/2R

(3)

2

number of spheres sphere index

a cutoff function

time

( ) ( )( ) ,1,1

trwN

tq iN

iaa

=

1

0 a r

( )rwa

( ) ( ) ( )trtrtr iii!!

+ ,

Mobility

Order parameter

We normalize the displacement to [0,1].

( ) ( )2/ara erw =( )( )!

"#

>

arar

01

or Overlap function

a1 0.5 0

( ) ( )taa

tqQ ,=Order parameter Qt(l, ) =

1

N

NX

i=1

!l(ri) (1)

ri = |ri(t+ ) ri(t)| (2)

!l(ri) = eri/l

(3)

2


antinodes

nodes

Experiment


antinodes

nodes

Experiment

Qt(l, ) =1

N

NX

i=1

!l(ri) (1)

ri = |ri(t+ ) ri(t)| (2)

!l(ri) = er2i /(2l)2

(3)

2


antinodes

nodes

Experiment

References References

[12] A. R. Abate and D. J. Durian, Topological persistence and dynamicalheterogeneities near jamming, Phys. Rev. E 76, 021306-19 (2007).

[13] F. Lechenault, O. Dauchot, G. Biroli, J. P. Bouchaud, Critical scaling andheterogeneous superdiffusion across the jamming/rigidity transition of agranular glass, Eur. Phys. Lett. 83, 46003-16 (2008).

[14] F. Lechenault, R. Candelier, O. Dauchot, J. P. Bouchaud, and G. Biroli,Super-diffusion around the rigidity transition: Levy and the Lil-liputians, Soft Matter 6, 30593064 (2010).

[15] P. Mayer, H. Bissig, L. Berthier, L. Cipelletti, J. P. Garrahan, P. Sollich, andV. Trappe, Heterogeneous Dynamics of Coarsening Systems, Phys. Rev.Lett. 93, 115701-14 (2004).

[16] D. Ertas and T. C. Halsey, Granular gravitational collapse and chuteflow, Europhys. Lett. 60, 931937 (2002).

[17] T. C. Halsey and D. Ertas, Coherent Structures in Dense GranularFlows, arXiv:cond-mat/0506170v1 (unpublished), 14 (2005).

[18] S. Douady, Experimental study of the Faraday instability, J. FluidMech. 221, 383408 (1990).

[19] C. Duez, C. Ybert, C. Clanet, L. Bocquet, Nat. Phys. 3, 180 (2007). Seethe section Methods for the cleaning protocol of both hydrophilic (wet-ting) glass material and hydrophobic teflon (non-wetting) particles. Notethat we immerse the hydrophilic glass container in piranha solution (1vol H2O2, 2 vol H2SO4) for 25 min. The cleaning procedure for ourhdrophilic polystyrene consists of rinsing the spheres with isopropanol.Then, both the container and the spheres are dried separately, and fur-ther heated to remove all the chemicals. Since the spheres are hydrophilicpolystyrene, the suggested heating temperature is 70 C at most. Weheated the spheres, as well as the container, at 65 C for 1 hour. Finally,both the container and the spheres are left to cool to ambient tempera-ture.

[20] T. Peng, Detect circles with various radiiin grayscale image via Hough Transform,http://www.mathworks.com/matlabcentral/fileexchange/9168, Copy-right (c) 2006.

128


antinodes

nodes

Experiment Qt(l, ) =

1

N

NX

i=1

!l(ri) (1)

ri = |ri(t+ ) ri(t)| (2)

!l(ri) = er2i /(2l)2

(3)

4(l, ) = NhQ2t (l, )it hQt(l, )i2t

(4)

2

A measure of mobility


Qt(l, ) =1

N

NX

i=1

!l(ri) (1)

ri = |ri(t+ ) ri(t)| (2)

!l(ri) = er2i /(2l)2

(3)

2

'''''''Four'point'dynamic'suscep+bility''''!'The'four9point'dynamic'suscep+bility,'''''4'!'Self9overlap'order'parameter:'

grain'index'

number'of'grains'

+me'

a'cutoff'func+on'

='1,'if'the'grain'''''travels'

Four-point dynamic susceptibility



antinodes

nodes

Experiment

Qt(l, ) =1

N

NX

i=1

!l(ri) (1)

ri = |ri(t+ ) ri(t)| (2)

!l(ri) = er2i /(2l)2

(3)

4(l, ) = NhQ2t (l, )it hQt(l, )i2t

(4)

2

To quantify the heterogeneous dynamics by the mobility (self-overlap order parameter):5.3. Dynamic susceptibility Chapter 5. Dense heterogenous flow...

0 10 20 30 40 50 60 70 80 900

5

10

15

20

25

4(l=

2R,

)

(s)

=0.652=0.725=0.741=0.749=0.753=0.755=0.760=0.761=0.762=0.766=0.770=0.771

(a)

4(l, )

h

4(l, )

0 10 20 30 40 50 60 70 80 900

5

10

15

20

25

4(l=

8R,

)

(s)

=0.652=0.725=0.741=0.749=0.753=0.755=0.760=0.761=0.762=0.766=0.770=0.771

(b)

0 10 20 30 40 50 60 70 80 900

5

10

15

20

25

4(l=

15R,

)

(s)

=0.652=0.725=0.741=0.749=0.753=0.755=0.760=0.761=0.762=0.766=0.770=0.771

(c)

Figure 5.7: The four-point dynamic susceptibility 4(l, ) as a function of the time difference forvarious . The distance l is equal to 2R (a), 8R (b), and 15R (c). The peak position 4 (l, ) and peak heighth4 (l, ) characterize the typical time scale of and the amount of floaters contributing to the dynamicalheterogeneity.

106

Quantifying the heterogeneous flow


5.3. Dynamic susceptibility Chapter 5. Dense heterogenous flow...

0.64 0.66 0.68 0.7 0.72 0.74 0.76 0.78

0

5

10

15

20

25

30

35

40

45

50

w Q(l,

)

(s)

l=2Rl=5Rl=8Rl=10Rl=15R =0.753

=0.761

0.64 0.66 0.68 0.7 0.72 0.74 0.76 0.78

0

5

10

15

20

25

30

35

40

45

50

4(l

, )

(s)

l=2Rl=5Rl=8Rl=10Rl=15R

0.64 0.66 0.68 0.7 0.72 0.74 0.76 0.78

0

5

10

15

20

25

30

h 4 (

l, )


Figure 5.8: The decay width wQ(l, ) of Q(l, ) (a), the peak position 4 (l, ) (b) and the height h4 (l, )(c), both of the dynamic susceptibility 4(l, ), as a function of . For each quantity, the distance l variesfrom 2R to 15R. The results indicate that l does not change the overall trend when increases. See maintext for further details.

107

5.3. Dynamic susceptibility Chapter 5. Dense heterogenous flow...

0.64 0.66 0.68 0.7 0.72 0.74 0.76 0.78

0

5

10

15

20

25

30

35

40

45

50

w Q(l,

)

(s)

l=2Rl=5Rl=8Rl=10Rl=15R =0.753

=0.761

0.64 0.66 0.68 0.7 0.72 0.74 0.76 0.78

0

5

10

15

20

25

30

35

40

45

50

4(l

, )

(s)


0.64 0.66 0.68 0.7 0.72 0.74 0.76 0.78

0

5

10

15

20

25

30

h

4 (l, )


Figure 5.8: The decay width wQ(l, ) of Q(l, ) (a), the peak position 4 (l, ) (b) and the height h4 (l, )(c), both of the dynamic susceptibility 4(l, ), as a function of . For each quantity, the distance l variesfrom 2R to 15R. The results indicate that l does not change the overall trend when increases. See maintext for further details.

107

time-scale of the heterogeneity

amount of the heterogeneity

Greetings,


[email protected]

[email protected]

[email protected]


[email protected]

Samenstelling promotiecommissie:

Prof. dr. G. van der Steenhoven (voorzitter) Universiteit TwenteProf. dr. Detlef Lohse (promotor) Universiteit TwenteDr. Devaraj van der Meer (assistent-promotor) Universiteit TwenteProf. dr. Karen E. Daniels North Carolina State UniversityProf. dr. J. Frits Dijksman Universiteit TwenteDr. Dirk van den Ende Universiteit TwenteProf. dr. Stefan Luding Universiteit TwenteDr. Peter Schall Universiteit van Amster

Education

Floaters on Faraday waves for Final Motimo Meeting - Toulouse Sep. 22-23, 2015