Anisotropic Particles Focusing Effect in Complex Flows
Severine Atis, Matthieu Leclair, Themistoklis Sapsis, Thomas Peacock
AAnisotropic Particles Focusing Effect in Complex Flows
S´everine Atis,
1, 2
Matthieu Leclair, Themistoklis Sapsis, and Thomas Peacock Massachusetts Institute of Technology, Department of Mechanical Engineering, Cambridge, Massachusetts 02139. University of Chicago, Department of Physics, Chicago, Illinois 60637
The dispersion of a tracer in a fluid flow is influenced by the Lagrangian motion of fluid elements.Even in laminar regimes, the irregular chaotic behavior of a fluid flow can lead to effective stirringthat rapidly redistributes a tracer throughout the domain. When the advected particles possess a fi-nite size and nontrivial shape, however, their dynamics can differ markedly from passive tracers, thusaffecting the dispersion phenomena. Here we investigate the behavior of neutrally buoyant particlesin 2-dimensional chaotic flows, combining numerical simulations and laboratory experiments. Weshow that depending on the particles shape and size, the underlying Lagrangian coherent structurescan be altered, resulting in distinct dispersion phenomena within the same flow field. Experimentsperformed in a two-dimensional cellular flow, exhibited a focusing effect in vortex cores of particleswith anisotropic shape. In agreement with our numerical model, neutrally buoyant ellipsoidal par-ticles display markedly different trajectories and overall organization than spherical particles, witha clustering in vortices that changes accordingly with the aspect ratio of the particles.
January 19, 2021The dynamics of finite-size particles in fluid flows canbe considerably different from infinitesimal fluid parti-cles, leading to distinct clustering and dispersion phe-nomena. From pollutant transport and plankton dis-tribution on the ocean surface, to accretion phenom-ena in astrophysics, predicting the dynamics of finite-sized objects in a given field of force matters in a widevariety of natural phenomena and industrial processes.The repercussions of finite-size effects on the transportof material particles are crucial in understanding manygeophysical and environmental processes, as well as forocean decision-making strategies, and presses the needfor improved predictive capability of material dispersionin fluid flows.In this context, Lagrangian coherent structures (LCS)provide a useful framework to understand the transportof material in complex, unsteady flow fields [1]. How-ever, depending on their size and buoyancy, objects thatpresent inertia can follow markedly different trajectoriesthan passive fluid particles [2–4]. The shape of the par-ticles; i.e their symmetry properties, can also affect theirresponse to fluid flows, leading to distinct behaviors be-tween anisotropic objects and their symmetric counter-part [5, 6]. For instance, in cloud formation process, theanisotropic shape of ice particles influences their trans-port [6] and aggregation [7, 8] in turbulent atmosphericflows. Fiber transport in cellular flows, on the otherhand, can lead to very different hydrodynamic disper-sion depending on the buckling properties of the material[9, 10]. Under marine conditions on the other hand, pref-erential alignment of elongated microorganisms in theflow local direction can also change the light backscat-tering and alter the rates of global carbon fixation inphytoplankton blooms [11]. In recent studies, predictivemodels suggested that the irregular shape of microplas-tic contaminants can influence the long term transporton the surface of the ocean [12–14].In this paper, we investigate experimentally the longterm dynamics of rod shape particles with various aspect ratio, in a quasi-2-dimensional Stokes flow. We presenta simple model that combines finite-size effects with anorientational-dependent dynamics, and perform in par-allel, numerical simulations under identical conditionsto the experiments. Our results suggest that particleswith anisotropic shape present stronger inertial effectsthan spherical particles, and are attracted to regions withhigher vorticity.Inertial effects arise when the advected particle’s ac-celeration is different from the carrier fluid’s accelera-tion; namely, the particles are responding to changes inthe underlying flow field over a finite time, for instancewhen the density of the advected particles and the den-sity of the fluid are different. Even for small densitydifferences, these effects can accumulate over time andeventually lead the particles to exhibit drastic deviationsfrom the fluid flow trajectories. To account for these ad-ditional forces, the particles dynamics can be describedby the Maxey-Riley equation [15, 16]. Assuming the par-ticle’s radius a is much smaller than the characteristicflow length scale L , the Fax´en correction terms and thememory term can be neglected, and after rescaling spaceby L , and time by T = L/U , with U the characteristicflow velocity, the simplified equation of motion becomes[17]: d v dt = 3 R D u Dt − R St ( v − u ) + (cid:18) − R (cid:19) g . (1)The particle’s velocity v variation now depends on threeforces described by the right hand side of the equation.The first term accounts for advection and added mass,with u the flow velocity, the second one is the Stokesdrag force, and the third is the buoyancy force, where g is gravity. Note that d./dt = ∂.∂t + ( v · ∇ ) . and D./Dt areconsidered different here since the particles are neutrallybuoyant but not infinitesimally small [18]. The differencebetween the particle density ρ p and the fluid density ρ f is characterize by the ratio R = 2 ρ f / ( ρ f + 2 ρ p ), and St a r X i v : . [ phy s i c s . f l u - dyn ] J a n is the particle’s Stokes number, given by:St = 29 ρ p ρ f (cid:16) aL (cid:17) Re, (2)wheres Re =
U L/ν is the flow Reynolds number and ν the fluid kinematic viscosity. This simplified Maxey-Riley equation has been commonly used in the litera-ture to describe small but finite-size particle’s dynam-ics [2, 16–19]. For instance, experiments conductedin 3-dimensional turbulent flows [4, 20–23], and in 2-dimensional chaotic flow [3, 18], have shown that theparticles behavior can be partly quantified by the par-ticle’s Stokes number.When the particles are neutrally buoyant, such that ρ p = ρ f , the equation (1) converges toward the fluid ele-ments motion for infinitesimal particles a (cid:28) L , such thatSt →
0. However, if the particle’s size is not negligible,their dynamics can exhibit a finite relaxation time dueto the interactions of the flow over the particle’s surface[2–4, 18]. Even for neutrally buoyant particles, these ef-fects can be responsible for additional forces acting onthe particle, and can result as well in particles’ trajecto-ries to deviate from the fluid flow. For neutrally buoyantparticles R = 2 /
3, and equation (1) becomes [17]: d v dt = D u Dt −
23 St − ( v − u ). (3)Compared with spheres, non-spherical particles ex-hibit an additional mechanisms due to the couplingbetween translational and rotational behavior. Theseparticles can then experience both a torque and anorientation dependent drag force. The motion of small,neutrally buoyant ellipsoidal particles in laminar flowhas been first studied by G. B. Jeffery in 1922 [24].His theoretical work on the torque experienced bynon-inertial ellipsoidal particles, and more recently,numerical simulations on anisotropic particles under-going orientation dependent drag force and relaxationtimes [5, 14, 25], suggest that the dynamics of evenvery small objects, i. e. St (cid:28)
1, can be stronglymodified due to their shape. Experiments in turbulentflow, have also indicated that the turbulent kineticenergy of a suspension is altered accordingly with thesuspended particles’ shape [26]. Finally, even withweak inertial effects coupled with orientation depen-dent forces, the transport of anisotropic particles canlead to distinct trajectories over long period of times [14].
Model
To describe the effects resulting from finite size andanisotropic shape with a minimal model, we present herean equation of motion that couples the particle’s orien-tation with the drag force. Neutrally buoyant ellipsoidalparticles experience a torque that tends to align theiraxes parallel to the surrounding fluid principal axis ofdistortion. When the particle’s Reynolds number is neg-ligible Re p = av/ν (cid:28)
1, they can be considered iner- tialess and their velocity are equal to the surroundingfluid v = u at all time, such that there is no resultantforce acting on the particle [24]. In laminar flow, thisresultant torque on the particle vanishes at first order,and the particle angular velocity ˙ θ can be simply ex-pressed as a function of the flow vorticity ω and strain-rate: [16, 24, 27, 28]:˙ θ = ω (cid:18) − (cid:15) (cid:15) + 1 (cid:19) × (cid:20) sin 2 θ (cid:18) ∂u x ∂x − ∂u y ∂y (cid:19) − cos 2 θ (cid:18) ∂u y ∂x + ∂u x ∂y (cid:19)(cid:21) (4)where (cid:15) = b/a is the particle aspect ratio, with a and b the semi-minor and semi-major axis respectively. In thisconfiguration θ represent the angle of the particle majoraxis with ˆx in the fixed frame, and u x and u y are thehorizontal and vertical components of the flow velocityin the same reference frame.In order to consider finite-size particles, we need totake into account inertial forces that will tend to sep-arate the particles trajectory from the fluid trajectory,such that now v (cid:54) = u . The dynamics of elongated par-ticles is not well approximated by Eq. (3), and need totake into account orientation specific dynamics [29, 30].For instance, the added mass and drag force terms canbecome dependent on the particle’s instantaneous orien-tation [5, 14, 25]. Here, we present a linear momentumequation with a the hydrodynamic drag force term cou-pled with the particle’s orientation [16, 31–34]: F = − πaµ ˆˆK ( v − u ), (5)where µ is the dynamic viscosity of the fluid , and ˆˆK represents the resistance tensor of an ellipsoidal particlein the co-moving frame [5, 16, 33]. The resistance tensor ˆK is initially determined in the particle frame , and re-lates to ˆˆK in the co-moving frame through the particlerotation matrix A : ˆˆK = A − ˆKA . (6)In the case of a prolate ellipsoidal particle, e.g. invariantby rotation along its major axis, the resistance tensorreduces to a diagonal matrix. We note ˆK = 8 / (cid:15) − / ˆk and two-dimensional flow, we have: ˆk = (cid:15) − (cid:15) − √ (cid:15) − ln( (cid:15) + √ (cid:15) − − (cid:15) (cid:15) − (cid:15) − √ (cid:15) − ln( (cid:15) + √ (cid:15) − (cid:15) .(7)Substituting the drag force term applying to spheres inequation (3), we obtain: d v dt = D u Dt −
169 ( (cid:15)
St) − ˆk ( v − u ) , (8) FIG. 1. Experimental setup of the rotor-oscillator flow: onerotor attached to a traverse, are respectively ensuring the ro-tation and translation of the rod. A mirror at 45 degrees isreflecting the recorded image of the interface from the bot-tom of the tank. The particles are trapped at the center of astratified glycerol layer. such that now, the equation of motion becomes coupledto the orientation of the particle given by Eq.(4) [16].Note that the drag force term’s prefactor is now pro-portional to ( (cid:15)
St) − , and indicates that the relaxationtime of the particle will now also depend on its aspectratio; i.e. the more elongated the particle the longer theresponse time. Methods
The experimental setup consists in a rectangular con-tainer of dimensions L × W × h = 100 × ×
200 mm. Tocircumvent any surface tension effect [36], the particlesare trapped at the interface between two miscible fluidsof different density. The tank is filled with two stratifiedlayers of pure glycerol of density ρ = 1 . ± .
001 g.cm − and a solution of salted glycerol and 4% water of den-sity ρ = 1 . ± .
01 g.cm − , as represented in Fig. 1.In order to prevent any viscous instability at the inter-face, the viscosity of the salted layer is matched withpure glycerol by adding water. Viscosity measurementswere performed with an Aton Paar MCR rheometer fordifferent concentrations of water to adjust the viscos-ity of the salted glycerol solutions. Particles of density ρ p = 1 . ± .
005 are then trapped at the miscible in-terface leading to a quasi-2-dimensional layer of particlesuspension. The bottom of the tank is filled with a 1 cmlayer of FC-40 to reduce the friction at the bottom of thetank and a layer of vegetable oil on the top to prevent wa-ter absorption by the glycerol. The experiment is lightedfrom the top with a series of led and the particles motionis recorded with a 45 degree tilted mirror at the bottomthe setup. Images are digitized with a 14 bit LaVisioncamera and particles tracking velocimetry (PTV) is per-formed from filtered images. 3d-printing ABS filament of1 .
75 mm diameter was cut to different length to create
FIG. 2. (a) Experimental flow velocity field determined fromPIV measurements, with a rotor speed at 100 rpm in a steadytransversal position. The color code corresponds to the normof the flow velocity. (b) Finite time Lyapunov exponent fielddetermined from the analytical approximation of the rotor-oscillator flow [35]. anisotropic particles with the desired aspect ratios. Tominimize particle-particle interactions, a small number ofparticles (32) is initially uniformely deposited across thedomain. For each particle’s aspect ratio, five indepen-dent realizations were prepared manually by randomlyredistributing the initial positions of the particles at theglycerol/glycerol-and-salt interface, as shown in the sup-plemental figure [16]. Because of the stable stratificationconfiguration and the high viscosity of the glycerol, theinterface remains well define during consecutive realiza-tions of the same experiment. The stratification is thenrenewed by changing the fluid mixture in the tank, anddifferent particles are deposited at the interface.The experimental flow is created by stirring the fluidwith a rotating rod attached to a longitudinal translationstage. The rotation is controlled with a Lexium NEMA34 motion control motor with a Plexiglas rod of 2 cmdiameter, and the translation of the rotor with a Parkerlead-screw traverse above the tank. The resulting flow isquasi-2-dimensional in the stratified fluid layer, and sim-ilar to the topology of the rotor-oscillator flow describedby Hackborn [37]. In the steady regime, when the po-sition of the rod is fixed, the rotation of the rod in thecentral region close to the bottom boundary creates a se-quence of two Moffatt eddies [38] and a hyperbolic stag-nation point in the center. Particle image velocimetry(PIV) is performed by seeding the upper glycerol layer
FIG. 3. Numerical particles positions after a transient regime in the steady vortex flow with A = 100 and ω = 2 π/
10. Forparticles inirially at rest and Stokes number St = 5 10 − (a) spheres and (b) ellispsoids with (cid:15) = 5, and St = 10 − (c) spheresand (d) ellispsoids with (cid:15) = 5. (e) Trajectories for two spherical (orange) and two ellispoidal (purple) particles with St = 10 − starting from the same initial positions. (f) Velocity difference v x − u x variation over time between the particle velocity andthe fluid velocity at the particle’s position, for a spherical (orange) and an ellispoidal (purple) particle. with 30 µ m diameter hollow glass beads. A YAG laser532 nm is used to illuminate a sheet in the horizontalplane to visualize the flow at the vicinity of the planecontaining the inertial particles. A typical flow velocityobtained from the PIV analysis is displayed on the Fig-ure 2. The inner region in the direct proximity of therotor exhibits the highest velocities (cid:107) u (cid:107) (cid:38)
10 mm.s − ,and rapidly decays within a rotor radius away from therotating boundary. The two co-rotating Moffatt eddiesare also visible, with an average flow speed in the outerregion of u (cid:39) ± − in their vicinity.We perform in parallel numerical simulations solvingthe governing equations (4) and (8) under identicalconditions with the experiments. In order to couple theequations (4) and (8) with the experimentally measuredflow field, the velocity field is nondimensionalized suchthat x ∗ = x/L with L = 100 mm corresponding to thewidth of the tank, and u ∗ = uT /L , where L/T is thecharacteristic flow velocity estimated from the averageflow in the outer region surrounding the vortices. Thesize and the aspect ratio of the particles are set by choos-ing identical Stokes number and (cid:15) with the experimentalparticles. The particles initial positions are on a gridand start with a slightly smaller initial velocity relativeto the fluid velocity, such that v ( t = 0) = 0 . u at theparticle’s initial position, and the initial orientation ofthe anisotropic particles θ are randomly distributed. Results
We first numerically investigate the effect of particlessize and shape on their trajectories over a large rangeof Stokes number 10 − (cid:54) St (cid:54) − , and aspect ratio1 (cid:54) (cid:15) (cid:54)
10. We consider a double gyre flow with periodicboundary conditions defined by the stream function ψ ( x, y, t ) = A sin[( πf ( x, t )] sin( πy ), (9)where A sets the flow amplitude, f ( x, t ) = x + B ( x −
2) sin ωt , and B sets the oscillation amplitude. In thesteady regime B = 0, and the stream function reducesto ψ ( x, y ) = A sin( πx ) sin( πy ), leading to a simple vortexflow. The resulting trajectories and spatial distributionof neutrally buoyant particles with different shapes areshown in Figure 3 for two different Stokes number and astream function amplitude of A = 100. At this flow am-plitude, the inertial particles are rapidly converging to asteady spatial distribution. As can be seen from the par-ticles distribution in Fig. 3(a)-(d), both types of particlesdisplay a preferential concentration in the vicinity of thevortex core, which increases with the St number. In addi-tion, ellipsoidal particles orientation aligns with the localflow streamlines, and systematically present a strongerfocusing behavior, converging toward smaller orbits radiiat equal St number with the spherical particles. Thiseffect can be attributed to the reduced prefactor in thedrag force term in Eq. (8) for anisotropic particles whichresults in an increasingly larger relaxation time with (cid:15) to FIG. 4. Particles FTLE field in simulations with spherical particles, left column, and elliptical particles with (cid:15) = 5, rightcolumn, after T = 20 time steps in a double-gyre flow. The corresponding flow parameters are: A = 100, e = 0 .
25, and ω = 2 πA with a particle inital velocity at v = 0 . u , (a) and (b), and v = 0, (e) and (f). Corresponding particles positions insimulations with spherical particles, left column, and elliptical particles with (cid:15) = 5, right column deviations from the flow velocity. Even at small St num-ber St ≈ − , when inertial effects are expected to benegligible, anisotropic particles’ trajectories continues toexhibit a preferential concentration at a given distancefrom the vortex center, while spherical particles trajec-tories remain more uniformly distributed.As can be seen in Figure 3(e) and (f), spherical par-ticles with the largest simulated St numbers are rapidlytrapped on a regular orbit, similar to the results showby Babiano et al. [18]. In contrast, anisotropic particlesat equal St number exhibit a more irregular trajectorywithin the same flow field, the particle’s velocity doesn’tconverge to the flow velocity field while their orbits arecloser to the vortex core.We now use the full time-dependent double gyre flow,and set e = 0 .
25. In time varying flows, attractive andrepulsive structures are only defined over a finite-timewindow period. Regions with a low FTLE value is in-dicative of attractive properties; i.e. initially neighbor-ing fluid particles remain close to each other over a giventime window, while high values of the FTLE field corre-spond to repulsive regions of the flow, where the spatialseparation over the same time window between neigh-boring fluid particles is maximized [1, 39]. Figures 4compares spatial distribution and corresponding FTLEfor spherical and ellipsoidal with two different St num-ber and initial conditions. In all configurations, after t = 2 /A , whith 1 /A corresponding to the gyre rotationperiod, the ellipsoidal particles distributions systemati-cally present more contrasted spacial density variationsthan the spherical ones. The corresponding FTLE fieldsare then computed from the same finite-size particle tra-jectories [40]. While the spherical particles FTLE fieldsresemble the flow FTLE field (see the supp. mat. [16]for comparison) while the elliptical particles FTLE fieldare more affected and exhibit additional structures local-ized in regions with stronger mixing. This suggests thatelliptical particles more complex dynamics lead to a de- parture from passive particles dynamics in time varyingflows even for very small St numbers.We now present a series of experiments with rod shapeparticles for three different aspect ratio (cid:15) = 1, 2 and5. The experimental flow has the geometry of a rotor-oscillator (RO) flow in the steady regime; an example ofthe flow velocity field is shown in Fig. 2(a) and moredetails can be found in [16]. We impose a constant rotorspeed of 200 rpm for all the experiments. In this configu-ration, the Re number typically range between 10 − and10 over the flow field in the region away from the directvicinity of the rotor. This leads to relatively small par-ticle Stokes number St ≈ − − − , such that, in theabsence of anisotropy induced effects, we expect the par-ticles to behave closer to tracers particles, and particleswith (cid:15) = 1 can be approximated with spherical particles.The particles are initially uniformly distributed acrossthe experimental domain with no preferential alignmentacross the quasi-2-dimensional plane defined by the two-layer stratification, see [ ? ] for details. Immediately afterthe flow is switched on, particles with (cid:15) > z axes and exhibit a preferential alignment of theirprincipal axes with the local flow direction, as shown inthe supplementary Figure [16] and supplemental movie 2.This suggests that the orientational behavior has a negli-gible response time to the changes in the flow field. Theelongated particles almost instantaneously align with thelocal flow direction, in good agreement with our assump-tion in Eq. (4) and previous observations in flows athigher Reynolds number, where the particles align withthe strongest Lagrangian stretching direction [28, 41].The particles remain in the same plane for the entireduration of the experiments, and the principal axes ofthe elongated particles doesn’t display off-plane motionsexcept near the direct vicinity of the rotor.After a time t (cid:38) τ , where τ = 0 . FIG. 5. Top row: experimental particles final positions at t = 3500s, middle row: PTV determined from the correspondingexperiments for t = 0 − t = 0 − strong dependence to aspect ratio, as can be seen on Fig.5(a)-(c) and supplemental video 3. The final position ofthe particles with aspect ratio (cid:15) = 1 do not exhibit anypreferential concentration, and remain uniformly spreadacross the domain, while particles with (cid:15) = 2 and (cid:15) = 5display a migration from the outer regions of the flowinto the vortex cores and their vicinity. Remarkably, for (cid:15) = 5, all the particles are focused in the central region.Note that for (cid:15) = 2 and (cid:15) = 5, a few particles can beattracted to the secondary vortices of the RO-flow nearthe edges of the tank and are not covered by our field ofview in Fig. 2(a) and Fig. 5).In order to quantify this behavior, we determine the in-dividual particles trajectories using particle tracking ve-locimetry (PTV), and monitor their spatial distributionover time. Figure 5(d)-(f) displays the particles posi-tion as a function of time for the entire duration of theexperiments corresponding to the snapshots in Figures5(a)-(c) respectively. As can be seen in Fig. 5(e) and5(f), in addition to be attracted to the central region ofthe flow field, some of the elongated particles’ final tra-jectories form closed orbits inside the vortices. While for (cid:15) = 2, the rest of the particles are evenly distributed inthe vicinity the vortices, particles with (cid:15) = 5 exhibit astrong concentration in a region encircling the two vor-tices. Figure 6 displays the measured particle densityvariation between 0 ≤ t ≤ τ in the central region ofthe flow shown by dashed white lines in Fig. 5(c). Ina first transient regime, the spatial density of the elon-gated particles increases linearly over time with a largermigration rate for particles with larger aspect ratio (cid:15) . Af- ter a time t = t ∗ , the migration ends and the density ofparticles in the central region of the flow remains con-stant over time. Interestingly, this saturation time scale t ∗ (cid:39) τ = 1500s seems to be identical for both testedaspect ratio. On the other hand, particles with aspectratio (cid:15) = 1 present a negligible focusing effect, and theirspatial distribution is stationary over time.To test the accuracy of our model, we perform in paral-lel numerical simulations using the experimentally mea-sured flow field shown in Fig. 2. The particles St numberand aspect ratio are chosen identical to the experimen-tal particles. The remaining free parameter, the charac-teristic time T of the flow field, is adjusted to give thebest match between the experimental and numerical par-ticles velocity for a given region of the flow, additionaldetails can be found in the Methods section [16]. Thebest agreement with the experimental particles trajecto-ries is obtained for a time scale of T = 20 s, and is ingood agreement with the largest flow structure’s averagevelocity u = L/T ≈ t (cid:46)
500 s, after
FIG. 6. Spatial particle density in variation as a functionof time in the central region of the flow indicated by whitedashed lines in Fig. 5c). Experimental data: lines representaverage value and shaded region the error. Numerical parti-cles with St = 5 10 − coupled with the same experimentalflow field: dots, the two series of data correspond to simula-tions with two different sets of parameters, see [16] for details.Blue : (cid:15) = 1, green: (cid:15) = 2, and red: (cid:15) = 5 which the numerical particles start entering the satura-tion regime around t (cid:39) τ = 750 s, a little earlierthan the experimental value t ∗ (cid:39) τ .Comparing the shape of high particle concentrationregions in the numerical simulations with the RO flowFTLE field shown in Fig ?? (b), we can identify to whichflow structures the particles are attracted to. The FTLEfield displays several ridges. Two minima are created inthe vortex cores, and correspond to the two ellipticalregions where the elongated particles are trapped inthe experiments. A third structure can be identifyand corresponds to the minima of the FTLE field thatforms a larger orbit encircling the two vortices. Bothexperimental and numerical particles with the largestaspect ratio (cid:15) = 5 are specifically attracted to this regionas can be seen in Fig 5f) and 5i). Discussion
In our experiments, the observed migration towardvortices for elongated particles with a small Stokes num-ber St ∼ − , suggests that even weak deviations frompassive trajectories can build up over time, and lead todramatically different dynamics over long time periods. These perturbations are not sufficient to significantly al-ter particles trajectories with an aspect ratio (cid:15) = 1, andparticles with (cid:15) = 2 exhibit a significantly weaker focus-ing effect than those with (cid:15) = 5. A direct comparisonof these observations with numerical simulations, showsthat our minimal model qualitatively describes this be-havior, although the particles focusing effect exhibits amore moderate dependence to the particle aspect ratio (cid:15) than in the experiments. This difference could be at-tributed to second order orientation dependent terms,such as the added mass [5, 14, 25, 29, 30], that are ne-glected here.These results suggest that anisotropic particles re-spond to the flow with a stronger lag, such that in Eq.8 St (cid:48) = (cid:15) St can be interpreted as an “effective” Stokesnumber St (cid:48) that increases with (cid:15) . When the inertial ef-fects are small, i. e. for smaller St number, the differencebetween St and St (cid:48) becomes relevant only over larger timescales, such that small differences in the particles dynam-ics lead to distinct trajectories after a long time period.This effect could potentially enhance the migration rateof irregularly shaped microplastic toward gyres of marinedebris [13, 14], and influence the garbage patches forma-tion dynamics [12, 42]. On the opposite length and timescales, in biology this interplay between the shape of theparticle and the flow field structures could influence theselective capture of microorganisms by larger animals.For instance, flows created by the beating of cilia cangenerate a local flow that attract only microorganismsof an optimal size [43]. Identical mechanisms could po-tentially select for an optimum shape, for instance, theaspect ratio of a given bacteria.In conclusion, even neutrally buoyant particles withsmall particle Stokes number St (cid:39) − exhibit trajec-tories that deviate from the fluid flow when they possessan anisotropic shape. Both experimental and numericalparticles exhibit a migration toward the regions in thevicinity of the vortices. The orientational dependenceof the drag term in our minimal model describes theparticles dynamics qualitatively, suggesting that thiseffect might be sufficient to capture the separationbetween isotropic and anisotropic particles trajectoriesin cellular flows. In addition, the effect of other orien-tation dependent terms should be studied in future work. Acknowledgements
The authors would like to thank A.W. Murray and E.Climent for stimulating discussions, G. Haller, T. Wittenand E. Kanso for their interesting comments. S. Atis wassupported by the French government DGA/DS - Missionpour la Recherche et l’Innovation Scientifique fellowshipduring her visit at MIT. [1] Thomas Peacock and George Haller, “Lagrangian coher-ent structures The hidden skeleton of fluid flows,” PhysicsToday , 41–47 (2013). [2] George Haller and Themistoklis Sapsis, “Where do in-ertial particles go in fluid flows?” Physica D-nonlinearPhenomena , 573–583 (2008). [3] Nicholas T Ouellette, P J J O’Malley, and J P Gollub,“Transport of Finite-Sized Particles in Chaotic Flow,”Physical Review Letters , 174504 (2008).[4] Rachel D Brown, Z Warhaft, and Greg A Voth, “Ac-celeration Statistics of Neutrally Buoyant Spherical Par-ticles in Intense Turbulence,” Physical Review Letters , 194501 (2009).[5] H F Zhang, G Ahmadi, F G Fan, and J B McLaughlin,“Ellipsoidal particles transport and deposition in turbu-lent channel flows,” International Journal of MultiphaseFlow , 971–1009 (2001).[6] Rui Ni, Nicholas T Ouellette, and Greg A Voth, “Align-ment of vorticity and rods with Lagrangian fluid stretch-ing in turbulence,” Journal of Fluid Mechanics , R3(2014).[7] Ewe Wei Saw, Raymond A Shaw, SathyanarayanaAyyalasomayajula, Patrick Y Chuang, and Armann Gyl-fason, “Inertial clustering of particles in high-reynolds-number turbulence,” Physical Review Letters ,214501 (2008).[8] K D SQUIRES and J K EATON, “Particle Responseand Turbulence Modification In Isotropic Turbulence,”Physics of Fluids A-fluid Dynamics , 1191–1203 (1990).[9] Elie Wandersman, Nawal Quennouz, Marc Fermigier,Anke Lindner, and O Du Roure, “Buckled in transla-tion,” Soft matter , 5715–5719 (2010).[10] N Quennouz, Michael Shelley, O Du Roure, and A Lind-ner, “Transport and buckling dynamics of an elastic fibrein a viscous cellular flow,” J Fluid Mech , 387–402(2015).[11] Marcos, Justin R Seymour, Mitul Luhar, William MDurham, James G Mitchell, Andreas Macke, and Ro-man Stocker, “Microbial alignment in flow changes oceanlight climate,” Proceedings of the National Academy ofSciences of the United States of America , 3860–3864(2011).[12] C Broday, D and Fichman, M and Shapiro, M andGutfinger, “Motion of diffusionless particles in verticalstagnation flows—II. Deposition efficiency of elongatedparticles,” Journal of Aerosol Science , 35–52 (1997).[13] I Chubarenko, A Bagaev, M Zobkov, and E Esiukova,“On some physical and dynamical properties of mi-croplastic particles in marine environment,” MPB ,105–112 (2016).[14] Michelle H Dibenedetto, Nicholas T Ouellette, and Jef-frey R Koseff, “Transport of anisotropic particles underwaves,” , 320–340 (2018).[15] M R Maxey and J J Riley, “Equation of Motion For ASmall Rigid Sphere In A Nonuniform Flow,” Physics ofFluids , 883–889 (1983).[16] Supplementary material, “No Title,” .[17] Themistoklis P Sapsis, Nicholas T Ouellette, Jerry PGollub, and George Haller, “Neutrally buoyant parti-cle dynamics in fluid flows: Comparison of experimentswith Lagrangian stochastic models,” Physics of Fluids , 93304 (2011).[18] Armando Babiano, Julyan H E Cartwright, Oreste Piro,and Antonello Provenzale, “Dynamics of a small neu-trally buoyant sphere in a fluid and targeting in Hamilto-nian systems,” Physical Review Letters , 5764 (2000).[19] E E Michaelides, “The Transient Equation of Motion forParticles, Bubbles, and Droplets,” Journal of fluids engi-neering , 233 (1997).[20] A La Porta, G A Voth, A M Crawford, J Alexander, and E Bodenschatz, “Fluid particle accelerations in fullydeveloped turbulence,” Nature , 1017–1019 (2001).[21] R Volk, E Calzavarini, E Leveque, and J . F Pinton, “Dy-namics of inertial particles in a turbulent von Karmanflow,” Journal of Fluid Mechanics , 223–235 (2011).[22] Enrico Calzavarini, Romain Volk, Emmanuel Leveque,Jean-Francois Pinton, and Federico Toschi, “Impact oftrailing wake drag on the statistical properties and dy-namics of finite-sized particle in turbulence,” Physica D-nonlinear Phenomena , 237–244 (2012).[23] Simon Klein, Mathieu Gibert, Antoine Berut, and Eber-hard Bodenschatz, “Simultaneous 3D measurement ofthe translation and rotation of finite-size particles andthe flow field in a fully developed turbulent water flow,”Measurement Science & Technology , 24006 (2013).[24] G B Jeffery, “The motion of ellipsoidal particles in a vis-cous fluid,” Proceedings of the Royal Society of LondonSeries A-containing Papers of A Mathematical and Phys-ical Character , 161–179 (1922).[25] P H Mortensen, H I Andersson, J J J Gillissen, andB J Boersma, “Dynamics of prolate ellipsoidal particlesin a turbulent channel flow,” Physics of Fluids , 93302(2008).[26] Gabriele Bellani, Margaret L Byron, Audric G Collignon,Colin R Meyer, and Evan A Variano, “Shape effects onturbulent modulation by large nearly neutrally buoyantparticles,” Journal of Fluid Mechanics , 41–60 (2012).[27] J A Olson and R J Kerekes, “The motion of fibres inturbulent flow,” Journal of Fluid Mechanics , 47–64(1998).[28] Shima Parsa, Jeffrey S Guasto, Monica Kishore,Nicholas T Ouellette, J P Gollub, and Greg A Voth, “Ro-tation and alignment of rods in two-dimensional chaoticflow,” Physics of Fluids , 43302 (2011).[29] I Gallily and A H Cohen, “Orderly Nature of the Mo-tion of Nonspherical Aerosol-particles .2. Inertial Colli-sion Between A Spherical Large Droplet and An AxiallySymmetrical Elongated Particle,” Journal of Colloid andInterface Science , 338–356 (1979).[30] D L Koch and E S G Shaqfeh, “The Instability of ADispersion of Sedimenting Spheroids,” Journal of FluidMechanics , 521–542 (1989).[31] H Brenner, “The Stokes Resistance of An Arbitrary Par-ticle .3. Shear Fields,” Chemical Engineering Science ,631–651 (1964).[32] H Brenner, “The Stokes Resistance of An Arbitrary Par-ticle .4. Arbitrary Fields of Flow,” Chemical EngineeringScience , 703–727 (1964).[33] Fa-Gung Fan and Goodarz Ahmadi, “Dispersion of el-lipsoidal particles in an isotropic pseudo-turbulent flowfield,” Journal of fluids engineering , 154–161 (1995).[34] E Loth, “Drag of non-spherical solid particles of regularand irregular shape,” Powder Technology , 342–353(2008).[35] M Weldon, T Peacock, G B Jacobs, M Helu, andG Haller, “Experimental and numerical investigation ofthe kinematic theory of unsteady separation,” Journal ofFluid Mechanics , 1–11 (2008).[36] Dominic Vella and L Mahadevan, “The “Cheerios ef-fect”,” American Journal of Physics , 817 (2005).[37] W W Hackborn, “Asymmetric Stokes flow between par-allel planes due to a rotlet,” Journal of Fluid Mechanics , 531–546 (1990).[38] H K Moffatt, “Viscous and resistive eddies near a sharp corner,” Journal of Fluid Mechanics , 1–18 (1964).[39] George Haller, “Lagrangian coherent structures,” AnnualReview of Fluid Mechanics , 137–162 (2015).[40] M Sudharsan, Steven L Brunton, and James J Riley,“Lagrangian coherent structures and inertial particle dy-namics,” Physical Review E , 033108 (2016).[41] R U I Ni, Nicholas T Ouellette, and Greg A Voth,“Alignment of vorticity and rods with Lagrangian fluid stretching in turbulence,” Journal of Fluid Mechanics (2014), arXiv:arXiv:1311.0739v1.[42] Erik van Sebille, Matthew H England, and Gary Froy-land, “Origin, dynamics and evolution of ocean garbagepatches from observed surface drifters,” EnvironmentalResearch Letters , 44040 (2012).[43] Yang Ding and Eva Kanso, “Selective particle captureby asynchronously beating cilia,” Physics of Fluids27