Dense turbulent suspensions at a liquid interface

Abstract The nexus between turbulence, particle interaction and interfacial tension is virtually unexplored, despite being highly relevant to a wealth of industrial and environmental settings. Here we investigate it by conducting experiments on non-Brownian spherical particles at the interface of turbulent liquid layers. The latter are electromagnetically stirred in a quasi-two-dimensional apparatus, while the particles are individually tracked. By systematically varying interfacial conditions, turbulence intensity, particle size and concentration from dilute to dense, we map the system behaviour over a wide parameter space. We reveal how the dynamics is governed by the balance of drag, capillarity and lubrication. Based on their scaling, we propose a phase diagram comprising three distinct regimes, characterized by widely different levels of clustering and fluctuating energy of the particles. This is quantitatively confirmed by the experimental results.


Introduction
Dense turbulent suspensions are ubiquitous in industrial and natural scenarios, from process engineering to sediment transport.However, due to the number of physical mechanisms and the daunting range of scales, reaching a predictive understanding of these systems remains a formidable and unmet challenge.Historically, research on solid-liquid mixtures has predominantly gravitated around two extrema: on one side, dense and viscous-dominated suspensions (Guazzelli, Morris & Pic 2011); on the other, turbulent and dilute ones (Balachandar & Eaton 2010).Significant progress has been made in the understanding of inertial effects in suspensions, focusing on instabilities and the onset of turbulence, e.g. in pipe flows (Matas, Morris & Guazzelli 2003;Hogendoorn & Poelma 2018;Agrawal, Choueiri & Hof 2019) and Taylor Couette flows (Majji, Banerjee & Morris 2018;Dash, Anantharaman & Poelma 2020;Morris 2020b).Only recently have researchers started bridging the gap and tackled fully turbulent dense suspensions: in experiments, through refractive-index matching and medical imaging (Zade et al. 2018;Baker & Coletti 2019;Alméras et al. 2021;Hogendoorn et al. 2023); and in simulations, through particle-resolved approaches (Picano, Breugem & Brandt 2015;Wang et al. 2016;Olivieri et al. 2020;Vowinckel et al. 2023).The nature of the measurements and the cost of the calculations, however, have imposed severe limitations on the particle material, size and number, as well as on the scales of the system.As such, the topic is still in its infancy.
An additional element, frequently encountered in solid-liquid mixtures, can further enrich the problem: the particles may be confined at fluid interface, where the dynamics is different from that in the bulk (Singh & Joseph 2005;Madivala, Fransaer & Vermant 2009;Magnaudet & Mercier 2020).The individual and collective behaviour of particles at interfaces has recently received significant attention, but the focus has been mostly on regimes where inertia is negligible or the particles are microscopic colloids (Hoekstra et al. 2003;Masschaele, Fransaer & Vermant 2011;Fuller & Vermant 2012;Garbin 2019).In all three classes of suspensions -dense, turbulent and interfacial -the particles tend to organize in clusters, but the underlying mechanisms are profoundly different.In dense suspensions, concentration fluctuations are driven by the hydrodynamic coupling between particles under shear (Wagner & Brady 2009;Brown & Jaeger 2014;Denn & Morris 2014).In turbulent suspensions, the particles distribute inhomogeneously when inertia prevents them from following the fluid trajectories (Bec et al. 2007;Balachandar & Eaton 2010;Monchaux, Bourgoin & Cartellier 2012;Brandt & Coletti 2022).At liquid interfaces, buoyancy and surface tension generate attractive forces leading to compact assemblies (Vella & Mahadevan 2005;Botto et al. 2012;Protière 2023).How do particles behave when these three situations intersect?
The present study focuses on the unexplored territory where turbulence, particle-particle interactions and interfacial effects are simultaneously at play.Examples of great practical relevance include microplastic pollution (van Sebille et al. 2020) and froth flotation in industrial processing (Quintanilla, Neethling & Brito-Parada 2021).Here we consider a seemingly simple system consisting of non-Brownian monodispersed spherical particles confined at the interface of quasi-two-dimensional (Q2D) turbulent layers.We perform extensive laboratory experiments in which turbulence intensity, interfacial tension, particle size, density and concentration are systematically varied.We investigate the propensity of the particles to form clusters and their response to the turbulent flow, both of which are shown to depend on just two non-dimensional parameters.These express the balance between the three primary forces: drag exerted by the fluid, capillarity at the interface, and lubrication between particles.We are thus able to establish a phase diagram that describes the widely different regimes exhibited by the system.

Methods
We employ shallow layers of conductive fluid (CuSO 4 , 10 % aqueous solution by mass, ρ f = 1.08 g ml −1 , ν = 1.0 × 10 −6 m 2 s −1 ) in a 320 mm × 320 mm tray placed above an 8 × 8 array of permanent magnets, alternating their polarity in a chequerboard arrangement.Two copper electrodes immersed at opposite sides run DC current through the fluid and drive the flow by Lorentz force, the centre-to-centre distance between neighbouring magnets (L f = 35 mm) being the forcing length scale.The apparatus, similar to those used in several investigations of Q2D turbulence (Kelley & Ouellette 2011;Boffetta & Ecke 2012), was fully characterized in a separate study (Shin, Coletti & Conlin 2023).We use two arrangements of the fluid layers and two sizes of millimetric polyethylene spheres (Cospheric WPMS-1.00,CPB-0.96), as summarized in table 1.In the single-layer configuration (SL), the particles float along the surface of (though are mostly submerged in) a 7 mm-thick conductive layer (air-liquid interface).In the double-layer configurations (DL1 and DL2), a 2 mm-thick layer of mineral oil (ρ f = 0.84 g ml −1 , ν = 1.9 × 10 −5 m 2 s −1 , Sigma Aldrich) is added on top of the 8 mm-thick conductive layer, with particles positioned at this conductive layer-oil interface (liquid-liquid interface).The double-layer configurations DL1 and DL2 employ two different particle types (table 1).To prevent surfactant contamination and maintain stable interfacial properties during measurements, each fluid layer arrangement is allowed to equilibrate overnight.Continuous agitation is ensured by the forced turbulent flow throughout the experiment, preserving consistency in fluid interface conditions.We vary the aerial fraction φ ≡ N p (πd 2 p /4)/A FOV between 1 % and 71 %, where N p is the average number of particles in the field of view (FOV) of area A FOV .
For each case, we analyse 100 s long videos acquired with a Phantom VEO 640 complementary metal oxide semiconductor camera operated between 100 and 240 Hz depending on the flow condition, ensuring inter-frame particle displacements around 5 pixels (∼1 mm).To mitigate potential boundary effects, we image exclusively on the central region of the flow tray, ensuring that our FOV remains at least one L f away from the sidewalls.We follow the particle centroids along Lagrangian trajectories with subpixel accuracy, using an in-house code for particle tracking velocimetry.Clusters, broadly defined as groups of adjacent particles, are detected with the DBSCAN algorithm (Ester et al. 1996).We set the search radius of 1.25d p around each centroid to seek for a minimum of four neighbours, and verify that the results are weakly dependent on the precise value of such parameters.
We characterize each level of hydrodynamic forcing by separate experiments, in which the fluids are laden only with tracer-like polyethylene microspheres (d p = 75-90 μm, Cospheric UVPMS-BG-1.00) whose motion is characterized by particle image velocimetry.Similarly to the millimetric spheres, the tracer particles also lie predominantly at the fluid interface.The root-mean-square flow velocity u rms yields the Reynolds number Re ≡ u rms L f /ν, where ν is the kinematic viscosity of the conductive fluid.We span a decade in Re by adjusting the DC current between 0.01 and 1.00 A. All forcing levels are in the turbulent regime (Shin et al. 2023) with Kolmogorov scales η comparable to d p .

Results and discussion
The present system shows a marked tendency to form clusters, as clear from figure 1, which displays two realizations at the same Re = 970 and φ = 0.28, but different interfacial conditions.For comparison, we define χ cl as the number of clustered particles normalized by the total number of particles.Remarkably, in the SL configuration (figure 1a), the fraction of clustered particles χ cl is more than five times larger than in the DL2 configuration (figure 1b).The sizes of the clusters, approximated as the diameter D cl of the circles encompassing them, also greatly differ between the two cases.To compare the dominant mechanisms at play, in the following we derive expressions for the main forces governing the particle motion: capillarity, drag and lubrication.In the considered range of parameters, electrostatic forces, e.g.due to asymmetrically distributed charges on the particles (Aveyard et al. 2002), are several orders of magnitude weaker.The expression of the capillary attraction is determined by the nature of the distortion of the liquid interface around the particles.Assuming small slopes, the shape of the interface can be decomposed into Fourier modes.These expand into decaying multipoles, with each multipole excited by its corresponding Fourier mode at the contact line (Stamou, Duschl & Johannsmann 2000).When body forces and the torque on the particle are negligible, the leading-order interfacial disturbance h at a distance r from its centre is the quadrupolar distortion (Liu, Sharifi-Mood & Stebe 2018) where h qp is the mode's amplitude.
Conversely, when body forces on the particle are significant, the interfacial profile is described by the modified Bessel function of the second kind and order zero, whose asymptotic behaviour is logarithmic for small r.To first order, the vertical balance between buoyancy and interfacial tension then yields (Vella & Mahadevan 2005) Here, l c = ( ρ f g/γ ) −1/2 is the capillary length between fluids with density difference ρ f under the action of gravity g and interfacial tension γ ; the Bond number Bo ≡ ((ρ f − ρ p )gd 2 p )/4γ , with ρ f the density of the fluid on which the particle of density ρ p is floating; and is a dimensionless buoyancy-subtracted weight of the particle with the contact angle θ.
In all the considered configurations, Bo 1 (table 1), which indicates that the distortion induced by body forces is negligibly small.For example, for a particle in the SL configuration, where it is predominantly immersed (θ < π/2), such distortion at the contact-line radius would be <1 μm, much smaller than the quadrupolar distortion of ∼14 μm.Thus, while the slower-decaying contribution from body forces becomes significant at large separations, the quadrupolar contribution remains dominant for the groups of adjacent particles considered here.Therefore, we write (Stamou et al. 2000;Liu et al. 2018) with r the centre-to-centre interparticle distance.
Rather than characterizing h qp (which requires precise measurements of the contact angles), we perform separate experiments in which particle pairs are placed in the fluid layer configurations and approach each other due to capillary attraction.In this creeping flow regime, F capillary = −F drag = −(3/2)πμṙd p , where ṙ/2 is the velocity of each particle in the pair and μ is the dynamic viscosity of the conductive fluid in which the particles are mostly submerged.This leads to the power-law relation (Loudet et al. 2005;Liu et al. 2018) where r 0 is the initial interparticle distance at time t = 0 and Θ = 12γ h 2 qp d 3 p /μ.We experimentally verify this relation by tracking particle pairs; see figure 2. The excellent agreement with (3.5) suggests that the interface deformation causes only marginal deviations from the Stokesian solution (Loudet et al. 2020).Least-squares fits to (3.5) yield values of Θ, and thus h qp , which are utilized for evaluating the capillary attraction in (3.4).
When the fluid flow is turbulent, the drag pulling a pair of particles away from each other is where u f is the velocity difference between two fluid locations corresponding to a pair of closely situated particles, evaluated along the separation vector between the pair.As we are interested in the interaction of adjacent particles, we approximate  The strain rate is estimated by schematizing the flow field as an array of Taylor-Green vortices, whose velocity components in directions x and y are, respectively, for which ˙ max ≈ (π/ √ 2)(u rms /L f ).While simplified, such a representation of the flow yields excellent quantitative agreement with the statistical properties of 2D turbulence (Shin et al. 2023).It follows that We then define the capillary number given by the ratio of F drag and F capillary evaluated for r = d p : In figures 1(a) and 1(b), Ca = 0.16 and 2.49, respectively, which accounts for the much stronger clustering in the former: drag from the straining flow dominates, and clusters are disrupted for Ca 1, and vice versa for Ca 1.This is confirmed in figure 3(a), showing χ cl versus Ca: collectively, the data adhere to a unified trend, confirming the role of Ca as the controlling parameter.This applies at aerial fractions φ ≤ 0.20, for which F drag and F capillary are the main forces.
For large φ, on the other hand, we observe stronger clustering and a reduction in the root-mean-square particle velocity, which we attribute to additional viscous dissipation by the lubrication force (Wagner & Brady 2009;Morris 2020a).For a pair of nearby particles moving at relative speed u p , this is Balancing F lubrication and F drag yields the separation r/d p = 9/8: lubrication prevails when neighbouring particles are, on average, closer than such distance, dissipating their kinetic energy.To obtain the corresponding aerial fraction, we employ random sequential adsorption (Evans 1993) to distribute in 2D space non-overlapping circles with a range of diameters comparable to those of our particles, i.e. assuming a normal distribution of diameters with the same standard deviations as in table 1.This synthetic method indicates that the average nearest-neighbour separation reaches (9/8)d p , for φ ∼ 0.4.This approximate threshold is confirmed in figure 3(b), showing the mean cluster diameter D cl normalized by the forcing length scale L f , and the particle kinetic energy E k,p normalized by the fluid kinetic energy E k,f (measured in the unladen case under the same forcing).For cases in which drag overcomes capillarity (Ca > 1), the trends for different interfacial conditions are quantitatively similar: for φ > 0.4, D cl grows rapidly and E k,p / E k,f dips.The cluster size appears to plateau at ∼8.5L f , corresponding to the diagonal of the square FOV of size 6L f , i.e. the maximum size detectable by the imaging system.That is, above approximately φ = 0.6, the system percolates forming a macrocluster of vanishing small kinetic energy that spans the entire domain.
In summary, our analysis suggests three distinct regimes depending on the balance of three factors: capillarity, viscous drag and lubrication.When Ca < 1, capillary attraction holds particles together and leads to the formation of tightly bound clusters.When Ca > 1, the strain field of the fluid acts on the particles through viscous drag and breaks up the clusters, as long as φ < 0.4; while lubrication preserves slowly moving clusters at higher concentrations.This is schematically illustrated in the phase diagram in figure 4(a) and by the snapshots in figure 4(b) corresponding to the three regimes: capillary-driven clustering, drag-driven break-up and lubrication-driven clustering.We remark that the identified values of the control parameters Ca and φ are not expected to be critical thresholds, as the considered process does not entail abrupt transitions.In fact, within each one of these regimes, quantitative changes are still observed as the control parameters are varied.For example, in the capillary-driven clustering regime, the aggregates grow bigger with increasing φ, as more particles stick to larger groups and these merge together; this is evident by comparing the two snapshots at Ca = 0.23 in figure 4(b).However, those differences appear quantitative rather than qualitative, and the main mechanisms responsible for the behaviour in each regime are not altered.The proposed phase diagram is supported by the contour maps over the Ca-φ space displayed in figure 5, which depicts the behaviour of the key observables discussed above for the parameter space we explored.In particular, in the map of χ cl (figure 5a), dense vertical contour lines at Ca < 1 give way to horizontal ones at Ca > 1, bounding the drag-driven break-up regime.The trend of the normalized mean cluster size D cl /L f (figure 5b) is similar, with a sharp increase for φ ≥ 0.4.Over this range of concentrations, larger clusters are often centred at the core of vortical structures, where the local strain rate is insufficient to tear them apart (see the supplementary movies available at https:// doi.org/10.1017/jfm.2024.246).Furthermore, comparing this map with that of the particle kinetic energy E k,p / E k,f , we notice that the motion of clustered particles is greatly inhibited when D cl becomes much larger than L f , which is the typical size of the forced turbulent eddies (Boffetta & Ecke 2012;Shin et al. 2023): as a cluster spans multiple counter-rotating eddies, the latter are not effective in moving it around.Lastly, the energy maps reveal that even giant clusters spanning the entire domain (e.g. at φ ≥ 0.6) have different dynamics depending on Ca: they are effectively rigid and static when capillarity dominates (Ca < 1), while they are softer and dynamic when drag takes over (Ca > 1); see the supplementary movies.
We remark that the fluid kinetic energy is not directly measured, as the millimetre-sized particles impede sufficient optical access.The values we use, evaluated at the same forcing and φ = 0, are, however, expected to be a close approximation of the fluid energy in the suspensions.The submerged particles only occupy a minor portion of the conductive fluid layer and therefore do not significantly alter the Lorentz force.The frictional dissipation on the particle surfaces is also a fraction of that experienced by the fluid on the floor of the tray, which dominates the dissipation rate (Shin et al. 2023), except when giant rigid clusters of vanishing kinetic energy are formed.

Conclusion
We have investigated the spatial distribution and motion of particles at the interface of Q2D turbulent liquid layers.Those are governed by the balance of capillarity, drag and lubrication, and two non-dimensional parameters rule the system: the capillarity number Ca and the aerial fraction φ (equivalently, the normalized interparticle distance r/d p ).When Ca < 1, capillary attraction effectively binds particles into tight clusters.When Ca > 1, drag from the straining fluid flow prevents the formation of large clusters as long as φ < 0.4; while short-range lubrication yields slow-moving clusters when φ > 0.4.The present analysis assumes that the governing forces are independent, while coupling is expected, e.g. as capillary-driven clustering causes changes in the effective drag force.The agreement between prediction and observation, however, indicates that such a simplification is appropriate to first order.This study opens several avenues for future investigations in related regimes.Clearly, the proposed phase diagram and its boundaries depend on the chosen configurations and the validity of the assumptions we made.For example, marine plastics include millimetre-sized particles but also micrometre-sized particles (van Sebille et al. 2020), and the latter may hardly aggregate into cohesive clusters of size comparable to the energetic eddies.On the other hand, for larger Bo (e.g.particles only partially submerged in free-surface flows), the monopole contribution dominates and the capillary attraction decays much more slowly, with F capillary ∼ r −1 (Vella & Mahadevan 2005), with significant implications for the prevalence of clustering.Under conditions of more intense turbulence close to the surface, large surface deformations may occur, influencing the transport of floating particles.Moreover, Q2D turbulence has peculiar characteristics which separate it from its classic three-dimensional (3D) counterpart, notably an inverse cascade of energy towards the larger scales (Tabeling 2002;Boffetta & Ecke 2012).Still, it possesses essential hallmarks including chaotic/nonlinear behaviour, rotationality, stretching, folding and mixing of fluid elements, and wide ranges of spatio-temporal scales, making it a valuable system to investigate fundamental turbulence dynamics (Suri et al. 2017;Ballouz & Ouellette 2020), particularly in the context of particle suspensions (Boffetta, De Lillo & Gamba 2004;Goto & Vassilicos 2008;Ouellette et al. 2008).
An important aspect of 3D interfacial flows is the non-solenoidal nature of the velocity field along the surface, which is known to play a significant role for colloids in viscous flows (Samaniuk & Vermant 2014;Pourali et al. 2021).Further research is warranted to explore this effect in dense suspensions of particles floating on the surface of a turbulent body of water.Overall, we expect that the important mechanisms and trends revealed by the considered configurations will extend beyond the present model problem, including regimes where additional forces, such as electrostatics (van Baalen, Vialetto & Isa 2023) and soft granular repulsion (Gupta et al. 2022), become significant.Finally, the present findings can also be relevant to systems in which the multi-scale, turbulent-like behaviour is rooted in the interaction and collective dynamics of active elements, such as bacteria (Dunkel et al. 2013;Peng, Liu & Cheng 2021) and self-propelled particles (Bourgoin et al. 2020).

Figure 1 .
Figure 1.Example snapshots with detected clusters obtained at the same Re = 970 and φ = 0.28 from different configurations: (a) SL, χ cl = 0.99; and (b) DL2, χ cl = 0.23.Scale bars correspond to L f = 35 mm.The particles in each cluster are represented with the same colour.Circles encompassing each cluster are displayed.

Figure 2 .
Figure 2. (a) A time series of a pair of particles in DL1 configuration.The scale bar indicates 5 mm.(b) Plot of r 6 0 − r 6 versus t in the three configurations.The dashed line indicates ∼ t scaling relation.

Figure 3 .
Figure 3. (a) The fraction of clustered particles (χ cl ) versus Ca at φ < 0.20.(b) The normalized mean cluster diameter ( D cl /L f , black) and the normalized particle kinetic energy ( E k,p / E k,f , red) as functions of φ at Ca > 1.

Figure 4
Figure 4. (a) Three predicted clustering/break-up regimes in Ca-φ space.(b) Snapshots from different configurations, illustrating varying degrees of clustering with χ cl values of 0.991 and 0.624 (i), (ii), and 0.990 and 0.249 (iii), (iv).Insets provide magnified views to highlight differences.Particles within each cluster are represented with the same colour.

Figure 5 .
Figure 5. Phase diagram in Ca-φ space to characterize the particle behaviour, mapping (a) χ cl , (b) D cl /L f and (c) E k,p / E k,f .
where ˙ max is the maximum principal strain rate.The Stokesian assumption is justified by the particle Reynolds number Re (Ouellette, O'Malley & Gollub 2008)ll.Indeed, as the density of the particles approximately matches that of the fluid, Re p is equivalent (within a geometric factor) to the Stokes number St ≡ τ p u rms /L f(Ouellette, O'Malley & Gollub 2008), which is listed in table 1.The particle response time is taken as τ p = ρ p d 2 p /(18μ), neglecting finite-Re p corrections.In turn, the values of St indicate that the particles have insignificant inertia with respect to the energetic scales of the flow.