Suspensions Of Spheres Research Articles

Taylor-Couette instability in sphere suspensions

We employ a rheological theory to show that circular Taylor-Couette flow of a suspension of non-Brownian spheres is less stable than that of a Newtonian fluid, at equal effective viscosity. The destabilization is related to the preferred orientation of the separation vector of the closely interacting spheres, in the compressive direction of the base flow. The results agree qualitatively with experimental observations from the literature.

Effects of Polydispersity on Structuring and Rheology in Flowing Suspensions

The size and distribution of particles suspended within a fluid influence the rheology of the suspension, as well as strength and other mechanical properties if the fluid eventually solidifies. An important motivating example of current interest is foamed cements used for carbon storage and oil and gas wellbore completion. In these applications, it is desired that the suspended particles maintain dispersion during flow and do not coalesce or cluster. This paper compares the role of mono- against polydispersity in the particle clustering process. The propensity of hard spherical particles in a suspension to transition from a random configuration to an ordered configuration, or to form localized structures of particles, due to flow is investigated by comparing simulations of monodisperse and polydisperse suspensions using Stokesian dynamics. The calculations examine the role of the polydispersity on particles rearrangements and structuring of particles due to flow and the effects of the particle size distribution on the suspension viscosity. A key finding of this work is that a small level of polydispersity in the particle sizes helps to reduce localized structuring of the particles in the suspension. A suspension of monodisperse hard spheres forms structures at a particle volume fraction of approximately 47% under shear, but a 47% volume fraction of polydisperse particles in suspension does not form these structures.

Shear-induced collective diffusivity down a concentration gradient in a viscous emulsion of drops

The shear-induced collective diffusivity down a concentration gradient in a viscous emulsion is computed using direct numerical simulation. A layer of randomly packed drops subjected to a shear flow, shows the layer width to increase with the $1/3$ power of time, consistent with a semi-dilute theory that assumes a diffusivity linear with concentration. This characteristic scaling and the underlying theory are used to compute the collective diffusivity coefficient. This is the first ever computation of this quantity for a system of deformable particles using fully resolved numerical simulation. The results match very well with previous experimental observations. The coefficient of collective diffusivity varies non-monotonically with the capillary number, due to the competing effects of increasing deformation and drop orientation. A phenomenological correlation for the collective diffusivity coefficient as a function of capillary number is presented. We also apply an alternative approach to compute collective diffusivity, developed originally for a statistically homogeneous rigid sphere suspension – computing the dynamic structure factor from the simulated droplet positions and examining its time variation at small wavenumber. We show that the results from this alternative approach qualitatively agree with our computation of collective diffusivity including the prediction of the non-monotonic variation of diffusivity with the capillary number.

A Potential Antitumor Effect of Dendritic Cells Fused with Cancer Stem Cells in Hepatocellular Carcinoma.

HCC stem cells were reported as posttreatment residual tumor cells that play a pivotal role in tumor relapse. Fusing dendritic cells (DCs) with tumor cells represents an ideal approach to effectively activate the antitumor immunity in vivo. DC/HCC stem cell vaccine provides a potential strategy to generate polyclonal immune response to multiple tumor stem cell antigens including those yet to be unidentified. To assess the potential capacity of DC/HCC stem cell vaccines against HCC, CD90+HepG2 cells were sorted from the HCC cell line HepG2. DC and CD90+HepG2 and DC and HepG2 fused cells were induced by polyethylene glycol (PEG). The influence of fusion cells on proliferation and immunological function transformation of lymphocytes was assessed by FCM and ELISA assay, respectively. The cytotoxicity assay of specific fusion cell-induced CTLs against HepG2 was conducted by CytoTox 96 Non-Radioactive Cytotoxicity Assay kit in vitro. At last, the prevention of HCC formation in vivo was described in a mouse model. The results of FCM analysis showed that the proportion of CD90+HepG2 cells in the spheral CD90+HepG2 enriched by suspension sphere culture was ranging from 98.7% to 99.5%, and 57.1% CD90+HepG2/DC fused cells were successfully constructed. The fusion cells expressed a higher level of costimulatory molecules CD80, CD83, CD86, and MHC-I and MHC-II molecules HLA-ABC and HLA-DR than did immature DCs (P < 0.05). And the functional analysis of fusion cell-induced CTLs also illustrated that CD90+HepG2/DC fusion cells showed a greater capacity to activate proliferation of lymphocytes in vitro (P < 0.05). The CD90+HepG2/DC-activated CTLs had a specific killing ability against CD90+HepG2 cells in vivo. These results suggested that CD90+HepG2/DC fusion cells could efficiently stimulate T lymphocytes to generate specific CTLs targeting CD90+HepG2 cells. It might be a promising strategy of immunotherapy for HCC.

Ovarian Cancer Stem Cells with High ROR1 Expression Serve as a New Prophylactic Vaccine for Ovarian Cancer.

Tumor vaccines offer a number of advantages for cancer treatment. In the study, the vaccination with cancer stem cells (CSCs) with high expression of the type I receptor tyrosine kinase-like orphan receptor (ROR1) was evaluated in a murine model for the vaccine's immunogenicity and protective efficacy against epithelial ovarian carcinoma (EOC). CD117+CD44+ CSCs were isolated from human EOC HO8910 cell line using a magnetic-activated cell sorting system; murine ID8 EOC suspension sphere cells, which are collectively known as cancer stem-like cells, were acquired from serum-free suspension sphere-forming culture. Mice were subcutaneously immunized with the repeat cycles of freezing and thawing whole HO8910 CD117+CD44+ CSCs and ID8 cancer stem-like cells, respectively, followed by a challenge with HO8910 or ID8 cells at one week after final vaccination. The results showed that the CSC vaccination significantly induced immunity against EOC growth and markedly prolonged the survival of EOC-bearing mice in the prophylactic setting compared with non-CSC vaccination. Flow cytometry showed significantly increased immunocyte cytotoxicities and remarkably reduced CSC counts in the CSC-vaccinated mice. Moreover, the protective efficacy against EOC was decreased when the ROR1 expression was downregulated by shRNA in CSC vaccines. The findings from the study suggest that CSC vaccines with high ROR1 expression were highly effective in triggering immunity against EOC in vaccinated mice and may serve as an effective vaccine for EOC immunoprophylaxis.

Sedimentation Velocity and Potential in Dilute Suspensions of Charge-Regulating Porous Spheres.

The sedimentation of a charge-regulating porous sphere surrounded by an arbitrary electric double layer, which usually models a permeable polyelectrolyte coil or an aggregate of nanoparticles, is analyzed for the first time. The hydrodynamic frictional segments and ionogenic functional groups uniformly distribute in the porous sphere, and a regulation mechanism for the dissociation and association reactions occurring at these functional groups linearly relates the local electric potential to fixed charge density. The linearized electrokinetic equations governing the ionic concentration (or electrochemical potential energy), electric potential, and fluid velocity fields are solved for the case of a small basic fixed charge density by the regular perturbation method. Analytical formulae for the sedimentation velocity of a porous sphere and sedimentation potential of a dilute suspension of porous spheres are then obtained. The charge regulation tends to reduce the electrokinetic retardation to sedimentation velocity and the sedimentation potential (can be as much as 50 and 25%, respectively) compared to the case that the fixed charge density is a constant. Both the electrokinetic retardation to sedimentation velocity and the sedimentation potential vanish at the isoelectric point of the particles. The increase in the bulk concentration of the potential-determining ions crossing the isoelectric point changes signs of the fixed charges and thus causes a reversal in the direction of the sedimentation potential. The effects of charge regulation on the sedimentation of porous particles differ substantially from those of hard particles.

Nonlinear optical properties of synthesized porous nanostructures PMMA via template-directed methods

Colloidal crystals (opals) made of close-packed polymethylmethacrylate (PMMA) were fabricated and grown by Template-Directed methods to obtain porous materials with well-ordered periodicity and interconnected pore systems to manufacture photonic crystals. Opals were made from aqueous suspensions of monodisperse PMMA spheres with diameters between 280 and 415 nm. SEM confirmed the PMMA spheres crystallized uniformly in a face-centered cubic (FCC) array. Optical properties of synthesized pores PMMA were characterized by UV–Visible spectroscopy. It shows that the colloidal crystals possess pseudo photonic band gaps in the visible region. A combination of Bragg’s law of diffraction and Snell’s law of refraction were used to calculate the sphere diameter. Finally, colloidal crystals were subjected to Z-scan experiment under pulsed Q-switched Nd:YAG laser illumination to characterize it for third order nonlinear optical properties. Z-scan results show the change in transmittance of a beam, and the nonlinear refractive index is n2 = 9.82787 x 10-12 (cm2/GW), while the nonlinear absorption coefficient β= 0.04673908 (cm/GW). These results were attributed to enhance the self-focusing arising from Kerr effect and the two-photon absorption.

Dynamic ordering of driven spherocylinders in a nonequilibrium suspension of small colloidal spheres.

The ordering effects of driven spherocylinder-shaped rods in a colloidal suspension of small spheres confined to a two-dimensional channel geometry are observed via Brownian dynamics simulations without hydrodynamics. To describe the ordering, an order parameter and an expression for a potential of mean force of an equivalent equilibrium system are defined and analyzed. By varying the application point of the external force along the rods and thus the resulting lever, a transition from a preferred orientation parallel to the direction of the force to a preferred orientation perpendicular to the direction of the force was observed. It is shown that this effect can only be found if the spheres and multiple rods are present. Furthermore, a dependency of the order parameter on the absolute value of the force was discovered. The analysis of the potential of mean force further indicates a transition between two different phases of mean orientation. An observation of the flow equilibrium mean velocity in channel direction led to a s-shaped progression regarding the lever dependency, also marking a transition between two states linked to the mean orientation of the rods. A finite size analysis was conducted. Its results indicate that the transition between the two orientation states is a general phenomenon of the observed rod-sphere mixture.

Optomechanical properties of optically self-arranged colloidal waveguides.

When a suspension of wavelength-sized polystyrene spheres is illuminated with non-interfering counter-propagating Gaussian beams, the particles self-arrange into a colloidal waveguide (CWG). Mutual force interaction among particles is mediated by scattered light, referred to as the optical binding. We analyzed the longitudinal and lateral motion of particles in such CWGs made of an increasing number of particles with diameters of either 520 or 657nm. We observed the enhancement of the binding stiffness of neighboring particles by more than an order of magnitude. This enhancement is done by optical means, mainly due to a local increase of optical intensity due to multiple light scattering in an optically bound structure.

Resolved simulations of sedimenting suspensions of spheres

We analyze resolved simulations of up to 2000 spherical particles suspended in an incoming fluid stream with single-particle Reynolds number from 40-110 and volume fraction 9-35%. Diffusivities, pair distributions, and other quantities are found. Tetrads (four particle groups) are also studied.

Effect of normal contact forces on the stress in shear rate invariant particle suspensions

We theoretically predict, that, in sphere suspensions, hydrodynamic interaction forces produce a negligible first normal stress difference, while contact forces produce a positive first normal stress difference.

Mechanisms of spontaneous chain formation and subsequent microstructural evolution in shear-driven strongly confined drop monolayers.

It was experimentally demonstrated by Migler and his collaborators [Phys. Rev. Lett., 2001, 86, 1023; Langmuir, 2003, 19, 8667] that a strongly confined drop monolayer sheared between two parallel plates can spontaneously develop a flow-oriented drop-chain morphology. Here we show that the formation of the chain-like microstructure is driven by far-field Hele-Shaw quadrupolar interactions between drops, and that drop spacing within chains is controlled by the effective drop repulsion associated with the existence of confinement-induced reversing streamlines, i.e., the swapping trajectory effect. Using direct numerical simulations and an accurate quasi-2D model that incorporates quadrupolar and swapping-trajectory contributions, we analyze microstructural evolution in a monodisperse drop monolayer. Consistent with experimental observations, we find that drop spacing within individual chains is usually uniform. Further analysis shows that at low area fractions all chains have the same spacing, but at higher area fractions there is a large spacing variation from chain to chain. These findings are explained in terms of uncompressed and compressed chains. At low area fractions most chains are uncompressed (spacing equals lst, which is the stable separation of an isolated pair). At higher area fractions compressed chains (with tighter spacing) are formed in a process of chain zipping along y-shaped structural defects. We also discuss the relevance of our findings to other shear-driven systems, such as suspensions of spheres in non-Newtonian fluids.

Path Selection of a Spherical Capsule in a Branched Channel

Capsules are liquid droplets enclosed by a thin membrane which can resist shear deformation. They are widely found in nature (e.g. red blood cells) and in numerous applications (e.g. food, cosmetic, biomedical and pharmaceutical industries [1]), where they often flow through a complicated network of tubes or channels: this is the case for RBCs in the human circulation or for artificial capsules flowing through microfluidic devices. Central to these flows is the dynamic motion of capsules at bifurcations, in particular the question of path selection. A good understanding of this problem is indeed needed to elucidate some intriguing phenomena in human circulation and to design multi-branched microfluidic devices to sort and enrich suspensions of artificial microcapsules or biological cells depending on their properties. <br/> Thanks to the extensive <i>in vivo</i> and in vitro experiments conducted on blood flows in branched capillaries and microchannels [2], it has been well established that the daughter branch with a higher flow rate receives a larger number of RBCs than the other branch: this is classically referred to as the Zweifach–Fung effect [3,4]. Similar phenomena have also been observed in experiments on suspensions, where the RBCs are modelled as flexible disks and the white blood cells as solid spheres [5], and in dilute suspensions of solid spheres [6]. This is a consequence of the plasma skimming effect due to the particle- free layer near the wall of the vessel [7] and of the particle screening effect due to the deviation of the particle trajectories from the background flow fluid streamlines as a result of the hydrodynamic interaction between the particles and vessel geometry at the bifurcation [8, 6]. <br/> In the dilute limit, the problem has, however, not been thoroughly studied experimentally, possibly due to the difficulty of manipulating individual cells. The problem has mostly been studied in recent years using two-dimensional numerical models [9, 10]. The results showed that, at equal flow rate between the two downstream channels, the capsule tends to flow into the side branch in particular when the capsule is highly deformable. But to what extent the results obtained from previous two-dimensional simulations can be applied to three-dimensional flows remains unclear. <br/> The objective of the present study is to conduct a systematic and in-depth three-dimensional numerical study of a deformable capsule in a branched tube and to determine the influence of inertia. Contrary to many biological systems, for which neglecting inertia is a good assumption, capsules are not necessarily small in size and the flow speed can be fast in some applications [11]. <br/> In the present work, we computationally study the motion of an initially spherical capsule flowing through a straight channel with an orthogonal lateral branch, using a three-dimensional immersed- boundary lattice Boltzmann method [12, 13]. Our primary focus is to study the influence of the geometry of the side branch on the capsule path selection. The capsule is enclosed by a strain-hardening membrane and contains an internal fluid of the same viscosity as the fluid in which it is suspended. It is initially centered on the axis of the feed channel. We impose the flow rate split ratio between the two downstream branches of the bifurcation. We analyze the reference cases wherein the side channel has the same cross- section as the main branch, and then compare the results to other geometries, in which the side branch has a half cross-sectional area as compared to the main branch. We consider different capsule-to-channel size ratio, flow Reynolds number (Re) and ratio of capillary to Reynolds numbers, and summarize the results in phase diagrams indicating the critical flow split ratio above which the capsule flows into the side branch. <br/> The capsule trajectory does not always obey the classical Fung's bifurcation law. For Re ≤ 5 and equal flow rate split between the two downstream branches, the capsule will enter a branch which is narrow in the spanwise direction, but will not enter a branch which is narrow in the flow direction. We show that this novel phenomenon results primarily from the background flow which is strongly influenced by the side branch cross-section geometry. For higher values of Re, the capsule relative size and deformability also play a specific role in the path selection: increasing the capillary number generally promotes cross-stream migration of the capsule towards the side branch. The present results obtained for dilute systems open new perspectives on the design of microfluidic systems with optimal channel geometries and flow conditions to enrich cell and particle suspensions.

Imaging Cryptococcus spp. Capsule by Differential Interference Contrast Microscopy Using Percoll®.

The most important virulence factor in the Cryptococcus genus is the polysaccharide capsule. This genus includes several species that cause life-threatening invasive disease. An increase in capsule thickness is important during fungal infection. The capsule is usually imaged using India ink, and crucial insights on the dynamics of its growth have been obtained using capsule-binding proteins such as specific antibodies or complement. We have developed an alternative method that allows both static and time-lapse imaging of the capsule using Percoll®, a suspension of nanometric spheres that do not penetrate the capsule. Given that these particles have a higher refractive index than the capsule, the latter can be imaged by differential interference contrast (DIC) microscopy. Static observation of the capsule with DIC and Percoll® results in capsule thickness measurements that match those made with India ink. Using capsule-inducing media, a glass-bottom incubation chamber and a live-imaging system equipped for DIC microscopy, this method allows time-lapse imaging of capsule growth. In contrast with India ink staining, DIC imaging of Percoll® exclusion halos result in crisp images. The greatest advantage of this method, though, is that unlike India ink, the Percoll® particles are non-toxic and unlike opsonins they do not bind the capsule, resulting in observations of capsule growth that are free from interference of bound proteins on capsule physiology.

Hydrodynamic dispersion and aggregation induced by shear in non-Brownian magnetic suspensions

In this paper, we examine a two-particle problem in order to study transport phenomena on magnetic suspensions such as the shear-induced hydrodynamic diffusion and the shear-induced aggregation in the regime of non-Brownian particles and creeping flow. New results are presented for the shear-induced hydrodynamic diffusivities and the rate of particle doublet formation, resulting from the diffusive and aggregative irreversible trajectories produced by particle magnetic interactions. The numerical computation of the particle shear-induced diffusivities and the rates of particle aggregation are performed by using a Monte Carlo integration scheme for different values of the relevant magnetic parameter α. This parameter represents the non-dimensional strength of the dipole-dipole magnetic interactions between the particles. For small values of α, the shear-induced self-diffusivity is remarkably described by a slight adaptation of the asymptotic theory for rough spherical particles interacting hydrodynamically [F. R. Cunha and E. J. Hinch, “Shear-induced dispersion in a dilute suspension of rough spheres,” J. Fluid Mech. 309, 211 (1996)]. We just replace the roughness parameter ϵ with α5/4, giving for the self-shear-induced diffusivity for small values of α, 0.156γ̇a2ϕα0.547log(1/α5/4)+1.347−0.701, where γ̇ is the applied shear rate, a is the radius of the magnetic spheres, and ϕ is the particle volume fraction. In addition, a root-square law dependence is obtained for the rate of particle aggregation as (1.830/π)ϕ1N1γ̇α1/2, where N1 and ϕ1 are, respectively, the number and the volume fraction of the isolated particles in the suspension. A comparison shows that the root-square law prediction is in excellent agreement with the results of the numerical simulations for all values of the parameter α investigated.

Coupling of sedimentation and liquid structure: Influence on hard sphere nucleation.

The discrepancy in nucleation rate densities between simulated and experimental hard spheres remains staggering and unexplained. Suggestively, more strongly sedimenting colloidal suspensions of hard spheres nucleate much faster than weakly sedimenting systems. In this work, we consider first the effect of sedimentation on the structure of colloidal hard spheres by tuning the density mismatch between solvent and colloidal particles. In particular, we investigate the effect on the degree of the fivefold symmetry present. Second, we study the size of density fluctuations in these experimental systems in comparison to simulations. The density fluctuations are measured by assigning each particle a local density, which is related to the number of particles within a distance of 3.25 particle diameters. The standard deviation of these local densities gives an indication of the fluctuations present in the system. Fivefold symmetry is suppressed by a factor of two when sedimentation is induced in our system. Density fluctuations are increased by a factor of two in experiments compared to simulations. The change in fivefold symmetry makes a difference to the expected nucleation rates, but we demonstrate that it is ultimately too small to resolve the discrepancy between experiment and simulation, while the fluctuations are shown to be an artefact of 3d particle tracking.

A polarization sensitive light scattering unit for high throughput screening

A simple, effective light scattering prototype sensitive to both polarized and depolarized scattering was constructed, and its performance was tested on a variety of pure liquids and optically isotropic and anisotropic polymer solutions and colloidal suspensions. The results, performance, and means to further improvement are reported here. Because of its simplicity and low cost, many identical units can be produced to construct a simultaneous multiple sample light scattering platform that can be used to monitor polymer and colloid solution stability, phase changes, aggregation, degradation, etc. Measurable depolarization was found for a variety of organic liquids and suspensions of both polyfluoroethylene and latex spheres. No detectable depolarization was found for various polymers.

Finite size effects in active microrheology in colloids

Active microrheology has emerged in recent years as a new technique to probe microscopically the mechanical properties of materials, particularly, viscoelastic ones. In this technique, a colloidal tracer is pulled through the material, and its dynamics is monitored. The interpretation of results usually relies on the Stokes–Einstein approximation, which is valid for a continuous medium in equilibrium. In this work, we have studied with simulations a suspension of quasi-hard colloidal spheres, where a large tracer is pulled by a constant force. The Navier–Stokes equation for a continuous bath predicts important finite size effects, decaying as the inverse box size, which require simulations of different systems to extract the microviscosity of a bulk system. A strategy to optimize the scheduling of the simulation tasks on a multi GPU–CPU cluster based on the adaptation of a genetic algorithm is presented here, and used to study the effect of different conditions on the friction experienced by the tracer (adding the tracer volume to the total system volume, fixing the center of mass of the system, varying the fluid friction coefficient and tracer size). It is observed that the theoretical prediction is not followed, but deviations are observed for large systems in all cases. These are attributed to the finite size of the bath particles, and the intrinsic dynamics of colloidal systems, as shown by the analysis of the velocity profile in the bath.

Brownian Dynamics of Particles "Dressed" by Chiral Director Configurations in Lyotropic Chromonic Liquid Crystals.

We study Brownian dynamics of colloidal spheres, with planar anchoring conditions, suspended in the nematic phase of the lyotropic chromonic liquid crystal disodium chromoglycate (DSCG). Unlike typical liquid crystals, the unusually small twist elastic modulus of DSCG permits two energetically distinct helical distortions (twisted tails) of the nematic director to "dress" the suspended spheres. Video microscopy is used to characterize the helical distortions versus particle size and to measure particle mean-square displacements. Diffusion coefficients parallel and perpendicular to the far-field director, and their anisotropy ratio, are different for the two twisted tail configurations. Moreover, the crossover from subdiffusive to diffusive behavior is anomalously slow for motion perpendicular to the director (>100 s). Simple arguments using Miesowicz viscosities and ideas about twist relaxation are suggested to understand the mean-square displacement observations.

Settling of finite-size particles in turbulence at different volume fractions

We study the settling of finite-size rigid spheres in quiescent fluid and in sustained homogeneous isotropic turbulence (HIT) by direct numerical simulations using an immersed boundary method to account for the dispersed solid phase. We consider semi-dilute and dense suspensions of rigid spheres with solid volume fractions $$\phi =0.5{-}10\%$$ , solid-to-fluid density ratio $$R=1.02$$ , and Galileo number (i.e., the ratio between buoyancy and viscous forces) $$Ga=145$$ . In HIT, the nominal Reynolds number based on the Taylor microscale is $$Re_{\lambda } \simeq 90$$ , and the ratio between the particle diameter and the nominal Kolmogorov scale is $$(2a)/\eta \simeq 12$$ (being a the particle radius). We find that in HIT the mean settling speed is less than that in quiescent fluid for all $$\phi $$ . For $$\phi =0.5\%$$ , the mean settling speed in HIT is $$8\%$$ less than in quiescent fluid. However, by increasing the volume fraction the difference in the mean settling speed between quiescent fluid and HIT cases reduces, being only $$1.7\%$$ for $$\phi =10\%$$ . Indeed, while at low $$\phi $$ the settling speed is strongly altered by the interaction with turbulence, at large $$\phi $$ this is mainly determined by the (strong) hindering effect. This is similar in quiescent fluid and in HIT, leading to similar mean settling speeds. On the contrary, particle angular velocities are always found to increase with $$\phi $$ . These are enhanced by the interaction with turbulence, especially at low $$\phi $$ . In HIT, the correlations of particle lateral velocity fluctuations oscillate around zero before decorrelating completely. The time period of the oscillation seems proportional to the ratio between the integral lengthscale of turbulence and the particle characteristic terminal velocity. Regarding the mean square particle displacement, we find that it is strongly enhanced by turbulence in the direction perpendicular to gravity, even at the largest $$\phi $$ . Finally, we investigate the collision statistics for all cases and find the interesting result that the collision frequency is larger in quiescent fluid than in HIT for $$\phi =0.5{-}1\%$$ . This is due to frequent drafting–kissing–tumbling events in quiescent fluid. The collision frequency becomes instead larger in HIT than in still fluid for $$\phi =5{-}10\%$$ , due to the larger relative approaching velocities in HIT, and to the less intense drafting–kissing–tumbling events in quiescent fluid. The collision frequency also appears to be almost proportional to the estimate for small inertial particles uniformly distributed in space, though much smaller. Concerning the turbulence modulation, we find that the mean energy dissipation increases almost linearly with $$\phi $$ , leading to a large reduction of $$Re_{\lambda }$$ .