Suspensions Of Deformable Particles Research Articles

Order–disorder transitions within deformable particle suspensions in planar Poiseuille flow

Three-dimensional simulations of the ordering of elastic capsule suspensions within planar Poiseuille flow channels are reported. The simulations utilize the immersed boundary method coupled with the lattice Boltzmann method to capture the complex flow-induced capsule deformations and hydrodynamic interactions within the suspensions. A parametric study is presented whereby the confinement ratio and the particle deformability are varied independently within a two-dimensional range relevant to this ordering phenomenon. The initial distribution of capsules is random, and the simulations evolve the system from a disordered state to an ordered one, while an order parameter that quantifies the fraction of capsules belonging to one-dimensional train assemblies is computed throughout time. A monotonic increase in ordering is observed with increasing deformability. However, an optimal confinement ratio is identified corresponding to a peak in the order parameter. This peak is attributed to the competition between increasing long-range capsule attractions and decreasing in-plane capsule density (with fixed volume fraction) as the confinement ratio increases. Simulations are also performed to understand how dispersity in capsule size and deformability impact the degree of ordering. It is shown that ordering is quite sensitive to dispersity in capsule size, and much less sensitive to dispersity in deformability. Overall, the results provide important insights for the design of microfluidic devices.

Micromorphic theory as a model for blood in the microcirculation: correction and analysis

This paper analyzes the applicability of Eringen’s Generalized Continuum Theories as a model for human blood in the microcirculation. The applied theory considers a fluid with a fully deformable substructure, namely a micromorphic fluid. This analysis is motivated by the fact that blood itself can be considered a suspension of deformable particles, i.e., red blood cells (RBCs), suspended in a Newtonian fluid, i.e., blood plasma. As a consequence, non-Newtonian phenomena such as shear-thinning are observed in blood. To test the micromorphic fluid as a model for blood, the solution for the velocity and the motion of substructure is determined for a cylindrical pipe flow and compared to experimental results of blood flow through narrow glass capillaries representing idealized blood vessels. A similar analysis was also conducted by Kang and Eringen in 1976, but it contains some misprints and minor errors regarding the mathematical expressions and subsequent discussion which are corrected in this paper. For certain material parameters, the micromorphic fluid models capture high-shear blood flow in narrow glass capillaries very well. This concerns both the velocity profiles and the shear-thinning behavior. Furthermore, a parameter study reveals that the flexibility of substructure governs the micromorphic shear-thinning. In this regard, parallels can be drawn to the shear-thinning of human blood, which is also induced by the deformability of RBCs. This makes the micromorphic fluid a complex but accurate model for human blood, at least for the considered experiments.

Integrated data-driven modeling and experimental optimization of granular hydrogel matrices

Integrated data-driven modeling and experimental optimization of granular hydrogel matrices

Rheology and structure of a suspension of deformable particles in plane Poiseuille flow

We present an experimental study of the rheology and structure of a confined suspension of deformable particles flowing in a quasi-two-dimensional Poiseuille flow. Thanks to a precise microfluidic viscosimetry technique combined with measurements of concentration profiles, our study provides the first experimental confirmation with three-dimensional particles of a strong relationship between structuring effects and rheology, previously only reported in numerical simulations of purely two-dimensional systems. In conditions where strong structuring effects take place due to confinement, the evolution of the effective viscosity with particle concentration (here, red blood cells) shows a remarkable succession of ranges of rapid growth and plateaus that are associated with qualitative transitions in the structure of the suspension.

Anisotropic Oldroyd-type models for non-colloidal suspensions of viscoelastic particles in Newtonian and yield-stress fluids via homogenization

This paper makes use of homogenization to generate a fairly general class of rheological models for non-colloidal suspensions of particles undergoing finite deformations and rotations under Stokes flow conditions. The particles can be nonlinear viscoelastic, or elasto-viscoplastic, while the suspending fluid can be Newtonian or viscoplastic. The microstructure is described by two microstructural ellipsoids, which are allowed to evolve with the flow, characterizing the shape and orientation of the inclusions and the angular dependence of their distribution in space. The models account for particle interactions through the two-point correlation function for the particle centers, in a way that reproduces exactly classical and more recent estimates for dilute suspensions. While special cases of the models have been considered in earlier publications, novel features of this work include the possibility of multiple inclusion families, physically motivated evolutions laws for the particle distributions and more general rheological behaviors for the matrix and inclusions. The resulting models can be viewed as anisotropic generalizations of Oldroyd’s invariant models accounting for statistical measures of the microstructure and their evolution. • Rheological models for suspensions of deformable particles in yield stress fluids. • Models are based on homogenization techniques for linear and nonlinear media. • The particles can be nonlinear viscoelastic or viscoplastic and the fluid Newtonian or viscoplastic. • Two ellipsoids characterize the evolution of the average particle shape and distribution. • The resulting models can be viewed as anisotropic generalizations of Oldroyd’s invariant models.

Suspensions of deformable particles in Poiseuille flows at finite inertia

We analyze a suspension of deformable particles in a pressure-driven flow. The suspension is composed of neutrally buoyant initially spherical particles and a Newtonian carrier fluid, and the flow is solved by means of direct numerical simulations, using a fully Eulerian method based on a one-continuum formulation. The solid phase is modeled with an incompressible viscous hyperelastic constitutive relation, and the flow is characterized by three main dimensionless parameters, namely the solid volume fraction, the Reynolds and capillary numbers. The dependency of the effective viscosity on these three quantities is investigated to study the inertial effects on a suspension of deformable particles. It can be observed that the suspension has a shear-thinning behavior, and the reduction in effective viscosity for high shear rates is emphasized in denser configurations. The separate analysis of the Reynolds and capillary numbers reveal that the effective viscosity depends more on the capillary than on the Reynolds number. In addition, our simulations exhibit a consistent tendency for deformable particles to move toward the center of the channel, where the shear rate is low. This phenomenon is particularly marked for very dilute suspensions, where a whole region near the wall is empty of particles. Furthermore, when the volume fraction is increased this near-wall region is gradually occupied, because of higher mutual particle interactions. Deformability also plays an important role in the process. Indeed, at high capillary numbers, particles are more sensitive to shear rate variations and can modify their shape more easily to accommodate a greater number of particles in the central region of the channel. Finally, the total stress budgets show that the relative particle-induced stress contribution increases with the volume fraction and Reynolds number, and decreases with the particle deformability.

On the effect of coalescence on the rheology of emulsions

We present a numerical study of the rheology of a two-fluid emulsion in dilute and semidilute conditions. The analysis is performed for different capillary numbers, volume fractions and viscosity ratios under the assumption of negligible inertia and zero buoyancy force. The effective viscosity of the system increases for low values of the volume fraction and decreases for higher values, with a maximum for approximately 20 % concentration of the disperse phase. When the dispersed fluid has lower viscosity, the normalised effective viscosity becomes smaller than 1 for high enough volume fractions. To single out the effect of droplet coalescence on the rheology of the emulsion we introduce an Eulerian force which prevents merging, effectively modelling the presence of surfactants in the system. When the coalescence is inhibited the effective viscosity is always greater than 1 and the curvature of the function representing the emulsion effective viscosity versus the volume fraction becomes positive, resembling the behaviour of suspensions of deformable particles. The reduction of the effective viscosity in the presence of coalescence is associated with the reduction of the total surface of the disperse phase when the droplets merge, which leads to a reduction of the interface tension contribution to the total shear stress. The probability density function of the flow topology parameter shows that the flow is mostly a shear flow in the matrix phase, with regions of extensional flow when the coalescence is prohibited. The flow in the disperse phase, instead, always shows rotational components. The first normal stress difference is positive, except for the smallest viscosity ratio considered, whereas the second normal difference is negative, with their ratio being constant with the volume fraction. Our results clearly show that the coalescence efficiency strongly affects the system rheology and that neglecting droplet merging can lead to erroneous predictions.

Dynamics, rheology, and applications of elastic deformable particle suspensions: a review

Suspensions with deformable solid inclusions have both scientific and industrial interest. Examples include microgel particle suspensions, biological fluids, and filled polymers. Due to the interplay of numerous factors (e.g., particle size, shape, and mechanical properties, suspending liquid rheology and flow conditions, flow cell shape, interactions between the particles and the flow channel and among the particles), these systems can show many different flow behaviors. The study of the flow behavior of soft particle suspensions is an open field and is attracting an increasing interest from the scientific community. As elastic beads are good models for biological cells, one reason for such interest is related to the possibility of exploiting particle mechanical-property-dependent flow behavior to diagnose diseases. On the other hand, in the recent years, suspensions of elastic microparticles are also proving useful in a completely different technological field, namely oil recovery from porous media. This paper gives an overview of the state of the art and envisages possible future perspectives on the dynamics, rheology, and applications of elastic particle suspensions.

Suspension of deformable particles in Newtonian and viscoelastic fluids in a microchannel

In this paper, we study a suspension of cells at a moderate volume fraction flowing in a microchannel filled with Newtonian or viscoelastic fluids and investigate the role of cell size, cell volume fraction, inertia, deformability, and fluid elasticity on the cell distribution. Our results suggest that the use of constant-viscosity viscoelastic fluid pushes the cells toward the channel centerline which can be used in microfluidic devices used for cell focusing such as cytometers. The cell-free layer increases which provides larger gap for separating rare cells in microfluidic devices. Furthermore, we show that the volumetric flow rate can be significantly enhanced with the addition of polymers in the suspending fluid. This effect enhances the processing speed which is of interest in designing microfluidic devices. This fundamental study can provide insight on the role of rheological properties of the fluid that can be tuned to control the motion of the cells for efficient design of microfluidic devices.

Immersed-finite-element method for deformable particle suspensions in viscous and viscoelastic media

Deformable elastic bodies in viscous and viscoelastic media constitute a large portion of synthetic and biological complex fluids. We present a parallelized 3D-simulation methodology which fully resolves the momentum balance in the solid and fluid domains. An immersed boundary algorithm is exploited known as the immersed finite element method (IFEM) which accurately determines the internal forces in the solid domain. The scheme utilized has the advantages of requiring no costly re-meshing, handling finite Reynold's number, as well as incorporating non-linear viscoelasticity in the fluid domain. Our algorithm is designed for computationally efficient simulation of multi-particle suspensions with mixed structure types. The internal force calculation in the solid domain in the IFEM is coupled with a finite volume based incompressible fluid solver, both of which are massively parallelized for distributed memory architectures. We performed extensive case studies to ensure the fidelity of our algorithm. Namely, a series of single particle simulations for capsules, red blood cells, and elastic solid deformable particles were conducted in viscous and viscoelastic media. All of our results are in excellent quantitative agreement with the corresponding reported data in the literature which are based on different simulation platforms. Furthermore, we assess the accuracy of multi-particle simulation of blood suspensions (red blood cells in plasma) with and without platelets. Finally, we present the results of a novel simulation of multiple solid deformable objects in a viscoelastic medium.

Suspensions of deformable particles in a Couette flow

We consider suspensions of deformable particles in a Newtonian fluid by means of fully Eulerian numerical simulations with a one-continuum formulation. We study the rheology of the visco-elastic suspension in plane Couette flow in the limit of vanishing inertia and examine the dependency of the effective viscosity μ on the solid volume-fraction Φ, the capillary number Ca, and the solid to fluid viscosity ratio K. The suspension viscosity decreases with deformation and applied shear (shear-thinning) while still increasing with volume fraction. We show that μ collapses to an universal function, μ(Φe), with an effective volume fraction Φe, lower than the nominal one owing to the particle deformation. This universal function is well described by the Eilers fit, which well approximate the rheology of suspension of rigid spheres at all Φ. We provide a closure for the effective volume fraction Φe as function of volume fraction Φ and capillary number Ca and demonstrate it also applies to data in literature for suspensions of capsules and red-blood cells. In addition, we show that the normal stress differences exhibit a non-linear behavior, with a similar trend as in polymer and filament suspensions. The total stress budgets reveals that the particle-induced stress contribution increases with the volume fraction Φ and decreases with deformability.

Modeling shear-induced particle ordering and deformation in a dense soft particle suspension

We apply the lattice Boltzmann method and the bead-spring network model of deformable particles (DPs) to study shear-induced particle ordering and deformation and the corresponding rheological behavior for dense DP suspensions confined in a narrow gap under steady external shear. The particle configuration is characterized with small-angle scattering intensity, the real-space 2D local order parameter, and the particle shape factors including deformation, stretching and tilt angles. We investigate how particle ordering and deformation vary with the particle volume fraction ϕ (=0.45–0.65) and the external shear rate characterized with the capillary number Ca (=0.003–0.191). The degree of particle deformation increases mildly with ϕ but significantly with Ca. Under moderate shear rate (Ca = 0.105), the inter-particle structure evolves from string-like ordering to layered hexagonal close packing (HCP) as ϕ increases. A long wavelength particle slithering motion emerges for sufficiently large ϕ. For ϕ = 0.61, the structure maintains layered HCP for Ca = 0.031–0.143 but gradually becomes disordered for larger and smaller Ca. The correlation in particle zigzag movements depends sensitively on ϕ and particle ordering. Layer-by-layer analysis reveals how the non-slippery hard walls affect particle ordering and deformation. The shear-induced reconfiguration of DPs observed in the simulation agrees qualitatively with experimental results of sheared uniform emulsions. The apparent suspension viscosity increases with ϕ but exhibits much weaker dependence compared to hard-sphere suspensions, indicating that particle deformation and unjamming under shear can significantly reduce the viscous stress. Furthermore, the suspension shear-thins, corresponding to increased inter-DP ordering and particle deformation with Ca. This work provides useful insights into the microstructure-rheology relationship of concentrated deformable particle suspensions.

Confined tube flow of low viscosity emulsions: Effect of matrix elasticity

This study is addressed to the effect of the elasticity of the continuous phase on confined flow of low viscosity emulsions, a topic of interest in several applications. A Newtonian silicone oil was used as the dispersed phase, while three different water-based fluids, a Newtonian and two Boger fluids, were used as the continuous phase. All the fluids have the same viscosity (ca. 0.05 Pa s). Emulsions were prepared by feeding the two phases to a T-junction followed by a membrane. When the continuous phase was a Boger fluid, the average drop size of the so obtained emulsions was higher (as compared to the Newtonian continuous phase) and a bimodal distribution was observed. Such lower mixing efficiency for a Boger continuous phase is in line with the result for an isolated droplet that breakup is hindered by matrix elasticity. High-speed microscopy visualization of the emulsions flowing through a microcapillary allowed one to determine the velocity profiles and the droplet spatial distribution as a function of the shear rate. In all the experiments, the formation of a droplet-rich core and of a droplet-devoid wall region Δ was observed, with a significant enhancement when the continuous phase was a Boger fluid (where values of Δ as high as 37% of capillary radius were found). So, confinement here refers to the fact that, although the droplets are much smaller than capillary radius, the droplet-free layer takes up a significant fraction of the capillary radius. The observed nonhomogeneous droplet spatial distribution was not due to a transient entry effect, as shown by imaging the emulsions flow at different sections from the capillary inlet, but can be explained by a balance between collision-induced drop diffusion and wall lift force. The former acts to homogenize droplet spatial distribution, while the latter tends to push droplets away from the wall and is enhanced by matrix elasticity. These effects lead to the accumulation of larger droplets around the center of the channel, while smaller droplets are pushed toward the periphery. The velocity profiles for the Boger continuous phase were flat in the droplet-rich core, thus suggesting that the confined emulsions behave as a Bingham fluid. The corresponding yield stress was determined by data fitting of the velocity profiles and rather large values were so obtained. This result is consistent with the high values, close to the maximum packing for spherical particles in binary dispersions, of the droplet volume fraction in the core, which were estimated by measuring the thickness of the droplet-devoid annular region as a function of the applied flow rate and making a volume balance. Overall, the results of this work bear strong analogies with the confined flow of other concentrated suspensions of deformable particles, such as red blood cells in microcirculation, where the formation of a cell-free layer at the wall, which is referred to as margination, has important pathophysiological implications. These findings suggest that low viscosity emulsions with an elastic matrix under confined flow can be used as a tunable model system of biomedical relevance.

Fluctuations and diffusion in sheared athermal suspensions of deformable particles

We analyze fluctuations of particle displacements and stresses in a sheared athermal suspension of elastic capsules (red blood cells). Upon variation of the volume fraction from the dilute up to the highly concentrated regime, our numerical simulations reveal different characteristic power-law regimes of the fluctuation variances and relaxation times. In the jammed phase and at high shear rates, anomalous scaling exponents are found that deviate from pure dimensional predictions. The observed behavior is rationalized via kinetic arguments and a dissipation balance model that takes into account the local fluid flows between the particles. Our findings support the view that the rheology of dense suspensions is essentially governed by the non-affine displacements.

A generalized equation for rheology of emulsions and suspensions of deformable particles subjected to simple shear at low Reynolds number

We present analyses to provide a generalized rheological equation for suspensions and emulsions of non-Brownian particles. These multiparticle systems are subjected to a steady straining flow at low Reynolds number. We first consider the effect of a single deformable fluid particle on the ambient velocity and stress fields to constrain the rheological behavior of dilute mixtures. In the homogenization process, we introduce a first volume correction by considering a finite domain for the incompressible matrix. We then extend the solution for the rheology of concentrated system using an incremental differential method operating in a fixed and finite volume, where we account for the effective volume of particles through a crowding factor. This approach provides a self-consistent method to approximate hydrodynamic interactions between bubbles, droplets, or solid particles in concentrated systems. The resultant non-linear model predicts the relative viscosity over particle volume fractions ranging from dilute to the the random close packing in the limit of small deformation (capillary or Weissenberg numbers) for any viscosity ratio between the dispersed and continuous phases. The predictions from our model are tested against published datasets and other constitutive equations over different ranges of viscosity ratio, volume fraction, and shear rate. These comparisons show that our model, is in excellent agreement with published datasets. Moreover, comparisons with experimental data show that the model performs very well when extrapolated to high capillary numbers (C a≫1). We also predict the existence of two dimensionless numbers; a critical viscosity ratio and critical capillary numbers that characterize transitions in the macroscopic rheological behavior of emulsions. Finally, we present a regime diagram in terms of the viscosity ratio and capillary number that constrains conditions where emulsions behave like Newtonian or Non-Newtonian fluids.

Accelerated boundary integral method for multiphase flow in non-periodic geometries

An accelerated boundary integral method for Stokes flow of a suspension of deformable particles is presented for an arbitrary domain and implemented for the important case of a planar slit geometry. The computational complexity of the algorithm scales as O(N) or O(NlogN), where N is proportional to the product of number of particles and the number of elements employed to discretize the particle. This technique is enabled by the use of an alternative boundary integral formulation in which the velocity field is expressed in terms of a single layer integral alone, even in problems with non-matched viscosities. The density of the single layer integral is obtained from a Fredholm integral equation of the second kind involving the double layer integral. Acceleration in this implementation is provided by the use of General Geometry Ewald-like method (GGEM) for computing the velocity and stress fields driven by a set of point forces in the geometry of interest. For the particular case of the slit geometry, a Fourier-Chebyshev spectral discretization of GGEM is developed. Efficient implementations employing the GGEM methodology are presented for the resulting single and the double layer integrals. The implementation is validated with test problems on the velocity of rigid particles and drops between parallel walls in pressure driven flow, the Taylor deformation parameter of capsules in simple shear flow and the particle trajectories in pair collisions of capsules in simple shear flow. The computational complexity of the algorithm is verified with results from several large scale multiparticle simulations.

Simulating 3D deformable particle suspensions using lattice Boltzmann method with discrete external boundary force

AbstractA method for direct numerical analysis of three‐dimensional deformable particles suspended in fluid is presented. The flow is computed on a fixed regular ‘lattice’ using the lattice Boltzmann method (LBM), where each solid particle is mapped onto a Lagrangian frame moving continuously through the domain. Instead of the bounce‐back method, an external boundary force (EBF) is used to impose the no‐slip boundary condition at the fluid–solid interface for stationary or moving boundaries. The EBF is added directly to the lattice Boltzmann equation. The motion and orientation of the particles are obtained from Newtonian dynamics equations. The advantage of this approach is outlined in comparison with the standard and higher‐order interpolated bounce‐back methods as well as the LBM immersed‐boundary and the volume‐of‐fluid methods. Although the EBF method is general, in this application, it is used in conjunction with the lattice–spring model for deformable particles. The methodology is validated by comparing with experimental and theoretical results. Copyright © 2009 John Wiley & Sons, Ltd.

Simulating deformable particle suspensions using a coupled lattice-Boltzmann and finite-element method

A novel method is developed to simulate suspensions of deformable particles by coupling the lattice-Boltzmann method (LBM) for the fluid phase to a linear finite-element analysis (FEA) describing particle deformation. The methodology addresses the need for an efficient method to simulate large numbers of three-dimensional and deformable particles at high volume fraction in order to capture suspension rheology, microstructure, and self-diffusion in a variety of applications. The robustness and accuracy of the LBM–FEA method is demonstrated by simulating an inflating thin-walled sphere, a deformable spherical capsule in shear flow, a settling sphere in a confined channel, two approaching spheres, spheres in shear flow, and red blood cell deformation in flow chambers. Additionally, simulations of suspensions of hundreds of biconcave red blood cells at 40% volume fraction produce continuum-scale physics and accurately predict suspension viscosity and the shear-thinning behaviour of blood. Simulations of fluid-filled spherical capsules which have red-blood-cell membrane properties also display deformation-induced shear-thinning behaviour at 40% volume fraction, although the suspension viscosity is significantly lower than blood.

Efficiency oriented, hybrid approach for modelling deformable particles in three dimensions

In this paper, we demonstrate a new, rigorous, accurate and efficient model for simulating the flow of a suspension of deformable particles. The model uses a hybrid system: an Eulerian approach is used for the Navier-Stokes solver and a Lagrangian approach for the evolution of the DPs mesh. We demonstrate in this manuscript the capabilities of the model by simulating more than 130 deformable RBC in a tube of 60 ? 30 ? 30 m; the transverse migration of 20 vesicles at a low concentration (25% by volume) in a capillary tube of dimensions 150 ? 40 ? 40 m, in a matter of hours.

Modeling the flow of dense suspensions of deformable particles in three dimensions.

We describe here a rigorous and accurate model for the simulation of three-dimensional deformable particles (DPs). The method is very versatile, easily simulating various types of deformable particles such as vesicles, capsules, and biological cells. Each DP is resolved explicitly and advects within the surrounding Newtonian fluid. The DPs have a preferred rest shape (e.g., spherical for vesicles, or biconcave for red blood cells). The model uses a classic hybrid system: an Eulerian approach is used for the Navier-Stokes solver (the lattice Boltzmann method) and a Lagrangian approach for the evolution of the DP mesh. Coupling is accomplished through the lattice Boltzmann velocity field, which transmits force to the membranes of the DPs. The novelty of this method resides in its ability (by design) to simulate a large number of DPs within the bounds of current computational limitations: our simple and efficient approach is to (i) use the lattice Boltzmann method because of its acknowledged efficiency at low Reynolds number and its ease of parallelization, and (ii) model the DP dynamics using a coarse mesh (approximately 500 nodes) and a spring model constraining (if necessary) local area, total area, cell volume, local curvature, and local primary stresses. We show that this approach is comparable to the more common - yet numerically expensive - approach of membrane potential function, through a series of quantitative comparisons. To demonstrate the capabilities of the model, we simulate the flow of 200 densely packed red blood cells - a computationally challenging task. The model is very efficient, requiring of the order of minutes for a single DP in a 50 microm x 40 microm x 40 microm simulation domain and only hours for 200 DPs in 80 microm x 30 microm x 30 microm . Moreover, the model is highly scalable and efficient compared to other models of blood cells in flow, making it an ideal and unique tool for studying blood flow in microvessels or vesicle or capsule flow (or a mixture of different particles). In addition to directly predicting fluid dynamics in complex suspension in any geometry, the model allows determination of accurate, empirical rules which may improve existing macroscopic, continuum models.