Non-dilute Suspensions Research Articles

Testing orientational closure approximations in dilute and non-dilute suspensions with Rheo-SANS

Testing orientational closure approximations in dilute and non-dilute suspensions with Rheo-SANS

On the fully coupled dynamics of flexible fibres dispersed in modulated turbulence

The present work investigates the mechanical behaviour of finite-size, elastic and inertial fibres freely moving in a homogeneous and isotropic turbulent flow at moderate Reynolds number. Four-way coupled, direct numerical simulations, based on a finite difference discretisation and the immersed boundary method, are performed to mutually couple the dynamics of fibres and fluid turbulence, allowing us to account for the backreaction of the dispersed phase to the carrier flow. An extensive parametric study is carried out in zero-gravity condition over the characteristic properties of the suspension, i.e. fibre's linear density (from iso-dense to denser-than-the-fluid fibres), length (from short fibres comparable with the dissipative scale to long fibres comparable with the integral scale) and bending stiffness (from highly flexible to almost rigid fibres), as well as the concentration (from dilute to non-dilute suspensions). Results reveal the existence of a robust turbulence modulation mechanism that is primarily controlled by the mass fraction of the suspension (with only a minor influence of the fibre's bending stiffness), which is characterised in detail by means of a scale-by-scale analysis in Fourier space. Despite such alteration with respect to the single-phase case due to the non-negligible backreaction, fibres experience only two possible flapping states (previously identified in the very dilute condition) while being transported and deformed by the flow. In addition, we show that the maximum curvature obeys different scaling laws that can be derived from the fibre dynamical equation. Finally, we explore the clustering and preferential alignment of fibres within the flow, highlighting the peculiar role of inertia and elasticity.

Simulation of non-dilute fibre suspensions using RBF-based macro\u2013micro multiscale method

The multiscale stochastic simulation method based on the marriage of the Brownian Configuration Field (BCF) and the Radial Basis Function mesh-free approximation for dilute fibre suspensions by our group, is further developed to simulate non-dilute fibre suspensions. For the present approach, the macro and micro processes proceeded at each time step are linked to each other by a fibre contributed stress formula associated with the used kinetic model. Due to the feature of non-dilute fibre suspensions, the interaction between fibres is introduced into the evolution equation to determine fibre configurations using the BCF method. The fibre stresses are then determined based on the fibre configuration fields using the Phan–Thien–Graham model. The efficiency of the simulation method is demonstrated by the analysis of the two challenging problems, the axisymmetric contraction and expansion flows, for a range of the fibre concentration from semi-dilute to concentrated regimes. Results evidenced by numerical experiments show that the present method would be potential in analysing and simulating various suspensions in food and medical industries.

Transient and steady shear rheology of particle-laden viscoelastic suspensions

We study the transient and steady shear rheology of rigid particle suspensions in Boger fluids via complete 3D numerical simulations and experiments. We calculate the transient per-particle extra viscosity and primary stress coefficients for suspensions at different particle volume fractions ϕ for a range of Weissenberg numbers (Wi). The per-particle viscosity ( ηp) and the primary normal coefficient ( ψ1p) increase monotonically to steady state in body-fitted (BF) simulations (for dilute suspensions) and immersed boundary (IB) simulations (for nondilute suspensions). We also present experimental measurements including small amplitude oscillatory shear, steady shear, and transient shear measurements at different particle volume fraction suspensions in a Boger fluid. The simulations and experiments suggest that longer strains are needed to achieve steady state at higher ϕ and Wi. We also show the comparison of the BF and the IB simulations with experimental data for the per-particle viscosity and find excellent quantitative agreement between simulations and experiments at Wi=3 but the IB simulations underpredict the steady values at higher Wi=6. Nevertheless, the IB simulations show an increase in the per-particle viscosity with ϕ as witnessed in the experiments. To understand this behavior, we examine the particle-induced fluid stress (PIFS) and the stresslet contributions using a novel method developed for the IB simulations in this work. We find that the PIFS is independent of ϕ but the stresslet values increase with ϕ. Thus, the particle-particle hydrodynamic interactions in nondilute suspensions affect the stresslet and, in turn, the per-particle viscosity at a given Wi.

Homogenization of Nondilute Suspension of Viscous Fluid with Magnetic Particles

This paper seeks to carry out the rigorous homogenization of a particulate flow consisting of a non-dilute suspension of a viscous Newtonian fluid with magnetizable particles. The fluid is assumed to be described by the Stokes flow, while the particles are either paramagnetic or diamagnetic, for which the magnetization field is a linear function of the magnetic field. The coefficients of the corresponding partial differential equations are locally periodic. A one-way coupling between the fluid domain and the particles is also assumed. The homogenized or effective response of such a suspension is derived, and the mathematical justification of the obtained asymptotics is carried out. The two-scale convergence method is adopted for the latter. As a consequence, the presented result provides a justification for the formal asymptotic analysis of Levy and Sanchez-Palencia for particulate steady-state Stokes flows.

Hydrodynamic interactions and the diffusivity of spheroidal particles.

It is intuitive that the diffusivity of an isolated particle differs from those in a monodisperse suspension, in which hydrodynamic interactions between the particles are operative. Batchelor [J. Fluid Mech. 74, 1-29 (1976) and J. Fluid Mech. 131, 155-175 (1983)] calculated how hydrodynamic interactions influenced the diffusivity of a dilute suspension of spherical particles, and Russel et al. [Colloidal Dispersions (Cambridge University Press, 1991)] and Brady [J. Fluid Mech. 272, 109-134 (1994)] treated nondilute (higher particle volume fraction) suspensions. Although most particles lack perfect sphericity, little is known about the effects of hydrodynamic interactions on the diffusivity of spheroidal particles, which are the simplest shapes that can be used to model anisotropic particles. Here, we calculate the effects of hydrodynamic interactions on the translational and rotational diffusivities of spheroidal particles of arbitrary aspect ratio in dilute monodisperse suspensions. We find that the translational and rotational diffusivities of prolate spheroids are more sensitive to eccentricity than for oblate spheroids. The origin of the hydrodynamic anisotropy is that found in the stresslet field for the induced-dipole interaction. However, in the dilute limit, the effects of anisotropy are at the level of a few percent. These effects have influence on a vast range of settings, from partially frozen colloidal suspensions to the dynamics of cytoplasm.

Stokes' second problem and reduction of inertia in active fluids

We study a generalized Navier-Stokes model describing the thin-film flows in non-dilute suspensions of ATP-driven microtubules or swimming bacteria that are enclosed by a moving ring-shaped container. Considering Stokes' second problem, which concerns the motion of an oscillating boundary, our numerical analysis predicts that a periodically rotating ring will oscillate at a higher frequency in an active fluid than in a passive fluid, due to an activity-induced reduction of the fluid inertia. In the case of a freely suspended fluid-container system that is isolated from external forces or torques, active fluid stresses can induce large fluctuations in the container's angular momentum if the confinement radius matches certain multiples of the intrinsic vortex size of the active suspension. This effect could be utilized to transform collective microscopic swimmer activity into macroscopic motion in optimally tuned geometries.

Mechanism of shear thickening in suspensions of rigid spheres in Boger fluids. Part II: Suspensions at finite concentration

The steady shear rheology of nondilute suspensions of noncolloidal, rigid spheres in highly elastic, constant viscosity fluids is studied via numerical simulation using three-dimensional, finite-volume methods and via experiments using cone-and-plate measurements. In the numerical study, we use an immersed boundary method to simulate an ensemble of particles as a function of time until they achieve steady average bulk properties, and the simulations include fully resolved particle-scale hydrodynamics and fluid stresses. The simulations show that for low volume fraction, nondilute suspensions, the viscosity thickens at high shear and this effect can be fully determined by considering a single particle’s interactions with the suspending fluid. In fact, we show that experimentally measured viscosities of suspensions up to ϕ∼0.25, where ϕ is the volume fraction of particles, can be characterized by a shift factor that determines the zero-shear viscosity and a master curve that describes the viscosity thickening. The master curve is determined by single particle–fluid interactions that depend critically on the extensional properties of the suspending fluid. Thus, the “per particle” thickening contribution to the viscosity, at large values of the Weissenberg number, is not dependent on particle–particle interactions when plotted versus the suspension stress. This “master curve” for the thickening of the shear viscosity is not only demonstrated in the simulations, but also we show that it is consistent with all available experimental data including our own. We show that the first normal stress difference coefficient can be described similarly by a shift factor and a master curve. Our numerical simulations do not quantitatively reproduce the experimental data and in general underpredict the experimentally measured shear-thickening of the rheological properties. We suggest, and provide supporting evidence, that this can be explained by the inability of the constitutive model employed in the simulations to adequately describe the extensional properties of the suspending fluid.

Random sets of stadiums in square and collective behavior of bacteria.

Collective motion of swimmers can be detected by hydrodynamic interactions through the eective (macroscopic) viscosity. It follows form the general hydrodynamics that the eective viscosity of non-dilute random suspensions depends on the shape of particles and of their spacial probabilistic distribution. Therefore, a comparative analysis of disordered and collectively interacting particles of the bacteria shape can be done in terms of the probabilistic geometric parameters which determine the eective viscosity. In this paper, we develop a quantitative criterion to detect the collective behavior of bacteria. This criterion is based on the basic statistic moments (e-sums or generalized Eisenstein-Rayleigh sums) which characterize the high-order correlation functions. The locations and the shape of bacteria are modeled by stadiums randomly embedded in medium without overlapping. This shape models can be considered as improvement of the previous segment model.We calculate the e-sums of the simulated disordered sets and of the observed experimental locations of bacteria subtilis. The obtained results show a dierence between these two sets that demonstrates the collective motion of bacteria.

Effects of shear rate, confinement, and particle parameters on margination in blood flow.

The effects of flow and particle properties on margination of particles in red blood cell (RBC) suspensions is investigated using direct numerical simulation (DNS) of cellar blood flow. We focus on margination of particles in the flow of moderately dense suspensions of RBCs. We hypothesize that margination rate in nondilute suspensions is mainly driven by the RBC-enhanced diffusion of marginating particles in the RBC-filled region. We derive a scaling law for margination length in a straight channel. Margination length increases cubically with channel height and is independent of shear rate. We verify this scaling law for margination length by DNS of flowing RBCs and marginating particles. We also show that rigidity and size both lead to particle margination with rigidity having a more significant effect compared to size within the range of parameters in this study.

Simulating flexible fiber suspensions using a scalable immersed boundary algorithm

We present an approach for numerically simulating the dynamics of flexible fibers in a three-dimensional shear flow using a scalable immersed boundary (IB) algorithm based on Guermond and Minev's pseudo-compressible fluid solver. The fibers are treated as one-dimensional Kirchhoff rods that resist stretching, bending, and twisting, within the generalized IB framework. We perform a careful numerical comparison against experiments on single fibers performed by S. G. Mason and co-workers, who categorized the fiber dynamics into several distinct orbit classes. We show that the orbit class may be determined using a single dimensionless parameter for low Reynolds flows. Lastly, we simulate dilute suspensions containing up to hundreds of fibers using a distributed- memory computer cluster. These simulations serve as a stepping stone for studying more complex suspension dynamics including non-dilute suspensions and aggregation of fibers (also known as flocculation).

The rheology of non-dilute dispersions of highly deformable viscoelastic particles in Newtonian fluids

AbstractWe present a model for the rheological behaviour of non-dilute suspensions of initially spherical viscoelastic particles in viscous fluids under uniform Stokes flow conditions. The particles are assumed to be neutrally buoyant Kelvin–Voigt solids undergoing time-dependent finite deformations and exhibiting generalized neo-Hookean behaviour in their purely elastic limit. We investigate the effects of the shape dynamics and constitutive properties of the viscoelastic particles on the macroscopic rheological behaviour of the suspensions. The proposed model makes use of known homogenization estimates for composite material systems consisting of random distributions of aligned ellipsoidal particles with prescribed two-point correlation functions to generate corresponding estimates for the instantaneous (incremental) response of the suspensions, together with appropriate evolution laws for the relevant microstructural variables. To illustrate the essential features of the model, we consider two special cases: (i) extensional flow and (ii) simple shear flow. For each case, we provide the time-dependent response and, when available, the steady-state solution for the average particle shape and orientation, as well as for the effective viscosity and normal stress differences in the suspensions. The results exhibit shear thickening for extensional flows and shear thinning for simple shear flows, and it is found that the volume fraction and constitutive properties of the particles significantly influence the rheology of the suspensions under both types of flows. In particular, for extensional flows, suspensions of particles with finite extensibility constraints are always found to reach a steady state, while this is only the case at sufficiently low strain rates for suspensions of (less realistic) neo-Hookean particles, as originally reported by Roscoe (J. Fluid Mech., vol. 28, 1967, pp. 273–293) and Gao et al. (J. Fluid Mech., vol. 687, 2011, pp. 209–237). For shear flows, viscoelastic particles with high viscosities can experience a damped oscillatory motion of decreasing amplitude before reaching the steady state.

Effective viscosity in a wave propagation model for ultrasonic particle sizing in non-dilute suspensions.

Estimates of particle size distributions (PSDs) in solid-in-liquid suspensions can be obtained from measurements of ultrasonic wave attenuation. The technique is based on adaptively fitting theoretical wave propagation models to the measured data across a frequency range. These models break down at high solid concentrations and it is believed that this failure is due to the effective viscosity of the mixture in the vicinity of the particles being different from that of the continuous phase. This paper discusses PSD estimation when a number of different viscosity formulations are incorporated into the wave propagation model. The viscosity model due to Happel provides the best estimate of PSD in suspensions of medium concentration.

On the use of interaction tensors to describe and predict rod interactions in rod suspensions

Recently, Ferec et al. (2009a) proposed a model for nondilute rod-like suspensions, where particle interactions are taking into account via a micromechanical approach. The derived governing equation used the well-known second- and fourth-order orientation tensors (a2 and a4) and novel second- and fourth-order interaction tensors (b2 and b4). To completely close the model, it is necessary to express a4, b2, and b4 in terms of a2. This paper gives the general framework to elaborate these new relations. Firstly, approximations for b2 are developed based on linear combinations of a2 and a4. Moreover, a new closure approximation is also derived for b4, based on the orthotropic fitted closure approach. Unknown parameters are determined by a least-square fitting technique with assumed exact solutions constructed from the probability distribution function (PDF). As numerical solutions for the PDF are difficult to obtain given the nonlinearity of the problem, a combination of steady state solutions is used to generate PDF designed to cover uniformly the entire domain of possible orientations. All these proposed approximations are tested against the particle-based simulations in a variety of flow fields. Improvements of the different approximations are observed, and the couple iORW-CO4P3 gives efficient results.

The dynamics of a non-dilute vesicle suspension in a simple shear flow

AbstractWe simulate multivesicle suspensions undergoing simple shear deformation by using the Stokes-flow boundary-integral equation method to quantify the effect of vesicle–vesicle interactions on the microstructure and rheology of the suspension. The binary collisions between vesicles are found to cause a vesicle’s instantaneous inclination angle to deviate considerably from its stationary tank-treading angle. The strength of the binary interaction becomes significant when the distance between centroids is less than three vesicle radii, which is consistent with observations from experiments. Vesicle interactions in the suspension delay the transition from tank-treading (TT) to trembling (TR)/tumbling (TU), which can be explained by the augmentation of the inclination angles in a binary collision when the viscosity ratio $\lambda $ is close to its critical value. Accordingly, in a non-dilute suspension the global minimum of the particle shear viscosity occurs at higher values of $\lambda $ due to the close correlation between the vesicle’s orientation and particle shear stress. In the high-$\lambda $ regime, collisions in a suspension disrupt the TU cycles of individual vesicles, resulting in intermittent TU/TR motions. In addition, an entangled state is observed to occur for an interacting vesicle pair in the TR/TU regime, where the centroids are aligned approximately parallel to the vorticity direction and rotates about it. The persistence of this close-range interaction results in two distinct peaks in the pair distribution of vesicles. However, their existence is limited to suspensions at low volume fraction, as they are otherwise disrupted by the three-body interactions that occur more frequently in a dense suspension.

Dynamic stability of non-dilute fiber shear suspensions

Temporal stability analysis of fiber suspended shear flow is performed. After introducing the second order structure tensor to determine the Folgar-Tucker inter-fiber interactions based on the Langevin?s equation, a system governing the flow stability is derived in conjunction with the fiber orientation closure. Effect of the inter-fiber interactions on the dynamic stability is studied by solving the general eigenvalue problem. Results show that fiber interaction has significant stabilizing effects on the flow. The most unstable wave number changes with the interaction coefficient. For given interaction coefficient, wave number and other relevant parameters, there is a Re number which corresponds to the critical flow. This Re number is related to the wave number.

The drainage of non-Newtonian fluids in the quasi-steady motion of a sphere towards a plane

In the lubrication limit, the time needed for the drainage of the liquid film between two particles or between particles and walls is of industrial importance, because it controls the dynamics and aggregation of nondilute suspensions. This problem is also of fundamental interest in the application of the dynamic surface force apparatus to nanorheology. Even if this problem has an exact solution in Newtonian fluid when the sphere moves steadily and slowly towards or away from a plane wall, this problem remains, to our knowledge, without any exact analytical solution in non-Newtonian fluids with negligible viscoelastic components. But Rodin, using the method of asymptotic expansions, gives an asymptotic solution to this problem in the lateral unbounded power-law fluid. Therefore, in this study, we give a numerical result using the dynamic mesh technique and an asymptotic analytical formula valid in the lubrication regime, for a fluidity index $$0.5 < n\, \leqslant \,1.8.$$ The comparison between the two results confirms their mutual validity.

Papermaking fibre-suspension flow simulations at multiple scales

Papermaking flows are extremely challenging for modelling and simulation, if one accepts their full complexity. A wide range of particles, including fibres, fibre fragments (fines) and fillers (non-organic particles), flow and interact with each other in a non-dilute suspension, a complex geometry and at a high flow rate. Different simulation approaches are reviewed from particle-level simulations, through meso-scale simulations to the full flow geometry of the papermaking line. Their application to papermaking and potential to provide fundamental understanding as well as direct process-optimization support are discussed.

Rheology, Microstructure and Migration in Brownian Colloidal Suspensions

We demonstrate that suspended spherical colloidal particles can be effectively modeled as single dissipative particle dynamics (DPD) particles provided that the conservative repulsive force is appropriately chosen. The suspension model is further improved with a new formulation, which augments standard DPD with noncentral dissipative shear forces between particles while preserving angular momentum. Using the new DPD formulation we investigate the rheology, microstructure and shear-induced migration of a monodisperse suspension of colloidal particles in plane shear flows (Couette and Poiseuille). Specifically, to achieve a well-dispersed suspension we employ exponential conservative forces for the colloid-colloid and colloid-solvent interactions but keep the conventional linear force for the solvent-solvent interactions. Our simulations yield relative viscosity versus volume fraction predictions in good agreement with both experimental data and empirical correlations. We also compute the shear-dependent viscosity and the first and second normal-stress differences and coefficients in both Couette and Poiseuille flow. Simulations near the close packingvolume-fraction (64%) at low shear rates demonstrate a transition to flow-induced string-like structures of colloidal particles simultaneously with a transition to a nonlinear Couette velocity profile in agreement with experimental observations. After a sufficient increase ofthe shear rate the ordered structure melts into disorder with restoration of the linear velocity profile. Migration effects simulated in Poiseuille flow compare well with experiments and model predictions. The important role of angular momentum and torque in nondilute suspensions is also demonstrated when compared with simulations by the standard DPD, which omits the angular degrees of freedom. Overall, the new method agrees very well with the Stokesian Dynamics method but it seems to have lower computational complexity and is applicable to general complex fluids systems.

Tornadoes in a microchannel

In non-dilute colloidal suspensions, gradients in particle volume fraction result in gradients in electrical conductivity and permittivity. An externally applied electric field couples with gradients in electrical conductivity and permittivity and, under some conditions, can result in electric body forces that drive the flow unstable forming vortices. The experiments are conducted in square 200 micron PDMS microfluidic channels. Colloidal suspensions consisted of 0.01 volume fraction of 2 or 3 micron diameter polystyrene particles in 0.1 mM Phosphate buffer and 409 mM sucrose to match particle-solution density. AC electric fields at 20 Hz and strength of 430 to 600 V/cm were used. We present a fluid dynamics video that shows the evolution of the particle aggregation and formation of vortical flow. Upon application of the field particles aggregate forming particle chains and three dimensional structures. These particles form rotating bands where the axis of rotation varies with time and can collide with other rotating bands forming increasingly larger bands. Some groups become vortices with a stable axis of rotation. Other phenomena showed include counter rotating vortices, colliding vortices, and non-rotating particle bands with internal waves.