Effective Viscosity Of Suspension Research Articles

Derivation of the Batchelor-Green formula for random suspensions

This paper is dedicated to the effective viscosity of suspensions without inertia, at low solid volume fraction ϕ. The goal is to derive rigorously a o(ϕ2) formula for the effective viscosity. In [17,19], such formula was given for rigid spheres satisfying the strong separation assumption dmin≥cϕ−13r, where dmin is the minimal distance between the spheres and r their radius. It was then applied to both periodic and random configurations with separation, to yield explicit values for the O(ϕ2) coefficient. We consider here complementary (and certainly more realistic) random configurations, satisfying softer assumptions of separation, and long range decorrelation. We justify in this setting the famous Batchelor-Green formula [3]. Our result applies for instance to hardcore Poisson point process with almost minimal hardcore assumption dmin>(2+ε)r, ε>0.

Structure and rheology of suspensions of spherical strain-hardening capsules

Abstract

Particle motion in pressure-driven suspension flow through a symmetric T-channel

Particle motion is studied in pressure-driven suspension flow through a bifurcating channel of square cross-section. Particles and suspending liquid are neutrally buoyant, i.e. of the same density ρ. Experimental particle trajectories are compared with the computed single-phase flow. The flow is in a symmetric T-junction, with each outgoing stream making a right angle relative to the incoming stream. All branches have the same cross-section and length, with L/D=29, where L is the length of each branch, and D is the side length. Particle trajectories are obtained by two-camera imaging for solid volume fraction 0 < ϕ ≤ 0.30. Inertia is characterized by the bulk, Re=ρDU/η, and particle scale Reynolds numbers, Rep=Re(d/D)2, where d is the diameter of the particles, U is the mean velocity in the inlet channel and η is the liquid viscosity. In this work, d/D=O(0.1) and Re=200. For the pure liquid, spiraling vortices are found in the outlet channels, emanating from near the bifurcation, and these flow structures are largely unchanged for ϕ ≤ 0.1, leading to spiraling motions of the particles. Particle tracks are compared against computed tracer motion in the pure fluid flow, and for dilute ϕ agree qualitatively well, with some deviation due to the particles’ finite size. For larger ϕ, the flow deviates strongly from that of pure fluid, keeping Re=200 based on the suspension effective viscosity. Vortices are damped for ϕ=0.2 and disappear for ϕ=0.30. Particle interactions with the wall and each other appear to dissipate energy at a higher rate than predicted by the effective viscosity.

The impact of porous walls on the rheology of suspensions

We study the effect of isotropic porous walls on a plane Couette flow laden with spherical and rigid particles. We perform a parametric study varying the volume fraction between 0 and 30%, the porosity between 0.3 and 0.9 and the non-dimensional permeability between 0 and 7.9×10-3 We find that the porous walls induce a progressive decrease in the suspension effective viscosity as the wall permeability increases. This behavior is explained by the weakening of the wall-blocking effect and by the appearance of a slip velocity at the interface of the porous medium, which reduces the shear rate in the channel. Therefore, particle rotation and the consequent velocity fluctuations in the two phases are dampened, leading to reduced particle interactions and particle stresses. Based on our numerical evidence, we provide a closed set of equations for the suspension viscosity, which can be used to estimate the suspension rheology in the presence of porous walls.

Analysis of the Viscosity of Dilute Suspensions Beyond Einstein’s Formula

We provide a mathematical analysis of the effective viscosity of suspensions of spherical particles in a Stokes flow, at low solid volume fraction $\phi$. Our objective is to go beyond the Einstein's approximation $\mu_{eff}=(1+\frac{5}{2}\phi)\mu$. Assuming a lower bound on the minimal distance between the $N$ particles, we are able to identify the $O(\phi^2)$ correction to the effective viscosity, which involves pairwise particle interactions. Applying the methodology developped over the last years on Coulomb gases, we are able to tackle the limit $N \rightarrow +\infty$ of the $O(\phi^2)$-correction, and provide explicit formula for this limit when the particles centers can be described by either periodic or stationary ergodic point processes.

Micromechanics-based assessment of the effective viscosity of suspensions of generalized-Newtonian fluids embedding noncolloidal angular/spheroidal pores and particles

The viscosity of suspensions depends on the properties of the fluid, i.e., the matrix material as well as on the volume fraction and shape of embedded inclusions. Regarding this dependence, a micromechanics-based homogenization approach is presented, giving access to the resulting effective viscous properties, starting from well established homogenization schemes originally proposed for the linear-elastic material response. In order to incorporate nonlinear matrix behavior, i.e., non-Newtonian fluids, the secant method of nonlinear homogenization is employed. The effect of nonsphericity of inclusions is considered by substituting the Eshelby tensor by a replacement tensor (replacement Eshelby tensor) determined by linear-elastic finite element simulations in Traxl and Lackner [Mech. Mater. 126, 126–139 (2018)]. In total, suspensions consisting of four different types of matrix materials, namely, (i) Newtonian, (ii) shear thickening, (iii) shear thinning power-law fluids, and (iv) Bingham or Herschel–Bulkley yield-stress fluids are considered embedding spherical, cubical, and tetrahedral inclusions (rigid particles and pores). The performance of the model is assessed by finite element simulations of respective representative volume elements, showing good agreement between model predictions and numerical results for small and moderate inclusion volume fractions.

Unravelling the effect of nano-heterogeneity on the viscosity of silicate melts: Implications for glass manufacturing and volcanic eruptions

Traditionally depicted as homogeneous materials, manmade and volcanic glasses are often subjected to nanostructuration during demixing and crystallization of their parental melt. While the controlled formation of nanocrystals has been exploited for decades in the industrial production of glass-ceramics to obtain materials with superior properties, nano-heterogeneities have been recently recognized in volcanic products erupted explosively. Here, a multi-pronged approach is adopted to highlight how the emergence of nano-heterogeneities exerts a previously unexplored influence on the viscosity of multicomponent silicate melts. Calorimetry and viscometry provide evidence of how the formation of TiO2- and FeO-bearing nanocrystals increases the effective viscosity of suspensions. Raman spectroscopy and transmission electron microscopy allow the correlation between the increase in viscosity and the degree of sample modification. This study has implications for glass and glass-ceramics manufacturing and for the development of better numerical models of volcanic eruptions, which are key for the operational assessment of volcanic risk.

A Local Version of Einstein's Formula for the Effective Viscosity of Suspensions

We prove a local variant of Einstein's formula for the effective viscosity of dilute suspensions, that is, $\mu^\prime=\mu ({{1+\frac 5 2\phi+o(\phi)}})$, where $\phi$ is the volume fraction of the suspended particles. Up to now rigorous justifications have only been obtained for dissipation functionals of the flow field. We prove that the formula holds on the level of the Stokes equation (with variable viscosity). We consider a regime where the number $N$ of particles suspended in the fluid goes to infinity while their size $R$ and the volume fraction $\phi=NR^3$ approach zero. We establish $L^\infty$ and $L^p$ estimates for the difference of the microscopic solution to the solution of the homogenized equation. Here we assume that the particles are contained in a bounded region and are well separated in the sense that the minimal distance is comparable to the average one. The main tools for the proof are a dipole approximation of the flow field of the suspension together with the so-called method of reflections and a coarse graining of the volume density.

Nonlinear microrheology of active Brownian suspensions.

The rheological properties of active suspensions are studied via microrheology: tracking the motion of a colloidal probe particle in order to measure the viscoelastic response of the embedding material. The passive probe particle with size R is pulled through the suspension by an external force Fext, which causes it to translate at some speed Uprobe. The bath is comprised of a Newtonian solvent with viscosity ηs and a dilute dispersion of active Brownian particles (ABPs) with size a, characteristic swim speed U0, and a reorientation time τR. The motion of the probe distorts the suspension microstructure, so the bath exerts a reactive force on the probe. In a passive suspension, the degree of distortion is governed by the Péclet number, Pe = Fext/(kBT/a), the ratio of the external force to the thermodynamic restoring force of the suspension. In active suspensions, however, the relevant parameter is Ladv/l = UprobeτR/U0τR∼Fext/Fswim, where Fswim = ζU0 is the swim force that propels the ABPs (ζ is the Stokes drag on a swimmer). When the external forces are weak, Ladv≪l, the autonomous motion of the bath particles leads to "swim-thinning," though the effective suspension viscosity is always greater than ηs. When advection dominates, Ladv≫l, we recover the familiar behavior of the microrheology of passive suspensions. The non-Newtonian behavior for intermediate values of Ladv/l is determined by l/Rc = U0τR/Rc-the ratio of the swimmer's run length l to the geometric length scale associated with interparticle interactions Rc = R + a. The results in this manuscript are approximate as they are based on numerical solutions to mean-field equations that describe the motion of the active bath particles.

Viscosity of nanofluids containing anisotropic particles: A critical review and a comprehensive model.

Compared to nanofluids with spherical particles, nanofluids with anisotropic particles possess higher thermal conductivity and present a better enhancement option in heat transfer applications. The viscosity variation of such nanofluids becomes of great importance in evaluating their pumping power in thermal systems. This paper presents a comprehensive review of the experimental and theoretical studies on the viscosity of nanofluids with anisotropic particles. The internal mechanisms of viscosity evolution are investigated considering three aspects: particle clustering, particle interactions, and Brownian motion. In experimental studies, important factors including classification and synthetic methods for particle preparation, base fluid, particle loading, particle shape and size, temperature, p H, shear stress and electric field are investigated in detail. Classical theoretical models and empirical relations of the effective viscosity of suspensions are discussed. Some crucial factors such as maximum particle packing fraction, fractal index and intrinsic viscosity models, are examined. A comparison of predictions and experimental results shows that the classical models underestimate suspension viscosity. A comprehensive combination of the modified Krieger-Dougherty (K-D) model with intrinsic viscosity relations for different aspect ratios is suggested for low particle loadings, and the modified Maron-Pierce model (M-D) is recommended for high particle loadings. Possible directions for future works are discussed.

Rheological signature of microswimmer phase-locking under flow

Unlike force-dipole swimmers, amoeboid swimmers affect the effective suspension viscosity in a nontrivial manner. Transitions between shear-thickening and shear-thinning behaviors, depending on applied shear rates, are due to the phase-locking of the swimmer dynamics with the shear flow.

Effective viscosity of a concentrated suspension of composite porous spherical particles

We present an analytical study of the effective viscosity of concentrated suspension of porous spherical particles with a rigid core, under the creeping flow conditions. It is assumed that the individual particle is surrounded by a hypothetical fluid envelope. This model is popularly known as cell model, which takes into account the interaction of neighboring particles. The flow fields inside the free flow region and porous region are governed by Stokes equation and Brinkman equation together with mass conservation, respectively. The effective viscosity of the suspension depends on various parameters such as, radii of the fluid envelope and porous particle, permeability of the porous particle and further on the type of boundary condition at the outer cell, volume fraction etc. Various limiting cases are obtained and compared with earlier existing results in literature both theoretical and experimental.

INVESTIGATION OF THE WATER-RETAINING CAPACITY OF THE CARBOHYDRATE COMPLEX OF RYE-WHEAT DOUGH WITH ADDITION OF POLYFUNCTIONAL FOOD SUPPLEMENT “MAGNETOFООD”

Today in Ukraine the share of low-quality bakery products is near 20–25 %, because they often don’t correspond to quality standards and sanitary norms, because of being produced of flour with low bakery properties. That is why new raw material sources, functional ingredients and so on are searched for in bread technologies. For using new raw material and food supplement types, it is necessary to know their functional-technological properties (FTP) that allows to prognosticate the behavior of powder-like raw materials and food supplements in food masses at technological processing and storage of ready products. The water-retaining capacity (WRC) is one of main functional-technological parameters of carbohydrate-containing raw materials, because it favors the outcome, structural-mechanical properties and quality characteristics of ready products. The authors introduced “Magnetofood” polyfunctional food supplement for increasing WRC of rye-wheat dough. For grounding the mechanism of forming supramolecular groups in carbohydrate food systems: Magnetofood-polysacharide-water, there was studied the influence of “Magnetofood” food supplement on processes of hydration, swelling and water-retention by rye-wheat starch and rye-wheat flour. It was established, that introduction of “Magnetofood” food supplement in rye-wheat starch and flour in amounts: 0,10; 0,15; 0,20 % to the mass of starch or flour increases swelling and water-retaining capacity (WRC) of starch and flour: swelling in 1,3–1,5 and 1,5–2,25 times, respectively; WRC in 1,10–1,15 and 1,1–1,3 times, respectively. The rational dose of “Magnetofood” food supplement – 0,15 % to the mass of dry raw material was experimentally set. There was studied the influence of “Magnetofood” food supplement on rheological properties of water suspensions of rye-wheat starch and flour at different temperatures and different speeds of shift. It was established, that adding “Magnetofood” in amount 0,15 % favors increasing the effective viscosity of suspensions of experimental samples of starch and flour at temperatures: (23±2) ºС and (40±2) ºС comparing with control samples in average by 29,0 % – for starch and by 22,0 % – for flour at 23 °С and by 16,0 % – for starch and by 10,0 % – for flour at 40 °С. There was studied the dynamics of changing viscosity of colloid solutions of starch and flour, enriched with “Magnetofood” in amount 0,15 % to the mass of dry raw material in the keeping process at different temperatures and shift speed 9 s-1. There was established the increase of viscosity of suspensions of rye-wheat starch and flour in the process of infusion (especially at adding “Magnetofood”) that is connected with continuing hydration and swelling process. It was demonstrated, that the temperature increase of the colloid system from (23±20) ° С to (40±2) °С favors the increase of its viscosity – both at the initial moment and in infusion process during 30 min in average by 20,0–25,0 % at the expanse of the aforesaid processes. Moreover, adding “Magnetofood” accelerates and intensifies the processes of swelling and hydration of experimental samples of rye-wheat starch and flour comparing with control ones. The received experimental data may be used at elaborating the technology of rye-wheat bread, enriched with “Magnetofood” food supplement; and also at elaborating innovative technologies of carbohydrate food systems. The results of this study may be used at elaborating recipes and technologies of carbohydrate-containing food products for increasing their viscosity, water-retaining and stabilizing capacities.

Coupling anisotropic viscosity and fiber orientation in applications to squeeze flow

The influence of anisotropic viscosity on fiber orientation evolution in classic squeeze flow is examined with a viscous fiber suspension of specified initial orientation state. The components of the anisotropic viscosity tensor for a well dispersed and collimated fiber suspension developed in micromechanical analyses earlier by one of the authors were combined in an orientation averaging approach to produce the effective suspension viscosity components for arbitrary orientation states. A coupled anisotropic viscosity constitutive relationship was numerically implemented into a displacement-based finite element code for nonlinear analysis of flow and fiber orientation. The numerical method was verified by agreement between predicted and observed responses for a single element subjected to canonical states of deformation. Evolution of fiber orientation during the deformation processes was confirmed by analytic predictions. This approach was validated by examining the squeeze flow of a fiber suspension studied earlier both analytically and experimentally. The present analysis was found to be in excellent agreement with experimental data and clearly demonstrated that evolution of fiber orientation and the resulting effective anisotropic viscosity tensor were essential to the correct predictions.The influence of anisotropic viscosity on fiber orientation evolution in classic squeeze flow is examined with a viscous fiber suspension of specified initial orientation state. The components of the anisotropic viscosity tensor for a well dispersed and collimated fiber suspension developed in micromechanical analyses earlier by one of the authors were combined in an orientation averaging approach to produce the effective suspension viscosity components for arbitrary orientation states. A coupled anisotropic viscosity constitutive relationship was numerically implemented into a displacement-based finite element code for nonlinear analysis of flow and fiber orientation. The numerical method was verified by agreement between predicted and observed responses for a single element subjected to canonical states of deformation. Evolution of fiber orientation during the deformation processes was confirmed by analytic predictions. This approach was validated by examining the squeeze flow of a fiber suspension stud...

Effects of the finite particle size in turbulent wall-bounded flows of dense suspensions

We use interface-resolved numerical simulations to study finite-size effects in turbulent channel flow of neutrally buoyant spheres. Two cases with particle sizes differing by a factor of two, at the same solid volume fraction of 20 % and bulk Reynolds number are considered. These are complemented with two reference single-phase flows: the unladen case, and the flow of a Newtonian fluid with the effective suspension viscosity of the same mixture in the laminar regime. As recently highlighted in Costa et al. (Phys. Rev. Lett., vol. 117, 2016, 134501), a particle–wall layer is responsible for deviations of the mesoscale-averaged statistics from what is observed in the continuum limit where the suspension is modelled as a Newtonian fluid with (higher) effective viscosity. Here we investigate in detail the fluid and particle dynamics inside this layer and in the bulk. In the particle–wall layer, the near-wall inhomogeneity has an influence on the suspension microstructure over a distance proportional to the particle size. In this layer, particles have a significant (apparent) slip velocity that is reflected in the distribution of wall shear stresses. This is characterized by extreme events (both much higher and much lower than the mean). Based on these observations we provide a scaling for the particle-to-fluid apparent slip velocity as a function of the flow parameters. We also extend the scaling laws in Costa et al. (Phys. Rev. Lett., vol. 117, 2016, 134501) to second-order Eulerian statistics in the homogeneous suspension region away from the wall. The results show that finite-size effects in the bulk of the channel become important for larger particles, while negligible for lower-order statistics and smaller particles. Finally, we study the particle dynamics along the wall-normal direction. Our results suggest that single-point dispersion is dominated by particle–turbulence (and not particle–particle) interactions, while differences in two-point dispersion and collisional dynamics are consistent with a picture of shear-driven interactions.

Heat transfer and shear-induced migration in dense non-Brownian suspension flows: modelling and simulation

Modelling and simulation are developed, generalized and validated for both heat transfer and shear-induced particle migration in dense non-colloidal laminar suspension flows. Past theory and measurements for the effective thermal conductivity in porous materials at zero shear rate are coupled to more recent effective thermal diffusivity measurements of sheared suspensions. The suspension effective heat transfer affected by the local shear rate ($\dot{\unicode[STIX]{x1D6FE}}$), the phenomenon of shear-induced particle migration (SIM), the buoyancy effects ($\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}$) and the thermal Péclet number ($Pe_{d_{p}}=\dot{\unicode[STIX]{x1D6FE}}d_{p}^{2}/\unicode[STIX]{x1D6FC}_{f}$, where $d_{p}$ is the diameter of rigid particles and $\unicode[STIX]{x1D6FC}_{f}$ is the fluid phase thermal diffusivity) at the particle scale are all considered in the present constitutive three-dimensional modelling. Moreover, the influence of the temperature, the shear rate and the particle volume fraction ($\unicode[STIX]{x1D719}$) on the suspension effective viscosity ($\unicode[STIX]{x1D702}_{S}$), the suspension effective thermal properties and the fluid density ($\unicode[STIX]{x1D70C}_{f}$) are taken also into account. The present contribution represents an emerging field of heat transfer applications of complex fluid flows and is very beneficial for many future applications where concentrated suspension laminar flows with conjugate heat transfer may be present (e.g. for designing more innovative and compact heat exchangers).

Effective viscosity of a two-dimensional suspension of interacting active particles.

Suspensions of hydrodynamical active particles exhibit interesting rheological properties. For a dilute suspension of microswimmers, it has been shown that the effective viscosity of the suspension depends on the volume fraction of swimmers, and it behaves differently for pushers and pullers. Here we develop a theoretical framework to study the rheological properties of an interacting suspension. Taking into account the hydrodynamic interaction between swimmers and considering the small Péclet number condition, we calculate the effective viscosity of a two-dimensional suspension. For a dilute suspension, a perturbative result is obtained up to the second order of the surface fraction of swimmers. Our results show that the effective viscosity for the suspension can be very different for pushers and pullers.

Drag reduction in turbulent channel flow laden with finite-size oblate spheroids

We study suspensions of oblate rigid particles in a viscous fluid for different values of the particle volume fractions. Direct numerical simulations have been performed using a direct-forcing immersed boundary method to account for the dispersed phase, combined with a soft-sphere collision model and lubrication corrections for short-range particle–particle and particle–wall interactions. With respect to the single-phase flow, we show that in flows laden with oblate spheroids the drag is reduced and the turbulent fluctuations attenuated. In particular, the turbulence activity decreases to lower values than those obtained by accounting only for the effective suspension viscosity. To explain the observed drag reduction, we consider the particle dynamics and the interactions of the particles with the turbulent velocity field and show that the particle–wall layer, previously observed and found to be responsible for the increased dissipation in suspensions of spheres, disappears in the case of oblate particles. These rotate significantly slower than spheres near the wall and tend to stay with their major axes parallel to the wall, which leads to a decrease of the Reynolds stresses and turbulence production and so to the overall drag reduction.

Universal Scaling Laws for Dense Particle Suspensions in Turbulent Wall-Bounded Flows

The macroscopic behavior of dense suspensions of neutrally buoyant spheres in turbulent plane channel flow is examined. We show that particles larger than the smallest turbulence scales cause the suspension to deviate from the continuum limit in which its dynamics is well described by an effective suspension viscosity. This deviation is caused by the formation of a particle layer close to the wall with significant slip velocity. By assuming two distinct transport mechanisms in the near-wall layer and the turbulence in the bulk, we define an effective wall location such that the flow in the bulk can still be accurately described by an effective suspension viscosity. We thus propose scaling laws for the mean velocity profile of the suspension flow, together with a master equation able to predict the increase in drag as a function of the particle size and volume fraction.

Large-scale calculation of hydrodynamic transport properties for random suspensions of hard-sphere particles

A numerical method based on fast multipole summation scheme is used to calculate hydrodynamic interactions in random suspensions of non-colloidal hard-sphere particles. The calculation is carried out for suspensions of 1,024 particles randomly placed in periodic unit cell to determine hydrodynamic transport properties such as permeability of a viscous flow through porous medium, effective viscosity of suspension, and sedimentation velocity of the suspended particles. The particle volume fraction o ranges from 0.01 to 0.25. Effect of particle number N on the transport properties was examined through the numerical calculations with N=64-1,024. It is shown that sedimentation velocity increases with N approaching an estimate for infinite N, and the finite N effect is negligible in effective viscosity and permeability problems. The present scheme is quite useful for obtaining a statistically-averaged quantity for random suspensions. As an example, ensemble-averaged velocity when position of one particle is fixed is numerically obtained in sedimentation problem. The numerical results are shown to be in excellent agreement with theoretical prediction.