Suspension Of Spherical Particles Research Articles

Non-Newtonian Flow of Structured Systems. XXXVIII. Viscosity of a Suspension of Spherical Particles with Electrostatic Stabilization

The structural rheological model is used to describe the flow of the aqueous styrene/acrylate suspension obtained by copolymerization of acrylic acid esters with styrene, whose particles have a double electric layer. It is assumed that there are large repulsive forces between the suspension particles, which prevent the particle aggregation. The theoretical analysis of viscosity curves and flow curves for the suspension of charged polymer particles at different concentrations and radii of the particles is presented. The generalized flow equation was used to approximate the flow curves at the separate areas of high and low shear rates. It is shown that the behavior of the generalized flow equation coefficients is consistent with the structural rheological model.

Fully optimized second-order estimates for the macroscopic behavior and field statistics of particle-reinforced viscoplastic composites

This paper is concerned with the characterization of the macroscopic behavior and statistics for the distribution of the stress and strain-rate fields in composites consisting of random and isotropic suspensions of rigid spherical particles in power-law viscoplastic materials. For this purpose, use is made of the Fully Optimized Second-Order (FOSO) homogenization method (Ponte Castañeda, 2016) in combination with recently developed estimates (Kammer and Ponte Castañeda, 2022) for the macroscopic properties of the associated ‘linear comparison composite’ (LCC). Special attention is devoted to the method’s ability to account for the dependence of the homogenized properties of the nonlinear composite on the Lode angle (third invariant) of the applied loading. It is found that, while, for large particle volume fractions c, the effective flow stress is only weakly dependent on the Lode angle, for dilute volume fractions, the dependence on the Lode angle becomes more pronounced. In the ideally plastic limit, as c tends to zero, the effective yield stress is shown to depend linearly on c for axisymmetric shear, while this dependence becomes weaker with a non-analytic leading-order correction of O(c/lnc) for pure shear loading. This strong dependence on the Lode angle at dilute concentrations is shown to be due to significant differences in the local deformation patterns, which become strongly anisotropic and localize for pure shear conditions, but do not for axisymmetric shear. In turn, the FOSO homogenization method is able to capture the statistical features of these different deformation patterns by providing consistent estimates for the covariance tensor of the strain-rate field fluctuations in the matrix phase, which tend to become more strongly anisotropic for the pure shear case. As c increases, the shear bands are deflected by the randomly dispersed spheres leading to a more isotropic distribution of the stress and strain-rate fields, which is consistent with a weaker Lode angle effect. The estimates can also capture the effect of strong particle interactions, including the existence of a rigidity threshold where the macroscopic flow stress and field fluctuations blow up.

Hydrodynamic irreversibility of non-Brownian suspensions in highly confined duct flow

The irreversible behaviour of a highly confined non-Brownian suspension of spherical particles at low Reynolds number in a Newtonian fluid is studied experimentally and numerically. In the experiment, the suspension is confined in a thin rectangular channel that prevents complete particle overlap in the narrow dimension and is subjected to an oscillatory pressure-driven flow. In the small cross-sectional dimension, particles rapidly separate to the walls, whereas in the large dimension, features reminiscent of shear-induced migration in bulk suspensions are recovered. Furthermore, as a consequence of the channel geometry and the development and application of a single-camera particle tracking method, three-dimensional particle trajectories are obtained that allow us to directly associate relative particle proximity with the observed migration. Companion simulations of a steadily flowing suspension highly confined between parallel plates are conducted using the force coupling method, which also show rapid migration to the walls as well as other salient features observed in the experiment. While we consider relatively low volume fractions compared to most prior work in the area, we nevertheless observe significant and rapid migration, which we attribute to the high degree of confinement.

Non-existence of Mean-Field Models for Particle Orientations in Suspensions

We consider a suspension of spherical inertialess particles in a Stokes flow on the torus T3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {T}^3$$\\end{document}. The particles perturb a linear extensional flow due to their rigidity constraint. Due to the singular nature of this perturbation, no mean-field limit for the behavior of the particle orientation can be valid. This contrasts with widely used models in the literature such as the FENE and Doi models and similar models for active suspensions. The proof of this result is based on the study of the mobility problem of a single particle in a non-cubic torus, which we prove to exhibit a nontrivial coupling between the angular velocity and a prescribed strain.

ANALYTICAL RELATIONSHIP BETWEEN THE COEFFICIENTS OF THE ADDED MASS AND THERMAL CONDUCTIVITY IN A SUSPENSION OF SPHERICAL PARTICLES

An analytical dependence between the dimensionless coefficients of the added mass and thermal conductivity was obtained in a suspension of spherical particles. The dependence is confirmed by comparison with the theoretical data of other authors, including at elevated concentrations of dispersed particles, taking into account their hydrodynamic interaction.

An investigation of the transient electrophoresis of conducting colloidal particles in porous media using a cell model

A comprehensive analytical study of the electrophoresis of a suspension of charged spherical particles in an arbitrary electrolyte solution through a porous medium is analyzed using a unit cell model. The unsteady Brinkman equation with the term of the electric force governing the fluid velocity fields is solved by means of the Laplace transform. The porous medium is uniformly charged by fixed charge density Q, and the embedded spherical particle is conductive and charged with constant zeta potential. Two different accurate concepts of the boundary conditions at the outer virtual surface, are presented to solve the governing equations for the flow field. Steady-state electrophoretic velocity is obtained analytically and displayed graphically for different physical parameters. Also, a closed form of the transient electrophoretic velocity versus the dimensionless elapsed time is plotted and discussed for different values of the Debye length parameter, particle volume fraction, density ratio, permeability of the porous medium, and for highly- and non-conducting particles. The steady/ transient electrophoretic velocity is a monotonic decreasing function of the permeability parameter, the particle-to-fluid density ratio, and the particle volume fraction, but it increases with an increase in the Debye length parameter with constant zeta potential. In general, the Kuwabara cell model predicts a smaller value for the electrophoresis velocity than the Happel model, but the difference is not significant. On the other hand, when the transient velocity is normalized by its steady state, the Happel model has smaller values than the Kuwabara model. The effects of the relevant parameters on the transient starting electrokinetic flow in the porous medium are interesting and significant. The results are found to be in excellent agreement with the exact numerical results obtained by Lai and Keh.

Continuum dynamics of suspensions at low Reynolds number

The dynamics of suspensions of particles has been an active area of research since Einstein first calculated the leading-order correction to the viscosity of a suspension of spherical particles (Einstein, Proc. R. Soc., vol. A102, 1906, pp. 161–179). Since then, researchers have strived to develop an accurate description of the behaviours of suspensions that goes beyond just leading order in the particle volume fraction. Here, we consider the low-Reynolds-number behaviour of a suspension of spherical particles. Working from the Green's functions for the flow due to a single particle, we derive a continuum-level description of the dynamics of suspensions. Our analysis corrects an error in the derivation of these equations in the work of Jackson (Chem. Engng Sci., vol. 52, 1997, pp. 2457–2469) and leads to stable equations of motion for the particles and fluid. In addition, our resulting equations naturally give the sedimentation speed for suspended particles and correct a separate error in the calculation by Batchelor (J. Fluid Mech., vol. 52, 1972, pp. 245–268). Using the pair-correlation function for hard spheres, we are able to compute the sedimentation speed out to seventh order in the volume fraction, which agrees with experimental data up to 30 %–35 %, and also get higher-order corrections to the suspension viscosity, which agree with experiments up to $\sim$ 15 %. Then, using the pair distribution for spheres in shear flow, we find alterations to both the first and second normal stresses.

Viscosity behavior for zirconia and γ-alumina nanoparticles in different electrolytic solutions: An experimental investigation

Viscosity modification of injected fluids is a vital mechanism for successful oil recovery from an oil reservoir. Various studies have demonstrated that nanoparticles (NPs) possess the tendency to alter the wettability of reservoir rocks, reduce the interfacial tension between injected fluids and crude oil, and modify the viscosity of injected fluids. Zirconia (ZrO2) and γ-alumina (γ-Al2O3) NPs have demonstrated huge prospects for enhanced oil recovery (EOR) in various studies. Recently, experimental investigations have attributed the viscosity behavior of silica nanofluids to the interaction between the ions of the silica NPs and electrolytes at the electrical double layer. Therefore, this study investigates the viscosity behavior of ZrO2 and γ-Al2O3 NPs with their increasing concentrations in different aqueous solutions present in reservoir brine by employing techniques such as dynamic light scattering and transmission electron microscopy in addition to viscosity and pH measurements at ambient conditions of approximately 25℃ and 1 atm. The viscosity behavior of spherical particles in suspensions is well suited for the viscosity profile of the nanofluids with increasing concentrations of ZrO2 and γ-Al2O3 NPs. The viscosity of the ZrO2 and γ-Al2O3 nanofluids containing the chloride salts increases in the order of Ca2+ > Na+ > K+ while those containing the sulfate salts increase in the order Mg2+ > Na+ > K+. Comparatively, the model proposed by Williams et al. predicts the viscosities of the electrolyte-based zirconia and γ-Al2O3 nanofluids with percentage absolute average deviation (%AAD) of 2.07 and 2.52 %, respectively, whereas those by Einstein’s model records %AAD of 3.15 and 3.68 %, respectively. The findings from this study provide experimental data and illuminate the understanding of the viscosity behavior of the nanofluids containing electrolytic ions and set the stage for further investigations, which are relevant for EOR purposes.

Dynamics and rheological properties of suspensions of paramagnetic spherical particles under constant magnetic fields

In this work, we simulated the coarsening of magnetic particles under a shear flow and for different magnetic fields. We found that particle assembly varied in fully three-dimensional simulations and constrained two-dimensional simulations, which affected the corresponding rheological properties.

Controlling extrudate volume fraction through poroelastic extrusion of entangled looped fibers

When a suspension of spherical or near-spherical particles passes through a constriction the particle volume fraction either remains the same or decreases. In contrast to these particulate suspensions, here we observe that an entangled fiber suspension increases its volume fraction up to 14-fold after passing through a constriction. We attribute this response to the entanglements among the fibers that allows the network to move faster than the liquid. By changing the fiber geometry, we find that the entanglements originate from interlocking shapes or high fiber flexibility. A quantitative poroelastic model is used to explain the increase in velocity and extrudate volume fraction. These results provide a new strategy to use fiber volume fraction, flexibility, and shape to tune soft material properties, e.g., suspension concentration and porosity, during delivery, as occurs in healthcare, three-dimensional printing, and material repair.

Effect of a double-chain surfactant on the stabilization of suspensions of silica and titania particles against agglomeration and sedimentation

Dispersions of vesicles of didodecyldimethylammonium bromide (DDAB) were prepared with magnetic stirring of multilamellar liposomes followed by ultrasonication (SS) in order to generate dispersions that contained only unilamellar vesicles, unlike the stirring-only (S) method which also produces many large liposomes. With the SS method, the vesicles had average diameters ranging from 68 to 80 nm, which were much smaller than the diameters generated with the S method. The new vesicle dispersions had nearly infinite viscosities at very low shear stresses at DDAB weight fractions wD from 0.025 to 0.027, even though they were not close-packed, in contrast to the S dispersions. The new SS vesicle dispersions stabilized completely against sedimentation polydisperse nonspherical crystalline titania particles with sizes ranging from ca. 96–156 nm, as well as suspensions of monodisperse spherical amorphous silica particles with diameters of dsed = 454 nm, 691 nm, and 826 nm. These results establish that DDAB vesicle dispersions can stabilize against sedimentation a broad range of particle types and sizes. In addition, at the low values of wD = 0.009 and 0.018, the effective viscosities, ηeff, of the DDAB dispersions, determined from the sedimentation velocities, ranged from 1.35-1.87 cP and 4.34–5.57 cP, respectively. At higher shear stresses, however, these dispersions were highly shear-thinning and still flowable in a capillary under gravity. This behavior is critical for the practical applications of such dispersions for inkjet printing and possibly paints.

Effective thermal conductivity simulations of suspensions containing non-spherical particles in shear flow

Suspensions of oblate spheroid particles can have different thermal characteristics from suspensions of spherical particles based on its orthotropic geometry. However, the effective thermal conductivity of non-spherical particle suspensions under shear flow conditions has not been reported yet. In a first step dilute suspensions are considered ignoring particle-particle interactions, and a representative volume is observed containing one single oblate spheroid rotating at the center coupled to the shear flow. The fluid-side convective heat and mass transfer is computed by the Lattice Boltzmann Method (LBM). And the Discrete Element Method (DEM) accounts for particle dynamics, while the Finite Element Method (FEM) for heat conduction inside particles. Three codes are coupled together. As predicted by Maxwell’s model and the model of Nan et al., the effective thermal conductivity of a water suspension would saturate if the conductivity of the particle exceeds 20∼30W/(mK) at stagnant state, because the thermal resistance of the particles becomes negligible in comparison to the fluid thermal resistance. The simulation results in this study confirm such saturation also to a shear flow. Besides, another saturation mechanism exists for spherical particles if the shear rate is increased to a point where particle rotation induces convective transport that supersedes conductive transport within the particle. However, this is not the case for the oblate ellipsoidal particles due to different rotating states. And comparing different rotating states, if the particle’s thermal conductivity is higher than the base fluid, the shear-induced rotation around the oblate minor axis (log-rolling) results in a higher effective thermal conductivity than around the oblate major axis (tumbling) or the rotation of a spherical particle of equivalent volume fraction. The results provide a design guideline for particle in suspensions with better heat transfer behaviors.

Particle orientation distribution in a rotating, dilute suspension of rod-shaped particles

We consider a theoretical model for the settling of rod-shaped particles of a dilute, initially homogeneous, suspension in rapid rotation. The particle Reynolds number and the particle Taylor number of the detailed flow around the particles are assumed small, representing a relevant limit for an industrial centrifugal separation process. By applying a statistical approach using the Fokker–Planck equation, and neglecting particle–particle interactions, we obtain an explicit, analytical solution for the time dependent, spatially uniform particle orientation distribution function. Not only does the volume fraction in the bulk of the suspension decrease with time due to the divergent centrifugal field, as similarly described in the literature for suspensions of spherical particles, the orientation of the rod particles also changes with time from an initially uniform distribution to one where the particles tend to align with a plane perpendicular to the axis of rotation. The corresponding particle trajectories, as also influenced by first-order effects from the Coriolis acceleration and gyroscopic effects, are obtained numerically for different initial particle orientation angles.

Collision efficiency of non-Brownian spheres in a simple shear flow – the role of non-continuum hydrodynamic interactions

We study the collisions in a gaseous medium of a dilute bidisperse suspension of non-Brownian spherical particles sedimenting along the flow axis of a simple shear flow. Continuum lubrication forces prevent particles from coming into contact in a finite time, thus collisions can occur only due to attractive interactions such as the van der Waals force. However, in a low-pressure medium, the lubrication forces are weaker than their continuum counterparts and allow particle pairs to collide, even without any attractive forces. The Knudsen number, defined as the ratio of the mean free path of the medium to the mean radius of the interacting spheres, captures the significance of non-continuum interactions. We use uniformly valid hydrodynamic mobility functions, reflecting non-continuum lubrication at small separations and full continuum hydrodynamic interactions at moderate to larger separations. Due to the nature of the pair trajectory topology, the collision efficiency vanishes at a critical Knudsen number when simple shear flow alone drives the dynamics. Thus we perform collision calculations where particles experience the combined effects of van der Waals attraction and non-continuum hydrodynamics; van der Waals interactions enable collisions below the critical Knudsen number. Next, we calculate the collision efficiency for coupled differential sedimentation and simple shear driven motion in the presence of van der Waals interaction and non-continuum hydrodynamics. Finally, we explore the role of small particle inertia on relative trajectories and collision efficiencies in a non-continuum gas subject to a simple shear flow, ignoring the van der Waals force and gravity.

Effects of coupling of mass transport and blood viscosity models for microchannel flows

In this study, various aspects of a continuum transport model for blood, which is treated as a suspension of red blood cells, are thoroughly investigated. Earlier, treating blood as a suspension of rigid spherical particles in plasma was reasonably successful in mimicking flow dynamics in microchannels of size 40 μ m , comparable to 4 μ m , the dimension of a red blood cell (RBC). However, it is still not entirely clear whether such success owes to the coupling of mass and momentum transport, or to a sophisticated viscosity model for blood. To understand this in detail, we obtain semi-analytical solutions to the continuum model, in the presence and absence of the coupling of mass transport of RBCs, that connects the hematocrit concentration to the local viscosity, for fully developed blood flow in circular and rectangular microchannels. In this study, the transport of RBCs in flowing fluid is modeled by the diffusive flux model (DFM). Thus, overall, we have a hierarchy of continuum models using various viscosity models for blood, with progressive inclusion of physical complexities. Our results clearly highlight the importance of such a coupled approach, without which the results are quantitatively, and sometimes even qualitatively, different. In addition, our study serves as a thorough comparison of different viscosity models for blood, especially the steady state and transient models, recently proposed by Apostolidis and Beris (2014) and Apostolidis et al. (2015), respectively. We observe that, though the transient model reduces exactly to the description by the steady model over a range of low to moderate shear rates, the transient model shows some differences, especially for the RBC concentration in the low shear rate region. In the semi-analytical framework, the Darcy friction factor and cell free layer thickness have also been computed for circular cross-sections and a range of aspect ratios of a rectangular microchannel. Our results also indicate that the cell free layer attains a constant thickness when the aspect ratio of a rectangular microchannel reaches a critical maximum value for a specified average hematocrit. • Rectangular microchannel flow is investigated by coupling with the transport of RBC. • Even a finely tuned viscosity model without RBC transport is shown to be inadequate. • Frictional characteristics are derived in the form of a semi-analytical solution. • The model is found to predict the cell free layer thickness quite well. • The steady viscosity model overpredicts WSS when compared to the transient model.

Effects of interactions, structure formation, and polydispersity on the dynamic magnetic susceptibility and magnetic relaxation of ferrofluids

Linear response theory relates the decay of equilibrium magnetisation fluctuations in a ferrofluid to the frequency-dependent response of the magnetisation to a weak ac external magnetic field. The characteristic relaxation times are strongly affected by interactions between the constituent particles. Similarly, the relaxation of an initially magnetised system towards equilibrium in zero field occurs on a range of timescales depending on the structure of the initial state, and the interactions between the particles. In this work, ferrofluids are modelled as colloidal suspensions of spherical particles carrying point dipole moments, and undergoing Brownian motion. Recent theoretical and simulation work on the relaxation and linear response of these model ferrofluids is reviewed, and the effects of interactions, structure formation, and polydispersity on the characteristic time scales are outlined. It is shown that: (i) in monodisperse ferrofluids, the timescale characterising the collective response to weak fields increases with increasing interaction strength and/or concentration; (ii) in monodisperse ferrofluids, the initial, short-time decay is independent of interaction strength, but the asymptotic relaxation time is the same as that characterising the collective response to weak fields; (iii) in the strong-interaction regime, the formation of self-assembled chains and rings introduces additional timescales that vary by orders of magnitude; and (iv) in polydisperse ferrofluids, the instantaneous magnetic relaxation time of each fraction varies in a complex way due to the role of interactions.

How do the finite-size particles modify the drag in Taylor–Couette turbulent flow

We experimentally investigate the drag modification by neutrally buoyant finite-size particles with various aspect ratios in a Taylor–Couette (TC) turbulent flow. The current Reynolds number, $Re$ , ranges from $6.5\times 10^3$ to $2.6\times 10^4$ , and the particle volume fraction, $\varPhi$ , is up to $10\,\%$ . Particles with three kinds of aspect ratio, $\lambda$ , are used: $\lambda =1/3$ (oblate), $\lambda =1$ (spherical) and $\lambda =3$ (prolate). Unlike the case of bubbly TC flow (van Gils et al., J. Fluid Mech., vol. 722, 2013, pp. 317–347; Verschoof et al., Phys. Rev. Lett., vol. 117, issue 10, 2016, p. 104502), we find that the suspended finite-size particles increase the drag of the TC system regardless of their aspect ratios. The overall drag of the system increases with increasing $Re$ , which is consistent with the literature. In addition, the normalized friction coefficient, $c_{f,\varPhi }/c_{f,\varPhi =0}$ , decreases with increasing $Re$ , the reason could be that in the current low volume fractions the turbulent stress becomes dominant at higher $Re$ . The particle distributions along the radial direction of the system are obtained by performing optical measurements at $\varPhi =0.5\,\%$ and $\varPhi = 2\,\%$ . As $Re$ increases, the particles distribute more evenly in the entire system, which results from both the greater turbulence intensity and the more pronounced finite-size effects of the particles at higher $Re$ . Moreover, it is found that the variation of the particle aspect ratios leads to different particle collective effects. The suspended spherical particles, which tend to cluster near the walls and form a particle layer, significantly affect the boundary layer and result in maximum drag modification. The minimal drag modification is found in the oblate case, where the particles preferentially cluster in the bulk region, and, thus, the particle layer is absent. Based on the optical measurement results, it can be concluded that, in the low volume fraction ranges ( $\varPhi =0.5\,\%$ and $\varPhi = 2\,\%$ here), the larger drag modification is connected to the near-wall particle clustering. The present findings suggest that the particle shape plays a significant role in drag modification, and the collective behaviours of rigid particles provide clues to understand the bubbly drag reduction.

Torque scaling at primary and secondary bifurcations in a Taylor–Couette flow of suspensions

The Taylor–Couette flow of non-colloidal, neutrally buoyant spherical particle suspensions in the $\phi =0\,\%-28\,\%$ concentration range and (17–250) Reynolds number ( $\mathcal {R})$ range is studied using synchronized flow visualization and torque measurements. Both methods are applied in ramp-up/down (acceleration/deceleration of the inner cylinder) experiments to detect the various flow structure states and bifurcation natures, their critical conditions and their lifetime in $\mathcal {R}$ range. Torque measurement allows us to discuss the evolution of the (pseudo) Nusselt number, $\mathcal {N}$ , and friction coefficient with $\mathcal {R}$ or alternatively the Taylor number, Ta. Flow visualization brings additional information on the unsteady dynamics of flow states. For concentrations higher than $\phi =6\,\%$ , two unsteady (spiral vortex flow, wavy vortex flow) and one steady (Taylor vortex flow) flow state are observed in both ramp-up/down experiments. Hysteretic behaviour is reported for some primary, secondary and tertiary bifurcations, which are thus found to be subcritical. A critical concentration is observed at $\phi =15\,\%$ for the range of $\mathcal {R}$ at which spiral vortex flow is encountered. Characteristic frequencies of unsteady flow state (spiral vortex flow, wavy vortex flow) for different suspension concentrations are evaluated. Finally, three hydrodynamic concentration subregimes are identified for the first time, with their distinct sets of concentration-dependent critical conditions, torque scaling exponents and friction coefficients.

Modulation of elasto-inertial transitions in Taylor–Couette flow by small particles

Particle suspensions in non-Newtonian liquid matrices are frequently encountered in nature and industrial applications. We here study the Taylor–Couette flow (TCF) of semidilute spherical particle suspensions (volume fraction $\leq 0.1$ ) in viscoelastic, constant-viscosity liquids (Boger fluids). We describe the influence of particle load on various flow transitions encountered in TCF of such fluids, and on the nature of these transitions. Particle addition is found to delay the onset of first- and second-order transitions, thus stabilising laminar flows. It also renders them hysteretic, suggesting an effect on the nature of bifurcations. The transition to elasto-inertial turbulence (EIT) is shown to be delayed by the presence of particles, and the features of EIT altered, with preserved spatio-temporal large scales. These results imply that particle loading and viscoelasticity, which are known to destabilise the flow when considered separately, can on the other hand compete with one another and ultimately stabilise the flow when considered together.

Electrophoretic Mobility and Electric Conductivity of Salt-Free Suspensions of Charged Soft Particles

A unit cell model is employed to analyze the electrophoresis and electric conduction in a concentrated suspension of spherical charged soft particles (each is a hard core coated with a porous polyelectrolyte layer) in a salt-free medium. The linearized Poisson–Boltzmann equation applicable to a unit cell is solved for the equilibrium electrostatic potential distribution in the liquid solution containing the counterions only surrounding a soft particle. The counterionic continuity equation and modified Stokes/Brinkman equations are solved for the ionic electrochemical potential energy and fluid velocity distributions, respectively. Closed-form formulas for the electrophoretic mobility of the soft particles and effective electric conductivity of the suspension are derived, and the effect of particle interactions on these transport characteristics is interesting and significant. Same as the case in a suspension containing added electrolytes under the Debye–Hückel approximation, the scaled electrophoretic mobility in a salt-free suspension is an increasing function of the fixed charge density of the soft particles and decreases with increases in the core-to-particle radius ratio, ratio of the particle radius to the permeation length in the porous layer, and particle volume fraction, keeping the other parameters unchanged. The normalized effective electric conductivity of the salt-free suspension also increases with an increase in the fixed charge density and with a decrease in the core-to-particle radius ratio, but is not a monotonic function of the particle volume fraction.