non-Brownian Suspensions Research Articles

Migration of rigid particles in non-Brownian viscous suspensions

There is an obvious need for obtaining a closed set of equations describing multiphase flows and the complexity of their configurations. Despite their “frozen” interfaces, suspensions of rigid particles are also concerned by this issue. The description of the relative motion between the two phases is still controversial, in particular, concerning the forces acting on the particles and involving their concentration gradient or a shear rate gradient. It is our purpose here (a) to develop a two-fluid model especially designed for particles dispersed in a viscous liquid and (b) to close the model for rigid particles with the help of low-Reynolds-number hydrodynamics. Besides the obvious role of gravity forces, the migration of particles relative to the fluid is shown to result from two different physical phenomena: (a) the inhomogeneity of the stress resulting from direct interparticle forces and (b) Fick-like terms in the hydrodynamic force acting on the particles.

Suspension rheology under oscillatory shear and its geophysical implications

Abstract Dynamic strain generated by seismic waves is often invoked to explain coseismic responses to earthquakes, such as liquefaction, dilatancy and the remote triggering of earthquakes and eruptions. In order to expand our understanding of the conditions under which such responses occur, we measure the rheology of non-Brownian particle suspensions under oscillatory shear at frequencies corresponding to the seismic band (0.1–10 Hz). We characterize the changes in rheology as a function of volumetric particle packing fraction (0.2–0.6), particle diameter (9, 39 μm), and the viscosity of the suspending fluid (0.97, 12.14 Pas). By varying the stress (corresponding to a strain range 10− 5

Collective diffusion in sheared colloidal suspensions

Collective diffusivity in a suspension of rigid particles in steady linear viscous flows is evaluated by investigating the dynamics of the time correlation of long-wavelength density fluctuations. In the absence of hydrodynamic interactions between suspended particles in a dilute suspension of identical hard spheres, closed-form asymptotic expressions for the collective diffusivity are derived in the limits of low and high Péclet numbers, where the Péclet number ${\it Pe}\,{=}\,\gamdot a^2/D_0$ with $\gamdot$ being the shear rate and D0 = kBT/6πη a is the Stokes–Einstein diffusion coefficient of an isolated sphere of radius a in a fluid of viscosity η. The effect of hydrodynamic interactions is studied in the analytically tractable case of weakly sheared (Pe ≪ 1) suspensions.For strongly sheared suspensions, i.e. at high Pe, in the absence of hydrodynamics the collective diffusivity Dc = 6 Ds∞, where Ds∞ is the long-time self-diffusivity and both scale as $\phi \gamdot a^2$, where φ is the particle volume fraction. For weakly sheared suspensions it is shown that the leading dependence of collective diffusivity on the imposed flow is proportional to D0 φPeÊ, where Ê is the rate-of-strain tensor scaled by $\gamdot$, regardless of whether particles interact hydrodynamically. When hydrodynamic interactions are considered, however, correlations of hydrodynamic velocity fluctuations yield a weakly singular logarithmic dependence of the cross-gradient-diffusivity on k at leading order as ak → 0 with k being the wavenumber of the density fluctuation. The diagonal components of the collective diffusivity tensor, both with and without hydrodynamic interactions, are of O(φPe2), quadratic in the imposed flow, and finite at k = 0.At moderate particle volume fractions, 0.10 ≤ φ ≤ 0.35, Brownian Dynamics (BD) numerical simulations in which there are no hydrodynamic interactions are performed and the transverse collective diffusivity in simple shear flow is determined via time evolution of the dynamic structure factor. The BD simulation results compare well with the derived asymptotic estimates. A comparison of the high-Pe BD simulation results with available experimental data on collective diffusivity in non-Brownian sheared suspensions shows a good qualitative agreement, though hydrodynamic interactions prove to be important at moderate concentrations.

Numerical simulation of planar elongational flow of concentrated rigid particle suspensions in a viscoelastic fluid

In this paper we study, by means of numerical simulation, planar elongational flow of concentrated, inertialess, non-Brownian rigid particle suspensions in a viscoelastic fluid. The simulation is 2D and follows the approach presented in our previous work [G. D’Avino, P.L. Maffettone, M.A. Hulsen, G.W.M. Peters, A numerical method for simulating concentrated rigid particle suspensions in an elongational flow using a fixed grid, J. Comput. Phys. 226 (2007) 688–711]. A small computational domain is obtained by considering a three-layer domain. The innermost part is used to calculate the suspension response, the outermost is used to set the elongational flow boundary conditions far from the particles, and an intermediate region is mandatory to allow the developing of viscoelastic stress fields around the particle before entering the innermost layer. This scheme allows to achieve steady state (in a statistical sense). A time-independent fixed grid is used to avoid difficulties due to deforming meshes and remeshing of the domain. A fictitious domain has been implemented in order to easily manage the rigid-body motion. The particles are described by their boundaries only (rigid-ring description) and the rigid-body motion is imposed through Lagrange multipliers. A DEVSS-G/SUPG finite element formulation is implemented, and the log-conformation representation of the constitutive equation is used in order to improve the numerical stability of the method. The suspending liquid is treated as a Giesekus fluid. The bulk properties are determined by using an averaging procedure. Bulk calculations show that the extensional viscosity increases, with respect to the unfilled fluid, with the particle area fraction as well as with the Weissenberg number. The strain hardening parameter is predicted to decrease with increasing particle area fraction. The relative increase of the strain hardening parameter with increasing Weissenberg numbers is greatly reduced for larger particle area fraction. The predicted reduction of strain hardening agrees with experimental evidence.

Rheology of concentrated carbon nanotube suspensions

The rheological properties of non-Brownian carbon nanotube suspensions are measured over a range of nanotube volume fractions spanning the transition from semidilute to concentrated. The polymer-stabilized nanotubes are "sticky" and form a quiescent elastic network with a well-defined shear modulus and yield stress that both depend strongly on nanotube volume fraction with different but related critical exponents. We compare controlled-strain-rate and controlled-stress measurements of yielding in shear flow, and we study the effect of slow periodic stress reversal on yielding and the arrest of flow. Our measurements support a universal scaling of both the linear viscoelastic and steady-shear viscometric response. The former allows us to extract the elastic shear modulus of semidilute nanotube networks for values that are near or below the resolution limit of the rheometers used, while the latter provides a similar extrapolation of the yield stress. A simple scaling argument is used to model the dependence of yield stress and elastic modulus on concentration.

Shape of the isotropic-(para)nematic coexistence curve in sheared nanotube suspensions

A geometrical argument is used to account for the exponent of 1/3 describing the isotropic-(para)nematic phase boundary of sheared carbon nanotube dispersions [E. K. Hobbie and D. J. Fry, Phys. Rev. Lett. 97, 036101 (2006)]. Through simplifying assumptions motivated by the effects of concentration and attractive interactions, the behavior is explained by relating the mean strain-induced alignment to the average distance between nanotubes for the scenario of limited anisotropy in non-Brownian suspensions.

Trajectory analysis for non-Brownian inertial suspensions in simple shear flow

We analyse pair trajectories of equal-sized spherical particles in simple shear flow for small but finite Stokes numbers. The Stokes number, $\mbox{\textit{St}} \,{=}\, \dot{\gamma} \tau_p$, is a dimensionless measure of particle inertia; here, $\tau_p$ is the inertial relaxation time of an individual particle and $\dot{\gamma}$ is the shear rate. In the limit of weak particle inertia, a regular small-$\mbox{\textit{St}}$ expansion of the particle velocity is used in the equations of motion to obtain trajectory equations to the desired order in $\mbox{\textit{St}}$. The equations for relative trajectories are then solved, to $O(\mbox{\textit{St}})$, in the dilute limit, including only pairwise interactions. Particle inertia is found to destroy the fore–aft symmetry of the zero-Stokes trajectories, and finite-$\mbox{\textit{St}}$ open trajectories suffer net transverse displacements in the velocity gradient and vorticity directions. The vorticity displacement remains $O(\mbox{\textit{St}})$, while the scaling of the gradient displacement increases from $O(\mbox{\textit{St}})$ for far-field open trajectories, to $O(\mbox{\textit{St}}^{{1}/{2}})$ for open trajectories with $O(\mbox{\textit{St}}^{{1}/{2}})$ upstream gradient offsets. The gradient displacement also changes sign, being negative close to the plane of the reference sphere (the shearing plane) on account of dominant lubrication interactions, and then becoming positive at larger off-plane separations. The transverse displacements accompanying successive pair interactions lead to a diffusive behaviour for long times. The shear-induced diffusivity in the vorticity direction is $O(\mbox{\textit{St}}^2\phi \dot{\gamma} a^2)$, while that in the gradient direction scales as $O(\mbox{\textit{St}}^2 \ln \mbox{\textit{St}}\,\phi \dot{\gamma} a^2)$ and $O(\mbox{\textit{St}}^2 \phi \ln (1/\phi) \dot{\gamma} a^2)$ in the limits $\phi \,{\ll}\, \mbox{\textit{St}}^{{1}/{3}}$ and $\mbox{\textit{St}}^{{1}/{3}} \,{\ll}\, \phi \,{\ll}\, 1$, respectively. Further, the region of zero-Stokes closed trajectories is destroyed, and there exists a new attracting limit cycle whose location in the shearing plane is, at leading order, independent of $\mbox{\textit{St}}$. The extension of the present analysis to include a generic linear flow, and the implications of the finite-$\mbox{\textit{St}}$ trajectory modifications for coagulating systems are discussed.

Irreversible Shear-Activated Aggregation in Non-Brownian Suspensions

We have studied the effect of shear on the stability of suspensions made of non-Brownian solid particles. We demonstrate the existence of an irreversible transition where the solid particles aggregate at remarkably low volume fractions (phi approximately 0.1). This shear-induced aggregation is dramatic and exhibits a very sudden change in the viscosity, which increases sharply after a shear-dependent induction time. We show that this induction time is related exponentially to the shear rate, reflecting the importance of the hydrodynamic forces in reducing the repulsive energy barrier that prevents the particles from aggregating.

Lattice Boltzmann simulation of 2D and 3D non-Brownian suspensions in Couette flow

In this study, the Lattice Boltzmann (LB) method is applied for computer simulation of suspension flow in Couette systems. Typical aspects of Couette flow such as wall effects and non-zero Reynolds numbers can be studied well with the LB method because of its time-dependent character. Couette flow of single, two and multi-particle systems was studied, where two-dimensional (2D) systems were compared with three-dimensional (3D) systems. Computations on multi-particle 3D suspensions, for instance to assess the viscosity or shear-induced diffusivity, were found to be very intensive. This was only partly a consequence of the 3D system size. The critical particle grid size, necessary for accurate results, was found to be relatively large, increasing the system to impractical sizes. It is however demonstrated that it is possible to carry out computer simulations on 2D suspensions and use relatively simple, linear scaling relations to translate these results to 3D suspensions, in this way avoiding intensive computations. By doing so, the LB method is shown to be well-suited for study of suspension flow in Couette systems, particularly for aspects as particle layering near solid walls, hydrodynamic particle interactions and viscous stresses at non-zero Reynolds numbers, which cannot be easily solved with alternative methods. It also opens the way to employ the LB method for other unexplored aspects, such as particle polydispersity and high Reynolds number flow, with large relevance to practical processing of suspensions.

Shear-induced self-diffusion and microstructure in non-Brownian suspensions at non-zero Reynolds numbers

This paper addresses shear-induced self-diffusion in a monodisperse suspension of non-Brownian particles in Couette flow by two-dimensional computer simulations following the lattice-Boltzmann method. This method is suited for the study of (many-particle) particulate suspensions and can not only be applied for Stokes flow, but also for flow with finite Reynolds number. At relatively low shear particle Reynolds numbers (up to 0.023), shear-induced diffusivity exhibited a linear dependence on the shear rate, as expected from theoretical considerations. Simulations at shear particle Reynolds numbers between 0.023 and 0.35, however, revealed that in this regime, shear-induced diffusivity did not show this linear dependence anymore. Instead, the diffusivity was found to increase more than linearly with the shear rate, an effect that was most pronounced at lower area fractions of 0.10 and 0.25. In the same shear regime, major changes were found in the flow trajectories of two interacting particles in shear flow (longer and closer approach) and in the viscosity of the suspension (shear thickening). Moreover, the suspended particles exhibited particle clustering. The increase of shear-induced diffusivity is shown to be directly correlated with this particle clustering. As for shear-induced diffusivity, the effect of increasing shear rates on particle clustering was the most intensive at low area fractions of 0.10 and 0.25, where the radius of the clusters increased from about 4 to about 7 particle radii with an increase of the shear Reynolds number from 0.023 to 0.35. The importance of particle clustering to shear-induced diffusion might also indicate the importance of other factors that can induce particle clustering, such as, for example, colloidal instability.

Shear-induced structure of concentrated suspensions of non-Brownian particles in a microgravity environment

We report measurements of the steady-state shear viscosity of non-Brownian particulate suspensions in which the particles were meticulously density matched to the surrounding liquid to simulate a microgravity environment. Four distinct types of behavior were found for the viscosity of the suspensions as the shear rate and concentration were varied. Possible interpretations of the data in term of the shear-induced structure and dynamics of the suspensions are discussed. In addition, the shear rate dependence of the average particle velocity and its variance has been measured using a new technique in ultrasonic correlation spectroscopy called Dynamic Sound Scattering, demonstrating the potential of this technique for investigating the dynamics of sheared particulate suspensions.

Sedimentation of Suspensions in Shear-Thinning Fluids

We present an experimental study of the sedimentation of non-Brownian monodisperse suspensions in polymeric shear-thinning fluids. In such media and contrary to what is observed in Newtonian fluids, no sedimentation front appears; particle aggregation moreover leads to the formation of vertical particle-rich structures and of an inhomogeneous particle volume fraction along the cell height. Once these structure formed, a very efficient settling occurs whereas the process further slows down at long times. We show that the temporal variations of the mean volume fraction at a given height can be described using two characteristic times. We have systematically studied the evolution of these times with the polymer concentration and the particle volume fraction of the suspension. Contrary to the well known results in Newtonian fluids, the more concentrated in particles is the suspension, the more efficient is the settling within intermediate times. The time needed for a complete separation of fluids and particles can however be longer than in a Newtonian fluid owing to the slow variations of the volume fraction at long times. Finally, the formation of the vertical structures is also studied according to the different parameters.

Flocculation in simulations of sheared fiber suspensions

We have developed a particle-level, dynamic simulation technique to probe the structural behavior of non-Brownian fiber suspensions in simple shear flow. The model incorporates a variety of realistic features including fiber flexibility, irregular equilibrium fiber shapes, and fiber interactions. Simulated suspensions exhibit heterogeneous structures, or flocculation, when the model fibers are flexible, have deformed equilibrium shapes, and interact through static friction forces, even in the absence of attractive forces between fibers. The addition of kinetic friction and weak attractive forces has little effect on flocculation behavior, while anisotropic fiber bending tends to shift the onset of flocculation to larger coefficients of friction.

Stripes in sheared non-Brownian suspensions with a free surface

The experiments of Tirumkudulu et al. (Phys. Fluids 11 (1999) 507; Phys. Fluids 12 (2000) 1615) showed spontaneous segregation and stripe formation in neutrally buoyant suspensions of particles in a partially filled, horizontal, rotating cylinder. Here, we propose a mechanism for this phenomenon, using two well-known properties of suspensions—shear-induced diffusion of particles and the concentration dependence of the viscosity.

Centrifugal instability of semidilute non-Brownian fiber suspensions

Linear stability of the Taylor–Couette (TC) flow of semidilute non-Brownian suspension is investigated by utilizing the fiber orientation model developed by Hinch and Leal [J. Fluid Mech. 76, 187 (1976)] in conjunction with a quadratic and hybrid closure proposed by Advani and Tucker [J. Rheol. 34, 367 (1990)]. It is found that irrespective of the closure approximation used the fiber additives suppress the centrifugal TC instability, i.e., the critical Reynolds number (Re) increases with the fiber volume fraction and aspect ratio as well as the interfiber interaction coefficient. This increase in the critical Re is significantly larger than that in the total viscosity, except for very small values of the volume fraction and the interaction coefficient. The enhanced stabilization can be attributed to the fact that the suspension develops negative first and second normal stresses in the TC flow when the inner cylinder rotates and the outer one is stationary, i.e., the fluid is in a state of compression. Moreover, the interfiber interactions result in alignment of the fiber orientation tensor with respect to the rate of deformation tensor. This coupling enhances the ability of the fluid elements to resist the amplification of radial velocity disturbances that give rise to the centrifugal instability. This mechanism is substantiated based on a rigorous energy analysis, demonstrating that the coupling between the fiber orientation and perturbation radial velocities gives rise to fiber-induced perturbation shear stresses that are dissipative. Specifically, the coupling of fiber-induced perturbation shear stresses with the base flow velocity leads to a compressive force that dissipates energy leading to the suppression of the centrifugal instability.

Rheology of non-Brownian rigid fiber suspensions with adhesive contacts

An experimental investigation is undertaken into the shear-thinning behavior of suspensions of non-Brownian rigid fibers in Newtonian fluids. In particular, the influence of the shear stress and the fiber concentration is investigated. The shear stress is adjusted by varying both the shear rate and the solvent viscosity. In the semidilute concentration regime, where direct mechanical contacts between fibers are rare, the suspension is found to be nearly Newtonian over the stress range investigated. In the concentrated regime, the suspension becomes shear thinning below a certain shear rate. The shear thinning increases with concentration and decreases with solvent viscosity. Although shear-thinning behavior of fiber suspensions has often been reported in the literature, its physical origins are not well understood. Our experiments are interpreted in terms of the formation and breakage of fiber flocs due to the competition between hydrodynamic and colloidal forces. Our interpretation is confirmed by measurements of the adhesive forces between two individual fibers.

Simulations of non-spherical particles suspended in a shear flow

The lattice-Boltzmann method was used to investigate the effects of the shape and concentration of the particles on the rheological properties of non-Brownian suspensions for non-zero Reynolds numbers. Several case studies were analyzed and the methods used were found to give accurate predictions for these systems. The viscosity of suspensions of both spherical and non-spherical particles was determined as functions of shear rate and concentration of particles. It was shown that, for high shear rates, shear thickening appears. This phenomenon is particularly pronounced for particles of irregular shape.

Normal stresses and microstructure in bounded sheared suspensions via Stokesian Dynamics simulations

We report the normal stresses in a non-Brownian suspension in plane Couette flow determined from Stokesian Dynamics simulations. The presence of normal stresses that are linear in the shear rate in a viscometric flow indicates a non-Newtonian character of the suspension, which is otherwise Newtonian. While in itself of interest, this phenomenon is also important because it is believed that normal stresses determine the migration of particles in flows with inhomogeneous shear fields. We simulate plane Couette flow by placing a layer of clear fluid adjacent to one wall in the master cell, which is then replicated periodically. From a combination of the traceless hydrodynamic stresslet on the suspended particles, the stresslet due to (non-hydrodynamic) inter-particle forces, and the total normal force on the walls, we determine the hydrodynamic and inter-particle force contributions to the isotropic ‘particle pressure’ and the first normal stress difference. We determine the stresses for a range of the particle concentration and the Couette gap. The particle pressure and the first normal stress difference exhibit a monotonic increase with the mean particle volume fraction ϕ. The ratio of normal to shear stresses on the walls also increases with ϕ, substantiating the result of Nott & Brady (1994) that this condition is required for stability to concentration fluctuations. We also study the microstructure by extracting the pair distribution function from our simulations; our results are in agreement with previous studies showing anisotropy in the pair distribution, which is the cause of normal stresses.

Structure, density, and velocity fluctuations in quasi-two-dimensional non-Brownian suspensions of spheres

Non-Brownian sedimenting suspensions exhibit density and velocity fluctuations. We have performed experiments on a quasi-two-dimensional counter-flow stabilized suspension of 2000 spherical particles, namely a liquid–solid fluidized bed in a Hele–Shaw cell. This two-dimensional suspension displays a uniform concentration but the particle radial distribution function and the fluctuations of the particle number in a subvolume of the suspension suggest that the microstructure is far from being random. We have also measured the velocity fluctuations of a test particle and the fluctuations of the mean particle velocity in a subvolume. It happens that the relation between velocity and concentration fluctuations in a subvolume can be deduced from a balance between buoyancy and parietal friction forces.

Segregation of a fine suspension in channel flow

This paper deals with three related problems. First, we consider the effective viscosity of a neutrally-buoyant concentrated Brownian suspension of identical hard spheres with allowance for shear-thinning of the suspension, and we also formulate a set of mean normal stresses due to particle thermal and shear-induced fluctuations. Secondly, the effect of these stresses on particle distribution in suspension flow is modeled, the resultant model being verified by comparing its predictions with particle concentration profiles experimentally found for rotational Couette flow. Thirdly, the model is applied to suspension flow in plane and cylindrical where the lateral segregation of particles affects the hydraulic flow characteristics to quite a considerable extent by making them essentially non-Newtonian. The theoretical results are compared with experimental evidence available for steady flow of non-Brownian suspensions.