Accelerated Stokesian Dynamics simulations

A new implementation of the conventional Stokesian Dynamics (SD) algorithm, called accelerated Stokesian Dynamics (ASD), is presented. The equations governing the motion of N particles suspended in a viscous fluid at low particle Reynolds number are solved accurately and efficiently, including all hydrodynamic interactions, but with a significantly lower computational cost of O(N ln N). The main differences from the conventional SD method lie in the calculation of the many-body long-range interactions, where the Ewald-summed wave-space contribution is calculated as a Fourier transform sum and in the iterative inversion of the now sparse resistance matrix. The new method is applied to problems in the rheology of both structured and random suspensions, and accurate results are obtained with much larger numbers of particles. With access to larger N, the high-frequency dynamic viscosities and short-time self-diffusivities of random suspensions for volume fractions above the freezing point are now studied. The ASD method opens up an entire new class of suspension problems that can be investigated, including particles of non-spherical shape and a distribution of sizes, and the method can readily be extended to other low-Reynolds-number-flow problems

Shear-induced self-diffusion in non-colloidal suspensions

Self-diffusion in a monodisperse suspension of non-Brownian particles in simple shear flow is studied using accelerated Stokesian dynamics (ASD) simulation. The availability of a much faster computational algorithm allows the study of large systems (typically of 1000 particles) and the extraction of accurate results for the complete shear-induced self-diffusivity tensor. The finite, and often large, autocorrelation time requires the mean-square displacements to be followed for very long times, which is now possible with ASD. The self-diffusivities compare favourably with the available experimental measurements when allowance is made for the finite strains sampled in the experiments. The relationship between the mean-square displacements and the diffusivities appearing in a Fokker–Planck equation when advection couples to diffusion is discussed

A Stability Diagram for Dense Suspensions of Model Colloidal Al2O3-Particles in Shear Flow

In Al2O3 suspensions, depending on the experimental conditions very different microstructures can be found, comprising fluid like suspensions, a repulsive structure, and a clustered microstructure. For technical processing in ceramics, the knowledge of the microstructure is of importance, since it essentially determines the stability of a workpiece to be produced. To enlighten this topic, we investigate these suspensions under shear by means of simulations. We observe cluster formation on two different length scales: the distance of nearest neighbors and on the length scale of the system size. We find that the clustering behavior does not depend on the length scale of observation. If inter-particle interactions are not attractive the particles form layers in the shear flow. The results are summarized in a stability diagram.Comment: 15 pages, 10 figures, revised versio

Fully coupled simulations of non-colloidal monodisperse sheared suspensions

In this work we investigate numerically the dynamics of sheared suspensions in the limit of vanishingly small fluid and particle inertia. The numerical model we used is able to handle the multi-body hydrodynamic interactions between thousands of particles embedded in a linear shear flow. The presence of the particles is modeled by momentum source terms spread out on a spherical envelop forcing the Stokes equations of the creeping flow. Therefore all the velocity perturbations induced by the moving particles are simultaneously accounted for. The statistical properties of the sheared suspensions are related to the velocity fluctuation of the particles. We formed averages for the resulting velocity fluctuation and rotation rate tensors. We found that the latter are highly anisotropic and that all the velocity fluctuation terms grow linearly with particle volume fraction. Only one off-diagonal term is found to be non zero (clearly related to trajectory symmetry breaking induced by the non-hydrodynamic repulsion force). We also found a strong correlation of positive/negative velocities in the shear plane, on a time scale controlled by the shear rate (direct interaction of two particles). The time scale required to restore uncorrelated velocity fluctuations decreases continuously as the concentration increases. We calculated the shear induced self-diffusion coefficients using two different methods and the resulting diffusion tensor appears to be anisotropic too. The microstructure of the suspension is found to be drastically modified by particle interactions. First the probability density function of velocity fluctuations showed a transition from exponential to Gaussian behavior as particle concentration varies. Second the probability of finding close pairs while the particles move under shear flow is strongly enhanced by hydrodynamic interactions when the concentration increases

Far-field approximation for hydrodynamic interactions in parallel-wall geometry

A complete analysis is presented for the far-field creeping flow produced by a multipolar force distribution in a fluid confined between two parallel planar walls. We show that at distances larger than several wall separations the flow field assumes the Hele-Shaw form, i.e., it is parallel to the walls and varies quadratically in the transverse direction. The associated pressure field is a two-dimensional harmonic function that is characterized by the same multipolar number m as the original force multipole. Using these results we derive asymptotic expressions for the Green's matrix that represents Stokes flow in the wall-bounded fluid in terms of a multipolar spherical basis. This Green's matrix plays a central role in our recently proposed algorithm [Physica A xx, {\bf xxx} (2005)] for evaluating many-body hydrodynamic interactions in a suspension of spherical particles in the parallel-wall geometry. Implementation of our asymptotic expressions in this algorithm increases its efficiency substantially because the numerically expensive evaluation of the exact matrix elements is needed only for the neighboring particles. Our asymptotic analysis will also be useful in developing hydrodynamic algorithms for wall-bounded periodic systems and implementing acceleration methods by using corresponding results for the two-dimensional scalar potential.Comment: 28 pages 5 figure

Frictionless bead packs have macroscopic friction, but no dilatancy

The statement of the title is shown by numerical simulation of homogeneously sheared packings of frictionless, nearly rigid beads in the quasistatic limit. Results coincide for steady flows at constant shear rate γ in the limit of small γ and static approaches, in which packings are equilibrated under growing deviator stresses. The internal friction angle ϕ, equal to 5.76 $\pm$ 0.22 degrees in simple shear, is independent on the average pressure P in the rigid limit. It is shown to stem from the ability of stable frictionless contact networks to form stress-induced anisotropic fabrics. No enduring strain localization is observed. Dissipation at the macroscopic level results from repeated network rearrangements, like the effective friction of a frictionless slider on a bumpy surface. Solid fraction Φ remains equal to the random close packing value ≃ 0.64 in slowly or statically sheared systems. Fluctuations of stresses and volume are observed to regress in the large system limit, and we conclude that the same friction law for simple shear applies in the large psystem limit if normal stress or density is externally controlled. Defining the inertia number as I = γ m/(aP), with m the grain mass and a its diameter, both internal friction coefficient $\mu$∗ = tan ϕ and volume 1/Φ increase as powers of I in the quasistatic limit of vanishing I, in which all mechanical properties are determined by contact network geometry. The microstructure of the sheared material is characterized with a suitable parametrization of the fabric tensor and measurements of connectivity and coordination numbers associated with contacts and near neighbors.Comment: 19 pages. Additional technical details may be found in v

Hydrodynamic interactions and Brownian forces in colloidal suspensions: Coarse-graining over time and length-scales

We describe in detail how to implement a coarse-grained hybrid Molecular Dynamics and Stochastic Rotation Dynamics simulation technique that captures the combined effects of Brownian and hydrodynamic forces in colloidal suspensions. The importance of carefully tuning the simulation parameters to correctly resolve the multiple time and length-scales of this problem is emphasized. We systematically analyze how our coarse-graining scheme resolves dimensionless hydrodynamic numbers such as the Reynolds number, the Schmidt number, the Mach number, the Knudsen number, and the Peclet number. The many Brownian and hydrodynamic time-scales can be telescoped together to maximize computational efficiency while still correctly resolving the physically relevant physical processes. We also show how to control a number of numerical artifacts, such as finite size effects and solvent induced attractive depletion interactions. When all these considerations are properly taken into account, the measured colloidal velocity auto-correlation functions and related self diffusion and friction coefficients compare quantitatively with theoretical calculations. By contrast, these calculations demonstrate that, notwithstanding its seductive simplicity, the basic Langevin equation does a remarkably poor job of capturing the decay rate of the velocity auto-correlation function in the colloidal regime, strongly underestimating it at short times and strongly overestimating it at long times. Finally, we discuss in detail how to map the parameters of our method onto physical systems, and from this extract more general lessons that may be relevant for other coarse-graining schemes such as Lattice Boltzmann or Dissipative Particle Dynamics.Comment: 31 pages, 14 figure

Many-particle hydrodynamic interactions in parallel-wall geometry: Cartesian-representation method

This paper describes the results of our theoretical and numerical studies of hydrodynamic interactions in a suspension of spherical particles confined between two parallel planar walls, under creeping-flow conditions. We propose a novel algorithm for accurate evaluation of the many-particle friction matrix in this system--no such algorithm has been available so far. Our approach involves expanding the fluid velocity field into spherical and Cartesian fundamental sets of Stokes flows. The interaction of the fluid with the particles is described using the spherical basis fields; the flow scattered with the walls is expressed in terms of the Cartesian fundamental solutions. At the core of our method are transformation relations between the spherical and Cartesian basis sets. These transformations allow us to describe the flow field in a system that involves both the walls and particles. We used our accurate numerical results to test the single-wall superposition approximation for the hydrodynamic friction matrix. The approximation yields fair results for quantities dominated by single particle contributions, but it fails to describe collective phenomena, such as a large transverse resistance coefficient for linear arrays of spheres

On discretization in time in simulations of particulate flows

We propose a time discretization scheme for a class of ordinary differential equations arising in simulations of fluid/particle flows. The scheme is intended to work robustly in the lubrication regime when the distance between two particles immersed in the fluid or between a particle and the wall tends to zero. The idea consists in introducing a small threshold for the particle-wall distance below which the real trajectory of the particle is replaced by an approximated one where the distance is kept equal to the threshold value. The error of this approximation is estimated both theoretically and by numerical experiments. Our time marching scheme can be easily incorporated into a full simulation method where the velocity of the fluid is obtained by a numerical solution to Stokes or Navier-Stokes equations. We also provide a derivation of the asymptotic expansion for the lubrication force (used in our numerical experiments) acting on a disk immersed in a Newtonian fluid and approaching the wall. The method of this derivation is new and can be easily adapted to other cases