Rigid Particles Of Arbitrary Shape Research Articles

A discrete element implementation for concrete: particle generation, contact calculations, packing under gravity and modeling material response

A three-dimensional discrete element modelling capability for concrete based on rigid particles of arbitrary shape and size has been developed. The novel particle generation algorithm allows control of particle size, angularity and flakiness. General rigid body kinematics including finite rotations is accounted for, and an explicit time integration algorithm that conserves energy and momentum is implemented. An efficient contact algorithm with several features to increase the efficiency of the contact computations has been developed. This enables the gravity packing problem for arbitrary shaped particles to be solved in reasonable run time. The proposed procedure is used to generate assemblies of concrete specimens of various sizes that are homogeneous and isotropic in the bulk, and can capture the wall effect due to the formwork. The calibrated specimens are seen to be capable of accurately capturing experimentally observed macro stress strain response and failure patterns. The influence of aggregate shape on texture formation in the packed specimen, and on macro strength and failure patterns in hardened concrete, is demonstrated and is seen to be consistent with experimental results.

Large scale Brownian dynamics of confined suspensions of rigid particles.

We introduce methods for large-scale Brownian Dynamics (BD) simulation of many rigid particles of arbitrary shape suspended in a fluctuating fluid. Our method adds Brownian motion to the rigid multiblob method [F. Balboa Usabiaga et al., Commun. Appl. Math. Comput. Sci. 11(2), 217-296 (2016)] at a cost comparable to the cost of deterministic simulations. We demonstrate that we can efficiently generate deterministic and random displacements for many particles using preconditioned Krylov iterative methods, if kernel methods to efficiently compute the action of the Rotne-Prager-Yamakawa (RPY) mobility matrix and its "square" root are available for the given boundary conditions. These kernel operations can be computed with near linear scaling for periodic domains using the positively split Ewald method. Here we study particles partially confined by gravity above a no-slip bottom wall using a graphical processing unit implementation of the mobility matrix-vector product, combined with a preconditioned Lanczos iteration for generating Brownian displacements. We address a major challenge in large-scale BD simulations, capturing the stochastic drift term that arises because of the configuration-dependent mobility. Unlike the widely used Fixman midpoint scheme, our methods utilize random finite differences and do not require the solution of resistance problems or the computation of the action of the inverse square root of the RPY mobility matrix. We construct two temporal schemes which are viable for large-scale simulations, an Euler-Maruyama traction scheme and a trapezoidal slip scheme, which minimize the number of mobility problems to be solved per time step while capturing the required stochastic drift terms. We validate and compare these schemes numerically by modeling suspensions of boomerang-shaped particles sedimented near a bottom wall. Using the trapezoidal scheme, we investigate the steady-state active motion in dense suspensions of confined microrollers, whose height above the wall is set by a combination of thermal noise and active flows. We find the existence of two populations of active particles, slower ones closer to the bottom and faster ones above them, and demonstrate that our method provides quantitative accuracy even with relatively coarse resolutions of the particle geometry.

On the Hydrodynamic Diffusion of Rigid Particles of Arbitrary Shape with Application to DNA

A general model for the diffusive dynamics of rigid particles in a viscous solvent is studied. The model applies to particles of arbitrary shape and allows for arbitrary cross- and self-coupling between translational and rotational degrees of freedom. Scaling and perturbation techniques are used to characterize the dynamics at time scales relevant to different classic experimental methods. It is shown that translational and rotational motion can be treated as independent at these time scales and can be described by simplified diffusion models, provided that certain geometric and hydrodynamic parameters associated with a particle are small. These parameters are estimated for DNA molecules of different length using a sequence-dependent geometric model based on x-ray crystallography and a numerical boundary element technique. Our results suggest that, for short DNA fragments up to about a persistence length, translational data can be accurately analyzed using a simplified model characterized by a scalar, ori...

A general theoretical model for the magnetohydrodynamic flow of micropolar magnetic fluids. Application to Stokes flow

Many practical applications, which have an inherent interest of physical and mathematical nature, involve the hydrodynamic flow in the presence of a magnetic field. Magnetic fluids comprise a novel class of engineering materials, where the coexistence of liquid and magnetic properties provides us with the opportunity to solve problems with high mathematical and technical complexity. Here, our purpose is to examine the micropolar magnetohydrodynamic flow of magnetic fluids by considering a colloidal suspension of ferromagnetic material (usually non-conductive) in a carrier magnetic liquid, which is in general electrically conductive. In this case, the ferromagnetic particles behave as rigid magnetic dipoles. Thus, the application of an external magnetic field, apart from the creation of an induced magnetic field of minor significance, will prevent the rotation of each particle, increasing the effective viscosity of the fluid and will cause the appearance of an additional magnetic pressure. Despite the fact that the general consideration consists of rigid particles of arbitrary shape, the assumption of spherical geometry is a very good approximation as a consequence of their small size. Our goal is to develop a general three-dimensional theoretical model that conforms to physical reality and at the same time permits the analytical investigation of the partial differential equations, which govern the micropolar hydrodynamic flow in such magnetic liquids. Furthermore, in the aim of establishing the consistency of our proposed model with the principles of both ferrohydrodynamics and magnetohydrodynamics, we take into account both magnetization and electrical conductivity of the fluid, respectively. Under this consideration, we perform an analytical treatment of these equations in order to obtain the three-dimensional effective viscosity and total pressure in terms of the velocity field, the total (applied and induced) magnetic field and the hydrodynamic and magnetic properties of the fluid, independently of the geometry of the flow. Moreover, we demonstrate the usefulness of our analytical approach by assuming a degenerate case of the aforementioned method, which is based on the reduction of the partial differential equations to a simpler shape that is similar to Stokes flow for the creeping motion of magnetic fluids. In view of this aim, we use the potential representation theory to construct a new complete and unique differential representation of magnetic Stokes flow, valid for non-axisymmetric geometries, which provides the velocity and total pressure fields in terms of easy-to-find potentials, via an analytical fashion. Copyright © 2009 John Wiley & Sons, Ltd.

Two-dimensional numerical simulations for the time-averaged acoustic forces acting on a rigid particle of arbitrary shape in a standing wave.

The time-averaged acoustic force can be applied to many practical fields such as contactless particle manipulation in biomedicine. It is necessary to accurately predict the mean forces on suspended obstacles to design ultrasonic particle manipulators. Although there have been many analytical solutions on this topic, it is difficult to determine the acoustic forces on obstacles under more complex system conditions such as proximity to the chamber wall, complex viscous function, acoustic streaming, and complicated particle shapes. Therefore, the numerical modeling may become a powerful tool. In this paper, the time-averaged forces, which act on rigid 2-D particles with different shapes in ideal and viscous fluids exerted by a standing sound wave field, are computed by solving the Navier-Stokes equations directly using the finite volume method (FVM) technique. The cylinder results agree well with Haydock’s theoretical prediction and his lattice Boltzmann simulations. Then, the force and torque acting on a needle shaped particle in a standing wave are calculated and discussed in detail. Furthermore, the viscous effects of the host medium are also investigated. Our program with the FVM algorithm proves to be quite suitable for calculating the acoustic forces in standing waves.

Stokes flow in ellipsoidal geometry

Particle-in-cell models for Stokes flow through a relatively homogeneous swarm of particles are of substantial practical interest, because they provide a relatively simple platform for the analytical or semianalytical solution of heat and mass transport problems. Despite the fact that many practical applications involve relatively small particles (inorganic, organic, biological) with axisymmetric shapes, the general consideration consists of rigid particles of arbitrary shape. The present work is concerned with some interesting aspects of the theoretical analysis of creeping flow in ellipsoidal, hence nonaxisymmetric domains. More specifically, the low Reynolds number flow of a swarm of ellipsoidal particles in an otherwise quiescent Newtonian fluid, that move with constant uniform velocity in an arbitrary direction and rotate with an arbitrary constant angular velocity, is analyzed with an ellipsoid-in-cell model. The solid internal ellipsoid represents a particle of the swarm. The external ellipsoid contains the ellipsoidal particle and the amount of fluid required to match the fluid volume fraction of the swarm. The nonslip flow condition on the surface of the solid ellipsoid is supplemented by the boundary conditions on the external ellipsoidal surface which are similar to those of the sphere-in-cell model of Happel (self-sufficient in mechanical energy). This model requires zero normal velocity component and shear stress. The boundary value problem is solved with the aim of the potential representation theory. In particular, the Papkovich–Neuber complete differential representation of Stokes flow, valid for nonaxisymmetric geometries, is considered here, which provides the velocity and total pressure fields in terms of harmonic ellipsoidal eigenfunctions. The flexibility of the particular representation is demonstrated by imposing some conditions, which made the calculations possible. It turns out that the velocity of first degree, which represents the leading term of the series, is sufficient for most engineering applications, so long as the aspect ratios of the ellipsoids remains within moderate bounds. Analytical expressions for the leading terms of the velocity, the total pressure, the angular velocity, and the stress tensor fields are obtained. Corresponding results for the prolate and the oblate spheroid, the needle and the disk, as well as for the sphere are recovered as degenerate cases. Novel relations concerning the ellipsoidal harmonics are included in the Appendix.

Shape factors in equations of state : Part II. Repulsion phenomena in multicomponent chain fluids

An equation of state for the multicomponent fluid phase of nonattracting rigid particles of arbitrary shape is presented. The equation is a generalization of a previously presented equation of state for pure fluids of rigid particles; the approach describes the volumetric properties of a pure fluid in terms of a shape factor, zeta, which can be back calculated by scaling the volumetric properties of pure fluids to that of a hard sphere. The performance of the proposed equation is tested against mixtures of chain fluids immersed in a "monomeric" solvent of hard spheres of equal and different sizes. Extensive new Monte Carlo simulation data are presented for 19 binary mixtures of hard homonuclear tangent freely-jointed hard sphere chains (pearl-necklace) of various lengths (three to five segments), with spheres of several size ratios and at various compositions. The performance of the proposed equation is compared to the hard-sphere SAFT approach and found to be of comparable accuracy. The equation proposed is further tested for mixtures of spheres with spherocylinders. In all cases, the equation proved to be accurate and simple to use.

Brownian Motion of a Particle of General Shape in Newtonian Fluid

We give a general formulation of handling the Brownian motion of a rigid particle of arbitrary shape in a Newtonian fluid subject to external force, torque and velocity field. We show how to includ...

Brownian Dynamics Simulation of Rigid Particles of Arbitrary Shape in External Fields

We have developed a Brownian dynamics simulation algorithm to generate Brownian trajectories of an isolated, rigid particle of arbitrary shape in the presence of electric fields or any other external agents. Starting from the generalized diffusion tensor, which can be calculated with the existing HYDRO software, the new program BROWNRIG (including a case-specific subprogram for the external agent) carries out a simulation that is analyzed later to extract the observable dynamic properties. We provide a variety of examples of utilization of this method, which serve as tests of its performance, and also illustrate its applicability. Examples include free diffusion, transport in an electric field, and diffusion in a restricting environment.

Shape factors and interaction parameters in equations of state Part I. Repulsion phenomena in rigid particle systems

An equation of state for the fluid phase of nonattracting rigid particles of arbitrary shape is presented. A particle shape factor is defined and its dependence on the packing fraction is evaluated from Monte Carlo and molecular dynamics simulation results for a variety of nonattracting asymmetric rigid particles, e.g., rigid-sphere polymers, ellipsoids, star particles, spherocylinders.

TRACTION SINGULARITIES ON SHARP CORNERS AND EDGES IN STOKES FLOWS

An integral identity developed by Brenner (1964) is used in the “reversed” context to derive a fixed point iterative scheme for the surface tractions on a rigid particle of arbitrary shape submerged in Stokes flows. The iterative approach facilitates the solution of very large systems of equations and thus the employment of high resolution discretization schemes. The utility of the approach is demonstrated by illustrative computations of the tractions on regular polyhedra, with special emphasis on the results near the edges and corners where (integrable) singular behavior is expected to provide a stringent test for the numerical method. Excellent agreement between our numerical estimates for the exponent of these singularities with the analytical result (local 2-D analysis) were obtained. The emphasis in this work is on validation of the approach, but references to applications, such as fluidic self-assembly of semiconductor microstructures, are provided.

Parallel Computational Strategies for Hydrodynamic Interactions Between Rigid Particles of Arbitrary Shape in a Viscous Fluid

A fast iterative algorithm is presented for the numerical solution of large linear systems that are encountered in multiparticle Stokes flows. It is applicable to solid particles of arbitrary shape, and finds the translation and rotation velocities when total forces and torques acting on these particles are given (mobility problems). An exact result for the stresslet is also given. The method is based on recently developed boundary integral equations, “the canonical equations for mobility and resistance problems.” These are well‐posed Fredholm equations of the second kind, for mobility problems or problems with arbitrary velocity boundary conditions. They are modified for the direct iterative solution of mobility problems, leading to fast numerical computations. For a single sphere the iteration operator is spectrally decomposed analytically. The convergence rate of the iterations is deduced, and supporting numerical observations are presented. Fast rate of convergence is numerically observed for multisphere problems even at small separations. Simulations in polymeric and suspension rheology and protein chemistry are envisioned as potential applications. Numerous solutions are needed when simulating such systems over some period of time. With nearly equal consecutive boundary configurations, each previous solution provides a good initial guess for the next iterative solution.

The effects of Brownian rotations in a dilute suspension of rigid particles of arbitrary shape

A set of constitutive equations is derived to describe the time-dependent flow of a dilute suspension of identical rigid particles of arbitrary shape which are influenced by Brownian couples. The form of the orientation probability distribution for small departures from isotropy is found. The case of weak flows with strong Brownian effects is studied in detail, and the viscoelastic approximation and second-order-fluid limit of the constitutive equation are derived for a general particle shape. A full numerical solution is given for ellipsoids. The general nearly spherical particle is also considered and constitutive equations for general flow strengths are obtained for this case.

Suspension rheology in the presence of rotary Brownian motion and external couples: elongational flow of dilute suspensions

A general dynamical theory is presented of the rheological properties of dilute suspensions of rigid particles of arbitrary shape dispersed in a Newtonian fluid, including rotary Brownian motion and external couples acting upon the suspended particles. Utilization of the theory for particles of specified shape requires knowledge of eight fundamental tensors that are intrinsic geometric properties of the external (i.e. wetted) surface of the particles. It is pointed out that the forms of these basic tensors can be deduced via group-theoretical arguments for particles of specified geometrical shape. These eight tensors are given explicitly for general triaxial ellipsoidal particles and for spherical dumbbells. The rheological behavior of suspensions of spheroidal and dumbbell particles in uniaxial elongational and compressive flows is worked out. Exact solutions of these problems are presented, and fundamental errors in the earlier work of a number of prior investigators are pointed out.

Coupling between the translational and rotational brownian motions of rigid particles of arbitrary shape

General phenomenological relations describing the diffusive and convective transport of small, rigid, anisotropic particles of any shape in physical-orientation space are developed. By employing a macroscopic hydrodynamic model, the 6 × 6 diffusion matrix arising in the theory is related to the quasi-static hydrodynamic resistance matrix of the particle, thereby furnishing the ultimate generalization of the Stokes-Einstein equations. The general properties of the diffusion matrix are systematically studied, especially as regards its positivity, symmetry, and dependence upon the choice of origin. The conceptual and computational advantages of partitioning this matrix are pointed out. The existence of coupling between the translational and rotational Brownian motions is shown to devolve upon the value of the “coupling diffusivity” submatrix at the “center of diffusion” of the particle. For particles devoid of screwlike geometric asymmetry, no coupling occurs.

The Stokes resistance of an arbitrary particle—Part V.: Symbolic operator representation of intrinsic resistance

It is demonstrated that the intrinsic hydrodynamic resistance of a rigid particle of arbitrary shape, immersed in an arbitrary quasistatic Stokes flow extending to infinity, may be represented by a pair of symbolic, dyadic operators. When multiplied by the viscosity coefficient, these symbolic entities “operate” on the algebraic difference between the particle velocity and the fluid velocity at infinity to yield the hydrodynamic force and torque, respectively, on the particle. These force and torque operators are intrinsic geometric properties of the particle and serve to uniquely characterize its viscous resistance, independently of the kinematical and dynamical properties and state of motion of the fluid. Their use provides an alternative and simpler method of describing hydrodynamic resistance than the polyadic scheme proposed in Part IV. Remarkably simple, closed-form expressions for these operators are obtained for spherical and ellipsoidal particles. The operator technique is extended to multiparticle systems and to systems which are partially bounded by container walls.

The Influence of Additional Orientation Mechanisms on the Theory of Streaming Birefringence

The problem of determining the orientation distribution function of rigid particles of arbitrary shape is formulated in a general stochastic approach to consider any acting orientation mechanism, stochastic or deterministic. The effect of the various orientation mechanisms on the partial differential equation of the problem is analysed. A particular orientation mechanism acting on rigid ellipsoidal macromolecules in addition to the hydrodynamic and the Brownian effects is considered in Couette flow: a force field in the radial direction x, varying linearly with x. It is shown that the effect of a uniform electric field in the radial direction on polarizable ellipsoidal particles is of this nature. The corresponding steady-state orientation distribution function is determined to the third order for the case of predominant Brownian influence, and the theory of streaming birefringence of a dilute suspension of rigid ellipsoidal macromolecules in Couette flow is generalised to include the influence of the additional orientation mechanism: the direction of the isocline and the amount of birefringence are calculated to the second order.

Coupling between the translational and rotational brownian motions of rigid particles of arbitrary shape I. Helicoidally isotropic particles

It is demonstrated that coupling exists between the translational and rotational Brownian movements of rigid macroscopic particles possessing screwlike geometric properties. The theory is developed for the case of helicoidally isotropic particles, these being the class of bodies for which the theory adopts its most elementary form. In addition to the classical translational and rotational diffusion coefficients characterizing the intensity of the Brownian motion, it is shown that there exists yet a third independent pseudoscalar diffusivity, which quantitatively describes the degree of coupling. The cross-effect associated with this coupling coefficient manifests itself by the appearance of cross-terms in the expressions for the components of the diffusion flux vector in a six-dimensional “configuration” space when projected onto its respective three-dimensional physical and orientation subspaces.