Transient settling of a dilute spherical suspension droplet at low Reynolds number

The transient settling in a viscous incompressible fluid of a spherical dilute cloud of particles starting from rest under the influence of a small constant applied force is studied in a continuum model on the basis of the linearized Navier-Stokes equations. Explicit expressions are derived for the motion of the cloud and for the flow velocity and pressure of the fluid. Equations of transient Stokesian dynamics are formulated that allow numerical study of the motion of a dilute cloud of particles of arbitrary initial configuration.     

Exponential Convergence to the Maxwell Distribution for Some Class of Boltzmann Equations

We consider a class of nonlinear Boltzmann equations describing return to thermal equilibrium in a gas of colliding particles suspended in a thermal medium. We study solutions in the space where is the one-particle phase space and is the Liouville measure on Γ⁽¹⁾. Special solutions of these equations, called “Maxwellians,” are spatially homogenous static Maxwell velocity distributions at the temperature of the medium. We prove that, for dilute gases, the solutions corresponding to smooth initial conditions in a weighted L ¹-space converge to a Maxwellian in , exponentially fast in time.     

Layering instability in a confined suspension flow

We show [J. Fluid Mech. 592, 447 (2007)] that swapping (reversing) trajectories in confined suspension flows prevent collisions between particles approaching each other in adjacent streamlines. Here we demonstrate that by inducing layering this hydrodynamic mechanism changes the microstructure of suspensions in a confined Couette flow. Layers occur either in the near-wall regions or span the whole channel width, depending on the strength of the swapping-trajectory effect. While our theory focuses on dilute suspensions, we postulate that this new hydrodynamic mechanism controls the formation of a layered microstructure in a wide range of densities.     

Experimental signature of the pair trajectories of rough spheres in the shear-induced microstructure in noncolloidal suspensions

The shear-induced microstructure in a semidilute noncolloidal suspension is studied. A high-resolution pair distribution function in the plane of shear is experimentally determined. It is shown to be anisotropic, with a depleted direction close to the velocity axis in the recession quadrant. The influence of roughness on the interaction between particles is quantitatively evidenced. The experimental results compare well with a model from particle pair trajectories.     

Elastic waves in suspensions

A model of heterogeneous medium taking into account the friction between the particles and liquid, as well as the relaxation of the small-size particles to the equilibrium on the stress, has been proposed to describe the propagation of the elastic waves in a suspension. A system of wave equations describing the propagation of a plane longitudinal wave has been formulated for the components of the medium. Analytical expressions for the sound velocity in a suspension has been obtained in the approximation in which the particles are completely carried away by liquid in the limiting cases in which the particles are in equilibrium under stress with the liquid or equilibrium is absent. The dependence of the sound velocity in the medium on the volumetric portion and the size of the inclusions has been studied. The obtained results agree with the experimental data and obtained analytical expressions for the sound velocity. The dynamics of the components of the medium at the propagation of the plane longitudinal monochromatic wave has been studied.     

Mathematical combustion model of nanoaluminum-air suspension

This paper presents a combustion model of a nano-aluminum-air (nAl-air) suspension. The special feature of the model is performing a local mathematical model of the oxidant diffusion through an aluminum oxide layer on the particle surface taking into account the aluminum-oxidant reaction to simulate the combustion of nano-size aluminum (nAl) particles. The oxidation rate of the aluminum particles and the associated with this process the rate of heat release are determined from the solution of the local combustion problems for the entire set of nAl particles in the suspension. To obtain the suspension state parameters we solve the equation system, which includes the energy conservation equations for the gas and particles, the mass-conservation equation for the gas-dispersed mixture and the motion equations for the gas and particles controlling for the particle velocity lag. The model considers gas expansion and thus gas and particle motion. The developed model does not require setting the ignition temperature of nAl particles. The study provides the calculated propagation rate of the combustion front in the nAl-air suspension depending on the nAl mass concentration and on the initial temperature of the suspension.     

Dynamics of freely cooling granular gases

We study dynamics of freely cooling granular gases in two dimensions using large-scale molecular dynamics simulations. We find that for dilute systems the typical kinetic energy decays algebraically with time, E(t) approximately t(-1), and velocity statistics are characterized by a universal Gaussian distribution in the long time limit. We show that in the late clustering regime particles move coherently as typical local velocity fluctuations, Deltav, are small compared with the typical velocity, Deltav/v approximately t(-1/4). Furthermore, locally averaged shear modes dominate over acoustic modes. The small thermal velocity fluctuations suggest that the system can be heuristically described by Burgers-like equations.     

Fluctuating hydrodynamics for driven granular gases

We study a granular gas heated by a stochastic thermostat in the dilute limit. Starting from the kinetic equations governing the evolution of the correlation functions, a Boltzmann-Langevin equation is constructed. The spectrum of the corresponding linearized Boltzmann-Fokker-Planck operator is analyzed, and the equation for the fluctuating transverse velocity is derived in the hydrodynamic limit. The noise term (Langevin force) is thus known microscopically and contains two terms: one coming from the thermostat and the other from the fluctuating pressure tensor. At variance with the free cooling situation, the noise is found to be white and its amplitude is evaluated.     

A new technique for fluid velocimetry based on near field scattering

We show that the time evolution of near-field scattering speckles, originated by a fluid suspension of particles, provides information about the velocity field in the fluid. This information can be extracted from a statistical analysis of speckle fields taken at different times, either by measuring their cross-correlation function or by recovering the power spectrum corresponding to the difference between the two speckle fields. Experimental data are in accordance to the expected behaviors. The results are independent of the scatterer's size, allowing one to exploit the technique also with sub-wavelength tracking particles.     

Lattice Boltzmann simulation of the two-dimensional flow of a magnetic suspension between two parallel walls under a non-uniform magnetic field

We have developed a lattice Boltzmann method based on fluctuation hydrodynamics that is applicable to the flow problem of a particle suspension. In this method, we have introduced the viscosity-modifying method, rather than the velocity-scaling method, in which a modified viscosity is used for generating random forces in lattice Boltzmann simulations. The viscosity-modifying method is found to be applicable to the simulation of a magnetic particle suspension. We have applied this method to the two-dimensional Poiseuille flow of a magnetic suspension between two parallel walls in order to investigate the behavior of magnetic particles in a non-uniform applied magnetic field. From the results of the snapshots, the pair correlation function between the magnetic pole and the magnetic particles and the averaged local particle velocity and magnetization distributions, it was observed that the behavior of the magnetic particles changes significantly depending upon which factor dominates the phenomenon in the balance between the magnetic particle–particle interaction, the non-uniform applied magnetic field and the translational and rotational Brownian motion.     

Free surface instability in a confined suspension jet

A suspension confined between two close parallel plates is studied in the Stokesian regime. The use of boundary integral equations and the lubrication approximation allows to compute the hydrodynamic forces acting on the particles. The forces are long ranged and depend on the orientation of the relative position and velocity of particles. This tensorial character predicts an “antidrag” that is observed in experiments. The effect of the computed hydrodynamic forces is studied in the dynamics of a jet of particles falling by a gravitational field, which shows a surface instability similar to the Kelvin–Helmholtz one. A theoretical model, based on hydrodynamic-like equations, is able to predict the instability that is produced by the interaction of the long-range forces and the free surface.     

Hybrid MC/RISM technique for simulation of polymer solutions: Monte Carlo+ RISM integral equations

A new (hybrid) method is reported for modelling complex macromolecular systems. The approach combines the traditional atomistic Monte Carlo (MC) computer simulation of flexible polymer chains with the numerical solution of the site-site Ornstein-Zernike-like (RISM) integral equations. The method is used for calculating properties of a linear polymer in dilute solution. Since the condensed-phase environment of a flexible macromolecule affects the equilibrium configuration probability distribution of the macromolecule, the site-site intramolecular correlation function and the intramolecular potential field are treated in a self-consistent manner. Briefly, the MC method is applied to generate the configurations of a single chain molecule. Using the coordinates of chain beads, the averaged intrapolymer correlation function is obtained. Then, solving the coupled RISM equations for a given density of solvent particles, we find the polymer-solvent correlation functions. This yields the medium-induced intrapolymer potential and the corresponding effective intramolecular energies, which are used in the standard Metropolis MC procedure. The structural properties of the polymer chain are computed by averaging over the statistically representative set of configurations. As a result of many such iterations, the intramolecular structure is determined self-consistently. Using the hybrid MC/RISM method, extensive studies have been made of static properties of flexible polymer chains surrounded by LJ particles with purely repulsive interactions between the particles and chain beads. Also, direct molecular dynamics simulations have been carried out and have demonstrated that the hybrid MC/RISM approach gives a quite accurate prediction for condensed-phase effects.     

Hydrodynamic Stress Tensor in Inhomogeneous Colloidal Suspensions: an Irving-Kirkwood Extension *

Based on statistical mechanics for classical fluids, general expressions for hydrodynamic stress in inhomogeneous colloidal suspension are derived on a molecular level. The result is exactly an extension of the Iving-Kirkwood stress for atom fluids to colloidal suspensions where dynamic correlation emerges. It is found that besides the inter-particle distance, the obtained hydrodynamic stress depends closely on the velocity of the colloidal particles in the suspension, which is responsible for the appearance of the solvent-mediated hydrodynamic force. Compared to Brady's stresslets for the bulk stress, our results are applicable to inhomogeneous suspension, where the inhomogeneity and anisotropy of the dynamic correlation should be taken into account. In the near-field regime where the packing fraction of colloidal particles is high, our results can reduce to those of Brady. Therefore, our results are applicable to the suspensions with low, moderate, or even high packing fraction of colloidal particles.     

Stress tensor of a dilute suspension of spheres in a viscoelastic liquid

The stress tensor for a dilute suspension of buoyancy-free, inertialess, non-Brownian, rigid spheres immersed in a viscoelastic liquid is determined via a perturbative expansion. The perturbation parameter is the Deborah number De, giving the ratio between the characteristic time of the liquid and the characteristic time of the imposed flow. The stress is also calculated from numerical simulations of continuity and momentum equations for the single sphere problem. Excellent agreement is found between the two predictions. Good agreement is found also with respect to experimental data found in the literature.     

Rheology of nanosized hairy grain suspensions

Suspensions of nanosized hairy grains have been prepared by grafting long polydimethylsiloxane chains (molecular weight ) onto silica particles (radius ), dispersed into a good solvent of PDMS. Depending on the particle volume fraction, different rheological behaviors are observed. In the very dilute regime, the suspensions are perfectly stable and the particles behave almost as hard spheres: flow penetration inside the corona is then very weak. When the particle volume fraction goes to the close packing volume fraction, the suspension viscosity does not diverge as for hard spheres due to the increase of flow penetration inside the corona and to corona entanglements. The particles have then the same behavior as polymer stars having an intermediate number of arms (). Finally, in the concentrated regime (), the suspensions form irreversible gels. We shown that this unexpected gelation phenomenon is related to the presence of the silica cores: grafted PDMS chains can adsorb onto different particles and form irreversible bonds between the cores. The viscosity and elastic modulus evolutions during gelation are well described by the scalar percolation model of sol-gel transition. Received 23 March 1998     

A theoretical model of turbulent fiber suspension and its application to the channel flow

A theoretical model of turbulent fiber suspension is developed by deriving the equations of Reynolds averaged Navier-Stokes,turbulence kinetic energy and turbulence dissipation rate with the additional term of fibers.In order to close the above equations,the equation of probability distribution function for mean fiber orientation is also derived.The theoretical model is applied to the turbulent channel flow and the corresponding equations are solved numerically.The numerical results are verified by comparisons with the experimental ones.The effects of Reynolds number,fiber concentration and fiber aspect-ratio on the velocity profile,turbulent kinetic energy and turbulent dissipation rate are analyzed.Based on the numerical data,the expression for the velocity profile in the turbulent fiber suspension channel flow,which includes the effect of Reynolds number,fiber concentration and aspect-ratio,is proposed.     

Perturbation theory for large Stokes number particles in random velocity fields

We derive a perturbative approach to study, in the large inertia limit, the dynamics of solid particles in a smooth, incompressible and finite-time correlated random velocity field. We carry on an expansion in powers of the inverse square root of the Stokes number, defined as the ratio of the relaxation time for the particle velocities and the correlation time of the velocity field. We describe in this limit the residual concentration fluctuations of the particle suspension, and determine the contribution to the collision velocity statistics produced by clustering. For both concentration fluctuations and collision velocities, we analyze the differences with the compressible one-dimensional case.     

Ultrasonic waves in diluted and densified suspensions

A theory of propagation of stress waves in diluted and densified suspensions is developed to make the theoretical basis for analysis of ultrasonic waves through these media. The formulae for the phase velocity and the attenuation coefficient are determined as the function of wave frequency and the suspension structure parameter, which is the volume or mass fraction of the solid phase. These formulae can be use, after suitable calibration, for determination of the solid volume fraction in diluted suspensions, and the solid mass fraction or the water content in densified suspensions, that is, parameters that characterize the structure of a suspension. These structure parameters can be determined by measuring the transition time of ultrasonic wave through a given distance of suspension. The phase velocity dispersion curves and the attenuation coefficients determined theoretically and experimentally are plotted as a function of the volume fraction of the solid phase for dilute suspension, or the solid mass fraction for densified suspension.     

Ultrasonic waves in diluted and densified suspensions

A theory of propagation of stress waves in diluted and densified suspensions is developed to make the theoretical basis for analysis of ultrasonic waves through these media. The formulae for the phase velocity and the attenuation coefficient are determined as the function of wave frequency and the suspension structure parameter, which is the volume or mass fraction of the solid phase. These formulae can be use, after suitable calibration, for determination of the solid volume fraction in diluted suspensions, and the solid mass fraction or the water content in densified suspensions, that is, parameters that characterize the structure of a suspension. These structure parameters can be determined by measuring the transition time of ultrasonic wave through a given distance of suspension. The phase velocity dispersion curves and the attenuation coefficients determined theoretically and experimentally are plotted as a function of the volume fraction of the solid phase for dilute suspension, or the solid mass fraction for densified suspension.