We investigate the lateral migration of a confined polymer under pressure driven and uniform shear flows. We employ a hybrid algorithm which couples point particles to a fluctuating lattice-Boltzmann fluid. We observe migration in both uniform shear and pressure driven flows, supporting the idea that migration is driven by a combination of shear and hydrodynamic interactions with the wall, rather than by the shear gradient. Recent numerical and theoretical investigations have suggested that polymers migrate toward the centerline when hydrodynamic interactions are included, but our simulations show that in sufficiently narrow channels there is a reversal of direction and the polymers move toward the wall.

Inertial migration of neutrally buoyant particles in a square duct has been investigated by numerical simulation in the range of Reynolds numbers from 100 to 1000. Particles migrate to one of a small number of equilibrium positions in the cross-sectional plane, located near a corner or at the center of an edge. In dilute suspensions, trains of particles are formed along the axis of the flow, near the planar equilibrium positions of single particles. At high Reynolds numbers (Re⩾750), we observe particles in an inner region near the center of the duct. We present numerical evidence that closely spaced pairs of particles can migrate to the center at high Reynolds number.

A numerical method to simulate the dynamics of polymer solutions in confined geometries has been implemented and tested. The method combines a fluctuating lattice-Boltzmann model of the solvent [Ladd, Phys. Rev. Lett. 70, 1339 (1993)] with a point-particle model of the polymer chains. A friction term couples the monomers to the fluid [Ahlrichs and Dunweg, J. Chem. Phys. 111, 8225 (1999)], providing both the hydrodynamic interactions between the monomers and the correlated random forces. The coupled equations for particles and fluid are solved on an inertial time scale, which proves to be surprisingly simple and efficient, avoiding the costly linear algebra associated with Brownian dynamics. Complex confined geometries can be represented by a straightforward mapping of the boundary surfaces onto a regular three-dimensional grid. The hydrodynamic interactions between monomers are shown to compare well with solutions of the Stokes equations down to distances of the order of the grid spacing. Numerical results are presented for the radius of gyration, end-to-end distance, and diffusion coefficient of an isolated polymer chain, ranging from 16 to 1024 monomers in length. The simulations are in excellent agreement with renormalization group calculations for an excluded volume chain. We show that hydrodynamic interactions in large polymers can be systematically coarse-grained to substantially reduce the computational cost of the simulation. Finally, we examine the effects of confinement and flow on the polymer distribution and diffusion constant in a narrow channel. Our results support the qualitative conclusions of recent Brownian dynamics simulations of confined polymers [Jendrejack et al., J. Chem. Phys. 119, 1165 (2003) and Jendrejack et al., J. Chem. Phys. 120, 2513 (2004)].

A rotating suspension of nonbuoyant particles can develop striking inhomogeneities in particle concentration, with regular bands of high and low concentration along the symmetry axis. We report Stokes-flow simulations showing that the formation of axial bands is correlated with an inhomogeneous particle distribution in the radial plane. An order parameter, based on the average angular velocity of the particles, characterizes two distinct phases: a low-frequency segregated phase and a high-frequency dispersed phase. The axial band structure develops during the transition between these two phases.

Lattice-Boltzmann simulations have been used to investigate low-Reynolds-number settling of monodisperse and polydisperse suspensions. We confirm the discovery that particle velocity fluctuations are strongly suppressed by no-slip walls at the top and bottom of the system, even in regions distant from the boundaries. We also show that a monodisperse suspension develops a strongly anisotropic long-range microstructure during the settling process, with vanishing density fluctuations in the horizontal plane. We find no numerical evidence that the particle concentration in the bulk is stratified; diffusive spreading of the suspension–supernatant interface is suppressed by hindered settling, as would be expected in moderately concentrated suspensions.Long-range correlations in particle density fluctuations are destroyed by polydispersity in particle size, and in this case density fluctuations are finite at all length scales and in all directions. However, in polydisperse suspensions there is significant stratification, due to differential settling rather than interface diffusion, which provides an alternative mechanism for screening the hydrodynamic interactions. It is possible that this is the dominant mechanism for hydrodynamic screening in several laboratory experiments.

We report direct observations of the structure factor in a settling suspension, using numerical simulations based on a lattice-Boltzmann model of the fluid. We find that the horizontal density fluctuations in bounded suspensions are strongly suppressed by the settling process, vanishing as k(2) at long wavelengths. Our measurements of the structure factor confirm the qualitative predictions of one of several competing theories, although this theory does not yet explain why container walls are so important. Our results contradict the idea that a settling suspension is inevitably stratified by hydrodynamic dispersion at the suspension-supernatent interface.

Simulations of heat and mass transport may require complex nonlinear boundary conditions to describe the flow of mass and energy across an interface. Although stochastic methods do not suffer from the numerical diffusion of grid-based methods, they typically lose accuracy in the vicinity of interfacial boundaries. In this work we introduce ideas and algorithms to account for mass (or energy) transfer at reactive interfaces, with accuracies comparable to the bulk phase. We show how to introduce particles into the system with the correct distribution near the interface, as well as the correct flux through the interface. The algorithms have been tested in a channel flow, for which accurate numerical solutions can be independently calculated.

Stochastic methods offer an attractively simple solution to complex transport-controlled problems, and have a wide range of physical, chemical, and biological applications. Stochastic methods do not suffer from the numerical diffusion that plagues grid-based methods, but they typically lose accuracy in the vicinity of interfacial boundaries. In this work we introduce some ideas and algorithms that can be used to implement boundary conditions in stochastic simulations of the convection-diffusion equation with accuracies comparable to the bulk phase. The algorithms have been tested in two-dimensional channel flows over a range of Peclet numbers, and compared with independent finite-difference calculations.

Numerical simulations that take explicit account of the hydrodynamic forces between the suspended particles are becoming useful tools for studying the dynamics of particle suspensions. This paper contains a summary of current theory and practice in simulating systems with hydrodynamic interactions.

The lattice-Boltzmann method has been refined to take account of near-contact interactions between spherical particles. First, we describe a comprehensive solution to the technical problems that arise when two discretized surfaces come into contact. Second, we describe how to incorporate lubrication forces and torques into lattice-Boltzmann simulations, and test our method by calculating the forces and torques between a spherical particle and a plane wall. Third, we describe an efficient update of the particle velocities, taking into account the possibility that some of the differential equations are stiff.

Simulations describing the settling of suspensions of solid particles bounded by a rigid container are presented. Results are compared with previous simulations of homogeneous suspensions with periodic boundary conditions. Velocity fluctuations in vertically inhomogeneous suspensions are found to saturate under conditions similar to those found in laboratory experiments, while in vertically homogeneous suspensions, with or without side walls, the velocity fluctuations diverge. A mechanism for the establishment of a correlation length in sedimenting suspensions is proposed.

We propose a theory for axial segregation of suspensions of non-neutrally buoyant particles in a rotating cylinder. The cylinder is oriented in the horizontal plane, so that any axial forces must arise from interparticle interactions. We show that the hydrodynamic interaction between pairs of particles produces a relative motion in the axial direction, independent of the gravitational force. If the particles are denser than the suspending fluid, differential centrifuging between particles at different radial positions leads to an at-tractive interaction, inducing a rapid growth of axial density perturbations. We suggest that this mecha-nism can explain the origin of band formation in rotating suspensions of non-neutrally buoyant particles.

We report on numerical simulations of acid erosion in a fractured specimen of Carrara marble. The simulations combine two recent advances in lattice-Boltzmann methodology to accurately and efficiently calculate the velocity field in the pore space. A tracer diffusion algorithm was then used to calculate the distribution of reactants in the fracture, and the local erosion rate was obtained from the flux of tracer particles across the surfaces. Our results show that at large length scales, erosion leads to increased heterogeneity via channel formation, whereas at small length scales it tends to smooth out the roughness in the local aperture.

A lattice-Boltzmann method has recently been developed to incorporate solid-fluid boundary conditions on length scales less than the grid spacing. By introducing a real numbered parameter, specified at each node and representing the fluid volume associated with that node, we were able to accurately simulate arbitrary geometries without the need to specify surface normals. In this paper a detailed description of the rules is presented and the accuracy and stability of the method is discussed, based on numerical results for flow in systems with planar surfaces and for flow through periodic arrays of disks and spheres.

We present a numerical method to solve the equations for low-Reynolds-number (Stokes) flow in porous media. The method is based on the lattice-Boltzmann approach, but utilizes a direct solution of time-independent equations, rather than the usual temporal evolution to steady state. Its computational efficiency is 1-2 orders of magnitude greater than the conventional lattice-Boltzmann method. The convergence of the permeability of random arrays of spheres has been analyzed as a function of mesh resolution at several different porosities. For sufficiently large spheres, we have found that the convergence is quadratic in the mesh resolution.

The sedimentation of monodisperse suspensions of rigid spheres has been studied by dynamical simulation; computational techniques are described and numerical results are reported. It has been found that there is a slow relaxation of the suspension microstructure during sedimentation, so that compared with the initial equilibrium distribution, there is an increased number of pairs of particles near contact; this leads to a 5%–10% increase in the average sedimentation velocity. Individual particle velocities fluctuate about the mean fall speed; these fluctuations are large and persist for long times. The resulting hydrodynamically induced dispersion of the particles can be characterized by strongly anisotropic diffusion coefficients; however, the dispersion process is non-Fickian at high solids concentrations.

We have used a novel Monte Carlo method to compute the gyration radius R(G) and the hydrodynamic radius R(H) of excluded-volume polymer chains. The hydrodynamic radius scales as N(almost-equal-to)0.55 (N is the number of bonds) over at least a decade of chain lengths, whereas the gyration radius exponent is close to the theoretical value of 0.59. The anomalous behavior of R(H) is well-known experimentally; it is commonly attributed to the belief that polymers in mediocre solvents are not swollen on short length scales. However, the polymer chains in our simulations are uniformly swollen on all length scales; we suggest that the discreteness of the polymer chain is sufficient to explain the behavior of R(H).

We present an accurate new method to compute absolute free energies of molecular solids by computer simulations. As a first application, we computed the thermodynamic phase transition between the fluid phase and the orientational disordered solid β phase of nitrogen at 300 K, using a well tested pair potential. The computed coexistence pressure and the volume change coincides within the error margins with the experimental values. The coexistence volume differed by 2% from the experimental value. To our knowledge these results constitutes the first numerical calculation of the thermodynamic stability for a model of a realistic molecular solid.

A three-dimensional lattice-gas model has been used to determine the dissipative hydrodynamic interactions between spherical particles. When the particles are close together the accuracy of the lattice-gas simulations is superior to typical integral-equation solutions of the creeping-flow equations. Moreover, the computational requirements scale linearly with system size, instead of quadratically or cubically. A new set of microrules have been implemented, which simulate a constant-velocity, no-slip boundary condition at the solid–fluid surfaces. Numerical results for various drag coefficients are reported.