Successive projections on hyperplanes

AbstractAny sequence of points in Rn obtained by successive projections of a point on elements of a finite set of hyperplanes is bounded

Correlated particle dynamics in concentrated quasi-two-dimensional suspensions

We investigate theoretically and experimentally how the hydrodynamically correlated lateral motion of particles in a suspension confined between two surfaces is affected by the suspension concentration. Despite the long range of the correlations (decaying as 1/r^2 with the inter-particle distance r), the concentration effect is present only at short inter-particle distances for which the static pair correlation is nonuniform. This is in sharp contrast with the effect of hydrodynamic screening present in unconfined suspensions, where increasing the concentration changes the prefactor of the large-distance correlation.Comment: 13 page

First-order virial expansion of short-time diffusion and sedimentation coefficients of permeable particles suspensions

For suspensions of permeable particles, the short-time translational and rotational self-diffusion coefficients, and collective diffusion and sedimentation coefficients are evaluated theoretically. An individual particle is modeled as a uniformly permeable sphere of a given permeability, with the internal solvent flow described by the Debye-Bueche-Brinkman equation. The particles are assumed to interact non-hydrodynamically by their excluded volumes. The virial expansion of the transport properties in powers of the volume fraction is performed up to the two-particle level. The first-order virial coefficients corresponding to two-body hydrodynamic interactions are evaluated with very high accuracy by the series expansion in inverse powers of the inter-particle distance. Results are obtained and discussed for a wide range of the ratio, x, of the particle radius to the hydrodynamic screening length inside a permeable sphere. It is shown that for x >= 10, the virial coefficients of the transport properties are well-approximated by the hydrodynamic radius (annulus) model developed by us earlier for the effective viscosity of porous-particle suspensions

The short-time self-diffusion coefficient of a sphere in a suspension of rigid rods

The short--time self diffusion coefficient of a sphere in a suspension of rigid rods is calculated in first order in the rod volume fraction. For low rod concentrations the correction to the Einstein diffusion constant of the sphere is a linear function of the rod volume fraction with the slope proportional to the equilibrium averaged mobility diminution trace of the sphere interacting with a single freely translating and rotating rod. The two--body hydrodynamic interactions are calculated using the so--called bead model in which the rod is replaced by a stiff linear chain of touching spheres. The interactions between spheres are calculated numerically using the multipole method. Also an analytical expression for the diffusion coefficient as a function of the rod aspect ratio is derived in the limit of very long rods. We show that in this limit the correction to the Einstein diffusion constant does not depend on the size of the tracer sphere. The higher order corrections depending on the applied model are computed numerically. An approximate expression is provided, valid for a wide range of aspect ratios.Comment: 11 pages, 6 figure

Hydrodynamic interactions of spherical particles in Poiseuille flow between two parallel walls

We study hydrodynamic interactions of spherical particles in incident Poiseuille flow in a channel with infinite planar walls. The particles are suspended in a Newtonian fluid, and creeping-flow conditions are assumed. Numerical results, obtained using our highly accurate Cartesian-representation algorithm [Physica A xxx, {\bf xx}, 2005], are presented for a single sphere, two spheres, and arrays of many spheres. We consider the motion of freely suspended particles as well as the forces and torques acting on particles adsorbed at a wall. We find that the pair hydrodynamic interactions in this wall-bounded system have a complex dependence on the lateral interparticle distance due to the combined effects of the dissipation in the gap between the particle surfaces and the backflow associated with the presence of the walls. For immobile particle pairs we have examined the crossover between several far-field asymptotic regimes corresponding to different relations between the particle separation and the distances of the particles from the walls. We have also shown that the cumulative effect of the far-field flow substantially influences the force distribution in arrays of immobile spheres. Therefore, the far-field contributions must be included in any reliable algorithm for evaluating many-particle hydrodynamic interactions in the parallel-wall geometry.Comment: submitted to Physics of Fluid

The intensity correlation function in evanescent wave scattering

As a first step toward the interpretation of dynamic light scattering with evanescent illumination from suspensions of interacting spheres, in order to probe their near wall dynamics, we develop a theory for the initial slope of the intensity autocorrelation function. An expression for the first cumulant is derived that is valid for arbitrary concentrations, which generalizes a well-known expression for the short-time, wave-vector dependent collective diffusion coefficient in bulk to the case where a wall is present. Explicit expressions and numerical results for the various contributions to the initial slope are obtained within a leading order virial expansion. The dependence of the initial slope on the components of the wave vector parallel and perpendicular to the wall, as well as the dependence on the evanescent-light penetration depth are discussed. For the hydrodynamic interactions between colloids and between the wall, which are essential for a correct description of the near-interface dynamics, we include both far-field and lubrication contributions. Lubrication contributions are essential to capture the dynamics as probed in experiments with small penetration depths. Simulations have been performed to verify the theory and to estimate the extent of the concentration range where the virial expansion is valid. The computer algorithm developed for this purpose will also be of future importance for the interpretation of experiments and to develop an understanding of near-interface dynamics, at high colloid concentrations

Diffusion, sedimentation, and rheology of concentrated suspensions of core-shell particles

Short-time dynamic properties of concentrated suspensions of colloidal core-shell particles are studied using a precise force multipole method which accounts for many-particle hydrodynamic interactions. A core-shell particle is composed of a rigid, spherical dry core of radius a surrounded by a uniformly permeable shell of outer radius b and hydrodynamic penetration depth κ(-1). The solvent flow inside the permeable shell is described by the Brinkman-Debye-Bueche equation, and outside the particles by the Stokes equation. The particles are assumed to interact non-hydrodynamically by a hard-sphere no-overlap potential of radius b. Numerical results are presented for the high-frequency shear viscosity, η(∞), sedimentation coefficient, K, and the short-time translational and rotational self-diffusion coefficients, D(t) and D(r). The simulation results cover the full three-parametric fluid-phase space of the composite particle model, with the volume fraction extending up to 0.45, and the whole range of values for κb, and a/b. Many-particle hydrodynamic interaction effects on the transport properties are explored, and the hydrodynamic influence of the core in concentrated systems is discussed. Our simulation results show that for thin or hardly permeable shells, the core-shell systems can be approximated neither by no-shell nor by no-core models. However, one of our findings is that for κ(b - a) ≳ 5, the core is practically not sensed any more by the weakly penetrating fluid. This result is explained using an asymptotic analysis of the scattering coefficients entering into the multipole method of solving the Stokes equations. We show that in most cases, the influence of the core grows only weakly with increasing concentration

Commutators, Lefschetz fibrations and the signatures of surface bundles

We construct examples of Lefschetz fibrations with prescribed singular fibers. By taking differences of pairs of such fibrations with the same singular fibers, we obtain new examples of surface bundles over surfaces with non-zero signature. From these we derive new upper bounds for the minimal genus of a surface representing a given element in the second homology of a mapping class group.Comment: 20 pages, 7 figures, accepted for publication in Topolog