B. Cichocki

Friction and mobility of many spheres in Stokes flow

An efficient scheme is presented for the numerical calculation of hydrodynamic interactions of many spheres in Stokes flow. The spheres may have various sizes, and are freely moving or arranged in rigid arrays. Both the friction and mobility matrix are found from the solution of a set of coupled equations. The Stokesian dynamics of many spheres and the friction and mobility tensors of polymers and proteins may be calculated accurately at a modest expense of computer memory and time. The transport coefficients of suspensions can be evaluated by use of periodic boundary conditions.

Friction and mobility for colloidal spheres in Stokes flow near a boundary: The multipole method and applications

We obtain the many-body hydrodynamic friction and mobility matrices describing the motion in a fluid of N hard-spheres with stick boundary conditions in the presence of a planar hard wall or free surface using (1) a multipole expansion of the hydrodynamic force densities induced on the spheres and (2) an image representation to account for the fluid boundary. The coupled multipole equations may be truncated at any order to give positive definite approximations to the exact friction and mobility matrices. An extension of the Bossis–Brady lubrication correction to the friction matrix is also discussed and included. The resulting method for computing the mobility matrix may be used for the Stokesian or Brownian dynamics simulation of N spheres subject to interparticle and external forces and imposed shear flow. We illustrate the method by performing Stokesian dynamics simulation of particles near a hard wall. The simulations exhibit the rapid convergence of the multipole truncation scheme including lubricati...

Lubrication corrections for three-particle contribution to short-time self-diffusion coefficients in colloidal dispersions

It is shown that the standard treatment of lubrication effects in many-particle hydrodynamic interactions leads to divergent three-particle contributions to the short-time translational self-diffusion coefficient. To resolve the problem the improved method to account for lubrication is proposed. The translational and rotational self-diffusion coefficients of the Brownian semidilute suspension are then evaluated up to terms of the second order in volume fraction.

Image representation of a spherical particle near a hard wall

The motion of a spherical colloidal particle suspended in a moving fluid near a planar hard wall or free surface is considered. The particle types include hard spheres with mixed slip-stick boundary conditions, droplets with high surface tension and porous particles. A general expression for the flow field is obtained in terms of a set of force multipoles induced on the particle and on its mirror image in the bounding surface. An earlier calculation of the friction and mobility functions for the particle is extended to include coupling to an incident linear shear flow. A numerical representation of all these functions is given which is accurate at all separations of particle and boundary. These results are used to study the position and orientation of a hard sphere falling towards a wall under gravity in the presence of a shear flow.

Kinetic theory of nonlinear viscous flow in two and three dimensions

On the basis of a nonlinear kinetic equation for a moderately dense system of hard spheres and disks it is shown that shear and normal stresses in a steady-state, uniform shear flow contain singular contributions of the form ¦X¦3/2 for hard spheres, or ¦X¦ log ¦X¦ for hard disks. HereX is proportional to the velocity gradient in the shear flow. The origin of these terms is closely related to the hydrodynamic tails t−d/2 in the current-current correlation functions. These results also imply that a nonlinear shear viscosity exists in two-dimensional systems. An extensive discussion is given on the range ofX values where the present theory can be applied, and numerical estimates of the effects are given for typical circumstances in laboratory and computer experiments.

Diffusion coefficients and effective viscosity of suspensions of sticky hard spheres with hydrodynamic interactions

We study the short‐time and long‐time diffusion coefficients and the high‐frequency and zero‐frequency effective viscosity of a dilute suspension of sticky hard spheres with hydrodynamic interactions. Due to the singular nature of the hydrodynamic interactions near touching the transport coefficients are strongly affected by the short‐range attractive potential. This suggests that the transport coefficients may be used as a test of the interaction potential.

Drag force on a sphere moving towards a corrugated wall

From the solution of the creeping-flow equations, the drag force on a sphere becomes infinite when the gap between the sphere and a smooth wall vanishes at constant velocity, so that if the sphere is displaced towards the wall with a constant applied force, contact theoretically may not occur. Physically, the drag is finite for various reasons, one being the particle and wall roughness. Then, for vanishing gap, even though some layers of fluid molecules may be left between the particle and wall roughness peaks, conventionally it may be said that contact occurs. In this paper, we consider the example of a smooth sphere moving towards a rough wall. The roughness considered here consists of parallel periodic wedges, the wavelength of which is small compared with the sphere radius. This problem is considered both experimentally and theoretically. The motion of a millimetre size bead settling towards a corrugated horizontal wall in a viscous oil is measured with laser interferometry giving an accuracy on the displacement of 0.1

Short-time dynamics of permeable particles in concentrated suspensions

\mu

ON THE MEMORY FUNCTION FOR THE DYNAMIC STRUCTURE FACTOR OF INTERACTING BROWNIAN PARTICLES

m. Several wedge-shaped walls were used, with various wavelengths and wedge angles. From the results, it is observed that the velocity of the sphere is, except for small gaps, similar to that towards a smooth plane that is shifted down from the top of corrugations. Indeed, earlier theories for a shear flow along a corrugated wall found such an equivalent smooth plane. These theories are revisited here. The creeping flow is calculated as a series in the slope of the roughness grooves. The cases of a flow along and across the grooves are considered separately. The shift is larger in the former case. Slightly flattened tops of the wedges used in experiments are also considered in the calculations. It is then demonstrated that the effective shift for the sphere motion is the average of the shifts for shear flows in the two perpendicular directions. A good agreement is found between theory and experiment.

Dynamic computer simulation of concentrated hard sphere suspensions: II. Re-analysis of mean square displacement data

We study short-time diffusion properties of colloidal suspensions of neutral permeable particles. An individual particle is modeled as a solvent-permeable sphere of interaction radius a and uniform permeability k, with the fluid flow inside the particle described by the Debye-Bueche-Brinkman equation, and outside by the Stokes equation. Using a precise multipole method and the corresponding numerical code HYDROMULTIPOLE that account for higher-order hydrodynamic multipole moments, numerical results are presented for the hydrodynamic function, H(q), the short-time self-diffusion coefficient, D(s), the sedimentation coefficient K, the collective diffusion coefficient, D(c), and the principal peak value H(q(m)), associated with the short-time cage diffusion coefficient, as functions of porosity and volume fraction. Our results cover the full fluid phase regime. Generic features of the permeable sphere model are discussed. An approximate method by Pusey to determine D(s) is shown to agree well with our accurate results. It is found that for a given volume fraction, the wavenumber dependence of a reduced hydrodynamic function can be estimated by a single master curve, independent of the particle permeability, given by the hard-sphere model. The reduced form is obtained by an appropriate shift and rescaling of H(q), parametrized by the self-diffusion and sedimentation coefficients. To improve precision, another reduced hydrodynamic function, h(m)(q), is also constructed, now with the self-diffusion coefficient and the peak value, H(q(m)), of the hydrodynamic function as the parameters. For wavenumbers qa>2, this function is permeability independent to an excellent accuracy. The hydrodynamic function of permeable particles is thus well represented in its q-dependence by a permeability-independent master curve, and three coefficients, D(s), K, and H(q(m)), that do depend on the permeability. The master curve and its coefficients are evaluated as functions of concentration and permeability.