An analysis of the far-field response to external forcing of a suspension in the Stokes flow in a parallel-wall channel

Abstract

The leading-order far-field scattered flow produced by a particle in a parallel-wall channel under creeping-flow conditions has a form of the parabolic velocity field driven by a two-dimensional dipolar pressure distribution. We show that in a system of hydrodynamically interacting particles, the pressure dipoles contribute to the macroscopic suspension flow in a similar way as the induced electric dipoles contribute to the electrostatic displacement field. Using this result we derive macroscopic equations governing suspension transport under the action of a lateral force, a lateral torque, or a macroscopic pressure gradient in the channel. The matrix of linear transport coefficients in the constitutive relations linking the external forcing to the particle and fluid fluxes satisfies the Onsager reciprocal relation. The transport coefficients are evaluated for square and hexagonal periodic arrays of fixed and freely suspended particles, and a simple approximation in a Clausius-Mossotti form is proposed for the channel permeability coefficient. We also find explicit expressions for evaluating the periodic Green's functions for the Stokes flow between two parallel walls.

Streaming potential studies of colloid, polyelectrolyte and protein deposition

“…These results were obtained by evaluating numerically the flow in the vicinity of adsorbed particles using the multipole expansion method [100][101][102], for the coverage range up to 0.5. Moreover, in the case of lower coverage range, analytical results have been derived by applying the cluster expansion method.…”

“…[67] exact numerical calculations were performed for particle configurations at planar interfaces characterized by desired coverage (reaching 0.5 as mentioned above) with periodic boundary conditions. The multipole algorithm [100][101][102] has been used. Results averaged over 300-400 configurations enabled one to formulate the following analytical functions, which approximated the exact numerical results with precision better than 1%.…”

Structure and dynamics of a layer of sedimented particles

“…The effective self-diffusion coefficient for suspensions at higher concentrations was evaluated using a periodic version [44] of the Cartesian-representation algorithm for a system of particles in a parallel-wall channel [45,46]. In our approach, periodic boundary conditions in the lateral directions are incorporated by splitting the flow reflected by the particles into a short-range near-field contribution and a long-range asymptotic Hele-Shaw component.…”

“…In our approach, periodic boundary conditions in the lateral directions are incorporated by splitting the flow reflected by the particles into a short-range near-field contribution and a long-range asymptotic Hele-Shaw component. The near-field contribution is summed explicitly over neighboring periodic cells, and the Hele-Shaw component is evaluated using Ewald summation method for a 2D harmonic potential [44,47].…”

The intensity correlation function in evanescent wave scattering

“…Evaluation of the hydrodynamic tensor -The hydrodynamic tensor H is determined using the periodic version [28] of the Cartesian-representation algorithm [29,30,31] for evaluation of the motion of spherical particles in Stokes flow in a parallel-wall channel. Our method combines spherical and Cartesian representations of Stokes flow in the system.…”