Far-field approximation for hydrodynamic interactions in parallel-wall geometry

A complete analysis is presented for the far-field creeping flow produced by a multipolar force distribution in a fluid confined between two parallel planar walls. We show that at distances larger than several wall separations the flow field assumes the Hele-Shaw form, i.e., it is parallel to the walls and varies quadratically in the transverse direction. The associated pressure field is a two-dimensional harmonic function that is characterized by the same multipolar number m as the original force multipole. Using these results we derive asymptotic expressions for the Green's matrix that represents Stokes flow in the wall-bounded fluid in terms of a multipolar spherical basis. This Green's matrix plays a central role in our recently proposed algorithm [Physica A xx, {\bf xxx} (2005)] for evaluating many-body hydrodynamic interactions in a suspension of spherical particles in the parallel-wall geometry. Implementation of our asymptotic expressions in this algorithm increases its efficiency substantially because the numerically expensive evaluation of the exact matrix elements is needed only for the neighboring particles. Our asymptotic analysis will also be useful in developing hydrodynamic algorithms for wall-bounded periodic systems and implementing acceleration methods by using corresponding results for the two-dimensional scalar potential.Comment: 28 pages 5 figure

Particle motion between parallel walls: Hydrodynamics and simulation

The low-Reynolds-number motion of a single spherical particle between parallel walls is determined from the exact reflection of the velocity field generated by multipoles of the force density on the particle’s surface. A grand mobility tensor is constructed and couples these force multipoles to moments of the velocity field in the fluid surrounding the particle. Every element of the grand mobility tensor is a finite, ordered sum of inverse powers of the distance between the walls. These new expressions are used in a set of Stokesian dynamics simulations to calculate the translational and rotational velocities of a particle settling between parallel walls and the Brownian drift force on a particle diffusing between the walls. The Einstein correction to the Newtonian viscosity of a dilute suspension that accounts for the change in stress distribution due to the presence of the channel walls is determined. It is proposed how the method and results can be extended to computations involving many particles and periodic simulations of suspensions in confined geometries

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

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 2D 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 Stokes flow between two parallel walls.Comment: 23 pages, 12 figure

Hydrodynamic interactions of spherical particles in suspensions confined between two planar walls

Hydrodynamic interactions in a suspension of spherical particles confined between two parallel planar walls are studied under creeping-flow conditions. The many-particle friction matrix in this system is evaluated using our novel numerical algorithm based on transformations between Cartesian and spherical representations of Stokes flow. The Cartesian representation is used to describe the interaction of the fluid with the walls and the spherical representation is used to describe the interaction with the particles. The transformations between these two representations are given in a closed form, which allows us to evaluate the coefficients in linear equations for the induced-force multipoles on particle surfaces. The friction matrix is obtained from these equations, supplemented with the superposition lubrication corrections. We have used our algorithm to evaluate the friction matrix for a single sphere, a pair of spheres, and for linear chains of spheres. The friction matrix exhibits a crossover from a quasi-two-dimensional behavior (for systems with small wall separation H) to the three-dimensional behavior (when the distance H is much larger than the interparticle distance L). The crossover is especially pronounced for a long chain moving in the direction normal to its orientation and parallel to the walls. In this configuration, a large pressure buildup occurs in front of the chain for small values of the gapwidth H, which results in a large hydrodynamic friction force. A standard wall superposition approximation does not capture this behavior

Many-particle hydrodynamic interactions in parallel-wall geometry: Cartesian-representation method

This paper describes the results of our theoretical and numerical studies of hydrodynamic interactions in a suspension of spherical particles confined between two parallel planar walls, under creeping-flow conditions. We propose a novel algorithm for accurate evaluation of the many-particle friction matrix in this system--no such algorithm has been available so far. Our approach involves expanding the fluid velocity field into spherical and Cartesian fundamental sets of Stokes flows. The interaction of the fluid with the particles is described using the spherical basis fields; the flow scattered with the walls is expressed in terms of the Cartesian fundamental solutions. At the core of our method are transformation relations between the spherical and Cartesian basis sets. These transformations allow us to describe the flow field in a system that involves both the walls and particles. We used our accurate numerical results to test the single-wall superposition approximation for the hydrodynamic friction matrix. The approximation yields fair results for quantities dominated by single particle contributions, but it fails to describe collective phenomena, such as a large transverse resistance coefficient for linear arrays of spheres

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

A mathematical model for top-shelf vertigo: the role of sedimenting otoconia in BPPV

Benign Paroxysmal Positional Vertigo (BPPV) is a mechanical disorder of the vestibular system in which calcite particles called otoconia interfere with the mechanical functioning of the fluid-filled semicircular canals normally used to sense rotation. Using hydrodynamic models, we examine the two mechanisms proposed by the medical community for BPPV: cupulolithiasis, in which otoconia attach directly to the cupula (a sensory membrane), and canalithiasis, in which otoconia settle through the canals and exert a fluid pressure across the cupula. We utilize known hydrodynamic calculations and make reasonable geometric and physical approximations to derive an expression for the transcupular pressure $\Delta P_c$ exerted by a settling solid particle in canalithiasis. By tracking settling otoconia in a two-dimensional model geometry, the cupular volume displacement and associated eye response (nystagmus) can be calculated quantitatively. Several important features emerge: 1) A pressure amplification occurs as otoconia enter a narrowing duct; 2) An average-sized otoconium requires approximately five seconds to settle through the wide ampulla, where $\Delta P_c$ is not amplified, which suggests a mechanism for the observed latency of BPPV; and 3) An average-sized otoconium beginning below the center of the cupula can cause a volumetric cupular displacement on the order of 30 pL, with nystagmus of order $2^\circ$/s, which is approximately the threshold for sensation. Larger cupular volume displacement and nystagmus could result from larger and/or multiple otoconia.Comment: 15 pages, 5 Figures updated, to be published in J. Biomechanic

Model microswimmers in channels with varying cross section

We study different types of microswimmers moving in channels with varying cross section and thereby interacting hydrodynamically with the channel walls. Starting from the Smoluchowski equation for a dilute suspension, for which interactions among swimmers can be neglected, we derive analytic expressions for the lateral probability distribution between plane channel walls. For weakly corrugated channels we extend the Fick--Jacobs approach to microswimmers and thereby derive an effective equation for the probability distribution along the channel axis. Two regimes arise dominated either by entropic forces due to the geometrical confinement or by the active motion. In particular, our results show that the accumulation of microswimmers at channel walls is sensitive to both, the underlying swimming mechanism and the geometry of the channels. Finally, for asymmetric channel corrugation our model predicts a rectification of microswimmers along the channel, the strength and direction of which strongly depends on the swimmer type.Comment: Added reference #4