68 research outputs found

### Phase equilibria in stratified thin liquid films stabilized by colloidal particles

Phase equilibria between regions of different thickness in thin liquid films
stabilized by colloidal particles are investigated using a
quasi-two-dimensional thermodynamic formalism. Appropriate equilibrium
conditions for the film tension, normal pressure, and chemical potential of the
particles in the film are formulated, and it is shown that the relaxation of
these parameters occurs consecutively on three distinct time scales. Film
stratification is described quantitatively for a hard-sphere suspension using a
Monte-Carlo method to evaluate thermodynamic equations of state. Coexisting
phases are determined for systems in constrained- and full-equilibrium states
that correspond to different stages of film relaxation.Comment: 7 page

### Hydrodynamic crystals: collective dynamics of regular arrays of spherical particles in a parallel-wall channel

Simulations of over $10^3$ hydrodynamically coupled solid spheres are
performed to investigate collective motion of linear trains and regular square
arrays of particles suspended in a fluid bounded by two parallel walls. Our
novel accelerated Stokesian-dynamics algorithm relies on simplifications
associated with the Hele--Shaw asymptotic far-field form of the flow scattered
by the particles. The simulations reveal propagation of particle-displacement
waves, deformation and rearrangements of a particle lattice, propagation of
dislocation defects in ordered arrays, and long-lasting coexistence of ordered
and disordered regions.Comment: 4 pages 6 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

### Swapping trajectories: a new wall-induced cross-streamline particle migration mechanism in a dilute suspension of spheres

Binary encounters between spherical particles in shear flow are studied for a
system bounded by a single planar wall or two parallel planar walls under
creeping flow conditions. We show that wall proximity gives rise to a new class
of binary trajectories resulting in cross-streamline migration of the
particles. The spheres on these new trajectories do not pass each other (as
they would in free space) but instead they swap their cross-streamline
positions. To determine the significance of the wall-induced particle
migration, we have evaluated the hydrodynamic self-diffusion coefficient
associated with a sequence of uncorrelated particle displacements due to binary
particle encounters. The results of our calculations quantitatively agree with
the experimental value obtained by \cite{Zarraga-Leighton:2002} for the
self-diffusivity in a dilute suspension of spheres undergoing shear flow in a
Couette device. We thus show that the wall-induced cross-streamline particle
migration is the source of the anomalously large self-diffusivity revealed by
their experiments.Comment: submited to JF

### Every mapping class group is generated by 6 involutions

Let Mod_{g,b} denote the mapping class group of a surface of genus g with b
punctures. Feng Luo asked in a recent preprint if there is a universal upper
bound, independent of genus, for the number of torsion elements needed to
generate Mod_{g,b}. We answer Luo's question by proving that 3 torsion elements
suffice to generate Mod_{g,0}. We also prove the more delicate result that
there is an upper bound, independent of genus, not only for the number of
torsion elements needed to generate Mod_{g,b} but also for the order of those
elements. In particular, our main result is that 6 involutions (i.e.
orientation-preserving diffeomorphisms of order two) suffice to generate
Mod_{g,b} for every genus g >= 3, b = 0, and g >= 4, b = 1.Comment: 15 pages, 7 figures; slightly improved main result; minor revisions.
to appear in J. Al

### Equilibrium and nonequilibrium thermodynamics of particle-stabilized thin liquid films

Our recent quasi-two-dimensional thermodynamic description of thin-liquid
films stabilized by colloidal particles is generalized to describe nonuniform
equilibrium states of films in external potentials and nonequilibrium transport
processes produced in the film by gradients of thermodynamic forces. Using a
Monte--Carlo simulation method, we have determined equilibrium equations of
state for a film stabilized by a suspension of hard spheres. Employing a
multipolar-expansion method combined with a flow-reflection technique, we have
also evaluated the short-time film-viscosity coefficients and collective
particle mobility.Comment: 16 pages, 10 figure

### 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

### 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

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