37 research outputs found
Convergence of the method of reflections for particle suspensions in Stokes flows
We study the convergence of the method of reflections for the Stokes
equations in domains perforated by countably many spherical particles with
boundary conditions typical for the suspension of rigid particles. We prove
that a relaxed version of the method is always convergent in under a
mild separation condition on the particles. Moreover, we prove optimal
convergence rates of the method in , and in
in terms of the particle volume fraction under a stronger separation
condition of the particles.Comment: 25 pages; relaxed assumptions for Corollary 2.
Fluctuations in the homogenization of the Poisson and Stokes equations in perforated domains
We study the homogenization problem of the Poisson and Stokes equations in
perforated by spherical holes, identically and independently
distributed. In the critical regime when the radii of the holes are of order
, we consider the fluctuations of the solutions around the
homogenization limit . In the central limit scaling, we show that these
fluctuations converge to a Gaussian field, locally in , with
an explicit covariance.Comment: 40 page
Hindered Settling of Well-Separated Particle Suspensions
We consider identical inertialess rigid spherical particles in a Stokes
flow in a domain . We study the average
sedimentation velocity of the particles when an identical force acts on each
particle. If the particles are homogeneously distributed in directions
orthogonal to this force, then they hinder each other leading to a mean
sedimentation velocity which is smaller than the sedimentation velocity of a
single particle in an infinite fluid. Under suitable convergence assumptions of
the particle density and a strong separation assumption, we identify the order
of this hindering as well as effects of small scale inhomogeneities and
boundary effects. For certain configurations we explicitly compute the leading
order corrections.Comment: All comments welcome
Non-existence of mean-field models for particle orientations in suspensions
We consider a suspension of spherical inertialess particles in a Stokes flow
on the torus . The particles perturb a linear extensional flow due
to their rigidity constraint. Due to the singular nature of this perturbation,
no mean-field limit for the behavior of the particle orientation can be valid.
This contrasts with widely used models in the literature such as the FENE and
Doi models and similar models for active suspensions. The proof of this result
is based on the study of the mobility problem of a single particle in a
non-cubic torus, which we prove to exhibit a nontrivial coupling between the
angular velocity and a prescribed strain.Comment: All comments welcom
Homogenization of the Navier–Stokes equations in perforated domains in the inviscid limit
We study the solution uε to the Navier–Stokes equations in R3 perforated
by small particles centered at (εZ)3 with no-slip boundary conditions at the
particles. We study the behavior of uε for small ε, depending on the diameter
εα, α>1, of the particles and the viscosity εγ, γ >0, of the fluid. We prove
quantitative convergence results for uε in all regimes when the local Reynolds
number at the particles is negligible. Then, the particles approximately exert a
linear friction force on the fluid. The obtained effective macroscopic equations
depend on the order of magnitude of the collective friction. We obtain (a) the
Euler–Brinkman equations in the critical regime, (b) the Euler equations in the
subcritical regime and (c) Darcy’s law in the supercritical regime