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 $\dot H^1$ under a mild separation condition on the particles. Moreover, we prove optimal convergence rates of the method in $\dot W^{1,q}$, $1 < q < \infty$ and in $L^\infty$ 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 $\mathbb{R}^3$ perforated by $m$ spherical holes, identically and independently distributed. In the critical regime when the radii of the holes are of order $m^{-1}$, we consider the fluctuations of the solutions $u_m$ around the homogenization limit $u$. In the central limit scaling, we show that these fluctuations converge to a Gaussian field, locally in $L^2(\mathbb{R}^3)$, with an explicit covariance.Comment: 40 page

Hindered Settling of Well-Separated Particle Suspensions

We consider $N$ identical inertialess rigid spherical particles in a Stokes flow in a domain $\Omega \subset \mathbb R^3$. 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 $\mathbb T^3$. 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