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Starting from microscopic $N$ particle systems, we study the derivation of Doi type models for suspensions of non-spherical particles in Stokes flows. While Doi models accurately describe the effective evolution of the spatial particle density to the first order in the particle volume fraction, this accuracy fails regarding the evolution of the particle orientations. We rigorously attribute this failure to the singular interaction of the particles via a $-3$-homogeneous kernel. In the situation that the particles are initially distributed according to a stationary ergodic point process, we identify the limit of this singular interaction term. It consists of two parts. The first corresponds to a classical term in Doi type models. The second new term depends on the (microscopic) $2$-point correlation of the point process. By including this term, we provide a modification of the Doi model that is accurate to first order in the particle volume fraction.

We consider the solutions $\rho_\varepsilon, \mathbf{u}_\varepsilon$ to the compressible Navier-Stokes equations (NSE) in a domain periodically perforated by holes of diameter $\varepsilon>0$. We focus on the case where the diameter of the holes is of the same order as the distance between neighboring holes. This is the same setting investigated in the paper by Masmoudi [\url{http://www.numdam.org/article/COCV_2002__8__885_0.pdf}], where convergence $\rho_\varepsilon, \mathbf{u}_\varepsilon$ of the system to the porous medium equation has been shown. We prove a quantitative version of this convergence result provided that the solution of the limiting system is sufficiently regular. The proof builds on the relative energy inequality satisfied by the compressible NSE.

We consider the sedimentation of $N$ spherical particles with identical radii $R$ in a Stokes flow in $\mathbb R^3$. The particles satisfy a no-slip boundary condition and are subject to constant gravity. The dynamics of the particles is modeled by Newton's law but with very small particle inertia as $N$ tends to infinity and $R$ to $0$. In a mean-field scaling we show that the evolution of the $N$-particle system is well approximated by the Vlasov-Stokes equation. In contrast to the transport-Stokes equation considered in the first part of this series, \cite{HoferSchubert23}, the Vlasov-Stokes equation takes into account the (small) inertia. Therefore we obtain improved error estimates. We also improve previous results on the Cauchy problem for the Vlasov-Stokes equation and on its convergence to the transport-Stokes equation in the limit of vanishing inertia. The proofs are based on relative energy estimates. In particular, we show new stability estimates for the Vlasov-Stokes equation in the $2$-Wasserstein distance. By combining a Lagrangian approach with a study of the energy dissipation, we obtain uniform stability estimates for arbitrary small particle inertia. We show that a corresponding stability estimate continues to hold for the empirical particle density which formally solves the Vlasov-Stokes equation up to an error. To this end we exploit certain uniform control on the particle configuration thanks to results in the first part \cite{HoferSchubert23}., Comment: All comments welcome

We study the multiscale viscoelastic Doi model for suspensions of Brownian rigid rod-like particles, as well as its generalization by Saintillan and Shelley for self-propelled particles. We consider the regime of a small Weissenberg number, which corresponds to a fast rotational diffusion compared to the fluid velocity gradient, and we analyze the resulting hydrodynamic approximation. More precisely, we show the asymptotic validity of macroscopic nonlinear viscoelastic models, in form of so-called ordered fluid models, as an expansion in the Weissenberg number. The result holds for zero Reynolds number in 3D and for arbitrary Reynolds number in 2D. Along the way, we establish several new well-posedness and regularity results for nonlinear fluid models, which may be of independent interest., Comment: 64 pages

We consider the sedimentation of $N$ spherical particles with identical radii $R$ in a Stokes flow in $\mathbb R^3$. The particles satisfy a no-slip boundary condition and are subject to constant gravity. The dynamics of the particles is modeled by Newton's law but with very small particle inertia as $N$ tends to infinity and $R$ to $0$. In a mean-field scaling, we show that the particle evolution is well approximated by the transport-Stokes system which has been derived previously as the mean-field limit of inertialess particles. In particular this justifies to neglect the particle inertia in the microscopic system, which is a typical modelling assumption in this and related contexts. The proof is based on a relative energy argument that exploits the coercivity of the particle forces with respect to the particle velocities in a Stokes flow. We combine this with an adaptation of Hauray's method for mean-field limits to $2$-Wasserstein distances. Moreover, in order to control the minimal distance between particles, we prove a representation of the particle forces. This representation makes the heuristic \enquote{Stokes law} rigorous that the force on each particle is proportional to the difference of the velocity of the individual particle and the mean-field fluid velocity generated by the other particles., Comment: V5: Corrected typo in the abstract. V4: Version as accepted by Duke Mathematical Journal. The main statement is now for the physically relevant situation of rotating particles (improved). Included a mean-field result for binary inertial interaction with asymptotically vanishing inertia. Stated more precisely bounds on Wasserstein distance and minimal distance. 59 pages

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!

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 welcome

We study the solution $u_\varepsilon$ to the Navier-Stokes equations in $\mathbb R^3$ perforated by small particles centered at $(\varepsilon \mathbb Z)^3$ with no-slip boundary conditions at the particles. We study the behavior of $u_\varepsilon$ for small $\varepsilon$, depending on the diameter $\varepsilon^\alpha$, $\alpha > 1$, of the particles and the viscosity $\varepsilon^\gamma$, $\gamma > 0$, of the fluid. We prove quantitative convergence results for $u_\varepsilon$ 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., Comment: Published version (in Nonlinearity)

We consider the long-time behavior of a fast, charged particle interacting with an initially spatially homogeneous background plasma. The background is modeled by the screened Vlasov-Poisson equations, whereas the interaction potential of the point charge is assumed to be smooth. We rigorously prove the validity of the \emph{stopping power theory} in physics, which predicts a decrease of the velocity $V(t)$ of the point charge given by $\dot{V} \sim -|V|^{-3} V$, a formula that goes back to Bohr (1915). Our result holds for all initial velocities larger than a threshold value that is larger than the velocity of all background particles and remains valid until (i) the particle slows down to the threshold velocity, or (ii) the time is exponentially long compared to the velocity of the point charge. The long-time behavior of this coupled system is related to the question of Landau damping which has remained open in this setting so far. Contrary to other results in nonlinear Landau damping, the long-time behavior of the system is driven by the non-trivial electric field of the plasma, and the damping only occurs in regions that the point charge has already passed., Comment: Published version (in Analysis&PDE)

We consider a microscopic model of $n$ identical axis-symmetric rigid Brownian particles suspended in a Stokes flow. We rigorously derive in the homogenization limit of many small particles a classical formula for the viscoelastic stress that appears in so-called Doi models which couple a Fokker-Planck equation to the Stokes equations. We consider both Deborah numbers of order $1$ and very small Deborah numbers. Our microscopic model contains several simplifications, most importantly, we neglect the time evolution of the particle centers as well as hydrodynamic interaction for the evolution of the particle orientations. The microscopic fluid velocity is modeled by the Stokes equations with given torques at the particles in terms of Stratonovitch noise. We give a meaning to this PDE in terms of an infinite dimensional Stratonovitch integral. This requires the analysis of the shape derivatives of the Stokes equations in perforated domains, which we accomplish by the method of reflections., Comment: V2: Corrected some typos

We investigate the dynamics of several slender rigid bodies moving in a flow driven by the three-dimensional steady Stokes system in presence of a smooth background flow. More precisely we consider the limit where the thickness of these slender rigid bodies tends to zero with a common rate $\varepsilon$, while their volumetric mass density is held fixed, so that the bodies shrink into separated massless curves. While for each positive $\varepsilon$, the bodies' dynamics are given by the Newton equations and correspond to some coupled second-order ODEs for the positions of the bodies, we prove that the limit equations are decoupled first-order ODEs whose coefficients only depend on the limit curves and on the background flow. These coefficients appear through appropriate renormalized Stokes resistance tensors associated with each limit curve, and through renormalized Fax\'en-type force and torque associated with the limit curves and the background flow. We establish a rate of convergence of the curves of order $O( \vert \log \varepsilon \vert^{-1/2})$. We also determine the limit effect due to the limit curves on the fluid, in the spirit of the immersed boundary method. Both for the convergence of the filament velocities and the fluid velocities we identify an initial exponential relaxation within a $O(\varepsilon^2 \vert \log \varepsilon \vert)$ time.

We investigate the sedimentation of identical inertialess spherical particles in a Stokes fluid in the limit of many small particles. It is known that the presence of the particles leads to an increase of the effective viscosity of the suspension. By Einstein's formula this effect is of the order of the particle volume fraction $\phi$. The disturbance of the fluid flow responsible for this increase of viscosity is very singular (like $|x|^{-2}$). Nevertheless, for well-prepared initial configurations and $\phi\to 0$, we show that the microscopic dynamics is approximated to order $\phi^2 |\log \phi|$ by a macroscopic coupled transport-Stokes system with an effective viscosity according to Einstein's formula. We provide quantitative estimates both for convergence of the densities in the $p$-Wasserstein distance for all $p$ and for the fluid velocity in Lebesgue spaces in terms of the $p$-Wasserstein distance of the initial data. Our proof is based on approximations through the method of reflections and on a generalization of a classical result on convergence to mean-field limits in the infinite Wasserstein metric by Hauray., Comment: 37 pages Added Appendix and small corrections

We consider the homogenization limit of the compressible barotropic Navier-Stokes equations in a three-dimensional domain perforated by periodically distributed identical particles. We study the regime of particle sizes and distances such that the volume fraction of particles tends to zero but their resistance density tends to infinity. Assuming that the Mach number is increasing with a certain rate, the rescaled velocity and pressure of the microscopic system converges to the solution of an effective equation which is given by Darcy's law. The range of sizes of particles we consider are exactly the same which lead to Darcy's law in the homogenization limit of incompressible fluids. Unlike previous results for the Darcy regime we estimate the deficit related to the pressure approximation via the Bogovski\u{{\i}} operator This allows for more flexible estimates of the pressure in Lebesgue and Sobolev spaces and allows to proof convergence results for all barotropic exponents $\gamma> \frac32$., Comment: Minor typos corrected. To appear in M3AS (Mathematical Models and Methods in Applied Sciences)

We study the homogenization of the Dirichlet problem for the Stokes equations in $\mathbb{R}^3$ perforated by $m$ spherical particles. We assume the positions and velocities of the particles to be identically and independently distributed random variables. In the critical regime, when the radii of the particles are of order $m^{-1}$, the homogenization limit $u$ is given as the solution to the Brinkman equations. We provide optimal rates for the convergence $u_m \to u$ in $L^2$, namely $m^{-\beta}$ for all $\beta < 1/2$. Moreover, we consider the fluctuations. In the central limit scaling, we show that these converge to a Gaussian field, locally in $L^2(\mathbb{R}^3)$, with an explicit covariance. Our analysis is based on explicit approximations for the solutions $u_m$ in terms of $u$ as well as the particle positions and their velocities. These are shown to be accurate in $\dot H^1(\mathbb{R}^3)$ to order $m^{-\beta}$ for all $\beta < 1$. Our results also apply to the analogous problem regarding the homogenization of the Poisson equations.

We consider the homogenization to the Brinkman equations for the incompressible Stokes equations in a bounded domain which is perforated by a random collection of small spherical holes. This problem has been studied by the same authors in [A. Giunti and R.M. H\"ofer, Homogenization for the Stokes equations in randomly perforated domains under almost minimal assumptions on the size of the holes] where convergence of the fluid velocity field towards the solution of the Brinkman equations has been established. In the present we consider the pressure associated to the solution of the Stokes equations in the perforated domain. We prove that it is possible to extend this pressure inside the holes and slightly modify it in a region of asymptotically negligible harmonic capacity such that it weakly converges to the pressure associated with the solution of the Brinkman equations., Comment: 14 pages. arXiv admin note: text overlap with arXiv:1809.04491

We provide a rigorous derivation of Einstein's formula for the effective viscosity of dilute suspensions of $n$ rigid balls, $n \gg 1$, set in a volume of size $1$. So far, most justifications were carried under a strong assumption on the minimal distance between the balls: $d_{min} \ge c n^{-\frac{1}{3}}$, $c > 0$. We relax this assumption into a set of two much weaker conditions: one expresses essentially that the balls do not overlap, while the other one gives a control of the number of balls that are close to one another. In particular, our analysis covers the case of suspensions modelled by standard Poisson processes with almost minimal hardcore condition., Comment: V3: In this new version we added references of results that have appeared after the first submission, and we refined the discussion on the assumptions (B1)-(B2) in Sections 5 and 6. To appear in Commun. Partial. Differ. Equ. V2: Assumption (B2) considerably weakened

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: Published version (in J Diff Eq)

We prove the homogenization to the Brinkman equations for the incompressible Stokes equations in a bounded domain which is perforated by a random collection of small spherical holes. The fluid satisfies a no-slip boundary condition at the holes. The balls generating the holes have centres distributed according to a Poisson point process and i.i.d. unbounded radii satisfying a suitable moment condition. We stress that our assumption on the distribution of the radii does not exclude that, with overwhelming probability, the holes contain clusters made by many overlapping balls. We show that the formation of these clusters has no effect on the limit Brinkman equations. Due to the incompressiblility condition and the lack of a maximum principle for the Stokes equations, our proof requires a very careful study of the geometry of the random holes generated by the class of probability measures considered., Comment: 37 pages, 4 figures. We have split the previous version of the paper. The second part is arXiv:2003.04724, containing convergence results for the pressure

This paper deals with the homogenization of the Poisson equation in a bounded domain of $\mathbb{R}^d$, $d>2$, which is perforated by a random number of small spherical holes with random radii and positions. We show that for a class of stationary short-range correlated measures for the centres and radii of the holes, we recover in the homogenized limit an averaged analogue of the "strange term" obtained by Cioranescu and Murat in the periodic case [D. Cioranescu and F. Murat, \textit{Un term \'etrange venu d'ailleurs} (1986)]. We stress that we only require that the random radii have finite $(d-2)$-moment, which is the minimal assumption in order to ensure that the average of the capacity of the balls is finite. Under this assumption, there are holes which overlap with probability one. However, we show that homogenization occurs and that the clustering holes do not have any effect in the resulting homogenized equation., Comment: 35 pages

We study the Vlasov-Stokes equations which macroscopically model the sedimentation of a cloud of particles in a fluid, where particle inertia are taken into account but fluid inertia are assumed to be negligible. We consider the limit when the inertia of the particles tends to zero, and obtain convergence of the dynamics to the solution of an associated inertialess system of equations. This system coincides with the model that can be derived as the homogenization limit of the microscopic inertialess dynamics., Comment: 31 pages

We investigate the sedimentation of a cloud of rigid, spherical particles of identical radii under gravity in a Stokes fluid. Both inertia and rotation of particles are neglected. We consider the homogenization limit of many small particles in the case of a dilute system in which interactions between particles are still important. In the relevant time scale, we rigorously prove convergence of the dynamics to the solution of a macroscopic equation. This macroscopic equation resembles the Stokes equations for a fluid of variable density subject to gravitation., Comment: 41 pages

We study the convergence of the Method of Reflections for the Dirichlet problem of the Poisson and the Stokes equations in perforated domains which consist in the exterior of balls. We prove that the method converges if the balls are contained in a bounded region and the density of the electrostatic capacity of the balls is sufficiently small. If the capacity density is too large or the balls extend to the whole space, the method diverges, but we provide a suitable modification of the method that converges to the solution of the Dirichlet problem also in this case. We give new proofs of classical homogenization results for the Dirichlet problem of the Poisson and the Stokes equations in perforated domains using the (modified) Method of Reflections., Comment: 47 pages, 1 figure