Suspensions are ubiquitous in nature (sediments, clouds,biological fluids ... etc.) and in industry such as civil engineering (paints, polymers ... etc.) among many others. The rigorous derivation of fluid-kinetic models for suspensions has attracted a lot of attention in the last decade. This lecture aims at presenting a review of the main results that have been obtained.
The first session aims at introducing both the microscopic and the limiting equation and giving a formal derivation of the former one. The second session aims at presenting the main early results concerning the derivation of an effective model starting from the microscopic model in which particle positions and velocities are fixed or given. Such a system takes the following form for example
\begin{equation}\label{eq:Stokes}
\left \{
\begin{array}{rcl}
-\Delta u+\nabla p &=& f, \text{ on } \Omega\setminus \overline{\underset{i=1}{\overset{N}{\bigcup}} B(x_i,r)} \\
\text{div } u&=& 0, \text{ on } \Omega\setminus \overline{\underset{i=1}{\overset{N}{\bigcup}} B(x_i,r)} \\
u&=& V_i, \text{ on } \partial B(x_i,r)\\
u&=& 0, \text{ on } \partial \Omega
\end{array}
\right.
\end{equation}
where $\Omega$ a smooth open set of $\mathbb{R}^3$, $x_1, x_2, \cdots, x_N$ are the particles position, $r$ their radius and $V_i$ the given velocity of the $i$th particle. The aim is then to perform an asymptotic analysis when the number of particles $N$ becomes large while their radius $r$ becomes small, first results have been obtained in [1,2,3] where the limit equations depend on the scale of the holes and their typical distance; Stokes equation, Darcy equation or Stokes-Brinkman equation. After recalling the recent contributions, we will present a short argument giving insights about the derivation of the Brinkman term in a simple case.
The last session of this mini-course aims at presenting the results regarding the rigorous derivation of fluid-kinetic models when taking into account the fluid-particle interactions and particle dynamics. This means that we consider the Stokes equation [1] coupled to Newton laws where we neglect particles inertia (balance of force and torque) and the motion of the center of the particles $\dot{x}_i=V_i$.
The rigorous derivation of a fluid-kinetic model in this setting have been obtained in [6,5,7] in the case $\Omega=\mathbb{R}^3$ under some separation assumptions on the particles. The obtained equation is a Transport-Stokes equation
\begin{equation}\label{eq:TS}\tag{TS}
\left\{
\begin{array}{rcl}
- \Delta u + \nabla p &=& \rho g,\\
\text{div } u&=& 0, \\
\partial_t \rho +\text{div }( ( u + \gamma^{-1} V_{\mathrm{St}})\rho) &=& 0,
\end{array}
\right.
\end{equation}
where $\gamma = \lim Nr \in (0,\infty]$.
This result is related to the mean field limit of many particles interacting through a kernel and has been extensively studied for several different problems. We present the main ideas for such a derivation using the method of reflections and stability estimates through Wasserstein distance following the approach by M. Hauray [4]. We finish by emphasizing new results based on a mean-field argument for the derivation of models of suspensions.
Several extensions have been made, we mention for instance [3] where authors considered steady Navier Stokes equation with non periodically distributed particles satisfying a minimal distance assumption and for general Dirichlet boundary conditions with uniform kinetic energy, they in particular characterized the convergence in terms of the limit of (marginals of) the empirical measure
$\rho^N(x)=\frac{1}{N} \underset{1 \leq i \leq n}{\sum} \delta_{x_i}
$
Several extensions have been then obtained. We mention [5] where the author extends the minimal distance assumption, for quantitative convergence estimates, [6] in the case of arbitrary shaped particles, the case of randomly distributed particles, fora simplified proof.
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English
