Collective behavior of particles in fluids

Paris, December 14-17, 2020

**Antoine Gloria** (Sorbonne Université and Université Libre de Bruxelles)

**Part I:** *Effective sedimentation speed of random suspensions*

We consider a random suspension of rigid particles in a steady Stokes flow.
If particles are heavier than the fluid, they experience gravity in a different way, and they fall in the fluid. For one single particle, the sedimentation speed is finite and can be computed explicitly. When several particles are considered, they interact via the Stokes fluid, and the picture is much less clear. In particular, this effective sedimentation could depend on the size of the sedimentation tank in physical experiments. Since the pioneering work of Batchelor it is known that one needs to rely on stochastic cancellations to define the notion of effective sedimentation speed and obtain a bound independent of the tank size (and therefore make quantitative assumptions on the law of the point set describing the suspension of particles). However, despite these strong cancellations captured in form of the central limit theorem scaling, formal calculations by Caflisch and Luke show that the variance of individual sedimentation velocities blows up with the size of the tank in dimension 3, which contradicts experiments. Physicists have argued that large-scale order of the suspension might screen hydrodynamic interactions at large distances and prevent this blow up. All the above calculations and arguments are made on a linear level, assuming that the solution linearly depends on the randomness, which it doesn't. In this talk I will present a rigorous analysis taking into account the nonlinear character of the hydrodynamic interaction, and revisit the Caflisch-Luke paradox and its resolution assuming large-scale order in form of hyperuniformity. The analysis relies on a quantitative version of the qualitative homogenisation of colloidal suspensions, and on a functional-analytic version of hyperuniformity. This is joint work with Mitia Duerinckx.

See slides and record.

**Part II:** *Effective viscosity of random suspensions*

For a random suspension of (colloidal) rigid particles in a steady Stokes flow, I will define the notion of effective viscosity in the homogenisation regime and discuss expansions of the effective viscosity with respect to the (higher-order) intensity of particles. This expansion relies on a cluster decomposition of the effective viscosity, and on the analysis of each term and of the remainder, which are both defined via borderline non integrable quantities. I will present the general strategy and describe the ingredients of the proof (combinatorial, probabilistic, and analytical). This is joint work with Mitia Duerinckx.

See slides and record.