Valeria Ricci

The Mean-Field Limit for Solid Particles in a Navier-Stokes Flow

AbstractWe propose a mathematical derivation of Brinkman’s force for a cloud of particles immersed in an incompressible viscous fluid. Specifically, we consider the Stokes or steady Navier-Stokes equations in a bounded domain Ω⊂ℝ3 for the velocity field u of an incompressible fluid with kinematic viscosity ν and density 1. Brinkman’s force consists of a source term 6πνj where j is the current density of the particles, and of a friction term 6πνρu where ρ is the number density of particles. These additional terms in the motion equation for the fluid are obtained from the Stokes or steady Navier-Stokes equations set in Ω minus the disjoint union of N balls of radius ε=1/N in the large N limit with no-slip boundary condition. The number density ρ and current density j are obtained from the limiting phase space empirical measure n

Empirical Measures and Vlasov Hierarchies

frac{1}{N}sum_{1le kle N}delta_{x_{k},v_{k}}

The Vlasov Limit for a System of Particles which Interact with a Wave Field

n, where xk is the center of the k-th ball and vk its instantaneous velocity. This can be seen as a generalization of Allaire’s result in [Arch. Ration. Mech. Anal. 113:209–259, [1991]] who considered the case of periodically distributed xks with vk=0, and our proof is based on slightly simpler though similar homogenization arguments. Similar equations are used for describing the fluid phase in various models for sprays.

DERIVATION OF A HOMOGENIZED TWO-TEMPERATURE MODEL FROM THE HEAT EQUATION

The present note reviews some aspects of the mean field limit for Vlasov type equations with Lipschitz continuous interaction kernel. We discuss in particular the connection between the approach involving the N-particle empirical measure and the formulation based on the BBGKY hierarchy. This leads to a more direct proof of the quantitative estimates on the propagation of chaos obtained on a more general class of interacting systems in [S.Mischler, C. Mouhot, B. Wennberg, arXiv:1101.4727]. Our main result is a stability estimate on the BBGKY hierarchy uniform in the number of particles, which implies a stability estimate in the sense of the Monge-Kantorovich distance with exponent 1 on the infinite mean field hierarchy. This last result amplifies Spohns uniqueness theorem [H. Spohn, Math. Meth. Appl. Sci. 3 (1981), 445-455].

A derivation of the Vlasov-Stokes system for aerosol flows from the kinetic theory of binary gas mixtures

In two recent publications, [Commun. PDE 22, 307–335 (1997), Commun. Math. Phys. 203, 1–19 (1999)], A. Komech, M. Kunze and H. Spohn studied the joint dynamics of a classical point particle and a wave type generalization of the Newtonian gravity potential, coupled in a regularized way. In the present paper the many-body dynamics of this model is studied. The Vlasov continuum limit is obtained in form equivalent to a weak law of large numbers. We also establish a central limit theorem for the fluctuations around this limit. c ©2005 The authors. Reproduction of this article, in its entirety, for non-commercial purposes is permitted.

A Derivation of the Vlasov-Navier-Stokes Model for Aerosol Flows from Kinetic Theory

This work studies the heat equation in a two-phase material with spherical inclusions. Under some appropriate scaling on the size, volume fraction and heat capacity of the inclusions, we derive a coupled system of partial differential equations governing the evolution of the temperature of each phase at a macroscopic level of description. The coupling terms describing the exchange of heat between the phases are obtained by using homogenization techniques originating from (D. Cioranescu, F. Murat, College de France Seminar, vol. II. Paris 1979-1980; vol. 60 of Res. Notes Math .P itman, Boston, London (1982) 98-138).

Large Number Asymptotics for Two-Component Systems with Self-Consistent Coupling

In this short paper, we formally derive the thin spray equation for a steady Stokes gas, i.e. the equation consists in a coupling between a kinetic (Vlasov type) equation for the dispersed phase and a (steady) Stokes equation for the gas. Our starting point is a system of Boltzmann equations for a binary gas mixture. The derivation follows the procedure already outlined in [Bernard-Desvillettes-Golse-Ricci, arXiv:1608.00422 [math.AP]] where the evolution of the gas is governed by the Navier-Stokes equation.

Modelling of Systems with a Dispersed Phase: “Measuring” Small Sets in the Presence of Elliptic Operators

This article proposes a derivation of the Vlasov-Navier-Stokes system for spray/aerosol flows. The distribution function of the dispersed phase is governed by a Vlasov-equation, while the velocity field of the propellant satisfies the Navier-Stokes equations for incompressible fluids. The dynamics of the dispersed phase and of the propellant are coupled through the drag force exerted by the propellant on the dispersed phase. We present a formal derivation of this model from a multiphase Boltzmann system for a binary gaseous mixture, involving the droplets/dust particles in the dispersed phase as one species, and the gas molecules as the other species. Under suitable assumptions on the collision kernels, we prove that the sequences of solutions to the multiphase Boltzmann system converge to distributional solutions to the Vlasov-Navier-Stokes equation in some appropriate distinguished scaling limit. Specifically, we assume (a) that the mass ratio of the gas molecules to the dust particles/droplets is small, (b) that the thermal speed of the dust particles/droplets is much smaller than that of the gas molecules and (c) that the mass density of the gas and of the dispersed phase are of the same order of magnitude.

Nonlocal Second Order Vehicular Traffic Flow Models And Lagrange-Remap Finite Volumes

We shall consider the large number asymptotics of particle models for partial differential equations describing two component mixtures with simplest kind of self-consistent couplings. We shall recall in particular two examples related to different classes of models, the first one having both particle-like components and the second one having only one particle-like component (the other being described as a fluid); for these examples, different techniques on the probabilistic and analytic point of view are to be used to rigorously prove the convergence to a limit of the self-consistent terms in a “mean-field”-like asymptotics. The two models were analysed resp. in Bernardin and Ricci (Kinet Relat Models 4(3), 633–668, 2011) and Desvillettes et al. (Derivation of a homogenized two-temperature model from the heat equation. Preprint hal–00827912, arXiv:1305.6920, 2013, to be published in Mathematical Modelling and Numerical Analysis).

Derivation of models for thin sprays from a multiphase Boltzmann model

When modelling systems with a dispersed phase involving elliptic operators, as is the case of the Stokes or Navier-Stokes problem or the heat equation in a bounded domain, the geometrical structure of the space occupied by the dispersed phase enters in the homogenization process through its capacity, a quantity which can be used to define the equivalence classes in (H^1). We shall review the relationship between capacity and homogenization terms in the limit when the number of inclusions becomes large, focusing in particular on the situation where the distribution of inclusions is not necessarily too regular (i.e. it is not periodic).