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

Abstract

We propose a mathematical derivation of Brinkman's force for a cloud of particles immersed in an incompressible fluid. Our starting point is the Stokes or steady Navier-Stokes equations set in a bounded domain with the disjoint union of N balls of radius 1/N removed, and with a no-slip boundary condition for the fluid at the surface of each ball. The large N limit of the fluid velocity field is governed by the same (Navier-)Stokes equations in the whole domain, with an additional term (Brinkman's force) that is (minus) the total drag force exerted by the fluid on the particle system. This can be seen as a generalization of Allaire's result in [Arch. Rational Mech. Analysis 113 (1991), 209-259] who treated the case of motionless, periodically distributed balls. Our proof is based on slightly simpler, though similar homogenization techniques, except that we avoid the periodicity assumption and use instead the phase-space empirical measure for the particle system. Similar equations are used for describing the fluid phase in various models for sprays.

Cited by in corpus (18)

• Large time behavior of the Vlasov-Navier-Stokes system on the torus
• The Method of Reflections, Homogenization and Screening for Poisson and Stokes Equations in Perforated Domains
• Analysis of the viscosity of dilute suspensions beyond Einstein's formula
• On the derivation of a Stokes-Brinkman problem from Stokes equations around a random array of moving spheres
• A Derivation of the Vlasov-Navier-Stokes Model for Aerosol Flows from Kinetic Theory
• A Derivation of the Vlasov-Stokes System for Aerosol Flows from the Kinetic Theory of Binary Gas Mixtures
• On hydrodynamic limits of the Vlasov-Navier-Stokes system
• Derivation of a homogenized two-temperature model from the heat equation
• A simple justification of effective models for conducting or fluid media with dilute spherical inclusions
• Derivation of Darcy's law in randomly punctured domains
• Concentration versus absorption for the Vlasov-Navier-Stokes system on bounded domains
• Gevrey regularity and analyticity for the solutions of the Vlasov-Navier-Stokes system
• Homogenization for the Stokes equations in randomly perforated domains under almost minimal assumptions on the size of the holes
• An Eulerian model for sea spray transport and evaporation
• Decay and absorption for the Vlasov-Navier-Stokes system with gravity in a half-space
• Homogenization of Poisson and Stokes equations in the whole space
• Global well-posedness of one-dimensional compressible Navier-Stokes-Vlasov system
• Hydrodynamic limit of a coupled Cucker-Smale system with strong and weak internal variable relaxation