We 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 Omega subset of R-3 for the velocity field u of an incompressible fluid with kinematic viscosity v and density 1. Brinkman's force consists of a source term 6 pi rvj where j is the current density of the particles, and of a friction term 6 pi vpu where rho 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 Omega minus the disjoint union of N balls of radius epsilon = 1/N in the large N limit with no-slip boundary condition. The number density p and current density j are obtained from the limiting phase space empirical measure 1/N Sigma(1 <= k <= N)delta(xk,vk), where x(k) is the center of the k-th hall and v(k) 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 x(k)S with v(k) = 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.

Desvillettes, L., Golse, F., Ricci, V. (2008). The mean-field limit for solid particles in a Navier-Stokes flow. JOURNAL OF STATISTICAL PHYSICS, 131(5), 941-967 [10.1007/s10955-008-9521-3].

The mean-field limit for solid particles in a Navier-Stokes flow

RICCI, Valeria
2008-01-01

Abstract

We 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 Omega subset of R-3 for the velocity field u of an incompressible fluid with kinematic viscosity v and density 1. Brinkman's force consists of a source term 6 pi rvj where j is the current density of the particles, and of a friction term 6 pi vpu where rho 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 Omega minus the disjoint union of N balls of radius epsilon = 1/N in the large N limit with no-slip boundary condition. The number density p and current density j are obtained from the limiting phase space empirical measure 1/N Sigma(1 <= k <= N)delta(xk,vk), where x(k) is the center of the k-th hall and v(k) 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 x(k)S with v(k) = 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.
2008
Desvillettes, L., Golse, F., Ricci, V. (2008). The mean-field limit for solid particles in a Navier-Stokes flow. JOURNAL OF STATISTICAL PHYSICS, 131(5), 941-967 [10.1007/s10955-008-9521-3].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10447/10845
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