# American Institute of Mathematical Sciences

February  2018, 11(1): 43-69. doi: 10.3934/krm.2018003

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

 1 IGN-LAREG, Université Paris Diderot, Bâtiment Lamarck A, 5 rue Thomas Mann, Case courrier 7071, 75205 Paris Cedex 13, France, 2 Université Paris Diderot, Sorbonne Paris Cité, Institut de Mathématiques de Jussieu — Paris Rive Gauche, UMR CNRS 7586, CNRS, Sorbonne Universités, UPMC Univ. Paris 06, 75013, Paris, France, 3 CMLS, Ecole polytechnique et CNRS, Université Paris-Saclay, 91128 Palaiseau Cedex, France, 4 Dipartimento di Matematica e Informatica, Universitá degli Studi di Palermo, Via Archirafi 34, I90123 Palermo, Italy

Received  October 2016 Revised  March 2017 Published  August 2017

In this 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, Commun.Math.Sci.,15 (2017), 1703-1741] wherethe evolution of the gas is governed by the Navier-Stokes equation.

Citation: Etienne Bernard, Laurent Desvillettes, Franç cois Golse, Valeria Ricci. A derivation of the Vlasov-Stokes system for aerosol flows from the kinetic theory of binary gas mixtures. Kinetic & Related Models, 2018, 11 (1) : 43-69. doi: 10.3934/krm.2018003
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##### References:
 Parameter Definition $L$ size of the container (periodic box) $\mathcal{N}_p$ number of particles$/L^3$ $\mathcal{N}_g$ number of gas molecules$/L^3$ $V_p$ thermal speed of particles $V_g$ thermal speed of gas molecules $S_{pp}$ average particle/particle cross-section $S_{pg}$ average particle/gas cross-section $S_{gg}$ average molecular cross-section $\eta=m_g/m_p$ mass ratio (molecules/particles) $\mu=(m_g \mathcal{N}_g)/(m_p \mathcal{N}_p)$ mass fraction (gas/dust or droplets) ${\epsilon}=V_p/V_g$ thermal speed ratio (particles/molecules)
 Parameter Definition $L$ size of the container (periodic box) $\mathcal{N}_p$ number of particles$/L^3$ $\mathcal{N}_g$ number of gas molecules$/L^3$ $V_p$ thermal speed of particles $V_g$ thermal speed of gas molecules $S_{pp}$ average particle/particle cross-section $S_{pg}$ average particle/gas cross-section $S_{gg}$ average molecular cross-section $\eta=m_g/m_p$ mass ratio (molecules/particles) $\mu=(m_g \mathcal{N}_g)/(m_p \mathcal{N}_p)$ mass fraction (gas/dust or droplets) ${\epsilon}=V_p/V_g$ thermal speed ratio (particles/molecules)
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