# 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
##### References:

show all references

##### 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)
 [1] Raffaele Esposito, Yan Guo, Rossana Marra. Stability of a Vlasov-Boltzmann binary mixture at the phase transition on an interval. Kinetic & Related Models, 2013, 6 (4) : 761-787. doi: 10.3934/krm.2013.6.761 [2] Jean Dolbeault. An introduction to kinetic equations: the Vlasov-Poisson system and the Boltzmann equation. Discrete & Continuous Dynamical Systems - A, 2002, 8 (2) : 361-380. doi: 10.3934/dcds.2002.8.361 [3] Renjun Duan, Tong Yang, Changjiang Zhu. Boltzmann equation with external force and Vlasov-Poisson-Boltzmann system in infinite vacuum. Discrete & Continuous Dynamical Systems - A, 2006, 16 (1) : 253-277. doi: 10.3934/dcds.2006.16.253 [4] Franco Flandoli, Marta Leocata, Cristiano Ricci. The Vlasov-Navier-Stokes equations as a mean field limit. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3741-3753. doi: 10.3934/dcdsb.2018313 [5] Ioannis Markou. Hydrodynamic limit for a Fokker-Planck equation with coefficients in Sobolev spaces. Networks & Heterogeneous Media, 2017, 12 (4) : 683-705. doi: 10.3934/nhm.2017028 [6] Robert T. Glassey, Walter A. Strauss. Perturbation of essential spectra of evolution operators and the Vlasov-Poisson-Boltzmann system. Discrete & Continuous Dynamical Systems - A, 1999, 5 (3) : 457-472. doi: 10.3934/dcds.1999.5.457 [7] Shuangqian Liu, Qinghua Xiao. The relativistic Vlasov-Maxwell-Boltzmann system for short range interaction. Kinetic & Related Models, 2016, 9 (3) : 515-550. doi: 10.3934/krm.2016005 [8] Stéphane Brull, Pierre Charrier, Luc Mieussens. Gas-surface interaction and boundary conditions for the Boltzmann equation. Kinetic & Related Models, 2014, 7 (2) : 219-251. doi: 10.3934/krm.2014.7.219 [9] Raffaele Esposito, Mario Pulvirenti. Rigorous validity of the Boltzmann equation for a thin layer of a rarefied gas. Kinetic & Related Models, 2010, 3 (2) : 281-297. doi: 10.3934/krm.2010.3.281 [10] Karsten Matthies, George Stone, Florian Theil. The derivation of the linear Boltzmann equation from a Rayleigh gas particle model. Kinetic & Related Models, 2018, 11 (1) : 137-177. doi: 10.3934/krm.2018008 [11] Laurent Bernis, Laurent Desvillettes. Propagation of singularities for classical solutions of the Vlasov-Poisson-Boltzmann equation. Discrete & Continuous Dynamical Systems - A, 2009, 24 (1) : 13-33. doi: 10.3934/dcds.2009.24.13 [12] Juhi Jang, Ning Jiang. Acoustic limit of the Boltzmann equation: Classical solutions. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 869-882. doi: 10.3934/dcds.2009.25.869 [13] Ling Hsiao, Fucai Li, Shu Wang. Combined quasineutral and inviscid limit of the Vlasov-Poisson-Fokker-Planck system. Communications on Pure & Applied Analysis, 2008, 7 (3) : 579-589. doi: 10.3934/cpaa.2008.7.579 [14] Renjun Duan, Shuangqian Liu, Tong Yang, Huijiang Zhao. Stability of the nonrelativistic Vlasov-Maxwell-Boltzmann system for angular non-cutoff potentials. Kinetic & Related Models, 2013, 6 (1) : 159-204. doi: 10.3934/krm.2013.6.159 [15] Stefan Possanner, Claudia Negulescu. Diffusion limit of a generalized matrix Boltzmann equation for spin-polarized transport. Kinetic & Related Models, 2011, 4 (4) : 1159-1191. doi: 10.3934/krm.2011.4.1159 [16] Karsten Matthies, George Stone. Derivation of a non-autonomous linear Boltzmann equation from a heterogeneous Rayleigh gas. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3299-3355. doi: 10.3934/dcds.2018143 [17] Fanghua Lin, Ping Zhang. On the hydrodynamic limit of Ginzburg-Landau vortices. Discrete & Continuous Dynamical Systems - A, 2000, 6 (1) : 121-142. doi: 10.3934/dcds.2000.6.121 [18] Xuwen Chen, Yan Guo. On the weak coupling limit of quantum many-body dynamics and the quantum Boltzmann equation. Kinetic & Related Models, 2015, 8 (3) : 443-465. doi: 10.3934/krm.2015.8.443 [19] Stéphane Mischler, Clément Mouhot. Stability, convergence to the steady state and elastic limit for the Boltzmann equation for diffusively excited granular media. Discrete & Continuous Dynamical Systems - A, 2009, 24 (1) : 159-185. doi: 10.3934/dcds.2009.24.159 [20] Xueke Pu, Boling Guo. Global existence and semiclassical limit for quantum hydrodynamic equations with viscosity and heat conduction. Kinetic & Related Models, 2016, 9 (1) : 165-191. doi: 10.3934/krm.2016.9.165

2018 Impact Factor: 1.38

## Metrics

• HTML views (119)
• Cited by (1)

• on AIMS