# 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 and Related Models, 2018, 11 (1) : 43-69. doi: 10.3934/krm.2018003
##### References:
 [1] G. Allaire, Homogenization of the Navier-Stokes equations in open sets perforated with tiny holes. I. Abstract framework, a volume distribution of holes, Arch. Rational Mech. Anal., 113 (1990), 209-259.  doi: 10.1007/BF00375065. [2] C. Bardos, F. Golse and C. D. Levermore, Fluid dynamic limits of kinetic equations. I. Formal derivations, J. Stat. Phys., 63 (1991), 323-344.  doi: 10.1007/BF01026608. [3] E. Bernard, L. Desvillettes, F. Golse and V. Ricci, A derivation of the Vlasov-Navier-Stokes model for aerosol flows from kinetic theory, Commun. Math. Sci., 15 (2017), 1703-1741.  doi: 10.4310/CMS.2017.v15.n6.a11. [4] J. A. Carrillo, Y.-P. Choi and T. K. Karper, On the analysis of a coupled kinetic-fluid model with local alignment forces, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 273-307.  doi: 10.1016/j.anihpc.2014.10.002. [5] C. Cercignani, Theory and Applications of the Boltzmann Equation, Elsevier, New York, 1975. [6] F. Charles, Kinetic modelling and numerical simulations using particle methods for the transport of dust in a rarefied gas, Proceedings of the 26th International Symposium on Rarefied Gas Dynamics, AIP Conf. Proc, 1084 (2009), 409-414.  doi: 10.1063/1.3076512. [7] F. Charles, Modélisation Mathématique et Étude Numérique d'un Aérosol dans un Gaz Raréfié. Application á la Simulation du Transport de Particules de Poussiére en Cas d'Accident de Perte de Vide dans ITER, Ph.D thesis, ENS Cachan, 2009. [8] F. Charles, S. Dellacherie and J. Segré, Kinetic modeling of the transport of dust particles in a rarefied atmosphere Math. Models Methods Appl. Sci. 22 (2012), 1150021, 60 pp. doi: 10.1142/S0218202511500217. [9] Y.-P. Choi, Finite-time blow-up phenomena of Vlasov/Navier-Stokes equations and related systems J. Math. Pures Appl. (2017). doi: 10.1016/j.matpur.2017.05.019. [10] Y.-P. Choi and B. Kwon, Global well-posedness and large-time behavior for the inhomogeneous Vlasov-Navier-Stokes equations, Nonlinearity, 28 (2015), 3309-3336.  doi: 10.1088/0951-7715/28/9/3309. [11] D. Cioranescu and F. Murat, Un terme étrange venu d'ailleurs, Nonlinear Partial Differential Equations and their Applications, 60 (1982), 98-138. [12] P. Degond and B. Lucquin-Desreux, The asymptotics of collision operators for two species of particles of disparate masses, Math. Models Meth. Appl. Sci., 6 (1996), 405-436.  doi: 10.1142/S0218202596000158. [13] B. Desjardins and M. J. Esteban, Existence of weak solutions for the motion of rigid bodies in a viscous fluid, Arch. Ration. Mech. Anal., 146 (1999), 59-71.  doi: 10.1007/s002050050136. [14] L. Desvillettes and F. Golse, A remark concerning the Chapman-Enskog asymptotics, Advances in Kinetic Theory and Computing, Series on Advances in Mathematics for Applied Sciences, 22 (1994), 191-203. [15] L. Desvillettes, F. Golse and V. Ricci, The mean-field limit for solid particles in a Navier-Stokes flow, J. Stat. Phys., 131 (2008), 941-967.  doi: 10.1007/s10955-008-9521-3. [16] L. Desvillettes and J. Mathiaud, Some aspects of the asymptotics leading from gas-particles equations towards multiphase flows equations, J. Stat. Phys., 141 (2010), 120-141.  doi: 10.1007/s10955-010-0044-3. [17] M. A. Gallis, J. R. Torczyinski and D. J. Rader, An approach for simulating the transport of spherical particles in a rarefied gas flow via the direct simulation Monte-Carlo method, Phys. Fluids, 13 (2001), 3482-3492.  doi: 10.1063/1.1409367. [18] D. Gérard-Varet and M. Hillairet, Regularity issues in the problem of fluid structure interaction, Arch. Ration. Mech. Anal., 195 (2010), 375-407.  doi: 10.1007/s00205-008-0202-9. [19] F. Golse, Fluid dynamic limits of the kinetic theory of gases, From Particle Systems to Partial Differential Equations, 75 (2013), 3-91.  doi: 10.1007/978-3-642-54271-8_1. [20] T. Goudon, P.-E. Jabin and A. Vasseur, Hydrodynamic limit for the Vlasov-Navier-Stokes equations. I. Light particles regime, Indiana Univ. Math. J., 53 (2004), 1495-1515.  doi: 10.1512/iumj.2004.53.2508. [21] T. Goudon, P.-E. Jabin and A. Vasseur, Hydrodynamic limit for the Vlasov-Navier-Stokes equations. II. Fine particles regime, Indiana Univ. Math. J., 53 (2004), 1517-1536.  doi: 10.1512/iumj.2004.53.2509. [22] K. Hamdache, Global existence and large time behaviour of solutions for the Vlasov-Stokes equations, Japan J. Indust. Appl. Math., 15 (1998), 51-74.  doi: 10.1007/BF03167396. [23] M. Hauray, Wasserstein distances for vortices approximation of Euler-type equations, Math. Models Methods Appl. Sci., 19 (2009), 1357-1384.  doi: 10.1142/S0218202509003814. [24] M. Hillairet, On the homogenization of the Stokes problem in a perforated domain, preprint, arXiv: 1604.04379 [math.AP]. [25] P.-E. Jabin and F. Otto, Identification of the dilute regime in particle sedimentation, Comm. Math. Phys., 250 (2004), 415-432.  doi: 10.1007/s00220-004-1126-3. [26] S. Klainerman and A. Majda, Compressible and incompressible fluids, Comm. Pure and Appl. Math., 35 (1982), 629-651.  doi: 10.1002/cpa.3160350503. [27] L. D. Landau and E. M. Lifshitz, Physical Kinetics. Course of Theoretical Physics, Vol. 10, Pergamon Press, 1981. [28] P.-L. Lions, Mathematical Topics in Fluid Mechanics, Vol. 1. Incompressible Models, Oxford University Press Inc., New York, 1996. [29] P.-L. Lions and N. Masmoudi, Incompressible limit for a compressible fluid, J. Math. Pures Appl., 77 (1998), 585-627.  doi: 10.1016/S0021-7824(98)80139-6. [30] V. A. L'vov and E. Ya. Khruslov, Perturbation of a viscous incompressible fluid by small particles, (Russian), Theor. Appl. Quest. Differ. Equ. Algebra, 267 (1978), 173-177. [31] G. de Rham, Differentiable Manifolds: Forms, Currents, Harmonic Forms Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-61752-2. [32] Y. Sone, Molecular Gas Dynamics. Theory, Techniques and Applications Birkhäuser, Boston, 2007. doi: 10.1007/978-0-8176-4573-1. [33] S. Taguchi, On the drag exerted on the sphere by a slow uniform flow of a rarefied gas, Proc. of the 29th Internat. Symp. on Rarefied Gas Dynamics, 1628 (2014), 51-59.  doi: 10.1063/1.4902574. [34] S. Taguchi, Asymptotic theory of a uniform flow of a rarefied gas past a sphere at low Mach numbers, J. Fluid Mech., 774 (2015), 363-394.  doi: 10.1017/jfm.2015.265. [35] S. Takata, Y. Sone and K. Aoki, Numerical analysis of a uniform flow of a rarefied gas past a sphere on the basis of the Boltzmann equation for hard-sphere molecules, Mathematical Analysis of Phenomena in Fluid and Plasma Dynamics, 824 (1993), 64-93.  doi: 10.1063/1.858655. [36] D. Wang and C. Yu, Global weak solution to the inhomogeneous Navier-Stokes-Vlasov equations, J. Diff. Equations, 259 (2015), 3976-4008.  doi: 10.1016/j.jde.2015.05.016. [37] C. Yu, Global weak solutions to the incompressible Navier-Stokes-Vlasov equations, J. Math. Pures Appl., 100 (2013), 275-293.  doi: 10.1016/j.matpur.2013.01.001.

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##### References:
 [1] G. Allaire, Homogenization of the Navier-Stokes equations in open sets perforated with tiny holes. I. Abstract framework, a volume distribution of holes, Arch. Rational Mech. Anal., 113 (1990), 209-259.  doi: 10.1007/BF00375065. [2] C. Bardos, F. Golse and C. D. Levermore, Fluid dynamic limits of kinetic equations. I. Formal derivations, J. Stat. Phys., 63 (1991), 323-344.  doi: 10.1007/BF01026608. [3] E. Bernard, L. Desvillettes, F. Golse and V. Ricci, A derivation of the Vlasov-Navier-Stokes model for aerosol flows from kinetic theory, Commun. Math. Sci., 15 (2017), 1703-1741.  doi: 10.4310/CMS.2017.v15.n6.a11. [4] J. A. Carrillo, Y.-P. Choi and T. K. Karper, On the analysis of a coupled kinetic-fluid model with local alignment forces, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 273-307.  doi: 10.1016/j.anihpc.2014.10.002. [5] C. Cercignani, Theory and Applications of the Boltzmann Equation, Elsevier, New York, 1975. [6] F. Charles, Kinetic modelling and numerical simulations using particle methods for the transport of dust in a rarefied gas, Proceedings of the 26th International Symposium on Rarefied Gas Dynamics, AIP Conf. Proc, 1084 (2009), 409-414.  doi: 10.1063/1.3076512. [7] F. Charles, Modélisation Mathématique et Étude Numérique d'un Aérosol dans un Gaz Raréfié. Application á la Simulation du Transport de Particules de Poussiére en Cas d'Accident de Perte de Vide dans ITER, Ph.D thesis, ENS Cachan, 2009. [8] F. Charles, S. Dellacherie and J. Segré, Kinetic modeling of the transport of dust particles in a rarefied atmosphere Math. Models Methods Appl. Sci. 22 (2012), 1150021, 60 pp. doi: 10.1142/S0218202511500217. [9] Y.-P. Choi, Finite-time blow-up phenomena of Vlasov/Navier-Stokes equations and related systems J. Math. Pures Appl. (2017). doi: 10.1016/j.matpur.2017.05.019. [10] Y.-P. Choi and B. Kwon, Global well-posedness and large-time behavior for the inhomogeneous Vlasov-Navier-Stokes equations, Nonlinearity, 28 (2015), 3309-3336.  doi: 10.1088/0951-7715/28/9/3309. [11] D. Cioranescu and F. Murat, Un terme étrange venu d'ailleurs, Nonlinear Partial Differential Equations and their Applications, 60 (1982), 98-138. [12] P. Degond and B. Lucquin-Desreux, The asymptotics of collision operators for two species of particles of disparate masses, Math. Models Meth. Appl. Sci., 6 (1996), 405-436.  doi: 10.1142/S0218202596000158. [13] B. Desjardins and M. J. Esteban, Existence of weak solutions for the motion of rigid bodies in a viscous fluid, Arch. Ration. Mech. Anal., 146 (1999), 59-71.  doi: 10.1007/s002050050136. [14] L. Desvillettes and F. Golse, A remark concerning the Chapman-Enskog asymptotics, Advances in Kinetic Theory and Computing, Series on Advances in Mathematics for Applied Sciences, 22 (1994), 191-203. [15] L. Desvillettes, F. Golse and V. Ricci, The mean-field limit for solid particles in a Navier-Stokes flow, J. Stat. Phys., 131 (2008), 941-967.  doi: 10.1007/s10955-008-9521-3. [16] L. Desvillettes and J. Mathiaud, Some aspects of the asymptotics leading from gas-particles equations towards multiphase flows equations, J. Stat. Phys., 141 (2010), 120-141.  doi: 10.1007/s10955-010-0044-3. [17] M. A. Gallis, J. R. Torczyinski and D. J. Rader, An approach for simulating the transport of spherical particles in a rarefied gas flow via the direct simulation Monte-Carlo method, Phys. Fluids, 13 (2001), 3482-3492.  doi: 10.1063/1.1409367. [18] D. Gérard-Varet and M. Hillairet, Regularity issues in the problem of fluid structure interaction, Arch. Ration. Mech. Anal., 195 (2010), 375-407.  doi: 10.1007/s00205-008-0202-9. [19] F. Golse, Fluid dynamic limits of the kinetic theory of gases, From Particle Systems to Partial Differential Equations, 75 (2013), 3-91.  doi: 10.1007/978-3-642-54271-8_1. [20] T. Goudon, P.-E. Jabin and A. Vasseur, Hydrodynamic limit for the Vlasov-Navier-Stokes equations. I. Light particles regime, Indiana Univ. Math. J., 53 (2004), 1495-1515.  doi: 10.1512/iumj.2004.53.2508. [21] T. Goudon, P.-E. Jabin and A. Vasseur, Hydrodynamic limit for the Vlasov-Navier-Stokes equations. II. Fine particles regime, Indiana Univ. Math. J., 53 (2004), 1517-1536.  doi: 10.1512/iumj.2004.53.2509. [22] K. Hamdache, Global existence and large time behaviour of solutions for the Vlasov-Stokes equations, Japan J. Indust. Appl. Math., 15 (1998), 51-74.  doi: 10.1007/BF03167396. [23] M. Hauray, Wasserstein distances for vortices approximation of Euler-type equations, Math. Models Methods Appl. Sci., 19 (2009), 1357-1384.  doi: 10.1142/S0218202509003814. [24] M. Hillairet, On the homogenization of the Stokes problem in a perforated domain, preprint, arXiv: 1604.04379 [math.AP]. [25] P.-E. Jabin and F. Otto, Identification of the dilute regime in particle sedimentation, Comm. Math. Phys., 250 (2004), 415-432.  doi: 10.1007/s00220-004-1126-3. [26] S. Klainerman and A. Majda, Compressible and incompressible fluids, Comm. Pure and Appl. Math., 35 (1982), 629-651.  doi: 10.1002/cpa.3160350503. [27] L. D. Landau and E. M. Lifshitz, Physical Kinetics. Course of Theoretical Physics, Vol. 10, Pergamon Press, 1981. [28] P.-L. Lions, Mathematical Topics in Fluid Mechanics, Vol. 1. Incompressible Models, Oxford University Press Inc., New York, 1996. [29] P.-L. Lions and N. Masmoudi, Incompressible limit for a compressible fluid, J. Math. Pures Appl., 77 (1998), 585-627.  doi: 10.1016/S0021-7824(98)80139-6. [30] V. A. L'vov and E. Ya. Khruslov, Perturbation of a viscous incompressible fluid by small particles, (Russian), Theor. Appl. Quest. Differ. Equ. Algebra, 267 (1978), 173-177. [31] G. de Rham, Differentiable Manifolds: Forms, Currents, Harmonic Forms Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-61752-2. [32] Y. Sone, Molecular Gas Dynamics. Theory, Techniques and Applications Birkhäuser, Boston, 2007. doi: 10.1007/978-0-8176-4573-1. [33] S. Taguchi, On the drag exerted on the sphere by a slow uniform flow of a rarefied gas, Proc. of the 29th Internat. Symp. on Rarefied Gas Dynamics, 1628 (2014), 51-59.  doi: 10.1063/1.4902574. [34] S. Taguchi, Asymptotic theory of a uniform flow of a rarefied gas past a sphere at low Mach numbers, J. Fluid Mech., 774 (2015), 363-394.  doi: 10.1017/jfm.2015.265. [35] S. Takata, Y. Sone and K. Aoki, Numerical analysis of a uniform flow of a rarefied gas past a sphere on the basis of the Boltzmann equation for hard-sphere molecules, Mathematical Analysis of Phenomena in Fluid and Plasma Dynamics, 824 (1993), 64-93.  doi: 10.1063/1.858655. [36] D. Wang and C. Yu, Global weak solution to the inhomogeneous Navier-Stokes-Vlasov equations, J. Diff. Equations, 259 (2015), 3976-4008.  doi: 10.1016/j.jde.2015.05.016. [37] C. Yu, Global weak solutions to the incompressible Navier-Stokes-Vlasov equations, J. Math. Pures Appl., 100 (2013), 275-293.  doi: 10.1016/j.matpur.2013.01.001.
 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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