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A non-relativistic model of plasma physics containing a radiation reaction term
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 |
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.
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. |
show all references
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 |
size of the container (periodic box) | |
| number of particles |
| number of gas molecules |
| thermal speed of particles |
| thermal speed of gas molecules |
| average particle/particle cross-section |
| average particle/gas cross-section |
| average molecular cross-section |
| mass ratio (molecules/particles) |
| mass fraction (gas/dust or droplets) |
| thermal speed ratio (particles/molecules) |
Parameter | Definition |
size of the container (periodic box) | |
| number of particles |
| number of gas molecules |
| thermal speed of particles |
| thermal speed of gas molecules |
| average particle/particle cross-section |
| average particle/gas cross-section |
| average molecular cross-section |
| mass ratio (molecules/particles) |
| mass fraction (gas/dust or droplets) |
| thermal speed ratio (particles/molecules) |
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