On the modeling of the transport of particles in turbulent flows
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 38 (2004) no. 4, p. 673-690

We investigate different asymptotic regimes for Vlasov equations modeling the evolution of a cloud of particles in a turbulent flow. In one case we obtain a convection or a convection-diffusion effective equation on the concentration of particles. In the second case, the effective model relies on a Vlasov-Fokker-Planck equation.

DOI : https://doi.org/10.1051/m2an:2004032
Classification:  35B25,  35Q99
Keywords: fluid-particles interaction, hydrodynamic limits, turbulence effects
@article{M2AN_2004__38_4_673_0,
     author = {Goudon, Thierry and Poupaud, Fr\'ed\'eric},
     title = {On the modeling of the transport of particles in turbulent flows},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {38},
     number = {4},
     year = {2004},
     pages = {673-690},
     doi = {10.1051/m2an:2004032},
     zbl = {1079.76037},
     mrnumber = {2087729},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2004__38_4_673_0}
}
Goudon, Thierry; Poupaud, Frédéric. On the modeling of the transport of particles in turbulent flows. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 38 (2004) no. 4, pp. 673-690. doi : 10.1051/m2an:2004032. http://www.numdam.org/item/M2AN_2004__38_4_673_0/

[1] P. Berthonnaud, Limites fluides pour des modèles cinétiques de brouillards de gouttes monodispersés. C. R. Acad. Sci. 331 (2000) 651-654. | Zbl 0965.35136

[2] M. Brassart, Limite semi-classique de transformées de Wigner dans des milieux périodiques ou aléatoires. Thèse Université de Nice-Sophia Antipolis (Novembre 2002).

[3] J.R. Brock and G.M. Hidy, The dynamics of aerocolloidal systems. Pergamon Press (1970).

[4] R. Caflisch and G. Papanicolaou, Dynamic theory of suspensions with Brownian effects. SIAM J. Appl. Math. 43 (1983) 885-906. | Zbl 0543.76133

[5] J.F. Clouet and K. Domelevo, Solutions of a kinetic stochastic equation modeling a spray in a turbulent gas flow. Math. Models Methods Appl. Sci. 7 (1997) 239-263. | Zbl 0868.60054

[6] L. Desvillettes, About the modeling of complex flows by gas-particles methods, Proceedings of the workshop “Trends in Numerical and Physical Modeling for Industrial Multiphase Flows”, Cargèse, France (2000).

[7] K. Domelevo and M.-H. Vignal, Limites visqueuses pour des systèmes de type Fokker-Planck-Burgers unidimensionnels. C. R. Acad. Sci. 332 (2001) 863-868. | Zbl 1067.76090

[8] K. Domelevo and P. Villedieu, Work in preparation. Personal communication.

[9] S. Gavrilyuck and V. Teshukhov, Kinetic model for the motion of compressible bubbles in a perfect fluid. Eur. J. Mech. B/Fluids 21 (2002) 469-491. | Zbl 1051.76620

[10] F. Golse, in From kinetic to macroscopic models in Kinetic equations and asymptotic theory, B. Perthame and L. Desvillettes Eds., Gauthier-Villars, Ser. Appl. Math. 4 (2000) 41-121. | Zbl 0979.82048

[11] T. Goudon, Asymptotic problems for a kinetic model of two-phase flow. Proc. Royal Soc. Edimburgh 131 (2001) 1371-1384. | Zbl 0992.35017

[12] T. Goudon, P.-E. Jabin and A. Vasseur, Hydrodymamic limit for the Vlasov-Navier-Stokes system: Light particles regime. Preprint.

[13] T. Goudon, P.-E. Jabin and A. Vasseur, Hydrodymamic limit for the Vlasov-Navier-Stokes system: Fine particles regime. Preprint.

[14] K. Hamdache, Global existence and large time behaviour of solutions for the Vlasov-Stokes equations. Japan J. Ind. Appl. Math. 15 (1998) 51-74. | Zbl 1306.76052

[15] H. Herrero, B. Lucquin-Desreux and B. Perthame, On the motion of dispersed balls in a potential flow: a kinetic description of the added mass effect. SIAM J. Appl. Math. 60 (1999) 61-83. | MR 1740835 | Zbl 0964.76085

[16] P.-E. Jabin, Large time concentrations for solutions to kinetic equations with energy dissipation. Comm. Partial Differential Equations 25 (2000) 541-557. | MR 1748358 | Zbl 0965.35014

[17] P.-E. Jabin, Macroscopic limit of Vlasov type equations with friction. Ann. IHP Anal. Non Linéaire 17 (2000) 651-672. | Numdam | MR 1791881 | Zbl 0965.35013

[18] P.-E. Jabin and B. Perthame, in Notes on mathematical problems on the dynamics of dispersed particles interacting through a fluid in Modeling in applied sciences, a kinetic theory approach, N. Bellomo and M. Pulvirenti Eds., Birkhäuser (2000) 111-147. | MR 1763153 | Zbl 0957.76087

[19] P. Kramer and A. Majda, Simplified models for turbulent diffusion: Theory, numerical modeling, and physical phenomena. Physics Reports 314 (1999) 237-574. | MR 1699757

[20] R. Kubo, Stochastic Liouville equations. J. Math. Phys. 4 (1963) 174-183. | MR 149885 | Zbl 0135.45102

[21] G. Loeper and A. Vasseur, Electric turbulence in a plasma subject to a strong magnetic field. Preprint. | MR 2096317 | Zbl 1080.35153

[22] P.J. O'Rourke, Statistical properties and numerical implementation of a model for droplets dispersion in a turbulent gas. J. Comp. Phys. 83 (1989) 345-360. | Zbl 0673.76065

[23] F. Poupaud and A. Vasseur, Classical and quantum transport in random media. J. Math. Pures Appl. 82 (2003) 711-748. | MR 1996779 | Zbl 1035.82037

[24] G. Russo and P. Smereka, Kinetic theory for bubbly flows I, II. SIAM J. Appl. Math. 56 (1996) 327-371. | Zbl 0857.76090

[25] C. Villani, A review of mathematical topics in collisional kinetic theory, in Handbook of mathematical fluid mechanics, S. Friedlander and D. Serre Eds., North-Holland (2002). | MR 1942465 | Zbl 1170.82369 | Zbl pre01942873

[26] F.A. Williams, Combustion theory. Benjamin Cummings Publ., 2nd edn. (1985).

[27] L.I. Zaichik, A statistical model of particle transport and heat transfer in turbulent shear flows. Phys. Fluids 11 (1999) 1521-1534. | Zbl 1147.76544