Angular moments models for rarefied gas dynamics. Numerical comparisons with kinetic and Navier-Stokes equations

Sébastien Guisset

doi:10.3934/krm.2020025

Previous Article
Slow flocking dynamics of the Cucker-Smale ensemble with a chemotactic movement in a temperature field
KRM Home
This Issue
Next Article
Strong solutions for the Alber equation and stability of unidirectional wave spectra

August 2020, 13(4): 739-758. doi: 10.3934/krm.2020025

Angular moments models for rarefied gas dynamics. Numerical comparisons with kinetic and Navier-Stokes equations

Sébastien Guisset ^,

CEA-DAM Ile-de-France, France

^* Corresponding author: Sébastien Guisset

Received August 2019 Revised January 2020 Published May 2020

Angular moments models based on a minimum entropy problem have been largely used to describe the transport of photons [14] or charged particles [18]. In this communication the $ M_1 $ and $ M_2 $ angular moments models are presented for rarefied gas dynamics applications. After introducing the models studied, numerical simulations carried out in various collisional regimes are presented and illustrate the interest in considering angular moments models for rarefied gas dynamics applications. For each numerical test cases, the differences observed between the angular moments models and the well-known Navier-Stokes equations are discussed and compared with reference kinetic solutions.

Keywords: Angular moments models, minimum entropy closure, rarefied gas dynamics, Navier-Stokes equations, kinetic equations.

Mathematics Subject Classification: Primary: 76P, 76N; Secondary: 65D, 65C.

Citation: Sébastien Guisset. Angular moments models for rarefied gas dynamics. Numerical comparisons with kinetic and Navier-Stokes equations. Kinetic & Related Models, 2020, 13 (4) : 739-758. doi: 10.3934/krm.2020025

References:

[1]	G. W. Alldredge, C. D. Hauck and A. L. Tits, High-order entropy-based closures for linear transport in slab geometry Ⅱ: A computational study of the optimization problem, SIAM J. Sci. Comput., 34 (2012), B361-B391. doi: 10.1137/11084772X. Google Scholar
[2]	C. Berthon, P. Charrier and B. Dubroca, An HLLC Scheme to Solve The M1 Model of Radiative Transfer in Two Space Dimensions, J. Sci. Comput., 31 (2007), 347-389. doi: 10.1007/s10915-006-9108-6. Google Scholar
[3]	F. Bouchut, Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws, and Well-Balanced Schemes for Sources, Birkhäuser Verlag, Basel, (2004). doi: 10.1007/b93802. Google Scholar
[4]	S. Brull and L. Mieussens, Local discrete velocity grids for deterministic rarefied flow simulations, J. Comput. Phys., 266 (2014), 22-46. doi: 10.1016/j.jcp.2014.01.050. Google Scholar
[5]	S. Chapman, On the law of distribution of molecular velocities, and on the theory of viscosity and thermal conduction, in a non-uniform simple monatomic gas, Phil. Trans. Roy. Soc. London Ser. A, 216 (1916), 538-548. doi: 10.1098/rsta.1916.0006. Google Scholar
[6]	S. Chapman and T. G. Cowling, The Mathematical Theory of Non-Uniform Gases, Cambridge University Press, Cambridge, 1939. Google Scholar
[7]	P. Charrier, B. Dubroca, G. Duffa, and R. Turpault, Multigroup model for radiating flows during atmospheric hypersonic re-entry, in Proceedings of International Workshop on Radiation of High Temperature Gases in Atmospheric Entry, Lisbonne, Portugal, 2003,103–110. Google Scholar
[8]	A. Decoster, Personal communication, (2018)., Google Scholar
[9]	B. Dubroca, J.-L. Feugeas and M. Frank, Angular moment model for the Fokker-Planck equation, Eur. Phys. J. D, 60, (2010) 302–307. doi: 10.1140/epjd/e2010-00190-8. Google Scholar
[10]	B. Dubroca and J. L. Feugeas, Étude théorique et numérique d'une hiéarchie de modèles aux moments pour le transfert radiatif, C. R. Acad. Sci. Paris Ser. I Math., 329 (1999), 915-920. doi: 10.1016/S0764-4442(00)87499-6. Google Scholar
[11]	R. Duclous, B. Dubroca and M. Frank, Deterministic partial differential equation model for dose calculation in electron radiotherapy, Phys. Med. Biol., 55 (2010), 3843-3857. doi: 10.1088/0031-9155/55/13/018. Google Scholar
[12]	D. Enskog, Kinetische Theorie der Vorgänge in Mässig Verdünnten Gasen, Uppsala, 1917. Google Scholar
[13]	F. Filbet and T. Rey, A hierarchy of hybrid numerical methods for multi-scale kinetic equation, SIAM J. Sci. Computing, 37 (2015), A1218–A1247. doi: 10.1137/140958773. Google Scholar
[14]	M. González, E. Audit and P. Huynh, Heracles: A three-dimensional radiation hydrodynamics code, A and A, 464 (2007), 429-435. Google Scholar
[15]	H. Grad, On the kinetic theory of rarefied gases, Commun. Pure Appl. Math., 2 (1949), 331-407. doi: 10.1002/cpa.3160020403. Google Scholar
[16]	E. P. Gross, P. L. Bathnagar and M. Krook, A model for collision processes in gases. Ⅰ. Small amplitude processes in charged and neutral one-component systems, Phys. Rev., 94 (1954), 511. doi: 10.1103/PhysRev.94.511. Google Scholar
[17]	S. Guisset, D. Aregba, S. Brull and B. Dubroca, The M1 angular moments model in a velocity-adaptive frame for rarefied gas dynamics applications, Multiscale Model. Simul., 15 (2017), 1719-1747. doi: 10.1137/16M1099327. Google Scholar
[18]	S. Guisset, S. Brull, E. d'Humières, B. Dubroca and V. Tikhonchuk, Classical transport theory for the collisional electronic M1 model, Phys. A, 446 (2016), 182-194. doi: 10.1016/j.physa.2015.12.001. Google Scholar
[19]	S. Guisset, S. Brull, B. Dubroca, E. d'Humières, S. Karpov and I. Potapenko, Asymptotic-preserving scheme for the M1-Maxwell system in the quasi-neutral regime, Commun. Comput. Phys., 19 (2016), 301-328. doi: 10.4208/cicp.131014.030615a. Google Scholar
[20]	S. Guisset, J. G. Moreau, R. Nuter, S. Brull, E. d'Humieres, B. Dubroca and V. T. Tikhonchuk, Limits of the M1 and M2 angular moments models for kinetic plasma physics studies, J. Phys. A, 48 (2015), 335501. doi: 10.1088/1751-8113/48/33/335501. Google Scholar
[21]	A. Harten, P. D. Lax and B. Van Leer, On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM Rev., 25 (1983), 35-61. doi: 10.1137/1025002. Google Scholar
[22]	P. Helluy, M. Massaro, L. Navoret, N. Pham and T. Strub, Reduced Vlasov-Maxwell modeling, PIERS Proceedings, (2014), 2622–2627. Google Scholar
[23]	F. Hermeline, A finite volume method for the approximation of the PN Boltzmann and similar systems of equations on general meshes, CEA Report, (2015). Google Scholar
[24]	M. Junk and A. Unterreiter, Maximum entropy moment systems and Galilean invariance, Contin. Mech. Thermodyn., 14 (2002), 563-576. doi: 10.1007/s00161-002-0096-y. Google Scholar
[25]	B. Van Leer, Towards the ultimate conservative difference scheme Ⅲ. Upstream-centered finite-difference schemes for ideal compressible flow, J. Comput. Phys., 23 (1977), 263-275. doi: 10.1016/0021-9991(77)90094-8. Google Scholar
[26]	C. D. Levermore, Moment closure hierarchies for kinetic theories, J. Stat. Phys., 83 (1996), 1021-1065. doi: 10.1007/BF02179552. Google Scholar
[27]	J. Mallet, S. Brull and B. Dubroca, General moment system for plasma physics based on minimum entropy principle, Kinet. Relat. Models, 8 (2015), 533-558. doi: 10.3934/krm.2015.8.533. Google Scholar
[28]	J. McDonald and M. Torrilhon, An affordable robust moment closures for CFD based on the maximum-entropy hierarchy, J. Comput. Phys., 251 (2013), 500-523. doi: 10.1016/j.jcp.2013.05.046. Google Scholar
[29]	J. G. McDonald and C. P. T. Groth, Towards realizable hyperbolic moment closures for viscous heat-conducting gas flows based on a maximum-entropy distribution, Contin. Mech. Thermodyn., 25 (2012), 573-603. doi: 10.1007/s00161-012-0252-y. Google Scholar
[30]	L. Mieussens, Discrete velocity model and implicit scheme for the BGK equation of rarefied gas dynamics, Math. Models Methods Appl. Sci., 10 (2000), 1121-1149. doi: 10.1142/S0218202500000562. Google Scholar
[31]	G. N. Minerbo, Maximum entropy eddington factors, J. Quant. Spectrosc. Radiat. Trans., 20 (1978), 541-545. doi: 10.1016/0022-4073(78)90024-9. Google Scholar
[32]	I. Muller and T. Ruggeri, Rational Extended Thermodynamics, Springer-Verlag, New York, NY, 1998. doi: 10.1007/978-1-4612-2210-1. Google Scholar
[33]	G. C. Pomraning, Maximum entropy Eddington factors and flux limited diffusion theory, J. Quant. Spectrosc. Radiat. Trans., 26 (1981), 385-388. doi: 10.1016/0022-4073(81)90101-1. Google Scholar
[34]	J.-F. Ripoll, An averaged formulation of the M1 radiation model with presumed probability density function for turbulent flows, J. Quant. Spectrosc. Radiat. Trans., 83 (2004), 493-517. Google Scholar
[35]	J.-F. Ripoll, B. Dubroca and E. Audit, A factored operator method for solving coupled radiation-hydrodynamics models, Transport Theory Statist. Phys., 31 (2002), 531-557. doi: 10.1081/TT-120015513. Google Scholar
[36]	J. Schneider, Entropic approximation in kinetic theory, M2AN Math. Model. Numer. Anal., 38 (2004), 541-561. doi: 10.1051/m2an:2004025. Google Scholar
[37]	H. Struchtrup, Macroscopic Transport Equations for Rarefied Gas Flows, Springer, Berlin, (2005). Google Scholar
[38]	B. Su, More on boundary conditions for differential approximations, J. Quant. Spectrosc. Radiat. Trans., 64 (2000), 409-419. doi: 10.1016/S0022-4073(99)00128-4. Google Scholar
[39]	T. Pichard, PhD thesis, 2016., Google Scholar
[40]	M. Torrilhon, Modeling nonequilibrium gas flow based on moment equations, Annu. Rev. Fluid Mech., 48 (2016), 429-458. Google Scholar
[41]	R. Turpault, A consistent multigroup model for radiative transfer and its underlying mean opacity, J. Quant. Spectrosc. Radiat. Trans., 94 (2005), 357-371. doi: 10.1016/j.jqsrt.2004.09.042. Google Scholar
[42]	R. Turpault, M. Frank, B. Dubroca and A. Klar, Multigroup half space moment approximations to the radiative heat transfer equations, J. Comput. Phys., 198 (2004), 363-371. Google Scholar

show all references

References:

[1]	G. W. Alldredge, C. D. Hauck and A. L. Tits, High-order entropy-based closures for linear transport in slab geometry Ⅱ: A computational study of the optimization problem, SIAM J. Sci. Comput., 34 (2012), B361-B391. doi: 10.1137/11084772X. Google Scholar
[2]	C. Berthon, P. Charrier and B. Dubroca, An HLLC Scheme to Solve The M1 Model of Radiative Transfer in Two Space Dimensions, J. Sci. Comput., 31 (2007), 347-389. doi: 10.1007/s10915-006-9108-6. Google Scholar
[3]	F. Bouchut, Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws, and Well-Balanced Schemes for Sources, Birkhäuser Verlag, Basel, (2004). doi: 10.1007/b93802. Google Scholar
[4]	S. Brull and L. Mieussens, Local discrete velocity grids for deterministic rarefied flow simulations, J. Comput. Phys., 266 (2014), 22-46. doi: 10.1016/j.jcp.2014.01.050. Google Scholar
[5]	S. Chapman, On the law of distribution of molecular velocities, and on the theory of viscosity and thermal conduction, in a non-uniform simple monatomic gas, Phil. Trans. Roy. Soc. London Ser. A, 216 (1916), 538-548. doi: 10.1098/rsta.1916.0006. Google Scholar
[6]	S. Chapman and T. G. Cowling, The Mathematical Theory of Non-Uniform Gases, Cambridge University Press, Cambridge, 1939. Google Scholar
[7]	P. Charrier, B. Dubroca, G. Duffa, and R. Turpault, Multigroup model for radiating flows during atmospheric hypersonic re-entry, in Proceedings of International Workshop on Radiation of High Temperature Gases in Atmospheric Entry, Lisbonne, Portugal, 2003,103–110. Google Scholar
[8]	A. Decoster, Personal communication, (2018)., Google Scholar
[9]	B. Dubroca, J.-L. Feugeas and M. Frank, Angular moment model for the Fokker-Planck equation, Eur. Phys. J. D, 60, (2010) 302–307. doi: 10.1140/epjd/e2010-00190-8. Google Scholar
[10]	B. Dubroca and J. L. Feugeas, Étude théorique et numérique d'une hiéarchie de modèles aux moments pour le transfert radiatif, C. R. Acad. Sci. Paris Ser. I Math., 329 (1999), 915-920. doi: 10.1016/S0764-4442(00)87499-6. Google Scholar
[11]	R. Duclous, B. Dubroca and M. Frank, Deterministic partial differential equation model for dose calculation in electron radiotherapy, Phys. Med. Biol., 55 (2010), 3843-3857. doi: 10.1088/0031-9155/55/13/018. Google Scholar
[12]	D. Enskog, Kinetische Theorie der Vorgänge in Mässig Verdünnten Gasen, Uppsala, 1917. Google Scholar
[13]	F. Filbet and T. Rey, A hierarchy of hybrid numerical methods for multi-scale kinetic equation, SIAM J. Sci. Computing, 37 (2015), A1218–A1247. doi: 10.1137/140958773. Google Scholar
[14]	M. González, E. Audit and P. Huynh, Heracles: A three-dimensional radiation hydrodynamics code, A and A, 464 (2007), 429-435. Google Scholar
[15]	H. Grad, On the kinetic theory of rarefied gases, Commun. Pure Appl. Math., 2 (1949), 331-407. doi: 10.1002/cpa.3160020403. Google Scholar
[16]	E. P. Gross, P. L. Bathnagar and M. Krook, A model for collision processes in gases. Ⅰ. Small amplitude processes in charged and neutral one-component systems, Phys. Rev., 94 (1954), 511. doi: 10.1103/PhysRev.94.511. Google Scholar
[17]	S. Guisset, D. Aregba, S. Brull and B. Dubroca, The M1 angular moments model in a velocity-adaptive frame for rarefied gas dynamics applications, Multiscale Model. Simul., 15 (2017), 1719-1747. doi: 10.1137/16M1099327. Google Scholar
[18]	S. Guisset, S. Brull, E. d'Humières, B. Dubroca and V. Tikhonchuk, Classical transport theory for the collisional electronic M1 model, Phys. A, 446 (2016), 182-194. doi: 10.1016/j.physa.2015.12.001. Google Scholar
[19]	S. Guisset, S. Brull, B. Dubroca, E. d'Humières, S. Karpov and I. Potapenko, Asymptotic-preserving scheme for the M1-Maxwell system in the quasi-neutral regime, Commun. Comput. Phys., 19 (2016), 301-328. doi: 10.4208/cicp.131014.030615a. Google Scholar
[20]	S. Guisset, J. G. Moreau, R. Nuter, S. Brull, E. d'Humieres, B. Dubroca and V. T. Tikhonchuk, Limits of the M1 and M2 angular moments models for kinetic plasma physics studies, J. Phys. A, 48 (2015), 335501. doi: 10.1088/1751-8113/48/33/335501. Google Scholar
[21]	A. Harten, P. D. Lax and B. Van Leer, On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM Rev., 25 (1983), 35-61. doi: 10.1137/1025002. Google Scholar
[22]	P. Helluy, M. Massaro, L. Navoret, N. Pham and T. Strub, Reduced Vlasov-Maxwell modeling, PIERS Proceedings, (2014), 2622–2627. Google Scholar
[23]	F. Hermeline, A finite volume method for the approximation of the PN Boltzmann and similar systems of equations on general meshes, CEA Report, (2015). Google Scholar
[24]	M. Junk and A. Unterreiter, Maximum entropy moment systems and Galilean invariance, Contin. Mech. Thermodyn., 14 (2002), 563-576. doi: 10.1007/s00161-002-0096-y. Google Scholar
[25]	B. Van Leer, Towards the ultimate conservative difference scheme Ⅲ. Upstream-centered finite-difference schemes for ideal compressible flow, J. Comput. Phys., 23 (1977), 263-275. doi: 10.1016/0021-9991(77)90094-8. Google Scholar
[26]	C. D. Levermore, Moment closure hierarchies for kinetic theories, J. Stat. Phys., 83 (1996), 1021-1065. doi: 10.1007/BF02179552. Google Scholar
[27]	J. Mallet, S. Brull and B. Dubroca, General moment system for plasma physics based on minimum entropy principle, Kinet. Relat. Models, 8 (2015), 533-558. doi: 10.3934/krm.2015.8.533. Google Scholar
[28]	J. McDonald and M. Torrilhon, An affordable robust moment closures for CFD based on the maximum-entropy hierarchy, J. Comput. Phys., 251 (2013), 500-523. doi: 10.1016/j.jcp.2013.05.046. Google Scholar
[29]	J. G. McDonald and C. P. T. Groth, Towards realizable hyperbolic moment closures for viscous heat-conducting gas flows based on a maximum-entropy distribution, Contin. Mech. Thermodyn., 25 (2012), 573-603. doi: 10.1007/s00161-012-0252-y. Google Scholar
[30]	L. Mieussens, Discrete velocity model and implicit scheme for the BGK equation of rarefied gas dynamics, Math. Models Methods Appl. Sci., 10 (2000), 1121-1149. doi: 10.1142/S0218202500000562. Google Scholar
[31]	G. N. Minerbo, Maximum entropy eddington factors, J. Quant. Spectrosc. Radiat. Trans., 20 (1978), 541-545. doi: 10.1016/0022-4073(78)90024-9. Google Scholar
[32]	I. Muller and T. Ruggeri, Rational Extended Thermodynamics, Springer-Verlag, New York, NY, 1998. doi: 10.1007/978-1-4612-2210-1. Google Scholar
[33]	G. C. Pomraning, Maximum entropy Eddington factors and flux limited diffusion theory, J. Quant. Spectrosc. Radiat. Trans., 26 (1981), 385-388. doi: 10.1016/0022-4073(81)90101-1. Google Scholar
[34]	J.-F. Ripoll, An averaged formulation of the M1 radiation model with presumed probability density function for turbulent flows, J. Quant. Spectrosc. Radiat. Trans., 83 (2004), 493-517. Google Scholar
[35]	J.-F. Ripoll, B. Dubroca and E. Audit, A factored operator method for solving coupled radiation-hydrodynamics models, Transport Theory Statist. Phys., 31 (2002), 531-557. doi: 10.1081/TT-120015513. Google Scholar
[36]	J. Schneider, Entropic approximation in kinetic theory, M2AN Math. Model. Numer. Anal., 38 (2004), 541-561. doi: 10.1051/m2an:2004025. Google Scholar
[37]	H. Struchtrup, Macroscopic Transport Equations for Rarefied Gas Flows, Springer, Berlin, (2005). Google Scholar
[38]	B. Su, More on boundary conditions for differential approximations, J. Quant. Spectrosc. Radiat. Trans., 64 (2000), 409-419. doi: 10.1016/S0022-4073(99)00128-4. Google Scholar
[39]	T. Pichard, PhD thesis, 2016., Google Scholar
[40]	M. Torrilhon, Modeling nonequilibrium gas flow based on moment equations, Annu. Rev. Fluid Mech., 48 (2016), 429-458. Google Scholar
[41]	R. Turpault, A consistent multigroup model for radiative transfer and its underlying mean opacity, J. Quant. Spectrosc. Radiat. Trans., 94 (2005), 357-371. doi: 10.1016/j.jqsrt.2004.09.042. Google Scholar
[42]	R. Turpault, M. Frank, B. Dubroca and A. Klar, Multigroup half space moment approximations to the radiative heat transfer equations, J. Comput. Phys., 198 (2004), 363-371. Google Scholar

Figure 1. Density (top left), speed (top right), temperature (bottom left), heat flux (bottom right) profiles in the case $ \tau = 1/\nu = 0 $ (fluid regime) at time $ t = 15 $

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2. Representation of the density (top left), speed (top right), temperature (bottom left), heat flux (bottom right) in the case $ \tau = 1/\nu = 1 $ (close fluid regime) at time $ t = 5 $

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3. Representation of the density (top left), speed (top right), temperature (bottom left), heat flux (bottom right) in the case $ \tau = 1/\nu = 100 $ (rarefied regime) at time $ t = 5 $

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4. Representation of the density (top left), speed (top right), temperature (bottom left), heat flux (bottom right) in the case $ \tau = 1/\nu = 10^{4} $ (strongly rarefied regime) at time $ t = 5 $

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5. Representation of the density (top left), speed (top right), temperature (bottom left), heat flux (bottom right) in the case $ \tau = 1/\nu = 0 $ (fluid regime) at time $ t = 0.25 $

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6. Representation of the density (top left), speed (top right), temperature (bottom left), heat flux (bottom right) in the case $ \tau = 1/\nu = 10^{-2} $ (close of fluid regime) at time $ t = 0.1 $

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 7. Representation of the density (top left), speed (top right), temperature (bottom left), heat flux (bottom right) in the case $ \tau = 1/\nu = 10^{-1} $ at time $ t = 0.1 $

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 8. Representation of the density (top left), speed (top right), temperature (bottom left), heat flux (bottom right) in the case $ \tau = 1/\nu = 1 $ (intermediate regimes) at time $ t = 0.1 $

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 9. Representation of the density (top left), speed (top right), temperature (bottom left), heat flux (bottom right) in the case $ \tau = 1/\nu = 0 $ (fluid regime) at time $ t = 0.2 $

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 10. Representation of the density (top left), speed (top right), temperature (bottom left), heat flux (bottom right) in the case $ \tau = 1/\nu = 10^{-1} $ at time $ t = 0.1 $

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 11. Representation of the density (top left), speed (top right), temperature (bottom left), heat flux (bottom right) in the case $ \tau = 1/\nu = 1 $ at time $ t = 0.1 $

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 12. Representation of the inverse of the shock thickness as function of the Mach number (speed of the in-going flow)

Figure Options

Download full-size image

Download as PowerPoint slide

[1]	Pavel I. Plotnikov, Jan Sokolowski. Optimal shape control of airfoil in compressible gas flow governed by Navier-Stokes equations. Evolution Equations & Control Theory, 2013, 2 (3) : 495-516. doi: 10.3934/eect.2013.2.495
[2]	Pavel I. Plotnikov, Jan Sokolowski. Compressible Navier-Stokes equations. Conference Publications, 2009, 2009 (Special) : 602-611. doi: 10.3934/proc.2009.2009.602
[3]	Jan W. Cholewa, Tomasz Dlotko. Fractional Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 2967-2988. doi: 10.3934/dcdsb.2017149
[4]	Alexander V. Bobylev, Sergey V. Meleshko. On group symmetries of the hydrodynamic equations for rarefied gas. Kinetic & Related Models, 2021, 14 (3) : 469-482. doi: 10.3934/krm.2021012
[5]	Yong Yang, Bingsheng Zhang. On the Kolmogorov entropy of the weak global attractor of 3D Navier-Stokes equations:Ⅰ. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2339-2350. doi: 10.3934/dcdsb.2017101
[6]	Hermenegildo Borges de Oliveira. Anisotropically diffused and damped Navier-Stokes equations. Conference Publications, 2015, 2015 (special) : 349-358. doi: 10.3934/proc.2015.0349
[7]	Hyukjin Kwean. Kwak transformation and Navier-Stokes equations. Communications on Pure & Applied Analysis, 2004, 3 (3) : 433-446. doi: 10.3934/cpaa.2004.3.433
[8]	Vittorino Pata. On the regularity of solutions to the Navier-Stokes equations. Communications on Pure & Applied Analysis, 2012, 11 (2) : 747-761. doi: 10.3934/cpaa.2012.11.747
[9]	C. Foias, M. S Jolly, I. Kukavica, E. S. Titi. The Lorenz equation as a metaphor for the Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2001, 7 (2) : 403-429. doi: 10.3934/dcds.2001.7.403
[10]	Igor Kukavica. On regularity for the Navier-Stokes equations in Morrey spaces. Discrete & Continuous Dynamical Systems, 2010, 26 (4) : 1319-1328. doi: 10.3934/dcds.2010.26.1319
[11]	Igor Kukavica. On partial regularity for the Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2008, 21 (3) : 717-728. doi: 10.3934/dcds.2008.21.717
[12]	Susan Friedlander, Nataša Pavlović. Remarks concerning modified Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2004, 10 (1&2) : 269-288. doi: 10.3934/dcds.2004.10.269
[13]	Hakima Bessaih, Benedetta Ferrario. Statistical properties of stochastic 2D Navier-Stokes equations from linear models. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 2927-2947. doi: 10.3934/dcdsb.2016080
[14]	Ben Duan, Zhen Luo. Dynamics of vacuum states for one-dimensional full compressible Navier-Stokes equations. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2543-2564. doi: 10.3934/cpaa.2013.12.2543
[15]	Fang Li, Bo You, Yao Xu. Dynamics of weak solutions for the three dimensional Navier-Stokes equations with nonlinear damping. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4267-4284. doi: 10.3934/dcdsb.2018137
[16]	Xin-Guang Yang, Rong-Nian Wang, Xingjie Yan, Alain Miranville. Dynamics of the 2D Navier-Stokes equations with sublinear operators in Lipschitz-like domains. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3343-3366. doi: 10.3934/dcds.2020408
[17]	Pan Zhang, Lan Huang, Rui Lu, Xin-Guang Yang. Pullback dynamics of a 3D modified Navier-Stokes equations with double delays. Electronic Research Archive, 2021, 29 (6) : 4137-4157. doi: 10.3934/era.2021076
[18]	Jean-Pierre Raymond. Stokes and Navier-Stokes equations with a nonhomogeneous divergence condition. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1537-1564. doi: 10.3934/dcdsb.2010.14.1537
[19]	Yoshikazu Giga. A remark on a Liouville problem with boundary for the Stokes and the Navier-Stokes equations. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1277-1289. doi: 10.3934/dcdss.2013.6.1277
[20]	Siegfried Maier, Jürgen Saal. Stokes and Navier-Stokes equations with perfect slip on wedge type domains. Discrete & Continuous Dynamical Systems - S, 2014, 7 (5) : 1045-1063. doi: 10.3934/dcdss.2014.7.1045

2020 Impact Factor: 1.432

Tools

Metrics

Tools

Article outline

Figures and Tables

2020 Impact Factor: 1.432

Tools

Figures and Tables

2020 Impact Factor: 1.432

American Institute of Mathematical Sciences

Angular moments models for rarefied gas dynamics. Numerical comparisons with kinetic and Navier-Stokes equations

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

Angular moments models for rarefied gas dynamics. Numerical comparisons with kinetic and Navier-Stokes equations

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors