A Fokker-Planck model of the Boltzmann equation with correct Prandtl number for polyatomic gases

Julien Mathiaud (1) , Luc Mieussens (2, 3)

1 CESTA - Centre d'études scientifiques et techniques d'Aquitaine
2 CARDAMOM - Certified Adaptive discRete moDels for robust simulAtions of CoMplex flOws with Moving fronts
3 IMB - Institut de Mathématiques de Bordeaux

We propose an extension of the Fokker-Planck model of the Boltzmann equation to get a correct Prandtl number in the Compressible Navier-Stokes asymptotics for polyatomic gases. This is obtained by replacing the diffusion coefficient (which is the equilibrium temperature) by a non diagonal temperature tensor, like the Ellipsoidal-Statistical model is obtained from the Bathnagar-Gross-Krook model of the Boltzmann equation, and by adding a diffusion term for the internal energy. Our model is proved to satisfy the properties of conservation and a H-theorem. A Chapman-Enskog analysis shows how to compute the transport coefficients of our model. Some numerical tests are performed to illustrate that a correct Prandtl number can be obtained.

Mots clés: polyatomic gas; Fokker-Planck model; Prandtl number; Ellipsoidal-Statistical model; H-theorem

Domaines: Mécanique des fluides [physics.class-ph]; Equations aux dérivées partielles [math.AP]

Julien Mathiaud, Luc Mieussens. A Fokker-Planck model of the Boltzmann equation with correct Prandtl number for polyatomic gases. Journal of Statistical Physics, 2017, 168 (5), pp.1031-1055. ⟨hal-01493376⟩