Derivation of BGK models.

Transcription

1 Derivation of BGK models. Stéphane BRULL, Vincent PAVAN, Jacques SCHNEIDER. University of Bordeaux, University of Marseille, University of Toulon September 27 th 2014 Stéphane BRULL (Bordeaux) Derivation of BGK models. September 27 th / 42

2 Motivations f(t, x, v) : distribution function, solution to the Boltzmann equation f t (t, x, v) + v xf(t, x, v) = Q(f, f)(t, x, v). Q(f, f) High complexity Construct a relaxation operator R(f) = λ(g f) Q(f, f) - Go beyond the BGK model, - As close as possible of Q(f, f), Generalization to mixtures : f i (t, x, v) ( f := (f 1,, f p )) k=p f i t (t, x, v) + v xf i (t, x, v) = Q ki (f k, f i ) λ(g i f i ). k=1 Stéphane BRULL (Bordeaux) Derivation of BGK models. September 27 th / 42

3 Motivations f(t, x, v) : distribution function, solution to the Boltzmann equation f t (t, x, v) + v xf(t, x, v) = Q(f, f)(t, x, v). Q(f, f) High complexity Construct a relaxation operator R(f) = λ(g f) Q(f, f) - Go beyond the BGK model, - As close as possible of Q(f, f), Generalization to mixtures : f i (t, x, v) ( f := (f 1,, f p )) k=p f i t (t, x, v) + v xf i (t, x, v) = Q ki (f k, f i ) λ(g i f i ). k=1 Stéphane BRULL (Bordeaux) Derivation of BGK models. September 27 th / 42

4 BGK models for gas mixtures Aim : Construct a relaxation operator for multi-species basing on (true) hydrodynamic limit and right kinetic coefficients (Fick, Soret, Duffour, Fourier, Newton). [Brull-Pavan-Schneider, 2012] Fick law. [Brull, 2015] ESBGK. Up to now : Approx. of moments exchanges of Boltzmann equation [Garzò-Santos-Brey, 1989] Pb : loss of positivity, no H theorem, uncorrect transport coefficients. One particular model : [Andries-Aoki-Perthame, 2002] Good mathematical properties : H theorem, positivity. Valid only for Maxwellian molecules uncorrect transport coefficients. ([Kosuge, 2009]) Application to reacting mixtures (Bisi, Groppi, Spiga). Stéphane BRULL (Bordeaux) Derivation of BGK models. September 27 th / 42

5 Plan 1 Monospecies case Setting of the problem Entropy minimisation problem Chapman-Enskog expansion 2 Polyatomic case 3 Mixture case Navier-Stokes for gas mixtures Chapman-Enskog expansion Derivation of the Fick-BGK model 4 Derivation an ESBGK model for gas mixtures 5 Conclusions and Perspectives Stéphane BRULL (Bordeaux) Derivation of BGK models. September 27 th / 42

6 Monospecies case Stéphane BRULL (Bordeaux) Derivation of BGK models. September 27 th / 42

7 BGK Models Relaxation operator Q(f, f) R(f) = ρ(m f). M : Maxwellian having the same moments as f w.r.t. (1, v, v 2 ). Entropy : H(g) = (g ln g g)dv. Minimization problem H(M) = min g C f H(g) C f = {g 0 s.t. R 3 1 v v 2 R g dv = 3 1 v v 2 f dv} Pb : BGK does not reproduce correct Prandtl number, Pr = 5 µ 2 κ Stéphane BRULL (Bordeaux) Derivation of BGK models. September 27 th / 42

8 Minimization principle. Aim : Find a systematic way to construct BGK models (λ(g f) ) correct transport coefficients up to Navier-Stokes. Minimization problem : m(v) given H(G) = min H(g), g C f C f = {g 0 / m(v)gdv = V( m(v)fdv)} V : R N R N Notations : P = span(m(v)), P 0 = span(1, v, v 2 ) Remark P 0 P. (P 0 = P and V = Id) BGK model Sonine polynoms : A(v) = v v 1 3 v 2 Id, B(v) = v( v ) Stéphane BRULL (Bordeaux) Derivation of BGK models. September 27 th / 42

9 Minimization principle. Aim : Find a systematic way to construct BGK models (λ(g f) ) correct transport coefficients up to Navier-Stokes. Minimization problem : m(v) given H(G) = min H(g), g C f C f = {g 0 / m(v)gdv = V( m(v)fdv)} V : R N R N Notations : P = span(m(v)), P 0 = span(1, v, v 2 ) Remark P 0 P. (P 0 = P and V = Id) BGK model Sonine polynoms : A(v) = v v 1 3 v 2 Id, B(v) = v( v ) Stéphane BRULL (Bordeaux) Derivation of BGK models. September 27 th / 42

10 Realisability problems Let V R N. Is there G 0 L 1 s.t. H(G) = min H(g) under the constraints R 3 g m(v)dv = V? CN : V corresponds to the moments of a nonnegative L 1 function Characterisation of realisability [M.Junk, 98], [J.Schneider, 2004 ] Stéphane BRULL (Bordeaux) Derivation of BGK models. September 27 th / 42

11 Realisability problems Let V R N. Is there G 0 L 1 s.t. H(G) = min H(g) under the constraints R 3 g m(v)dv = V? CN : V corresponds to the moments of a nonnegative L 1 function Characterisation of realisability [M.Junk, 98], [J.Schneider, 2004 ] Stéphane BRULL (Bordeaux) Derivation of BGK models. September 27 th / 42

12 Realisability problems Let V R N. Is there G 0 L 1 s.t. H(G) = min H(g) under the constraints R 3 g m(v)dv = V? CN : V corresponds to the moments of a nonnegative L 1 function Characterisation of realisability [M.Junk, 98], [J.Schneider, 2004 ] Stéphane BRULL (Bordeaux) Derivation of BGK models. September 27 th / 42

13 New approach with relaxation coefficients Relaxation coefficients : [Levermore, J.S.P., 1996] Problem : We obtain only Pr 1. R(f) = Σ i λ i (G i f) New approach : One unique relaxation coefficient λ > 0 and different relaxation rates (λ) i=1 N 0 s.t. λ(g f) m i (v)dv = λ i f m i (v)dv, m i P Stéphane BRULL (Bordeaux) Derivation of BGK models. September 27 th / 42

14 New approach with relaxation coefficients Relaxation coefficients : [Levermore, J.S.P., 1996] Problem : We obtain only Pr 1. R(f) = Σ i λ i (G i f) New approach : One unique relaxation coefficient λ > 0 and different relaxation rates (λ) i=1 N 0 s.t. λ(g f) m i (v)dv = λ i f m i (v)dv, m i P Stéphane BRULL (Bordeaux) Derivation of BGK models. September 27 th / 42

15 P = P 0 + v v P = P 0 A(v u), for the scalar product ϕ, ψ = Mϕψ dv Minimization problem G = min g C f H(g). (1) C f = {g 0 s.t. (1, v, v 2 ) gdv = (1, v, v 2 ) fdv, (2) R 3 R 3 λ(g f)a(v u) dv = λ 1 fa(v u)dv}. (3) R 3 R 3 Setting ν = 1 λ 1 λ (3) can be written 1 c c gdv = νθ + (1 ν)tid = T, ρ R 3 Θ = 1 ρ R 3 c c fdv. (4) Stéphane BRULL (Bordeaux) Derivation of BGK models. September 27 th / 42

16 P = P 0 + v v P = P 0 A(v u), for the scalar product ϕ, ψ = Mϕψ dv Minimization problem G = min g C f H(g). (1) C f = {g 0 s.t. (1, v, v 2 ) gdv = (1, v, v 2 ) fdv, (2) R 3 R 3 λ(g f)a(v u) dv = λ 1 fa(v u)dv}. (3) R 3 R 3 Setting ν = 1 λ 1 λ (3) can be written 1 c c gdv = νθ + (1 ν)tid = T, ρ R 3 Θ = 1 ρ R 3 c c fdv. (4) Stéphane BRULL (Bordeaux) Derivation of BGK models. September 27 th / 42

17 Main result Theorem Let f 0, f 0 s.t. (1 + v 2 ) f < + and ν [ 1 2, 1[, the problem (1, 17, 3) has a unique solution G ( ρ G(v) = exp 1 ) det(2πt ) 2 c, T 1 c. Conversely, if the problem (1, 17, 3) has a solution for any f 0 s.t. f (1 + v ) 2 < +, then ν [ 1 2, 1[. Arguments : C f. Ex : G ES C f. M.Junk, J.Schneider a solution to the minimization problem as soon as C f. Stéphane BRULL (Bordeaux) Derivation of BGK models. September 27 th / 42

18 Chapman-Enskog expansion f is expanded as Computation of λ and λ 1 Prandtl number ( ) t + v x f = λ (G f), ε f = M(1 + εf (1) + ε 2 f (2) +... ). λ 1 = ρt µ, λ = 5 ρ T 2 κ. Pr = 5 µ 2 κ = λ = 1 λ 1 1 ν. Pr = 2 3 ν = 1 2 Result : Ellipsoidal Statistical Model ([Holway, 1964]). + New proof of H theorem. Stéphane BRULL (Bordeaux) Derivation of BGK models. September 27 th / 42

19 Chapman-Enskog expansion f is expanded as Computation of λ and λ 1 Prandtl number ( ) t + v x f = λ (G f), ε f = M(1 + εf (1) + ε 2 f (2) +... ). λ 1 = ρt µ, λ = 5 ρ T 2 κ. Pr = 5 µ 2 κ = λ = 1 λ 1 1 ν. Pr = 2 3 ν = 1 2 Result : Ellipsoidal Statistical Model ([Holway, 1964]). + New proof of H theorem. Stéphane BRULL (Bordeaux) Derivation of BGK models. September 27 th / 42

20 Chapman-Enskog expansion f is expanded as Computation of λ and λ 1 Prandtl number ( ) t + v x f = λ (G f), ε f = M(1 + εf (1) + ε 2 f (2) +... ). λ 1 = ρt µ, λ = 5 ρ T 2 κ. Pr = 5 µ 2 κ = λ = 1 λ 1 1 ν. Pr = 2 3 ν = 1 2 Result : Ellipsoidal Statistical Model ([Holway, 1964]). + New proof of H theorem. Stéphane BRULL (Bordeaux) Derivation of BGK models. September 27 th / 42

21 Polyatomic case Stéphane BRULL (Bordeaux) Derivation of BGK models. September 27 th / 42

22 Polyatomic modeling Microscopic model : [Borgnakke, Larsen, 1975] variables : t, x, v, I. (I 0) : internal energy variable ε(i) = I 2 δ = internal energy, δ = internal degrees number Mesoscopic model : [Bourgat, Desvillettes, Le Tallec, Perthame, 94]. Distribution function : f = f(t,x,v,i) Collisional model of Boltzmann type : Equilibrium States, H theorem,... Conserved moments :(1, v, 1 2 v 2 + I 2 δ ), P0 = Vect[1, v, 1 2 v 2 + I 2 δ ]. Euler limit : [Desvillettes, 1997]. Polyatomic gas mixture [Desvillettes, Monaco, Salvarani, 2005] Discrete energy variable : [Giovangigli] Stéphane BRULL (Bordeaux) Derivation of BGK models. September 27 th / 42

23 Hydrodynamic quantities Hydrodynamic quantities : ρ, u, like in monoatomic case. Entropy : like in the monoatomic case. Specific internal energy : e = 1 ( 1 ρ R 3 R + 2 v u 2 + I 2 δ ) f dvdi. e = e tr + e int, e tr = 1 2ρ R 3 R + v u 2 f dvdi, e int = 1 ρ R 3 R + I 2 δ f dvdi. Associated temperatures : e = 3 + δ 2 T eq, e tr = 3 2 T tr, e int = δ 2 T int. Stéphane BRULL (Bordeaux) Derivation of BGK models. September 27 th / 42

24 Space of constraints B(c, I) = 1 3 c δ ( c I 2 δ ), P = P0 A(c) B(c, I) Minimization problem G = min g C f H(g). (5) C f = {g 0 s.t. g (1, v, 1 R 3 R + 2 c 2 + I 2 δ ) dvdi = f (1, v, 1 R 3 R + 2 c 2 + I 2 δ ) dvdi, (6) R 3 R + B(c, I) λ(g f) dvdi = λ 2 R 3 R + A(c) λ(g f) dvdi = λ 1 R 3 R + B(c, I) f dvdi, (7) R 3 R + A(c) f dvdi} (8) Stéphane BRULL (Bordeaux) Derivation of BGK models. September 27 th / 42

25 Construction of G. Change of variables : θ = 1 λ 2 λ, λ 1 λ = 1 ν(1 θ). (8) can be rewritten T = 1 c c g dv di = (1 θ) ((1 ν) T tr Id + νθ) + θ T eq Id ρ R 3 Relaxation temperature : T rel = θt eq + (1 θ)t int Pressure tensor : Θ = 1 c c f dv di. ρ Comparisons with the Ellipsoidal Statistical Model in the polyatomic case : [Andries-LeTallec-Perlat-Perthame, 2000] Stéphane BRULL (Bordeaux) Derivation of BGK models. September 27 th / 42

26 Main theorem Theorem Let f (f 0), f 0 s.t. f (1 + v 2 + I 2 δ ) dvdi < +, ν [ 1 2, 1[ and θ [0, 1]. Then the minimization problem (5, 6, 7, 8) has a unique solution, G = ρλ δ det(2πt )(Teq ) δ 2 ( exp 1 2 c, T 1 c I 2 ) δ. T rel Conversely, if the minimization problem (5, 6, 7, 8) has a unique solution for all f 0 s.t. f (1 + v 2 + I 2 δ ) dvdi < +, then ν [ 1 2, 1[ and θ [0, 1]. [S.B-J.Schneider, 2009] Argument : [M.Junk, 1998] a solution to the minimization problem as soon as C f. Stéphane BRULL (Bordeaux) Derivation of BGK models. September 27 th / 42

27 Definition of λ, λ 1, λ 2. Tensor of constraints for the polyatomic Navier-Stokes system 2 viscosities : µ, α σ ij = µ ( xj u i + xi u j αdiv(u)δ ij ). Chapmann-Enskog expansion exact up to order 1 Definition of λ(ρ, T, κ), λ 1 (ρ, T, µ) et λ 2 (ρ, T, µ, α). Result : Ellipsoidal Statistical Model for polyatomic gases [Andries-LeTallec-Perlat-Perthame, 2000] Stéphane BRULL (Bordeaux) Derivation of BGK models. September 27 th / 42

28 Mixture case Stéphane BRULL (Bordeaux) Derivation of BGK models. September 27 th / 42

29 Aim 1 Clarify the Chapman-Enskog expansion for gas mixtures [Chapman, Cowling] 2 Derive a Relaxation model for gas mixtures Fick Matrix Relaxation on non conserved moments Solve the entropy minimization problem Determine the relaxation coefficients 3 Prove the fundamental properties for the BGK operator : i=p f, f i 0, φ, R i (f) φ i dv = 0 φ P 0, (9) i=1 R 3 i=p f, f i 0, R i (f) ln (f i ) dv 0, (10) R 3 i=1 R (f) = 0 n i, u, T s.t. i [1, p], f i = M i (11) Stéphane BRULL (Bordeaux) Derivation of BGK models. September 27 th / 42

30 Navier-Stokes system for a mixture of p components. Navier-Stokes system : i [1, p], t n i + (n i u + J i ) = 0, t (ρ u) + (P + ρ u u + J u ) = 0, t E + (Eu + P [u] + J u [u] + J q ) = 0, J i, J u J q : mass, momentum and heat fluxes. Thermodynamics of Irreversible Processes assumptions. J i = j=p j=1 L ij ( µ j ) T + L iq ( ) 1 T, J q = j=p j=1 L qj ( µ j ) T + L qq ( ) 1 T, J u = L uu D (u), µ i : chemical potential : µ ( i T = k B ln (n i ) 3 ( )) 2 ln 2πkB T. m i Stéphane BRULL (Bordeaux) Derivation of BGK models. September 27 th / 42

31 Properties of the matrix L i,j. Onsager relations : L := L ij L iq 0 L qi L qq L uu is symmetric and non positive. Total mass conservation : i=p m i J i = 0 j [1, p], i=1 i=p m i L ij = 0 rank( L ij ) p 1. i=1 Stéphane BRULL (Bordeaux) Derivation of BGK models. September 27 th / 42

32 Fick, Dufour, Soret, Fourier coefficients Phenomenological point of view ([Chapman-Cowling]). J i = J q = j=p D ij n j + D it T, j=1 j=p D qj n j D qq T. j=1 D ij : Fick coefficient : Diffusion D it : Soret coefficient : Thermal diffusion D qj : Duffour coefficient : Diffusion thermo-effect D qq : Fourier coefficient Stéphane BRULL (Bordeaux) Derivation of BGK models. September 27 th / 42

33 Notations Distribution function : f := (f 1,, f p ) n i, u i, T i. Maxwellian : M := (M 1,, M p ). i=p Scalar product : f, g = f i g i M i dv Euclidean norm :. R 3 i=1 Collisional invariants P 0 of L 2 (M) spanned by : ,, 0 0 m 1 v x m 2 v x m 1 v y m 2 v y m 1 v z m 2 v z,,, m p v x m p v y m p v z, m 1 v 2 m 2 v 2. m p v 2. Notation : (C i ) j = δ ij (v u). Stéphane BRULL (Bordeaux) Derivation of BGK models. September 27 th / 42

34 Chapman-Enskog and space C. Chapman-Enskog expansion : f = f 0 + εf 1 + ε 2... f 1 = Mg P P0 = orthogonal projection on P 0 and I the identity operator. L B (g) = j=p j=1 ( 1 (I P P0 ) (C j ) k B µ j T ) + 1 K B T A : D (u) 1 k B B A and B : Sonine polynomials : (A) i = m i [(v u) (v u) 1 ] 3 (v u)2 I, [ 1 (B) i = (v u) 2 m i (v u) 2 5 ] 2 k BT. New space C = span (I P P0 ) (C i ), i [1, p]. (I P P0 ) (C i ), i [1, p 1] basis of C = dim(c) = 3 (p 1). ( ) 1, T Stéphane BRULL (Bordeaux) Derivation of BGK models. September 27 th / 42

35 Fluxes and transport coefficients. Fluxes : J i = g, C i = g, (I P P0 ) (C i ), J u = g, A, J q = g, B. Transport coefficients : L ij I = k 1 B L 1 B [(I P P 0 ) (C i )], (I P P0 ) (C j ) L 1 L iq I = L qi I = k 1 B L uu I I = 1 k B T L qq I = k 1 B B (B), (I P P 0 ) (C i ) L 1 B (A), A L 1 B (B), B L B self adjoint and negative on P 0 L satisfies Ohnsager properties Stéphane BRULL (Bordeaux) Derivation of BGK models. September 27 th / 42

36 Entropy minimization problem. Basis of P 0 : (φ l ) l {1,p+4} Basis of C : (w r ) r {1;p 1} Entropy : H(f) = p i=1 R 3 (f i ln(f i ) f i ) dv. Minimization problem G = min g C f H(f) (12) i=p C f = {g 0, l [1, p + 4], φ l i (g i f i ) dv = 0, (13) i=1 R 3 i=p i=p r [1, p 1], w r i (g i f i )dv = λ r w r i f idv}, (14) R 3 R 3 i=1 i=1 Stéphane BRULL (Bordeaux) Derivation of BGK models. September 27 th / 42

37 Result of the minimization problem The minimization problem (12, 13, 14) has a unique solution G n i i [1, p], G i = exp (2πk B T 3/2 /m i ) m i (v u i ) 2 2k B T. Argument : [M.Junk, 1998] a solution to the minimization problem. Notations : U = (u 1,..., u p ) T f, U = (u 1,..., u p ) T G N : diagonal matrix ( ρ 1,..., ρ p ). Λ diagonal matrix (λ 1,..., λ p ). Definition of u i U U = N 1 W T ( I 1 ν Λ ) WN ( U U ). (15) Choice of T Energy conservation T 0 if ν max r λ r. Stéphane BRULL (Bordeaux) Derivation of BGK models. September 27 th / 42

38 Effective construction of the model Introduction of L ij Diagonalization of L Theorem i, j [1, p], L ij = L ij C i C j. spectrum of L : (l r, w r ) r {1,...,p 1} (0, w p ) λ r = l r 1 Fick laws, λ p = 0 Conservation of impulsion. j=p ( µj ) Density fluxes : J i = L ij T j=1 Stéphane BRULL (Bordeaux) Derivation of BGK models. September 27 th / 42

39 Properties of the BGK model. The Fick relaxation operator satisfies the fundamental properties : i=p f, f i 0, φ, R i (f) φ i dv = 0 φ P 0, i=1 R 3 i=p f, f i 0, R i (f) ln (f i ) dv 0, R 3 i=1 R (f) = 0 n i, u, T s.t. i [1, p], f i = M i, ( L = ν (P P0 + Λ P C I), Λ (w r ) = 1 λ r ν is self adjoint and negative on P 0 and KerL = P 0. ) w r, r {1, p 1} Stéphane BRULL (Bordeaux) Derivation of BGK models. September 27 th / 42

40 Computation of transport coefficients. L ij (R) = L ij (Boltzmann or experimental) = 1 L 1 (I P P0 ) (C i ), (I P P0 ) (C j ), k B L 1 (B), (I P P0 ) (C i ) = L iq = L qi = 0, 1 L 1 (A), A = 1 k B T ν A, A = L uui I = nk BT I I. ν µ = L uu = nk BT. ν Correct Fick coefficients and correct viscosity if ν max r λ r. Stéphane BRULL (Bordeaux) Derivation of BGK models. September 27 th / 42

41 Derivation an ESBGK model for gas mixtures Stéphane BRULL (Bordeaux) Derivation of BGK models. September 27 th / 42

42 Construction of G Minimization problem G = min g C f H(g) (16) i=p C f = {g 0 s.t. l [1, p + 4], φ l i (g i f i ) dv = 0, (17) R 3 i=p i=1 i=1 R 3 (g i f i )A i (v u) dv = λ 1 Pressure tensor : Θ = p i=1 p i=1 R 3 f i A i (v u) dv} (18) R 3 m i (v u) (v u) f i dv. (19) Change of variable : ν = 1 λ 1 λ 18 can be rewritten as 1 p m i g i (v u) (v u) dv = νθ + (1 ν)t = T (20) ρ R 3 i=1 Stéphane BRULL (Bordeaux) Derivation of BGK models. September 27 th / 42

43 Main results Theorem f = (f 1,..., f p ) 0 and ν [ 1 2, 1[, T is symmetric defined positif and the problem (16, 17, 20) has a unique solution G = (G 1,..., G p ) G i = ( ρ i exp 1 ) 2π det(t ) 2 v u, T 1 (v u). (21) Conversely, if the problem (16, 17, 20) has a unique solution f L 2 1, then ν [ 1 2, 1[. [Brull, CMS 2015] Argument : [M.Junk, 1998] a solution to the minimization problem as soon as C f. Stéphane BRULL (Bordeaux) Derivation of BGK models. September 27 th / 42

44 Chapman-Enskog expansion t f i + v f i = λ ε (G i f i ), i {1,..., p}, Computation of λ and λ 1 Prandtl number λ 1 = nk BT µ, λ = 5 2L qq k 2 B T 3 p i=1 n i m i. Pr = 1 1 ν 1, m = p n i m n i=1 m i p i=1 n i [ 2 n m i, Range : 3 1 p n i m n i m i [, +. Remark : Situation of indifferentiability, m i = m, Pr = 2/3 Stéphane BRULL (Bordeaux) Derivation of BGK models. September 27 th / 42

45 Properties of the model Properties of the relaxation operator i=p f, f i 0, φ, R i (f) φ i dv = 0 φ P 0, i=1 R 3 i=p f, f i 0, R i (f) ln (f i ) dv 0, R 3 Transport coefficients i=1 R (f) = 0 n i, u, T s.t. i [1, p], f i = M i. L ij = 3Tni k B m i δ ij, L iq = L qi = 0, L qq = 5 2λ k 2 B T 3 p i=1 n i m i, L uu = µ. Stéphane BRULL (Bordeaux) Derivation of BGK models. September 27 th / 42

46 Conclusions and Perspectives Stéphane BRULL (Bordeaux) Derivation of BGK models. September 27 th / 42

47 Conclusions New methodology to derive BGK operators based on the true transport coefficients Results Rigorous derivation of the Ellipsoidal Statistical Model (monoatomic and polyatomic case) New BGK model Fick matrix + fundamental properties. Simplicity of the model : Requires only the diagonalization of L ESBGK model for gas mixtures [Brull, CMS 2015] Related results BGK model for slow reactive mixtures [Brull-Schneider, CMS 2014]. Fick matrix i {1; 4}, t f ε i + v x f ε i R CE : [Groppi, Rjasanow, Spiga, 2009]. = 1 ε RME i (f ε ) + R CE i (f ε ). Stéphane BRULL (Bordeaux) Derivation of BGK models. September 27 th / 42

48 Bibliography S.Brull, J.Schneider A new approach of the Ellipsoidal Statistical Model. Cont.Mech.Thermodyn. 20 (2008), no.2, S.Brull, J.Schneider On the Ellipsoidal Statistical Model for polyatomic gases. Cont.Mech.Thermodyn. 20 (2009), no.8, S.Brull, V.Pavan, J.Schneider Derivation of BGK models for mixtures. European Journal of Mechanics/B Fluids (2012), S.Brull, J.Schneider Derivation of a BGK model for reacting gas mixtures. Comm. Math. Sci., vol 12, No 7, (2014), S.Brull An Ellipsoidal Statistical Model for gas mixtures. Comm. Math. Sci. vol 13, no1, (2015), Stéphane BRULL (Bordeaux) Derivation of BGK models. September 27 th / 42

49 Perspectives Fit in the same time all the laws (Fick, Soret, Dufour, Newton, Fourier) Increase the number of constraints (Realizability problems). Numerical implementation of the new BGK models Generalization to polyatomic gas mixtures for concrete applications (air N0 2 ) Construction of ESBGK for polyatomic gas mixtures Generalization of the Fick-BGK operator to a polyatomic setting Clarify Chapman-Enskog expansion for polyatomic gas mixtures (Giovangigli : discrete energy variable). Continous energy variable : [Bisi, Brull], in progress Stéphane BRULL (Bordeaux) Derivation of BGK models. September 27 th / 42

50 THANKS FOR YOUR ATTENTION! Stéphane BRULL (Bordeaux) Derivation of BGK models. September 27 th / 42