Recent advances in kinetic theory for mixtures of polyatomic gases

Transcription

1 Recent advances in kinetic theory for mixtures of polyatomic gases Marzia Bisi Parma University, Italy Conference Problems on Kinetic Theory and PDE s Novi Sad (Serbia), September 25 27, 2014 M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 1 / 25

2 Summary Kinetic Boltzmann model for (inert or reactive) polyatomic gas mixtures M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 2 / 25

3 Summary Kinetic Boltzmann model for (inert or reactive) polyatomic gas mixtures BGK relaxation model (joint work with M.J. Cáceres) M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 2 / 25

4 Summary Kinetic Boltzmann model for (inert or reactive) polyatomic gas mixtures BGK relaxation model (joint work with M.J. Cáceres) Hydrodynamic limit leading to incompressible Navier Stokes equations (based on a joint work with L. Desvillettes) M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 2 / 25

5 Summary Kinetic Boltzmann model for (inert or reactive) polyatomic gas mixtures BGK relaxation model (joint work with M.J. Cáceres) Hydrodynamic limit leading to incompressible Navier Stokes equations (based on a joint work with L. Desvillettes) Work in progress and open problems M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 2 / 25

6 Kinetic Boltzmann approach to polyatomic gases Motivation: It is well known that gas mixtures involved in physical applications are usually composed also of polyatomic species (for instance in simple dissociation and recombination processes, or in the evolution of powders in the atmosphere) M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 3 / 25

7 Kinetic Boltzmann approach to polyatomic gases Motivation: It is well known that gas mixtures involved in physical applications are usually composed also of polyatomic species (for instance in simple dissociation and recombination processes, or in the evolution of powders in the atmosphere) In kinetic approaches, each gas is endowed with a suitable internal energy variable to mimic non translational degrees of freedom Groppi, Spiga, J. Math. Chem. (1999): discrete internal energy levels Desvillettes, Monaco, Salvarani, Europ. J. Mech. B/Fluids (2005): continuous internal energy variable M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 3 / 25

8 Kinetic Boltzmann approach to polyatomic gases Motivation: It is well known that gas mixtures involved in physical applications are usually composed also of polyatomic species (for instance in simple dissociation and recombination processes, or in the evolution of powders in the atmosphere) In kinetic approaches, each gas is endowed with a suitable internal energy variable to mimic non translational degrees of freedom Groppi, Spiga, J. Math. Chem. (1999): discrete internal energy levels Desvillettes, Monaco, Salvarani, Europ. J. Mech. B/Fluids (2005): continuous internal energy variable Our physical frame: Mixture of M polyatomic gases G s, s = 1,...,M M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 3 / 25

9 Kinetic Boltzmann approach to polyatomic gases Motivation: It is well known that gas mixtures involved in physical applications are usually composed also of polyatomic species (for instance in simple dissociation and recombination processes, or in the evolution of powders in the atmosphere) In kinetic approaches, each gas is endowed with a suitable internal energy variable to mimic non translational degrees of freedom Groppi, Spiga, J. Math. Chem. (1999): discrete internal energy levels Desvillettes, Monaco, Salvarani, Europ. J. Mech. B/Fluids (2005): continuous internal energy variable Our physical frame: Mixture of M polyatomic gases G s, s = 1,...,M Each gas G s is considered as a mixture of Q monatomic components A i, i = s, s + M, s + 2M, s + M(Q 1), each one characterized by a different internal energy E i M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 3 / 25

10 In the frame of each species, energies are monotonically increasing with their index: i, j s, i< j E i < E j M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 4 / 25

11 In the frame of each species, energies are monotonically increasing with their index: i, j s, i< j E i < E j Besides classical elastic scattering, particles may undergo also inelastic transitions in which they change their internal energy level A i + A j A h + A k h i, k j M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 4 / 25

12 In the frame of each species, energies are monotonically increasing with their index: i, j s, i< j E i < E j Besides classical elastic scattering, particles may undergo also inelastic transitions in which they change their internal energy level A i + A j A h + A k h i, k j Boltzmann equations for distribution functions of single components f i + v x f i = K ijhk [f](v, w, ˆn )dwdˆn 1 i QM t i (j, h, k) D i ( ) [ K ijhk [f] = Θ g 2 2 Ehk ij gσ hk (g, ˆn ˆn ) f h( ) ( ) v hk i µ ij ij ij f k w hk ij f i (v)f (w)] j M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 4 / 25

13 In the frame of each species, energies are monotonically increasing with their index: i, j s, i< j E i < E j Besides classical elastic scattering, particles may undergo also inelastic transitions in which they change their internal energy level A i + A j A h + A k h i, k j Boltzmann equations for distribution functions of single components f i + v x f i = K ijhk [f](v, w, ˆn )dwdˆn 1 i QM t i (j, h, k) D i ( ) [ K ijhk [f] = Θ g 2 2 Ehk ij gσ hk (g, ˆn ˆn ) f h( ) ( ) v hk i µ ij ij ij f k w hk ij f i (v)f (w)] j Here the set D i includes all possible collisions, v hk, w hk are post collision velocities, g = v w = g ˆn, ij ij Θ is the Heaviside function providing a threshold for all endothermic interactions in which E hk = E h + E k E i E j > 0 ij M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 4 / 25

14 Collision invariants: number density of each gas N s = global velocity u, global energy n i, i s s = 1,...,M, 3 QM 2 nkt + E i n i i=1 M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 5 / 25

15 Collision invariants: number density of each gas N s = global velocity u, global energy n i, i s s = 1,...,M, 3 QM 2 nkt + E i n i Collision equilibria of the Boltzmann equations: ( ) m f i s 3/2 ] M (v) = ni exp[ ms 2πKT 2KT v u 2 i s, s = 1,...,M, i=1 M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 5 / 25

16 Collision invariants: number density of each gas N s = global velocity u, global energy n i, i s s = 1,...,M, 3 QM 2 nkt + E i n i Collision equilibria of the Boltzmann equations: ( ) m f i s 3/2 ] M (v) = ni exp[ ms 2πKT 2KT v u 2 i s, s = 1,...,M, with equilibrium number densities related by the constraints where i=1 n i = N s ψ(e i, T) i s ψ(e i, T) = exp( ) Ei Es KT i s ( ) = exp( exp Ei E s Z s (T) KT ) Ei Es KT M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 5 / 25

17 Collision invariants: number density of each gas N s = global velocity u, global energy n i, i s s = 1,...,M, 3 QM 2 nkt + E i n i Collision equilibria of the Boltzmann equations: ( ) m f i s 3/2 ] M (v) = ni exp[ ms 2πKT 2KT v u 2 i s, s = 1,...,M, with equilibrium number densities related by the constraints where i=1 n i = N s ψ(e i, T) i s ψ(e i, T) = exp( ) Ei Es KT i s ( ) = exp( exp Ei E s Z s (T) KT ) Ei Es KT Remark:ψ(E i, T) represents the fraction of particles G s (s i) belonging to the component A i in any equilibrium configuration; for any i, j with i j and i< j, we haveψ(e i, T)>ψ(E j, T) M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 5 / 25

18 Generalization to chemically reactive frames (4 species) Besides elastic scattering and inelastic transitions (with transfer of internal energy), particles may undergo the binary and reversible chemical reaction G 1 + G 2 G 3 + G 4, hence, for single components, A i + A j A h + A k i 1, j 2, h 3, k 4 M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 6 / 25

19 Generalization to chemically reactive frames (4 species) Besides elastic scattering and inelastic transitions (with transfer of internal energy), particles may undergo the binary and reversible chemical reaction G 1 + G 2 G 3 + G 4, hence, for single components, A i + A j A h + A k i 1, j 2, h 3, k 4 Basic properties: For reactive ( encounters K ijhk [f] = Θ i ) g 2 2 Ehk ij gσ hk (g, ˆn ˆn ) µ ij ij [( µ ij ) 3f ( ) ( ) h v hk µ hk ij f k w hk ij f i (v)f (w)] j M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 6 / 25

20 Generalization to chemically reactive frames (4 species) Besides elastic scattering and inelastic transitions (with transfer of internal energy), particles may undergo the binary and reversible chemical reaction G 1 + G 2 G 3 + G 4, hence, for single components, A i + A j A h + A k i 1, j 2, h 3, k 4 Basic properties: For reactive ( encounters K ijhk [f] = Θ i ) g 2 2 Ehk ij gσ hk (g, ˆn ˆn ) µ ij ij [( µ ij ) 3f ( ) ( ) h v hk µ hk ij f k w hk ij f i (v)f (w)] j Collision invariants: N 1 + N 3, N 1 + N 4, N 2 + N 4, global momentum, and total energy M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 6 / 25

21 Generalization to chemically reactive frames (4 species) Besides elastic scattering and inelastic transitions (with transfer of internal energy), particles may undergo the binary and reversible chemical reaction G 1 + G 2 G 3 + G 4, hence, for single components, A i + A j A h + A k i 1, j 2, h 3, k 4 Basic properties: For reactive ( encounters K ijhk [f] = Θ i ) g 2 2 Ehk ij gσ hk (g, ˆn ˆn ) µ ij ij [( µ ij ) 3f ( ) ( ) h v hk µ hk ij f k w hk ij f i (v)f (w)] j Collision invariants: N 1 + N 3, N 1 + N 4, N 2 + N 4, global momentum, and total energy Collision equilibria: Maxwellian distributions f i = f i M (ni, u, T) with n i = N s ψ(e i, T) plus the mass action law of chemistry N 1 N 2 ( ) µ 12 3/2 Z 1 N 3 N = (T)Z 2 (T) 4 Z 3 (T)Z 4 (T) e E34 12 KT E 34 = 12 E3 + E 4 E 1 E 2 > 0 µ 34 M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 6 / 25

22 BGK approximation for polyatomic INERT mixtures f i + v x f i =ν i (M i f i ) i = 1,...,QM t M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 7 / 25

23 BGK approximation for polyatomic INERT mixtures f i + v x f i =ν i (M i f i ) i = 1,...,QM t where attractorsm i =M i (ñ i, ũ, T) take the form ( M i (v) = ñ i m i ) 3/2 ] exp[ mi 2 2πK T 2K T v ũ M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 7 / 25

24 BGK approximation for polyatomic INERT mixtures f i + v x f i =ν i (M i f i ) i = 1,...,QM t where attractorsm i =M i (ñ i, ũ, T) take the form ( M i (v) = ñ i m i ) 3/2 ] exp[ mi 2 2πK T 2K T v ũ with densities ñ i bound together as ñ i = Ñ s ψ(e i, T) M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 7 / 25

25 BGK approximation for polyatomic INERT mixtures f i + v x f i =ν i (M i f i ) i = 1,...,QM t where attractorsm i =M i (ñ i, ũ, T) take the form ( M i (v) = ñ i m i ) 3/2 ] exp[ mi 2 2πK T 2K T v ũ with densities ñ i bound together as ñ i = Ñ s ψ(e i, T) Key points: Only one relaxation operator for each component [Andries, Aoki, Perthame, J. Stat. Phys. (2002)] M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 7 / 25

26 BGK approximation for polyatomic INERT mixtures f i + v x f i =ν i (M i f i ) i = 1,...,QM t where attractorsm i =M i (ñ i, ũ, T) take the form ( M i (v) = ñ i m i ) 3/2 ] exp[ mi 2 2πK T 2K T v ũ with densities ñ i bound together as ñ i = Ñ s ψ(e i, T) Key points: Only one relaxation operator for each component [Andries, Aoki, Perthame, J. Stat. Phys. (2002)] AttractorsM i fulfill the equilibrium conditions [Bisi, Groppi, Spiga, Proceedings RGD26 (2009)] M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 7 / 25

27 BGK approximation for polyatomic INERT mixtures f i + v x f i =ν i (M i f i ) i = 1,...,QM t where attractorsm i =M i (ñ i, ũ, T) take the form ( M i (v) = ñ i m i ) 3/2 ] exp[ mi 2 2πK T 2K T v ũ with densities ñ i bound together as ñ i = Ñ s ψ(e i, T) Key points: Only one relaxation operator for each component [Andries, Aoki, Perthame, J. Stat. Phys. (2002)] AttractorsM i fulfill the equilibrium conditions [Bisi, Groppi, Spiga, Proceedings RGD26 (2009)] Ñ s, ũ, T are M + 4 independent free parameters to be properly determined as functions of the actual macroscopic fields n i, u i, T i M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 7 / 25

28 Strategy: we impose that the BGK model preserves the correct collision invariants M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 8 / 25

29 Strategy: we impose that the BGK model preserves the correct collision invariants ν i (M i f i )dv = 0 s = 1,...,M (a) i s M ν i s=1 i s m s v(m i f i )dv = 0 (b) M ( ) 1 ν i 2 ms v 2 + E i (M i f i )dv = 0 (c) s=1 i s M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 8 / 25

30 Strategy: we impose that the BGK model preserves the correct collision invariants ν i (M i f i )dv = 0 s = 1,...,M (a) i s M ν i s=1 i s m s v(m i f i )dv = 0 (b) M ( ) 1 ν i 2 ms v 2 + E i (M i f i )dv = 0 (c) s=1 i s For any s = 1,...,M, condition (a) provides ν i ñ i = ν i n i from which, bearing in mind the constraint ñ i = Ñ s ψ(e i, T) we get / Ñ s = ν i n i ν i ψ(e i, T) i s i s i s i s M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 8 / 25

31 Strategy: we impose that the BGK model preserves the correct collision invariants ν i (M i f i )dv = 0 s = 1,...,M (a) i s M ν i s=1 i s m s v(m i f i )dv = 0 (b) M ( ) 1 ν i 2 ms v 2 + E i (M i f i )dv = 0 (c) s=1 i s For any s = 1,...,M, condition (a) provides ν i ñ i = ν i n i from which, bearing in mind the constraint ñ i = Ñ s ψ(e i, T) we get / Ñ s = ν i n i ν i ψ(e i, T) i s i s M / Momentum conservation (b) yields ũ = m s M ν i n i u i m s ν i n i s=1 i s s=1 i s i s i s M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 8 / 25

32 Energy conservation (c) provides M [ 1 ν i 2 ms ñ i ũ ñi K T + Eiñ i 1 2 ms n i u i 2 3 ] 2 ni KT i E i n i = 0 s=1 i s M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 9 / 25

33 Energy conservation (c) provides M [ 1 ν i 2 ms ñ i ũ ñi K T + Eiñ i 1 2 ms n i u i 2 3 ] 2 ni KT i E i n i = 0 s=1 i s that, recalling the explicit expressions for ñ i, ũ, may be written as a transcendental equation for T: M F( T) = Λ where F( T) = ν j n j 3 2 K T + i sν i E i ψ(e i, T) j sν j ψ(e j, T) s=1 and Λ is a known explicit function of the actual macroscopic fields that turns out to fulfill Λ M ( s=1 i sν i n i) E s j s M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 9 / 25

34 Lemma: For any Λ, the equation F( T) = Λ has a unique positive solution M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 10 / 25

35 Lemma: For any Λ, the equation F( T) = Λ has a unique positive solution Steps of the proof: by direct computations, we check that F( T) is a monotonically increasing function of its argument; M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 10 / 25

36 Lemma: For any Λ, the equation F( T) = Λ has a unique positive solution Steps of the proof: by direct computations, we check that F( T) is a monotonically increasing function of its argument; recalling that, for each gas G s, energy levels are such that E s < E s+m < E s+2m < <E s+qm, we get i sν i E i ψ(e i, T) ν s E s + i s i s = j sν j ψ(e j, T) lim T 0 ν s + i s i s therefore M F( T) = ν i n i Es, s=1 i s ν ( ) i E i exp Ei E s ν i exp ( Ei E s K T K T ) min i s Ei = E s lim F( T) = + T + M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 10 / 25

37 Lemma: For any Λ, the equation F( T) = Λ has a unique positive solution Steps of the proof: by direct computations, we check that F( T) is a monotonically increasing function of its argument; recalling that, for each gas G s, energy levels are such that E s < E s+m < E s+2m < <E s+qm, we get i sν i E i ψ(e i, T) ν s E s + i s i s = j sν j ψ(e j, T) lim T 0 ν s + i s i s therefore M F( T) = ν i n i Es, s=1 i s ν ( ) i E i exp Ei E s ν i exp ( Ei E s K T K T ) min i s Ei = E s lim F( T) = + T + Since for T> 0 the function F( T) monotonically increases from the minimum admissible value for Λ to +, existence and uniqueness of solution to F( T) = Λ is guaranteed M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 10 / 25

38 Remarks and basic properties We have thus proved that the proposed BGK model is well defined, in the sense that all auxiliary parameters are uniquely determined in terms of the actual fields M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 11 / 25

39 Remarks and basic properties We have thus proved that the proposed BGK model is well defined, in the sense that all auxiliary parameters are uniquely determined in terms of the actual fields Collision equilibria: f i (v) =M i (v) v R 3 n i = ñ i, u i = ũ, T i = T hence also the actual number densities fulfill the constraints n i = N s ψ(e i, T) (i s) M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 11 / 25

40 Remarks and basic properties We have thus proved that the proposed BGK model is well defined, in the sense that all auxiliary parameters are uniquely determined in terms of the actual fields Collision equilibria: f i (v) =M i (v) v R 3 n i = ñ i, u i = ũ, T i = T hence also the actual number densities fulfill the constraints n i = N s ψ(e i, T) (i s) It may be proved that the usual H functional H = M s=1 i s f i log f i dv is a Lyapunov functional even for the BGK kinetic model M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 11 / 25

41 BGK approximation for polyatomic REACTING mixtures (4 species) f i + v x f i =ν i (M i f i ) i = 1,...,4Q t with parameters of the Maxwellian attractorsm i =M i (ñ i, ũ, T) bound together as ñ i = Ñ s ψ(e i, T), Ñ 1 Ñ 2 Ñ 3Ñ4 = ( µ 12 µ 34 ) 3/2 Z 1 ( T)Z 2 ( T) Z 3 ( T)Z 4 ( T) e E K T (a) M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 12 / 25

42 BGK approximation for polyatomic REACTING mixtures (4 species) f i + v x f i =ν i (M i f i ) i = 1,...,4Q t with parameters of the Maxwellian attractorsm i =M i (ñ i, ũ, T) bound together as ñ i = Ñ s ψ(e i, T), Ñ 1 Ñ 2 Ñ 3Ñ4 = ( µ 12 µ 34 ) 3/2 Z 1 ( T)Z 2 ( T) Z 3 ( T)Z 4 ( T) e E K T (a) 7 independent free parameters (ũ, T, three among Ñ s ) to be determined imposing the preservation of correct collision invariants M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 12 / 25

43 BGK approximation for polyatomic REACTING mixtures (4 species) f i + v x f i =ν i (M i f i ) i = 1,...,4Q t with parameters of the Maxwellian attractorsm i =M i (ñ i, ũ, T) bound together as ñ i = Ñ s ψ(e i, T), Ñ 1 Ñ 2 Ñ 3Ñ4 = ( µ 12 µ 34 ) 3/2 Z 1 ( T)Z 2 ( T) Z 3 ( T)Z 4 ( T) e E K T (a) 7 independent free parameters (ũ, T, three among Ñ s ) to be determined imposing the preservation of correct collision invariants Main difference with respect to the inert mixture: the constraint coming from energy conservation and mass action law (a) are two transcendental equations for the unknowns (Ñ 1, T) M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 12 / 25

44 Incompressible hydrodynamic limit of the Boltzmann equations (four reacting species) ε t f i ε + v x f i ε = 1 ε 4Q Q ij (fε, i fε) j +ε J i j=1 i = 1,...,4Q where Q ij (f i ε, f j ε) is the classical elastic operator and J i takes into account all non conservative collisions (inelastic transitions and chemical reactions) M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 13 / 25

45 Incompressible hydrodynamic limit of the Boltzmann equations (four reacting species) ε t f i ε + v x f i ε = 1 ε 4Q Q ij (fε, i fε) j +ε J i j=1 i = 1,...,4Q where Q ij (f i ε, f j ε) is the classical elastic operator and J i takes into account all non conservative collisions (inelastic transitions and chemical reactions) As in [Bardos, Golse, Levermore, J. Stat. Phys. (1991)] and in [Bisi, Desvillettes, ESAIM - Math. Model. Numer. Anal. (2014)], we look for solutions in the form f i ε =ρi M i (1,0,1) (1 +ε gi ε ) (perturbations of collision equilibria) M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 13 / 25

46 Here M i are absolute normalized Maxwellians (1,0,1) ( ) m M i i 3/2 (v) = e mi 2 v2 2π andρ i > 0 are constants (without loss of generalityρ= 4Q i=1 = 1) such thatρ i M i are equilibria even of the non conservative (1,0,1) operators J i M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 14 / 25

47 Here M i are absolute normalized Maxwellians (1,0,1) ( ) m M i i 3/2 (v) = e mi 2 v2 2π andρ i > 0 are constants (without loss of generalityρ= 4Q i=1 = 1) such thatρ i M i are equilibria even of the non conservative (1,0,1) operators J i In other words, if we denote N s = ρ i, Z s = e (Ei E s), s = 1,...,4, i s i s for any i s the constantρ i has to be related to N s and Z s as ρ i = Ns E s ) Z s e (Ei and global number densities have to fulfill the mass action law N 1 N 2 Z 1 Z 2 = ( µ 12 µ 34 ) 3/2 e E34 12 N 3 N 4 Z 3 Z 4 M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 14 / 25

48 Incompressible Navier Stokes equations By inserting the ansatz f i ε =ρ i M i (1,0,1) (1 +ε gi ε) into the scaled Boltzmann equations we get 4Q j=1 ] ρ i ρ [Q j ij (gε i Mi, M j ) + Q ij (M i, g j εm j ) = O(ε) M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 15 / 25

49 Incompressible Navier Stokes equations By inserting the ansatz f i ε =ρ i M i (1,0,1) (1 +ε gi ε) into the scaled Boltzmann equations we get hence 4Q j=1 ] ρ i ρ [Q j ij (gε i Mi, M j ) + Q ij (M i, g j εm j ) = O(ε) ( 1 gε i (v) =αi + m i v u+ 2 mi v 2 3 ) T + O(ε) 2 M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 15 / 25

50 Incompressible Navier Stokes equations By inserting the ansatz f i ε =ρ i M i (1,0,1) (1 +ε gi ε) into the scaled Boltzmann equations we get hence 4Q j=1 ] ρ i ρ [Q j ij (gε i Mi, M j ) + Q ij (M i, g j εm j ) = O(ε) ( 1 gε i (v) =αi + m i v u+ 2 mi v 2 3 ) T + O(ε) 2 The parametersα i, u, T (depending on t and x) are perturbations of number densities, velocity and temperature fε(v) i dv =ρ i (1 +εα i ) + O(ε 2 ) v fε(v) i dv =ερ i u + O(ε 2 ) m i v 2 fε(v) i dv = 3ρ i +ε 3ρ i (α i + T) + O(ε 2 ) M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 15 / 25

51 Incompressible Navier Stokes equations By inserting the ansatz f i ε =ρ i M i (1,0,1) (1 +ε gi ε) into the scaled Boltzmann equations we get hence 4Q j=1 ] ρ i ρ [Q j ij (gε i Mi, M j ) + Q ij (M i, g j εm j ) = O(ε) ( 1 gε i (v) =αi + m i v u+ 2 mi v 2 3 ) T + O(ε) 2 The parametersα i, u, T (depending on t and x) are perturbations of number densities, velocity and temperature fε(v) i dv =ρ i (1 +εα i ) + O(ε 2 ) v fε(v) i dv =ερ i u + O(ε 2 ) m i v 2 fε(v) i dv = 3ρ i +ε 3ρ i (α i + T) + O(ε 2 ) We look for evolution equations for α i (i = 1,...,4Q), u, T M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 15 / 25

52 We consider interactions of Maxwell molecules type: gσ ij (g,χ) =ϑ ij (χ) M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 16 / 25

53 We consider interactions of Maxwell molecules type: gσ ij (g,χ) =ϑ ij (χ) and define κ ij = 2π π 0 ϑ ij (χ)(1 cosχ) sinχdχ ν ij = 2π π 0 ϑ ij (χ)(1 cos 2 χ) sinχ dχ M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 16 / 25

54 We consider interactions of Maxwell molecules type: and define κ ij = 2π π 0 ϑ ij (χ)(1 cosχ) sinχdχ ν ij = 2π π 0 gσ ij (g,χ) =ϑ ij (χ) ϑ ij (χ)(1 cos 2 χ) sinχ dχ We formally get that parameters α i (i = 1,...,4Q), u, T satisfy the following Navier Stokes system for polyatomic mixtures M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 16 / 25

55 We consider interactions of Maxwell molecules type: and define κ ij = 2π π 0 ϑ ij (χ)(1 cosχ) sinχdχ ν ij = 2π π 0 gσ ij (g,χ) =ϑ ij (χ) ϑ ij (χ)(1 cos 2 χ) sinχ dχ We formally get that parameters α i (i = 1,...,4Q), u, T satisfy the following Navier Stokes system for polyatomic mixtures Incompressibility condition: x u = 0 M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 16 / 25

56 We consider interactions of Maxwell molecules type: and define κ ij = 2π π 0 ϑ ij (χ)(1 cosχ) sinχdχ ν ij = 2π π 0 gσ ij (g,χ) =ϑ ij (χ) ϑ ij (χ)(1 cos 2 χ) sinχ dχ We formally get that parameters α i (i = 1,...,4Q), u, T satisfy the following Navier Stokes system for polyatomic mixtures Incompressibility condition: x u = 0 Boussinesq identity: 4Q ( x ρ i α i) + T = 0 i=1 [Such constraints follow from conservation of total number density and of global momentum, respectively] M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 16 / 25

57 [ t j i ( + j i Convection-diffusion equations for the densities of the components (main difference with respect to the single species frame): ] [ ] [ ] ρ j µ ij κ ij (α i α j ) + u x ρ j µ ij κ ij (α i α j ) = x ρ j (α i α j ) j i j i ρ j ) µ ij κ ij ρ i J i (1) dv µ ij κ ij j i J j dv i = 1,...,4Q 1, (1) whereµ ij = mi m j is the reduced mass and J i is the O(ε) part of m i +m j (1) the operator J i (contributions will be made explicit for Maxwell molecules) M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 17 / 25

58 [ t j i ( + j i Convection-diffusion equations for the densities of the components (main difference with respect to the single species frame): ] [ ] [ ] ρ j µ ij κ ij (α i α j ) + u x ρ j µ ij κ ij (α i α j ) = x ρ j (α i α j ) j i j i ρ j ) µ ij κ ij ρ i J i (1) dv µ ij κ ij j i J j dv i = 1,...,4Q 1, (1) whereµ ij = mi m j is the reduced mass and J i is the O(ε) part of m i +m j (1) the operator J i (contributions will be made explicit for Maxwell molecules) More precisely, if D i in and Di denote the sets of all inelastic ch transitions and chemical reactions involving particles A i, we have [ ] = {ρ h ρ k Jijhk+ i (gh ε Mh, M k ) + Jijhk+ i (Mh, gε k Mk ) J i (1) (j,h,k) D i in Di ch ]} ρ i ρ [J j ijhk i (gi ε Mi, M j ) + Jijhk i (Mi, g j εm j ) M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 17 / 25

59 Convection-diffusion equation for the momentum t u + u x u + x p = d 1 x u M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 18 / 25

60 Convection-diffusion equation for the momentum t u + u x u + x p = d 1 x u Convection-diffusion equation for the temperature 4Q ( ) 1 t T + u x T = d 2 x T + 5 mi v 2 1 J i (1) dv where diffusion coefficients d 1, d 2 are the unique solutions of suitable linear systems and depend on masses and on averaged collision frequenciesκ ij,ν ij i=1 M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 18 / 25

61 Convection-diffusion equation for the momentum t u + u x u + x p = d 1 x u Convection-diffusion equation for the temperature 4Q ( ) 1 t T + u x T = d 2 x T + 5 mi v 2 1 J i (1) dv where diffusion coefficients d 1, d 2 are the unique solutions of suitable linear systems and depend on masses and on averaged collision frequenciesκ ij,ν ij Remarks: The final system is not strongly coupled, evolution equation for u could be solved separately i=1 M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 18 / 25

62 Convection-diffusion equation for the momentum t u + u x u + x p = d 1 x u Convection-diffusion equation for the temperature 4Q ( ) 1 t T + u x T = d 2 x T + 5 mi v 2 1 J i (1) dv where diffusion coefficients d 1, d 2 are the unique solutions of suitable linear systems and depend on masses and on averaged collision frequenciesκ ij,ν ij Remarks: The final system is not strongly coupled, evolution equation for u could be solved separately i=1 Velocities or temperatures specific to each species would appear only if we considered higher orders in expansions of distributions, or if we took as dominant operator (of order 1/ε) only Q ii (f i ε, f i ε ) M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 18 / 25

63 In the one-species case concentration α is completely known from the equation for T, while for a mixture 4Q 1 additional independent evolution equations forα i are needed M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 19 / 25

64 In the one-species case concentration α is completely known from the equation for T, while for a mixture 4Q 1 additional independent evolution equations forα i are needed For a mixture of only two monatomic species, the additional equation is simply provided by the difference of the two kinetic equations M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 19 / 25

65 In the one-species case concentration α is completely known from the equation for T, while for a mixture 4Q 1 additional independent evolution equations forα i are needed For a mixture of only two monatomic species, the additional equation is simply provided by the difference of the two kinetic equations Computation of diffusion coefficients d 1 and d 2 and the proof that they are strictly positive is not a trivial extension of the one-species case M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 19 / 25

66 In the one-species case concentration α is completely known from the equation for T, while for a mixture 4Q 1 additional independent evolution equations forα i are needed For a mixture of only two monatomic species, the additional equation is simply provided by the difference of the two kinetic equations Computation of diffusion coefficients d 1 and d 2 and the proof that they are strictly positive is not a trivial extension of the one-species case Contributions due to the polyatomic nature of gases (and to chemical reactions) affect equations for number densities and global temperature M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 19 / 25

67 Some steps of the derivation Equations for concentrations (i = 1,...,4Q) ε t (gε i Mi )(v) dv + x v(gε i Mi )(v) dv =ε 1 ρ i J i (1) dv We need a closure for the streaming terms to O(ε) accuracy M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 20 / 25

68 Some steps of the derivation Equations for concentrations (i = 1,...,4Q) ε t (gε i Mi )(v) dv + x v(gε i Mi )(v) dv =ε 1 ρ i J i (1) dv We need a closure for the streaming terms to O(ε) accuracy We resort to momentum equations of each component ε 2 ρ i t v(gε i Mi )(v) dv +ερ i x v v(gε i Mi )(v) dv = [ ] = {ρ i ρ j v Q ij (gε i Mi, M j ) + Q ij (M i, g j εm j ) dv j i +ερ i ρ j v Q ij (g i ε Mi, g j εm j )dv } +ε 2 v J i (1) dv M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 20 / 25

69 Some steps of the derivation Equations for concentrations (i = 1,...,4Q) ε t (gε i Mi )(v) dv + x v(gε i Mi )(v) dv =ε 1 ρ i J i (1) dv We need a closure for the streaming terms to O(ε) accuracy We resort to momentum equations of each component ε 2 ρ i t v(gε i Mi )(v) dv +ερ i x v v(gε i Mi )(v) dv = [ ] = {ρ i ρ j v Q ij (gε i Mi, M j ) + Q ij (M i, g j εm j ) dv j i +ερ i ρ j v Q ij (g i ε Mi, g j εm j )dv } +ε 2 v J i (1) dv We find an explicit relation between the terms in red M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 20 / 25

70 ρ i ρ j v j i = ρ j µ ij κ ij j i [ ] Q ij (gε i Mi, M j ) + Q ij (M i, g j εm j ) dv ρ i m i v (gε i Mi )(v) dv + ρi ρ j µ ij κ ij m i j i v (g j εm j )(v) dv hence we can insert the i th momentum equation into a suitable linear combination of number densities equations, and we evaluate then each term recalling that g i ε(v) =α i + m i v u+ ( 1 2 mi v 2 3 2) T + O(ε) M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 21 / 25

71 ρ i ρ j v j i = ρ j µ ij κ ij j i [ ] Q ij (gε i Mi, M j ) + Q ij (M i, g j εm j ) dv ρ i m i v (gε i Mi )(v) dv + ρi ρ j µ ij κ ij m i j i v (g j εm j )(v) dv hence we can insert the i th momentum equation into a suitable linear combination of number densities equations, and we evaluate then each term recalling that g i ε(v) =α i + m i v u+ ( 1 2 mi v 2 3 2) T + O(ε) Analogously for closure of momentum and temperature equations: 4Q integrals of streaming terms x ρ i B i n (v)(gi ε Mi )(v) dv i=1 with B i 1 (v v (v) = 1 ) ( 1 mi 3 v2 I, B i 2 (v) = 2 mi v 2 5 ) v 2 are proportional to 4Q [ ] ρ i ρ j θn i Bn(v) i Q ij (gε i Mi, M j )+Q ij (M i, g j εm j ) dv (n = 1, 2) i,j=1 M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 21 / 25

72 Inelastic collision contributions J i (1) dv = K i ijhk (j,h,k) D i in Di ch wherek i i represents the net production of particles of species A ijhk due to the interaction A i + A j A h + A k M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 22 / 25

73 Inelastic collision contributions J i (1) dv = K i ijhk (j,h,k) D i in Di ch wherek i i represents the net production of particles of species A ijhk due to the interaction A i + A j A h + A k Obvious symmetry property: K i ijhk =K j ijhk = K h ijhk = K k ijhk M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 22 / 25

74 Inelastic collision contributions J i (1) dv = K i ijhk (j,h,k) D i in Di ch wherek i i represents the net production of particles of species A ijhk due to the interaction A i + A j A h + A k Obvious symmetry property: K i ijhk =K j ijhk = K h ijhk = K k ijhk We adopt a Maxwell molecule assumption for any option (i, j, h, k) corresponding to an endothermic direct interaction (i.e. E hk > 0): ij ν hk ij = gσ hk ij (g,χ) dˆn M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 22 / 25

75 Inelastic collision contributions J i (1) dv = K i ijhk (j,h,k) D i in Di ch wherek i i represents the net production of particles of species A ijhk due to the interaction A i + A j A h + A k Obvious symmetry property: K i ijhk =K j ijhk = K h ijhk = K k ijhk We adopt a Maxwell molecule assumption for any option (i, j, h, k) corresponding to an endothermic direct interaction (i.e. E hk > 0): ij ν hk ij = gσ hk ij (g,χ) dˆn The relation for the cross section of the reverse exothermic interactionσ ij follows from the microreversibility condition hk M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 22 / 25

76 Inelastic collision contributions J i (1) dv = K i ijhk (j,h,k) D i in Di ch wherek i i represents the net production of particles of species A ijhk due to the interaction A i + A j A h + A k Obvious symmetry property: K i ijhk =K j ijhk = K h ijhk = K k ijhk We adopt a Maxwell molecule assumption for any option (i, j, h, k) corresponding to an endothermic direct interaction (i.e. E hk > 0): ij ν hk ij = gσ hk ij (g,χ) dˆn The relation for the cross section of the reverse exothermic interactionσ ij follows from the microreversibility condition hk J i (1) dv = K i ijhk K h hkij (j,h,k) D i En (j,h,k) D i Ex M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 22 / 25

77 Recalling that gε(v) i =α i + m i v u+ ( 1 2 mi v 2 2) 3 T + O(ε) and the assumptions on the leading order number densitiesρ i we get J i (1) dv = 2 [ ] ( 3 ) Λ hk π ij α h +α k α i α j T E hk ij Γ 2, E hk ij (j,h,k) D i in Di ch where Λ hk ij = ν hk ij ρ i ρ j if (j, h, k) D i En ν ij hk ρh ρ k if (j, h, k) D i Ex M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 23 / 25

78 Recalling that gε(v) i =α i + m i v u+ ( 1 2 mi v 2 2) 3 T + O(ε) and the assumptions on the leading order number densitiesρ i we get J i (1) dv = 2 [ ] ( 3 ) Λ hk π ij α h +α k α i α j T E hk ij Γ 2, E hk ij (j,h,k) D i in Di ch where Λ hk ij = ν hk ij ρ i ρ j if (j, h, k) D i En ν ij hk ρh ρ k if (j, h, k) D i Ex The content of the square brackets is the linearization (i.e., the O(ε) terms) of the mass action law for global distribution functions (f i ε, f j ε, f h ε, f k ε ) M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 23 / 25

79 Recalling that gε(v) i =α i + m i v u+ ( 1 2 mi v 2 2) 3 T + O(ε) and the assumptions on the leading order number densitiesρ i we get J i (1) dv = 2 [ ] ( 3 ) Λ hk π ij α h +α k α i α j T E hk ij Γ 2, E hk ij (j,h,k) D i in Di ch where Λ hk ij = ν hk ij ρ i ρ j if (j, h, k) D i En ν ij hk ρh ρ k if (j, h, k) D i Ex The content of the square brackets is the linearization (i.e., the O(ε) terms) of the mass action law for global distribution functions (fε, i fε, j fε, h fε k ) Suitable combinations of J i (1) dv complete the derivation of incompressible Navier Stokes equations for number densities and global temperature M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 23 / 25

80 Work in progress and open problems Work in progress (joint with S. Brull): formal compressible hydrodynamic limit (at Navier Stokes level) for polyatomic gas mixtures, owing to the Chapman Enskog method M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 24 / 25

81 Work in progress and open problems Work in progress (joint with S. Brull): formal compressible hydrodynamic limit (at Navier Stokes level) for polyatomic gas mixtures, owing to the Chapman Enskog method Open problem: Fredholm alternative for the linearized Boltzmann operator for polyatomic gases (with discrete or continuous internal energy) M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 24 / 25

82 Work in progress and open problems Work in progress (joint with S. Brull): formal compressible hydrodynamic limit (at Navier Stokes level) for polyatomic gas mixtures, owing to the Chapman Enskog method Open problem: Fredholm alternative for the linearized Boltzmann operator for polyatomic gases (with discrete or continuous internal energy) Open problem: more appropriate descriptions for chains of chemical reactions occurring in physical applications M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 24 / 25

83 Thank you for your attention M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 25 / 25