Non-equilibrium mixtures of gases: modeling and computation

Size: px

Start display at page:

Download "Non-equilibrium mixtures of gases: modeling and computation"

Kerry Wilkerson
5 years ago
Views:

1 Non-equilibrium mixtures of gases: modeling and computation Lecture 1: and kinetic theory of gases Srboljub Simić University of Novi Sad, Serbia

2 Aim and outline of the course Aim of the course To present modelling and computational issues in extended thermodynamics of multi-temperature mixtures. To make comparison with classical and kinetic approach. Outline of the course and kinetic theory of gases Extended thermodynamics of monatomic gases and mixtures Kinetic modeling of mixtures Shock structure in gaseous mixtures and other problems Reacting mixtures of gases

3 Outline 1 Single-component fluids Mixtures of fluids 2

4 Single-component fluids Mixtures of fluids Outline 1 Single-component fluids Mixtures of fluids 2

5 Single-component fluids Mixtures of fluids Equations of continuum theories of fluids Balance laws Govern the rate of change of the basic fields (ρ, v, T ). Balance of mass Balance of momentum Balance of energy Constitutive relations Provide mathematical description of material response by relating (ε, t, q) to the basic fields. material objectivity thermodynamic restrictions

6 Single-component fluids Mixtures of fluids Balance laws of continuum mechanics Mass Momentum Energy ρ t + div(ρv) = 0 (ρv) t + div(ρv v t) = 0 ( 1 2 ρv2 + ρε ) {( ) } 1 + div t 2 ρv2 + ρε v tv + q = 0

7 Single-component fluids Mixtures of fluids Balance laws of continuum mechanics Euler gas dynamics equations p = ρ kb m T ε = k B (γ 1)m T t = p I q = 0 ρ t + div(ρv) = 0 (ρv) t + div(ρv v + p) = 0 ( 1 2 ρv2 + ρε ) {( ) } 1 + div t 2 ρv2 + ρε + p v = 0

8 Single-component fluids Mixtures of fluids Entropy inequality and TIP Entropy balance law (ρs) t + div(ρsv + Φ) 0 Basic assumptions of TIP Assumption of local equilibrium (Gibbs relation) ds = 1 (dε pρ ) T dρ 2 Structure of entropy flux Φ = q T

9 Single-component fluids Mixtures of fluids Entropy inequality and TIP Entropy balance law (ρs) t + div(ρsv + Φ) 0 Basic assumptions of TIP Assumption of local equilibrium (Gibbs relation) ds = 1 (dε pρ ) T dρ 2 Structure of entropy flux Φ = q T

10 Single-component fluids Mixtures of fluids Entropy inequality and TIP Balance laws ρ + ρ div v = 0 ρ v div t = 0 ρ ε t : grad v + div q = 0 (a) Gibbs relation ṡ = 1 T ( ε p ρ 2 ρ ) ( ). = d/dt = / t + v x (b) Entropy balance law from (a) and (b) ( q ) ρṡ + div = Σ T Σ = 1 T 2 q grad T + 1 T td : D D + 1 T ( ) 1 3 tr t + p div v 0 D = 1 2 ( grad v + (grad v) T ) D D = D 1 (tr D)I 3

11 Single-component fluids Mixtures of fluids Constitutive relations Constitutive relations heat flux q = κ grad T κ 0 stress deviator t D = 2µ D D µ 0 ( ) 1 dynamic pressure π = 3 tr t + p = λ div v λ 0 Entropy production Σ = κ T (grad T 2 )2 + 2µ ( D D) 2 λ + T T (div v)2 0

12 Single-component fluids Mixtures of fluids Paradox of infinite speeds Case 1 ρ = const., v = 0 T t = κ T ρε T Case 2 ρ = const., T = const., v = (0, v(x 1, t), 0) v t = µ 2 v ρ x 2 1

13 Single-component fluids Mixtures of fluids Paradox of infinite speeds Case 1 ρ = const., v = 0 T t = κ T ρε T Case 2 ρ = const., T = const., v = (0, v(x 1, t), 0) v t = µ 2 v ρ x 2 1

14 Single-component fluids Mixtures of fluids Outline 1 Single-component fluids Mixtures of fluids 2

15 Single-component fluids Mixtures of fluids Balance laws for mixtures State variables (ρ α, v, T ); α = 1,..., n Mass Momentum Energy ρ α + div(ραvα) = τα; t (ρv) + div(ρv v t) = 0 t ( 1 2 ρv2 + ρε ) + div t vα = v + uα {( 1 2 ρv2 + ρε ) } v tv + q = 0 Gibbs relation ṡ = 1 T ( ) ε p n 1 ρ ρ (µ 2 b µ n)ċ b b=1 µ α chemical potentials; c α = ρα ρ mass concentrations

16 Single-component fluids Mixtures of fluids Constitutive relations for mixtures Constitutive relations (τ α = 0 non-reacting mixtures) heat flux q = 1 n 1 T L grad T ( µc µ ) n L 2 c grad T c=1 diffusion flux J b = 1 n 1 T L 2 b grad T J b = ρ b (v b v) c=1 L bc grad ( µc µ ) n T stress deviator dynamic pressure t D = 2µ D D ( ) 1 π = 3 tr t + p = λ div v

17 Outline 1 Single-component fluids Mixtures of fluids 2

18 State of the atom State of the atom (molecule) at time t is determined by its position x and its velocity ξ, (t, x, ξ). Collision transformation Assumptions Only binary collisions between atoms are considered. Collisions are assumed to be elastic. Interaction time is much shorter that the mean free time. Change of state is governed by the momentum and energy conservation laws.

19 Collision transformation Conservation laws (ξ, ξ ) outgoing velocities of molecules (ξ, ξ ) incoming velocities of molecules Velocity transformation ξ + ξ = ξ + ξ ξ 2 + ξ 2 = ξ 2 + ξ 2 ω notation σ notation ξ = ξ ω(ω (ξ ξ )) ξ = 1 2 (ξ + ξ ) ξ ξ σ ξ = ξ + ω(ω (ξ ξ )) ξ = 1 2 (ξ + ξ ) 1 2 ξ ξ σ

20 Velocity distribution function f(t, x, ξ) f(t, x, ξ)dxdξ number of atoms in elementary volume dxdξ in phase space Collision integral Q(f, f) = f t + R 3 3 i=1 ξ i f x i = Q(f, f) S 2 ( f f ff ) B(ξ ξ, σ)dσdξ Collision cross-section B(ξ ξ, σ) describes model of interaction between the atoms (hard spheres, repulsive potential,...)

21 Collision invariants Collision invariants R 3 ψ(ξ)q(f, f)dξ = 0 Elementary collision invariants ψ(ξ ) + ψ(ξ ) = ψ(ξ) + ψ(ξ ) ψ 0 = 1; ψ i = ξ i (i = 1, 2, 3); ψ 4 = ξ 2 Macroscopic quantities ρ ρv i ρ v 2 + 2ρε = R 3 m 1 ξ i ξ 2 f(t, x, ξ)dξ

22 Macroscopic equations Peculiar velocity C i = ξ i v i, i = 1, 2, 3 Internal energy, pressure tensor and heat flux 1 internal energy ρε = R 3 2 m C 2 f(t, x, v + C)dC pressure tensor p ij = mc ic jf(t, x, v + C)dC = t ij R 3 1 heat flux q i = R 3 2 m C 2 C if(t, x, v + C)dC Pressure, internal energy and temperature in monatomic gases 3p = tr{p ij} = 2ρε T = 2m 3k B ε

23 Macroscopic equations Macroscopic equations Mass ψ αf dξ + t R 3 ρ t + 3 i=1 3 ξ iψ αf dξ = 0 α = 0,..., 4 x i R 3 i=1 x i (ρv i) = 0 Momentum 3 t (ρvj) + (ρv jv i + p ji) = 0 x i i=1 ( ) 1 Energy t 2 ρ v 2 + ρε { 3 ( ) 1 + x i 2 ρ v 2 + ρε v i + i=1 } 3 p jiv j + q i = 0 j=1

24 H-theorem f f = ff = log f + log f = log f + log f Equilibrium (Maxwellian) distribution f M = ρ m log f M = ( m 2πk BT 4 a αψ α(ξ) α=0 ) 3/2 exp { ξ v 2 2(k B/m)T Entropy production functional D(f) := log f Q(f, f) dξ 0 R 3 }

25 H-theorem H-theorem Assume that cross section B is positive a.e. and f 0 is such that Q(f, f) and D(f) are well defined. Then: Entropy production is non-positive, D(f) 0. The following statements are equivalent 1 For any ξ R 3, Q(f, f) = 0; 2 Entropy production vanishes, D(f) = 0; 3 There exists ρ > 0, T > 0 and v R 3 such that f = ρ m ( ) m 3/2 exp { 2πk B T ξ v 2 2(k B /m)t }.

26 H-theorem Entropy inequality H = f log f dξ J i = ξ if log f dξ R 3 R 3 If f is any solution of B.e., then H t + 3 i=1 J i x i = D(f) 0.

27 Outline 1 Single-component fluids Mixtures of fluids 2

28 Scaled Scaled Asymptotic expansion Compatibility condition f ɛ t + ɛ Kn 3 i=1 ξ i f ɛ x i = 1 ɛ Q(f ɛ, f ɛ ) small parameter f ɛ = f (0) + ɛf (1) + ɛ 2 f (2) + = ɛ k f (k) k=0 R 3 mψ jf (k) dξ = 0; k 1

29 Scaled More expansions Non-convective fluxes Material derivative p ij = mc ic jf ɛ dξ = R 3 k=1 m q i = R 3 2 C 2 f ɛ dξ = k=1 ɛ k p (k) ij ɛ k q (k) i Df ɛ + 3 f ɛ C i = 1 x i ɛ Q(f ɛ, f ɛ ) D = 3 t + i=1 D = D 0 + ɛd 1 + ɛ 2 D 2 + = ɛ k D k k=0 i=1 v i x i

30 Formal expansion Expansion of conservation laws D 0ρ + iv i = 0 D k ρ = 0 ρd 0v i + ip = 0 ρd k v i + jp (k) ij = 0 k ρ kb m D0T + p ivi = ρ kb m D kt + p (k) ij jvi = 0 Expansion of the Q(f (0), f (0) ) = 0 2Q(f (0), f (1) ) = D 0f (0) + 3 i=1 C i f (0) x i

31 Formal expansion Expansion of conservation laws D 0ρ + iv i = 0 D k ρ = 0 ρd 0v i + ip = 0 ρd k v i + jp (k) ij = 0 k ρ kb m D0T + p ivi = ρ kb m D kt + p (k) ij jvi = 0 Expansion of the Q(f (0), f (0) ) = 0 2Q(f (0), f (1) ) = D 0f (0) + 3 i=1 C i f (0) x i

32 Formal expansion First order approximation Non-convective fluxes f (0) = ρ ( ) 3/2 } m ξ v 2 exp { m 2πk BT 2(k B/m)T { f (1) = f (0) φ = f (0) A } T v i T Ci 2βBC ic j x i x j p (1) ij = 2µ v i q (1) i = λ T x j x i

33 Formal expansion First order approximation Non-convective fluxes f (0) = ρ ( ) 3/2 } m ξ v 2 exp { m 2πk BT 2(k B/m)T { f (1) = f (0) φ = f (0) A } T v i T Ci 2βBC ic j x i x j p (1) ij = 2µ v i q (1) i = λ T x j x i

34 Formal expansion First order approximation Non-convective fluxes f (0) = ρ ( ) 3/2 } m ξ v 2 exp { m 2πk BT 2(k B/m)T { f (1) = f (0) φ = f (0) A } T v i T Ci 2βBC ic j x i x j p (1) ij = 2µ v i q (1) i = λ T x j x i

35 Outline 1 Single-component fluids Mixtures of fluids 2

36 Moment equations Moments of distribution function F i1 i n = mξ i1 ξ in f dξ R 3 F i1 i nk = mξ k ξ i1 ξ in f dξ P i1 i n = mξ i1 ξ in Q(f, f) dξ R 3 R 3 Transfer equations for moments t Fi 1 i n + 3 k=1 x k F i1 i nk = P i1 i n P = 0, P i = 0, P ii = 0 due to collision invariants

37 Moment equations Hierarchical structure of moment equations t F + t Fi 1 + t Fi 1i 2 +. t Fi 1 i n +. 3 k=1 3 k=1 3 k=1 3 k=1 x k F k = 0 x k F i1 k = 0 x k F i1 i 2 k = P i1 i 2 x k F i1 i nk = P i1 i n Fluxes of order m become densities of order m + 1. Hierarchy is infinite. If the system is truncated at order n, fluxes of order n + 1 and source terms remain undetermined. Closure problem fluxes and source terms should be expressed in terms of densities.

38 Moment equations Hierarchical structure of moment equations t F + t Fi 1 + t Fi 1i 2 +. t Fi 1 i n +. 3 k=1 3 k=1 3 k=1 3 k=1 x k F k = 0 x k F i1 k = 0 x k F i1 i 2 k = P i1 i 2 x k F i1 i nk = P i1 i n Fluxes of order m become densities of order m + 1. Hierarchy is infinite. If the system is truncated at order n, fluxes of order n + 1 and source terms remain undetermined. Closure problem fluxes and source terms should be expressed in terms of densities.

39 Grad s method Approximation of non-equilibrium distribution function f = f M (a + a iξ i + a ijξ iξ j + a ijk ξ iξ jξ k + ) Expansion in terms of Hermite polynomials f M = ρ ( ) 3/2 } m ξ v 2 exp { = ρ ( m ) 3/2 ω(c) m 2πk BT 2(k B/m)T m kt ω(c) = 1 ( 2 /2 m ) 1/2 e c c (2π) 3/2 i = Ci kt H i1 i n (c) = ( 1)N ω N ω c i1 c in

40 Grad s method Approximation of non-equilibrium distribution function f = f M (a + a iξ i + a ijξ iξ j + a ijk ξ iξ jξ k + ) Expansion in terms of Hermite polynomials f M = ρ ( ) 3/2 } m ξ v 2 exp { = ρ ( m ) 3/2 ω(c) m 2πk BT 2(k B/m)T m kt ω(c) = 1 ( 2 /2 m ) 1/2 e c c (2π) 3/2 i = Ci kt H i1 i n (c) = ( 1)N ω N ω c i1 c in

41 Grad s method Expansion in terms of Hermite polynomials (cont.) H i1 i n (c) = ( 1)N N ω ω c i1 c in f = f M (ah + a ih i + 1 2! aijhij ) N! ai 1 i n H i1 i n + H = 1 H i = c i H ij = c ic j δ ij H ijk = c ic jc k (c iδ jk + c jδ ki + c k δ ij)

42 Grad s method Expansion in terms of Hermite polynomials (cont.) H i1 i n (c) = ( 1)N N ω ω c i1 c in f = f M (ah + a ih i + 1 2! aijhij ) N! ai 1 i n H i1 i n + H = 1 H i = c i H ij = c ic j δ ij H ijk = c ic jc k (c iδ jk + c jδ ki + c k δ ij)

43 Grad s method Moments of distribution function ρ = 0 i = ρε = 1 2 R 3 m R 3 m ( ) 3/2 kt fh dc m ( ) 2 kt fh i dc m ( ) 5/2 kt f(h ii + 3H) dc m R 3 m p ij = p ij 1 3 p kkδ ij= q i= 1 2 R 3 m R 3 m ( ) 5/2 kt f (H ij 13 ) m Hrrδij dc ( ) 3 kt f(h ijj + 5H i) dc m

44 Grad s method Moments of distribution function ρ = 0 i = ρε = 1 2 R 3 m R 3 m ( ) 3/2 kt fh dc m ( ) 2 kt fh i dc m ( ) 5/2 kt f(h ii + 3H) dc m R 3 m p ij = p ij 1 3 p kkδ ij= q i= 1 2 R 3 m R 3 m ( ) 5/2 kt f (H ij 13 ) m Hrrδij dc ( ) 3 kt f(h ijj + 5H i) dc m

45 Grad s method 13 moments approximation Velocity distribution with 13 moments a finite-dimensional approximation of non-equilibrium distribution function. ( f 13 = f M a + a ih i aijhij + 1 ) 10 arrihssi a = 1 a i = 0 a rr = 0 a ij = p ( ij a rri = 2qi m ) 1/2 p p kt { f 13 = f M ( m ) [ 2 p ij C ic j + 4 ( m ρ 2kT 5 qici 2kT C 2 5 )]} 2

46 Grad s method 13 moments approximation Velocity distribution with 13 moments a finite-dimensional approximation of non-equilibrium distribution function. ( f 13 = f M a + a ih i aijhij + 1 ) 10 arrihssi a = 1 a i = 0 a rr = 0 a ij = p ( ij a rri = 2qi m ) 1/2 p p kt { f 13 = f M ( m ) [ 2 p ij C ic j + 4 ( m ρ 2kT 5 qici 2kT C 2 5 )]} 2

47 Grad s method 13 moments equations Conservation laws for mass, momentum and energy tρ + i(ρv i) = 0 t(ρv i) + j(ρv iv j + p ij) = 0 ( ) {( ) } 1 1 t 2 ρ v 2 + ρε + i 2 ρ v 2 + ρε v i + p ijv j + q i = 0 Balance laws for momentum and energy flux t (ρv iv j + p ij) + k {ρv iv jv k + v ip jk + v jp ki + v k p ij + p ijk } = P ij {( ) } {( ) 1 1 t 2 ρ v 2 + ρε v i + p ijv j + q i + j 2 ρ v 2 + ρε v iv j +v iv k p jk + v jv k p ik ρ v 2 p ij + q iv j + q jv i + p ijk v k + q ij } = Q i

48 Grad s method 13 moments equations Conservation laws for mass, momentum and energy tρ + i(ρv i) = 0 t(ρv i) + j(ρv iv j + p ij) = 0 ( ) {( ) } 1 1 t 2 ρ v 2 + ρε + i 2 ρ v 2 + ρε v i + p ijv j + q i = 0 Balance laws for momentum and energy flux t (ρv iv j + p ij) + k {ρv iv jv k + v ip jk + v jp ki + v k p ij + p ijk } = P ij {( ) } {( ) 1 1 t 2 ρ v 2 + ρε v i + p ijv j + q i + j 2 ρ v 2 + ρε v iv j +v iv k p jk + v jv k p ik ρ v 2 p ij + q iv j + q jv i + p ijk v k + q ij } = Q i

49 Grad s method Non-convective fluxes p ijk = mc ic jc k f 13 dc = 2 R 3 5 (qiδ jk + q jδ ki + q k δ ij) q ij = 1 mc ic j C 2 f 13 dc = 7 p p2 pij 2 R 3 2 ρ ρ δij Source terms (linearized) P ij = mξ iξ jq(f 13, f 13) dξ = 1 R 3 τ p ij Q i = 1 m ξ 2 ξ iq(f 13, f 13) dξ = 2 2 R 3 3τ qi τ = 5m ( m ) 1/2 1 16ρ πkt Ω (2,2)

50 Grad s method Non-convective fluxes p ijk = mc ic jc k f 13 dc = 2 R 3 5 (qiδ jk + q jδ ki + q k δ ij) q ij = 1 mc ic j C 2 f 13 dc = 7 p p2 pij 2 R 3 2 ρ ρ δij Source terms (linearized) P ij = mξ iξ jq(f 13, f 13) dξ = 1 R 3 τ p ij Q i = 1 m ξ 2 ξ iq(f 13, f 13) dξ = 2 2 R 3 3τ qi τ = 5m ( m ) 1/2 1 16ρ πkt Ω (2,2)

51 Entropy and entropy flux Entropy and entropy flux in equilibrium ρs E = k ρ { ( ρ ( m ) ) 3/2 log 3 } m m 2πkT 2 {dε pρ } dρ 2 φ ie = 0 ds E = 1 T Entropy and entropy flux in 13 moments approximation ρs = ρs 1 k ρ E 4 m p p ij p 2 ij 1 k 5 m φ i = qi T 2 k ρ 5 m p p ij q 2 j ρ 2 p 3 qiqi

52 Entropy and entropy flux Entropy and entropy flux in equilibrium ρs E = k ρ { ( ρ ( m ) ) 3/2 log 3 } m m 2πkT 2 {dε pρ } dρ 2 φ ie = 0 ds E = 1 T Entropy and entropy flux in 13 moments approximation ρs = ρs 1 k ρ E 4 m p p ij p 2 ij 1 k 5 m φ i = qi T 2 k ρ 5 m p p ij q 2 j ρ 2 p 3 qiqi

53 Outline 1 Single-component fluids Mixtures of fluids 2

54 Modelling polyatomic gases molecules constituted of two or more atoms presence of internal degrees of freedom other than translational internal states discrete sequence of values E i continuous variable I during collision total energy is redistributed between translational and internal degrees of freedom

55 Velocity distribution function f(t, x, ξ, I); I 0 n(t, x) = Collision transformation Conservation laws R 3 0 f(t, x, ξ, I) ϕ(i)di dξ ξ + ξ = ξ + ξ, 1 2 m ξ m ξ 2 + I + I = 1 2 m ξ m ξ 2 + I + I, e = 1 4 m ξ ξ 2 + I + I = 1 4 m ξ ξ 2 + I + I.

56 Redistribution of energy Re = 1 4 m ξ ξ 2, (1 R)e = I + I ; R [0, 1] I = r (1 R)e, I = (1 r)(1 R)e; r [0, 1] Collision transformation [ ] ξ ξ 4Rε ξ = ξ m Tω ; T ωz = z 2(ω z)ω, z R 3 ξ ξ ξ = ξ + ξ 2 + [ ] Rε ξ ξ m Tω ξ ξ ξ = ξ + ξ 2 [ ] Rε ξ ξ m Tω ξ ξ

57 Redistribution of energy Re = 1 4 m ξ ξ 2, (1 R)e = I + I ; R [0, 1] I = r (1 R)e, I = (1 r)(1 R)e; r [0, 1] Collision transformation [ ] ξ ξ 4Rε ξ = ξ m Tω ; T ωz = z 2(ω z)ω, z R 3 ξ ξ ξ = ξ + ξ 2 + [ ] Rε ξ ξ m Tω ξ ξ ξ = ξ + ξ 2 [ ] Rε ξ ξ m Tω ξ ξ

58 Collision invariants ( ψ(ξ, I) = m 1, ξ i, ξ I ) T m Internal energy decomposition ρε T = ρε I = R 3 0 R m C 2 f(t, x, C, I)ϕ(I) di dc If(t, x, C, I)ϕ(I) di dc

59 Equilibrium distribution f E = ρ ( m m q(t ) q(t ) = 2πk BT Internal energy in equilibrium 0 ) 3/2 { exp 1 k BT ( exp I k BT ( )} 1 2 m C 2 +I ) ϕ(i) di ϕ(i) = I α ε E = ( 5 ) k 2 + α m T α > 1

60 Non-equilibrium distribution 14 moments approximation { f 14 = f E 1 ρ p 2 qici + ρ 2p 2 3 ρ 2(1 + α) p Π 2 ( 1 2 C 2 + I m [ ( 5 p ij + ) + ) ] 2 + α (1 + α) 1 Πδ ij C ic j ( 7 ) 1 ρ α p 3 qi ( 1 2 C 2 + I ) } C i m

61 Momentum hierarchy tρ + i(ρv i) = 0, t(ρv i) + j(ρv iv j + p ij) = 0, t (ρv iv j + p ij) + k {ρv iv jv k + v ip jk + v jp ki + v k p ij + (7/2 + α) 1 (q iδ jk + q jδ ki + q k δ } ij) = P ij,

62 Energy hierarchy ( ) {( ) } 1 1 t 2 ρ v 2 + ρε + i 2 ρ v 2 + ρε v i + p ijv j + q i = 0, {( ) } {( ) 1 1 t 2 ρ v 2 + ρε v i + p ijv j + q i + j 2 ρ v 2 + ρε v iv j +v iv k p jk + v jv k p ik ρ v 2 p ij + v iq j + v jq i ( ) 1 ( ) } 7 9 p α (v iq j + v jq i + v k q k δ ij) α p2 pij ρ ρ δij = Q i.

63 Source terms Pij 14 = 22s+4 ρ (k T ) 2 π 15 m q(t ) 2 K ( p ij + = 1 τ s p ij + 1 τ Π Πδ ij ( ) s [ k T Γ s + 3 ] m 2 20 (2s + 5)(2s + 7) ( α + 5 ) ) (α + 1) 1 Πδ ij 2 ( ) 1 Q 14 i = u k Pik α K 22s+5 (s (2s + 15) + 30) ρ (k T ) 2 π 9 (2s + 5) (2s + 7) m q(t ) 2 = u k P 14 ik 1 τ q q i ( ) s [ k T Γ s + 3 ] q i m 2

64 Summary Macroscopic approach balance laws and constitutive relations (CR) Entropy inequality imposes restrictions on CR Local equilibrium state Gibbs relation; structure of entropy flux Linear CR are determined from non-negativity of entropy production Mesoscopic approach velocity distribution function (BE); collision invariants and macroscopic equations Chapman-Enskog method closure via asymptotic analysis of BE; derivation of phenomenological equations Moment equations hierarchical structure Grad s moment method balance laws for non-convective fluxes Closure via approximate form of distribution function

65 Summary Macroscopic approach balance laws and constitutive relations (CR) Entropy inequality imposes restrictions on CR Local equilibrium state Gibbs relation; structure of entropy flux Linear CR are determined from non-negativity of entropy production Mesoscopic approach velocity distribution function (BE); collision invariants and macroscopic equations Chapman-Enskog method closure via asymptotic analysis of BE; derivation of phenomenological equations Moment equations hierarchical structure Grad s moment method balance laws for non-convective fluxes Closure via approximate form of distribution function

The Methods of Chapman-Enskog and Grad and Applications

The Methods of Chapman-Enskog and Grad and Applications Gilberto Medeiros Kremer Departamento de Física Universidade Federal do Paraná 81531-980 Curitiba, BRAZIL Contents 1 Introduction 3 Boltzmann and