Extended Thermodynamics of Real Gas: Maximum Entropy Principle for Rarefied Polyatomic Gas

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1 Tommaso Ruggeri Mathematical Departement and Research Center of Applied Mathematics University of Bologna International Conference in Nonlinear Analysis: Evolutionary P.D.E. and Kinetic Theory Taipei October 29 - November 2, 2012

2 Motivation In an endeavor to understand ubiquitous nonequilibrium phenomena, a number of thermodynamic theories have been proposed and developed. In particular Extended Thermodynamics (ET) which describe a nonequilibrium phenomena beyond the assumption of local equilibrium shows successful result and full agreement with Kinetic Theory. Nevertheless the weak point of ET and KT is that the applicable range is limited to rarefied monatomic gas. Figure: semiconductor(copyright c itak (international) Limited.) Atmospheric ReentryJAXA)

3 ommaso Ruggeri Co-Workers Phenomenological ET or dense gas : Takashi Arima, Shigeru Taniguchi, and Masaru Sugiyama (Nagoya) Maximum Entropy Principle - Polyatomic Gas : Milana Pavić and Srboljub Simić (Novi Sad)

4 The Extended Thermodynamics of Rarefied Monoatomic Gas The kinetic theory describes the state of a rarefied gas through the phase density f (x, t, ξ), where f (x, t, ξ)dξ is the number density of atoms at point x and time t that have velocities between ξ and ξ + dξ. The phase density obeys the Boltzmann equation f t + ξ f i = Q (1) x i where Q represents the collisional terms. Most macroscopic thermodynamic quantities are identified as moments of the phase density F k1k 2 k j = f ξ k1 ξ k2 ξ kj dξ, (2) R 3 and due to the Boltzmann equation (1), the moments satisfy an infinity hierarchy of balance laws in which the flux in one equation becomes the density in the next one:

5 ommaso Ruggeri t F + i F i = 0 t F k1 + i F ik1 = 0 t F k1k 2 + i F ik1k 2 = P <k1k 2> t F k1k 2k 3 + i F ik1k 2k 3 = P k1k 2k 3. t F k1k 2...k n + i F ik1k 2...k n = P k1k 2...k n. The hierarchy structure of the system 1 The tensorial rank of the equations increases one by one. 2 The flux in one equation becomes the density in the next equation.

6 ommaso Ruggeri Taking into account that P kk = 0, the first five equations are conservation laws and coincides with the mass, momentum and energy conservation respectively, while the remaining ones are balance laws. Remark: ET with this hierarchy structure is valid only for rarefied monatomic gases. In fact due to the previous structure we have 3p = 2ρε that implies i.e. monoatomic gas. γ = c p c V = 5 3, The Closure Problem When we cut the hierarchy at the density with tensor of rank n, we have the problem of closure because the last flux end the production terms are not in the list of the densities.

7 The first idea of Rational Extended Thermodynamics Müller and Ruggeri (Springer - Verlag 1993, 1998) was to view the truncated system as a phenomenological system of continuum mechanics and then we consider the new quantities as constitutive functions: F k1k 2...k nk n+1 F k1k 2...k nk n+1 (F, F k1, F k1k 2,... F k1k 2...k n ) P k1k 2...k j P k1k 2...k j (F, F k1, F k1k 2,... F k1k 2...k n ) 2 j n. According with the continuum theory, the restrictions on the constitutive equations come only from universal principles, i.e.: Entropy principle, Objectivity Principle and Causality and Stability (convexity of the entropy).

8 13 Moments ET of rarefied monatomic gas Basic field ET of rarefied monatomic gas with 13 independent fields; mass density: F (= ρ), momentum density: F i (= ρv i ), momentum flux: F ij, energy flux: F ppi. In the case of rarefied monatomic gas, F ii (= ρv 2 + 3p) is, except for a factor 1/2, equal to the energy density, ρv 2 /2 + ρε, and therefore 3p = 2ρε, where v i, ε and p are the velocity, specific internal energy density and pressure, respectively.

9 ET of rarefied monatomic gas Balance equations The structure of balance laws is of the following type: F t + F k = 0, the conservation law of mass F i t + F ik = 0, the conservation law of momentum F ij t F ppi t + F ijk = P ij, (trace part:)the conservation law of energy + F ppik = P ppi, F ijk, F ppik :the fluxes of F ij, F ppi respectively. P ij, P ppi :the productions of F ij, F ppi respectively.

10 ET of rarefied monatomic gas The closed field equations The closed system of field equations is obtained as follows: ρ + ρ v i x i = 0 ρ v i (S ij pδ ij ) = 0 x j ρ 3 k 2 m Ṫ + q j (S ij pδ ij ) v i = 0 x j x j v j v n Ṡ ij + 2S n i + S ij 4 q i 2p v i = 1 S ij x n x n 5 x j x j τ S v i q i + q k q v k i q k S ij ρ S jk v (i ) k m T S ij x j 7 k 2 m S ij + S ij p + 5 k x j ρ x j 2 m p x i = 1 τ q q i where τ S, τ q are the relaxation time which relate to the viscosity and heat conductivity. Tommaso Ruggeri

11 ommaso Ruggeri Extended Thermodynamics Problems The Navier-Stokes Fourier theory comes out as a limiting case of ET through carrying out the so-called Maxwellian iteration. In this respect, the Navier-Stokes Fourier theory can be seen as an approximation of ET where the relaxation times of dissipative fluxes (viscous stress and heat flux) are negligible (Navier-Stokes Fourier limit).

12 ommaso Ruggeri Extended Thermodynamics Problems The Navier-Stokes Fourier theory comes out as a limiting case of ET through carrying out the so-called Maxwellian iteration. In this respect, the Navier-Stokes Fourier theory can be seen as an approximation of ET where the relaxation times of dissipative fluxes (viscous stress and heat flux) are negligible (Navier-Stokes Fourier limit). However, within its validity range, the classical Navier-Stokes Fourier theory is applicable to any fluids that are not necessarily limited to rarefied gases nor to monatomic gases.

13 ommaso Ruggeri Extended Thermodynamics Problems The Navier-Stokes Fourier theory comes out as a limiting case of ET through carrying out the so-called Maxwellian iteration. In this respect, the Navier-Stokes Fourier theory can be seen as an approximation of ET where the relaxation times of dissipative fluxes (viscous stress and heat flux) are negligible (Navier-Stokes Fourier limit). However, within its validity range, the classical Navier-Stokes Fourier theory is applicable to any fluids that are not necessarily limited to rarefied gases nor to monatomic gases. Therefore, after the successful establishment of ET for rarefied monatomic gases, there appeared many studies of ET for rarefied polyatomic gases and also for real gases (or dense gases).

14 ommaso Ruggeri 1 The Extended Thermodynamics of Rarefied Monoatomic Gas 2 Model of dense gases 3 Relationship between ET and Navier-Stokes-Fourier theory 4 Extended Thermodynamics of dense gas 5 Rarefied-gas limit 6 Extended Thermodynamics of Moments and Maximum Entropy Principle MEP 7 Equilibrium distribution function for polyatomic gases 8 The hierarchy of moment equations for polyatomic gases 9 The 14 moments system for polyatomic gases Production Terms and relaxation times 10 Qualitative Analysis 11 The 6 Moments case

15 ommaso Ruggeri Model of dense gases Heuristic viewpoint In order to grasp the structure of the basic system appropriate for ET of dense gases, first of all, let us reconsider the structure of the Navier-Stokes Fourier system. ρ t + (ρv k ) = 0, t (ρv i) + (ρv i v k t ik ) = 0, x ( k ) ρv 2 t 2 + ρε + [( ) ] ρv ρε v k t kj v j + q k = 0, S ij = 2µ v i, (v i δ jk + v j δ ik 23 ) x j x v kδ ij = S ij k µ, Π = ν v n x n, v k = Π ν, q i = κ x i, = q k κ.

16 ommaso Ruggeri Model of dense gases Heuristic viewpoint In order to grasp the structure of the basic system appropriate for ET of dense gases, first of all, let us reconsider the structure of the Navier-Stokes Fourier system. ρ t + (ρv k ) = 0, t (ρv i) + (ρv i v k t ik ) = 0, x ( k ) ρv 2 t 2 + ρε + [( ) ] ρv ρε v k t kj v j + q k = 0, S ij = 2µ v i, (v i δ jk + v j δ ik 23 ) x j x v kδ ij = S ij k µ, Π = ν v n x n, v k = Π ν, q i = κ x i, = q k κ.

17 Model of dense gases Heuristic viewpoint The system composed of the conservation equations and above equations can be seen as a system of 14 equations for the 14 unknown variables: ρ, v i, ε, q i, S ij and Π. Its mathematical structure is in the form of balance type, but, in the above equations, we have no term with time derivative. Therefore the system is not hyperbolic but parabolic. It is, therefore, natural to assume that the mathematical structure of balance laws in ET of dense gases must be of the following type:

18 ommaso Ruggeri The balance equations of Navier-Stokes Fourier system ρ t + (ρv k ) = 0,. The balance equations of ET of dense gases F t + F k = 0,

19 ommaso Ruggeri The balance equations of Navier-Stokes Fourier system ρ t + (ρv k ) = 0, t (ρv i ) + (ρv i v k t ik ) = 0,. The balance equations of ET of dense gases F t + F k = 0, F i t + F ik = 0,

20 ommaso Ruggeri The balance equations of Navier-Stokes Fourier system ρ t + (ρv k ) = 0, t (ρv i ) + (ρv i v k t ik ) = 0, ( ρv 2 ) t 2 + ρε + [( ρv 2 ) ] 2 + ρε v k t kj v j + q k = 0,. The balance equations of ET of dense gases F t + F k = 0, F i t + F ik = 0, G ii t + G iik = 0,

21 ommaso Ruggeri The balance equations of Navier-Stokes Fourier system ρ t + (ρv k ) = 0, t (ρv i ) + (ρv i v k t ik ) = 0, (v i δ jk + v j δ ik 23 ) x v kδ ij = S ij k µ,. ( ρv 2 ) t 2 + ρε + [( ρv 2 ) ] 2 + ρε v k t kj v j + q k = 0, The balance equations of ET of dense gases F t + F k = 0, F i t + F ik = 0, F ij + F ij k t = P ij, G ii t + G iik = 0,

22 ommaso Ruggeri The balance equations of Navier-Stokes Fourier system ρ t + (ρv k ) = 0, t (ρv i ) + (ρv i v k t ik ) = 0, (v i δ jk + v j δ ik 23 ) x v kδ ij = S ij k µ, v k = Π ν. The balance equations of ET of dense gases ( ρv 2 ) t 2 + ρε + [( ρv 2 ) ] 2 + ρε v k t kj v j + q k = 0, F t + F k = 0, F i t + F ik = 0, F ij + F ij k t = P ij, F ii + F iik t = P ii, G ii t + G iik = 0,

23 ommaso Ruggeri The balance equations of Navier-Stokes Fourier system ρ t + (ρv k ) = 0, t (ρv i ) + (ρv i v k t ik ) = 0, (v i δ jk + v j δ ik 23 ) x v kδ ij = S ij k µ, v k = Π ν. ( ρv 2 ) t 2 + ρε + [( ρv 2 ) ] 2 + ρε v k t kj v j + q k = 0, = q k κ, The balance equations of ET of dense gases F t + F k = 0, F i t + F ik = 0, F ij + F ij k t = P ij, F ii + F iik t = P ii, G ii t G ppi t + G iik = 0, + G ppik = Q ppi.

24 ommaso Ruggeri The balance equations of ET of dense gases F t + F k = 0, F i t + F ik = 0, G ii t + G iik = 0, F ij t + F ijk = P ij, G ppi t + G ppik = Q ppi, where F is the mass density, F i is the momentum density, G ii is the energy density, F ij is the momentum flux, and G ppi is the energy flux. And F ijk and G ppik are the fluxes of F ij and G ppi, respectively, and P ij and Q ppi are the productions with respect to F ij and G ppi, respectively. We observe that the structure of the balance equations is much more restrictive than that adopted in the previous works, and moreover the system does not have the fourth-rank tensor in the set of densities.

25 To sum up, the hierarchy structure of the system of the balance equations is composed of two parallel series: The one is the series starting from the mass and momentum balance equations (F -series) and the other is from the energy balance equation (G-series). In each series, the flux in one equation becomes the density in the next equation. Such a structure is also completely consistent with the structure of the set of balance equations derived from the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy of many-body distribution functions in statistical mechanics, which is valid not only for rarefied gases but also for dense gases and liquids.

26 ET of dense gases Field equations The closed system of field equations is obtained as follows: ρ + ρ v k = 0, ρ v i + p x i + Π S ij = 0, x i x j ( ) [ ( ) ] ε ρ Ṫ + p + Π ρ 2 ε vk v i S ik + q k = 0, ρ ρ T v i v v k i v i K q i Ṡ ij 2p + S ij 2Π + 2 S j k q i δ j k K = 1 S ij, x j x j x j τ S Π 2h 2 T q j + ( v k K 2 ) p ρ ε Π v k 2 3 p ρ ε v i L S ik + q k + L v k q j + K ( v k q k L K ) v j q k h 4 2 x j 3 2T 2 + h 4 L x j 4h 2 [ ε + Π p ρ ρ 2 + ρ h 4 4h 2 [ { ε S jk p ρ ρ 2 + ρ L ( h4 4h 3 K p ρ ε )} ρ + ρ ε + x j { ε p ρ ε q k = 1 Π, τ Π p ρ ε + h 4 L 4h 2 + ( )} h4 K 4h 3 where τ S, τ Π and τ q are the relaxation times given by τ S = 2h 3, τ σ Π = 2h 2, ζ τq = 2h 4. τ Tommaso Ruggeri Π x j h 4 4h 3 K p ρ ε Π ρ ρ x i S jk 1 Π + 1 S ] jk x j ρ x j ρ S ] ik = 1 q j, τq

27 ET of dense gases Concavity of the entropy density and causality The system of balance equations must be symmetric hyperbolic so as to ensure the causality. Near equilibrium this requirement corresponds to the condition of the concavity of the entropy density. Considering the second derivative of the entropy density h near equilibrium, the condition is satisfied when h E is a concave function with respect to {ρ, T } and the following inequalities are fulfilled: h 2 < 0, h 3 < 0, h 4 < 0. Then we obtain the concavity condition expressed as follows: ( ) ( ) ε p p > 0, > 0, > 0, ρ 5 6 Tp + ρt 2 ( 2T 2 ε + p ρ ρ ( ) p ρ ) ) ( p T ρ + T ( T 2 p ) 2 2ρ ρ ( ε ) ρ T 2 ( β1 ) < 0, ρ < 0.

28 ommaso Ruggeri 1 The Extended Thermodynamics of Rarefied Monoatomic Gas 2 Model of dense gases 3 Relationship between ET and Navier-Stokes-Fourier theory 4 Extended Thermodynamics of dense gas 5 Rarefied-gas limit 6 Extended Thermodynamics of Moments and Maximum Entropy Principle MEP 7 Equilibrium distribution function for polyatomic gases 8 The hierarchy of moment equations for polyatomic gases 9 The 14 moments system for polyatomic gases Production Terms and relaxation times 10 Qualitative Analysis 11 The 6 Moments case

29 ommaso Ruggeri Relationship between ET and Navier-Stokes-Fourier theory Maxwellian iteration 1 We carry out the Maxwellian iteration in the field equations. 1 The 0th iterates: equilibrium values S (0) ij = 0, Π(0) = 0 and q (0) i = 0 2 The 1st iterates: Π (1), S (1) ij and q(1) i are obtained by the substitution of the 0th iterates S (0) ij, Π(0) and q (0) i into the left hand side of the field equations. v v k i v i K q i v i Ṡ ij + S ij 2Π + 2 S j k q i δ j k K 2p = 1 S ij, x j x j x j τ S Π + 5 p 3 ρ ε Π v k 2 p 3 ρ ε v p i L S ik + q k + L ρ ε q k 2h 2 v k = 1 Π, T τ Π q j + ( 1 + K 2 ) [ ε + Π ρ v k q j + K ( v k q k L K ) v j q k + h 4 L 2 x j 3 4h 2 p ρ 2 + h p 4 ρ ε L ρ 4h 2 ρ ε + + x j [ { ε S jk p ρ ρ 2 + ρ ( h4 4h 3 K )} ρ + { ε p ρ ε h 4 4h 2 Π x j h 4 4h 3 K L + ( )} h4 K 1 4h 3 ρ p ρ ε Π + 1 ρ S jk 1 Π + 1 S ] jk x j ρ x j ρ S ] ik h 4 2T 2 = 1 q j, x j τq x i

30 ommaso Ruggeri Relationship between ET and Navier-Stokes-Fourier theory Maxwellian iteration 1 We carry out the Maxwellian iteration in the field equations. 1 The 0th iterates: equilibrium values S (0) ij = 0, Π(0) = 0 and q (0) i = 0 2 The 1st iterates: Π (1), S (1) ij and q(1) i are obtained by the substitution of the 0th iterates S (0) ij, Π(0) and q (0) i into the left hand side of the field equations. v v k i v i K q i v i Ṡ ij + S ij 2Π + 2 S j k q i δ j k K 2p = 1 S ij, x j x j x j τ S Π + 5 p 3 ρ ε Π v k 2 p 3 ρ ε v p i L S ik + q k + L ρ ε q k 2h 2 v k = 1 Π, T τ Π q j + ( 1 + K 2 ) [ ε + Π ρ v k q j + K ( v k q k L K ) v j q k + h 4 L 2 x j 3 4h 2 p ρ 2 + h p 4 ρ ε L ρ 4h 2 ρ ε + + x j [ { ε S jk p ρ ρ 2 + ρ ( h4 4h 3 K )} ρ + { ε p ρ ε h 4 4h 2 Π x j h 4 4h 3 K L + ( )} h4 K 1 4h 3 ρ p ρ ε Π + 1 ρ S jk 1 Π + 1 S ] jk x j ρ x j ρ S ] ik h 4 2T 2 = 1 q j, x j τq x i

31 Relationship between ET and Navier-Stokes-Fourier theory Maxwellian iteration 2 Therefore we obtain the following relation: ij = 2pτ S, Π (1) = 2h 2 x j T τ v k Π, q (1) i = h 4 2T 2 τ q. x i S (1) v i On the other hand, we have the laws of Navier-Stokes and Fourier: S ij = 2µ v i x j, Π = ν v k, q i = κ x i. The comparison between 1st Maxwellian iterates of ET and Navier-Stokes-Fourier laws reveals that µ = pτ S, ν = 2h 2 T τ Π, κ = h 4 2T 2 τ q. We can therefore estimate the values of the relaxation times τ S, τ q and τ Π from the experimental data of the coefficients µ, ν and κ. 3 The 2nd iterates and higher iterates are obtained in a similar way.

32 ommaso Ruggeri Relationship between ET and Navier-Stokes-Fourier theory Maxwellian iteration 2 Therefore we obtain the following relation: ij = 2pτ S, Π (1) = 2h 2 x j T τ v k Π, q (1) i = h 4 2T 2 τ q. x i S (1) v i On the other hand, we have the laws of Navier-Stokes and Fourier: S ij = 2µ v i x j, Π = ν v k, q i = κ x i. The comparison between 1st Maxwellian iterates of ET and Navier-Stokes-Fourier laws reveals that µ = pτ S, ν = 2h 2 T τ Π, κ = h 4 2T 2 τ q. We can therefore estimate the values of the relaxation times τ S, τ q and τ Π from the experimental data of the coefficients µ, ν and κ. 3 The 2nd iterates and higher iterates are obtained in a similar way.

33 ET of dense gases Summary With this procedure, the system can be certainly closed except for some nonessential constants, provided that we know the thermal and caloric equations of state and the viscosity and heat conductivity coefficients. This surprising result, which could not be achieved in previous works on this subject, shows clearly the power of our theory.

34 ommaso Ruggeri 1 The Extended Thermodynamics of Rarefied Monoatomic Gas 2 Model of dense gases 3 Relationship between ET and Navier-Stokes-Fourier theory 4 Extended Thermodynamics of dense gas 5 Rarefied-gas limit 6 Extended Thermodynamics of Moments and Maximum Entropy Principle MEP 7 Equilibrium distribution function for polyatomic gases 8 The hierarchy of moment equations for polyatomic gases 9 The 14 moments system for polyatomic gases Production Terms and relaxation times 10 Qualitative Analysis 11 The 6 Moments case

35 Rarefied-gas limit Purpose It is important to study the rarefied-gas limit of the ET of dense gases in order to check its consistency with the results from the kinetic theory of gases. The dependence of the limit on the degrees of freedom of a molecule is also made clear.

36 Rarefied-gas limit Equations of state From the ideal gas law with the equipartition law of energy, the thermal and caloric equations of state for a classical rarefied gas are expressed as p = k B m ρt, ε = D k B 2 m T, where k B and m being, respectively, the Boltzmann constant and the mass of a molecule, and D is the degrees of freedom of a molecule, i.e., D = 3 + f where 3 corresponds to the translational motion in the 3-dimensional space and f is the internal degrees of freedom.

37 Rarefied-gas limit Field equations We obtain the closed system of field equations: ρ + ρ v k = 0, ρ v i + p x i 2 Ṫ + D k B ρ m + Π S ij = 0, x i x j (p + Π) v k 2 v i 2 D k S ik + B ρ x m k D k B ρ m q k = 0, v i v v k i v i Ṡ ij 2p + S ij 2Π + 2 S j k 4 q i = 1 S ij, x j x j D + 2 x j τ S 2(D 3) Π + p v k + 5D 6 3D 3D Π v k 2(D 3) 3D v i S ik + 4(D 3) 3D(D + 2) q k = 1 Π, τ Π q i + D + 4 v k 2 v k q i + q k + D + 4 v i q k + D + 2 ( ) kb 2 ρt D + 2 D + 2 x i D m x i + k B T Π k k S Bm B ik T ρ T + Π + D + 2 k B 1 Π + 1 S ik m x i m ρ x i 2 m x i ρ x i ρ k Bm T ρ S ik + D + 2 k B 1 Π + 1 S pk = 1 q i, ρ 2 m ρ ρ xp τq

38 Rarefied-gas limit Relaxation time where τ S, τ q and τ Π are the relaxation times given by τ S = 2pT σ, τ 2(D 3)pT Π = 3Dζ, τ q = 2(D + 2) ( ) k Bm 2 ρt 3. τ The relaxation times are related to the shear and bulk viscosities and the heat conductivity: µ = pτ S, ν = 2(D 3) 3D pτ Π, κ = D p 2 ρt τ q.

39 Rarefied-gas limit Concavity of entropy density The requirement of the concavity of the entropy density are expressed as ( ) ( ) ε p p > 0, > 0, > 0, ρ ρ D (D 3) k B m ρt > 0, 1 ( ) 2 2 > 0. (D + 2) kb m ρt 3 It is easy to see that all inequalities are identically satisfied for classical polyatomic gases with D > 3. However, as will be discussed, we should be careful for monatomic gases with D = 3. T

40 Extended Thermodynamics of Moments and MEP In the case of large number of moments it is too difficult to proceed with a pure macroscopic theory (as the 13-moments theory) and it is necessary to recall that the F s are moments of a distribution function f : F i1...i n = m ξ i1 ξ in f dξ, (i k {1, 2, 3}, k). R 3 Transfer equations for moments have the form of balance laws: t F + j F j = P, (3) where, the densities F, the fluxes F j and the productions P are: F(t, x) = Ψ(ξ)f dξ, R 3 F j (t, x) = ξ j Ψ(ξ)f dξ, (4) R 3 P(t, x) = Ψ(ξ)Q(f ) dξ, R 3

41 with 1 ξ i1 ξ i1 ξ i2 Ψ(ξ) = m.. ξ i1 ξ in. It is well known that every solution f of the Boltzmann equation satisfies the supplementary balance law: where h = k f log f dξ, R 3 t h + j h j = Σ 0, (5) h j = k ξ j f log f dξ, R 3 k being the Boltzmann constant. This is the so called H-Theorem. Σ = k Q(f ) log f dξ, R 3

42 The hierarchy of macroscopic equations (3) is usually truncated at some order of moments, say N, and accuracy of the solution of truncated system strongly depends upon proper choice of the distribution function. It was shown that local Maxwellian distribution (with hydrodynamic variables which depend on t and x), as well as Grad s distribution, maximizes physical entropy h. These results motivated the formulation of the so called maximum entropy principle MEP (Dreyer 1987, Müller & Ruggeri 1993).

43 It states that actual distribution function f is the one which maximizes the physical entropy: h = k f log f dξ max, (6) R 3 under the constraints that its moments are prescribed: F (N) (t, x) = Ψ (N) (ξ)f (t, x, ξ) dξ, (7) R 3 where F (N) is truncated vector of moments up to order N, and Ψ (N) is the vector Ψ truncated at order N. Thus, the approximate distribution function comes out as solution of variational problem with constraints. Variational problem (6) with constraints (7) is solved by the method of Lagrange multipliers and yields the following solution f (t, x, ξ) = exp ( 1 χ/k), χ = λ (N) (t, x) Ψ (N) (ξ). (8)

44 For both mathematical (convergence) and physical reasons (small Knudsen number), the maximizer (8) is expanded in the neighborhood of local equilibrium where all the moments except hydrodynamic ones can be regarded as small. Therefore, the distribution function can be approximated as: ( f = f E 1 1 ) k λ (N) Ψ (N), λ (N) = λ (N) λ (N) E (9) where f E denotes Maxwellian distribution in local equilibrium (the local Maxwellian) and λ (N) E are Lagrange multipliers evaluated at local equilibrium state.

45 Plugging (9) into (7) we obtain a linear algebraic system that permits to evaluate the Lagrange multipliers λ (N) in terms of the densities F (N) : F (N) (t, x) F (N) E (t, x) = 1 ( λ (N) (t, x) Ψ (ξ)) (N) Ψ (N) (ξ)f E (t, x, ξ) dξ, k (F (N) E R 3 denotes the equilibrium values). Therefore in this way we can have the explicit dependence of the truncating fluxes and source terms F (N) j (t, x) = ξ j Ψ (N) (ξ)f dξ, R 3 P (N) (t, x) = ξ j Ψ (N) (ξ)q(f ) dξ, R 3 on the densities and, as a consequence, the truncated system of moment equations: t F (N) + j F (N) j = P (N), (10) is closed system in which the only unknown are the densities.

46 It was shown by Boillat and Ruggeri (1997) that maximum entropy solution (9) of order N generates the equivalent set of balance laws as extended thermodynamics of the same order in which it is required that the truncated system (10) satisfy the entropy principle (5) with entropy, entropy density and entropy production which depend on the truncated densities. It is also proved that the closed balance laws system can be put into symmetric hyperbolic form using the main field u which coincides with Lagrange multipliers λ (N). Moreover, the same authors put in evidence that the moments closure gives a nesting theories of principal subsystems. In fact, it was proved that closure obtained with two different truncation orders M < N, then the system of order M is a principal subsystem of the system of order N. The main consequence is that maximum characteristic velocity increase with the index N and when N tends to in a classical framework and to the light velocity in a relativistic theory.

47 Equilibrium distribution function for polyatomic gases Our first aim is to recover the equilibrium distribution function and appropriate field equations for hydrodynamic variables (i.e. transfer equations for moments) via MEP. We shall, therefore, briefly describe the kinetic model for polyatomic gases and point out the important consequences related to internal energy density. As a consequence of considering one additional parameter, velocity distribution function f (t, x, ξ, I ) is defined on extended domain [0, ) R 3 R 3 [0, ). Its rate of change is determined by the Boltzmann equation which has the same form as for monoatomic gas but collision integral Q(f ) takes into account the influence of internal degrees of freedom through collisional cross section (Bourgat, Desvillettes, Le Tallec and Perthame, see also Borgnakke and Larsen)

48 Collision invariants for this model form a 5-vector: ( ψ(ξ, I ) = m 1, ξ i, ξ I ) T, (11) m which lead to hydrodynamic variables in the form: ρ ρu i = ψ(ξ, I )f (t, x, ξ, I )ϕ(i ) di dξ, (12) ρ u 2 + 2ρε R 3 0 where ρ, u and ε are mass density, hydrodynamic velocity and internal energy, respectively. A non-negative measure ϕ(i ) di is property of the model aimed at recovering classical caloric equation of state for polyatomic gases in equilibrium. Entropy is defined by the following relation: h = k f log f ϕ(i ) di dξ. (13) R 3 0

49 We shall introduce the peculiar velocity: C = ξ u (14) and rewrite the Eq. (12) in terms of it. Then: ρ 1 0 i = m C i f (t, x, C, I )ϕ(i ) di dc. (15) 2ρε R 3 0 C 2 + 2I /m Note that the internal energy density can be divided into the translational part ρε T and part related to the internal degrees of freedom ρε I : 1 ρε T = R m C 2 f (t, x, C, I )ϕ(i ) di dc, ρε I = If (t, x, C, I )ϕ(i ) di dc. (16) R 3 0 The former can be related to kinetic temperature in the following way: ε T = 3 k 2 m T, (17) whereas the latter should determine the contribution of internal degrees of freedom to internal energy of a polyatomic gas. Tommaso Ruggeri

50 ommaso Ruggeri The maximum entropy principle is expressed in terms of the following variational problem: determine the velocity distribution function f (t, x, C, I ) such that h max, subject to the constraints (12), or equivalently, due to Galilean invariance, to the constraints (15). The solution is given as follows: Theorem The distribution function which maximizes the entropy subject to constraints of prescribed moments has the form: ρ ( m ) { 3/2 f E = exp 1 ( )} 1 m q(t ) 2πkT kt 2 m C 2 + I, (18) where q(t ) = 0 ( exp I ) ϕ(i ) di. (19) kt The structure of weighting function ϕ(i ) is determined by the need to recover caloric equation of state for polyatomic gases and we obtain ϕ(i ) = I α.

51 In fact if D is the number of degrees of freedom of a molecule, it can be shown that ϕ(i ) = I α leads to appropriate caloric equation in equilibrium provided: α = D 5. (20) 2 In fact if the weighting function is chosen to be ϕ(i ) = I α, internal energy of a polyatomic gas in equilibrium reads: ( ) 5 k ε E = 2 + α T, α > 1. (21) m The relation between α and D (20) follows directly from comparison between (21) and well-know caloric equation for polyatomic gases: ε E = D 2 k m T. Observe that model for a monatomic gas (D = 3) cannot be recovered from the one with continuous internal energy, since the value of parameter α in monatomic case violates the overall restriction α > 1.

52 ommaso Ruggeri The hydrodynamic variables moments of distribution function cannot be arbitrary: they have to satisfy the transfer equations for moments. As a matter of fact, they are the Euler gas dynamics equations for polyatomic gases. Theorem If (18) is local equilibrium distribution with ρ ρ(t, x), u u(t, x) and T T (t, x), then the hydrodynamic variables ρ, u and T satisfy the following systems of equations: t ρ + j (ρu j ) = 0, t (ρu i ) + j (ρu i u j + p δ ij ) = 0, (22) ( ) {( )} 1 12 t 2 ρ u 2 + ρε + j ρ u 2 + ρε + p u j = 0, where δ ij is Kronecker delta, p = k m T and ε = D 2 k m T.

53 The hierarchy of moment equations for polyatomic gases In non-equilibrium motivated by the idea of phenomenological ET, we shall generalize the moment equations for polyatomic gases by constructing two independent hierarchies. One will be much alike classical momentum hierarchy of monatomic gases (F hierarchy); the other one, energy hierarchy, commences with the moment related to energy collision invariant and proceeds with standard increase of the order through multiplication by velocities (G hierarchy). They read: t F + j F j = P, t G + j G j = Q.

54 Moments, fluxes and productions of F hierarchy are defined as: F(t, x) = Ψ(ξ)f ϕ(i ) di dξ, R 3 0 F j (t, x) = ξ j Ψ(ξ)f ϕ(i ) di dξ, R 3 0 P(t, x) = Ψ(ξ)Q(f ) ϕ(i ) di dξ, with: R 3 0 Ψ(ξ) = m 1 ξ i1 ξ i1 ξ i2.. ξ i1 ξ in..

55 Moments, fluxes and productions of G hierarchy are defined as: G(t, x) = Θ(ξ, I )f ϕ(i ) di dξ, R 3 0 G j (t, x) = ξ j Θ(ξ, I )f ϕ(i ) di dξ, R 3 0 Q(t, x) = Θ(ξ, I )Q(f ) ϕ(i ) di dξ, with: R 3 Θ(ξ, I ) = m 0 ( ξ I m) ( ξ I m) ξk1 ξ I m ξk1 ξ k2 (. ) ξ I m ξk1 ξ km., Note that minimal order of the moment in F hierarchy is 0, while minimal order in G hierarchy is 2.

56 ommaso Ruggeri The 14 moments system for polyatomic gases In the case of 14 moments it is possible to deduce using the previous hierarchy Theorem The non equilibrium distribution function which maximizes the entropy subject to the constraints of prescribed 14 moments, for the choice of weighting function ϕ(i ) = I α, has the form: { f = f E 1 ρ p 2 q ic i + ρ ( ) ] 5 [p p 2 ij + (1 + α) 1 Πδ ij C i C j (23) 3 ρ 2(1 + α) p 2 Π ( 1 2 C 2 + I ) + m 2 + α ( 7 ) 1 ρ α p 3 q i ( 1 2 C 2 + I ) } C i m where f E is equilibrium distribution (18) and q(t ) is auxiliary function (19);

57 In this way we have a closed explicit form of 14 moments equations. Equations of F hierarchy have the form: t ρ + i (ρu i ) = 0, t (ρu i ) + j (ρu i u j + p ij ) = 0, (24) t (ρu i u j + p ij ) + k {ρu i u j u k + u i p jk + u j p ki + u k p ij } + (7/2 + α) 1 (q i δ jk + q j δ ki + q k δ ij ) = P ij,

58 while equations of G hierarchy read: ( ) {( ) } 1 1 t 2 ρ u 2 + ρε + i 2 ρ u 2 + ρε u i + p ij u j + q i = 0, {( ) } {( ) 1 1 t 2 ρ u 2 + ρε u i + p ij u j + q i + j 2 ρ u 2 + ρε u i u j (25) +u i u k p jk + u j u k p ik + 1 ( ) } 9 p 2 ρ u 2 p ij α ρ p ij p2 ρ δ ij = Q i, where the production terms are given by: P ij = mξ i ξ j Q(f ) ϕ(i ) di dξ, (26) R 3 0 ( Q i = m ξ I ) ξ i Q(f ) ϕ(i ) di dξ. (27) m R 3 0 In this way we obtain the same result of macroscopic approach of ET and we prove the equivalence between MEP and Entropy Principle also in the case of polyatomics gas.

59 Production Terms and relaxation times The closure of the moments system needs the explicitely calculation of production terms P ij, Q i given by (26) and (27). Zero entries appear due to collision invariants. Typical obstacle in calculation of production terms is complicated structure of collision integral Q(f ). In this study, as example, we shall assume the simple structure of collision integral given in [5] and [21]: Q(f )(ξ, I ) = [f (ξ, I ) f (ξ, I ) f (ξ, I ) f (ξ, I )] R 3 R + [0,1] 2 S 2 B(ξ, ξ, I, I, r, R, ω)(1 R) ξ ξ 1 1 ϕ(i ) dω dr dr di dξ, where r [0, 1] and R [0, 1] are parameters describing the exchange of internal energy during molecular collision and ω S 2 is a unit sphere vector. As in the classical case, the model of interaction between molecules is reflected on collision cross section B, and its choice is of utmost importance in the study of macroscopic transport processes.

60 The more realistic is the model of interaction, the more complicated are the calculations of collision integral and production terms. To get the impression about the structure of production terms we shall recourse to the following form of the cross section: B = KR 1/2 ξ ξ ω ξ ξ ξ ξ, where K is appropriate dimensional constant. This cross section resembles the variable hard spheres model and permits the derivation of production terms in the closed form. It tought to be regarded only as a simplified model of interaction of polyatomic molecules, without any intention to give the general results. Furthermore, we shall assume that state of the gas during processes is not far from local equilibrium. Therefore, products of non-equilibrium distribution functions which appear in collision integral will be linearized with respect to moments of distribution functions, i.e. stress tensor p ij and heat flux q i.

61 ommaso Ruggeri Theorem Up to first order terms in densities, production term in second order moments equation in F hierarchy reads: P ij = 4 Kmπ p 2 { 15 q 2 p ij + 4 ( ) } 5 (T ) ρ α (1 + α) 1 Πδ ij (28) whereas production term in third order moments equation in G hierarchy has the following form: Q i = 8 Kmπ p 2 {( 15 q 2 p ij + 4 ( ) ) 5 (T ) ρ α (1 + α) 1 Πδ ij u j + 10 ( ) } α q i, (29) where p ij denotes traceless part of the pressure tensor.

62 The statement that maximization of entropy is equivalent to extended thermodynamics of moments permit that the relaxation times τ S for viscous stress, τ Π for dynamic pressure and τ q for heat flux, of the ET model can be explicitly calculated from our production terms: 1 = 4 Kmπ τ S 15 q 2 (T ) ρ, 1 = 16 ( ) 5 τ Π α 1 = 8 ( ) 1 7 τ q α Kmπ q 2 (T ) p 2 1 Kmπ (1 + α) q 2 (T ) In [1] it was also proved, by means of so-called Maxwellian iteration, that the classical equations of Navier-Stokes and Fourier appear as limit cases of the balance laws (24) 3 and (25) 2. p 2 ρ. p 2 ρ,

63 Taking into account these results one may obtain explicit form of transport coefficients, i.e. shear viscosity µ, bulk viscosity ν and heat conductivity κ: µ = p τ S = 15 Γ 2 (1 + α) (kt ) 1+2α, 4 Kπ 2(D 3) ν = p τ Π = 35 ( ) 2 5 (1 + α)2 3D α Γ 2 (1 + α) (kt ) 1+2α, (30) Kπ κ = D + 2 p 2 2 ρt τ q = 21 ( ) α k Γ 2 (1 + α) (kt ) 1+2α. m Kπ Note that the transport coefficients can be evaluated explicitly, up to a dimensional constant K, from the model with internal energy, in contrast to ET where only the relation to the relaxation times can be determined.

64 Qualitative Analysis As we proved the equivalence between ET and MEP for rarefied polyatomic gas, we can apply the same results concerning hyperbolic system endowed with a supplementary convex entropy. According with the general results due to Godunov (1961), Boillat (1974) and Ruggeri & Strumia (1981), the original system can be put in symmetric form using the main field u, which coincide with Lagrange multipliers multipliers of balance laws. Therefore, we can conclude that the system (24), (25) is symmetric hyperbolic (at least in neighborhood of an equilibrium state) in the main field components. Symmetric structure of the system ensures local well posedness for Cauchy problems.

65 Systems of mixed type Usual conservation laws of hyperbolic type cannot have global smooth solutions but our systems belong to the case of hyperbolic systems of mixed type, some equations are conservation laws and other ones are real balance laws, i.e., we are in the case in which u t + i F i (u) = F(u) with F(u) ( 0 g(u) ) ; g R N M. In this case the coupling condition discovered for the first time by Kawashima and Shizuta (K-condition) (1985) such that the dissipation present in the second block have effect also to the first block of equation plays a very important role in this case for global existence of smooth solutions.

66 ommaso Ruggeri Global Existence of smooth solutions In fact, if the system of balance law is endowed with a convex entropy law, and it is dissipative, then the K-condition becomes a sufficient condition for the existence of global smooth solutions provided that the initial data are sufficiently smooth (Hanouzet and Natalini 2003, Wen-An Yong 2004, Bianchini, Hanouzet and Natalini 2007) Theorem (Global Existence) Assume that the system of balance laws is strictly dissipative and the K-condition is satisfied. Then there exists δ > 0, such that, if u(x, 0) 2 δ, there is a unique global smooth solution, which verifies u C 0 ( [0, ); H 2 (R) C 1 ( [0, ); H 1 (R). )

67 ommaso Ruggeri Moreover Ruggeri and Serre (2004) have proved in the one-dimensional case that the constant states are stable: Theorem (Stability of Constant State) Under natural hypotheses of strongly convex entropy, strict dissipativeness, genuine coupling and zero mass initial for the perturbation of the equilibrium variables, the constant solution stabilizes ( u(t) 2 = O t 1/2). Lou and Ruggeri (2006) have observed that the weaker K-condition in which we require the K-condition only for the right eigenvectors corresponding to genuine non linear is a necessary (but not sufficient) condition for the global existence of smooth solutions.

68 The 6 Moments case The most simple case of dissipative polyatomic gas is the one in which we neglet heat conductivity and shear viscosity and we suppose that only the bulk viscosity is not negligibile. In this case we have the most simpe model after Euler in the presence of dissipation due to the role of the dynamical pressure (Arima, Taniguchi, Ruggeri and Sugiyama Physics Lett. A -2012). The balance equations for one-dimensional problems are ρ t + ρv x = 0, ρv + ( p + Π + ρv 2 ) = 0, t x ( 2ρε + ρv 2 ) + ( 2(p + Π)v + 2ρεv + ρv 3 ) = 0, t x ( 3(p + Π) + ρv 2 ) + ( 5(p + Π)v + ρv 3 ) = 3DΠ t x (D 3)τΠ. (31) We notice that when D = 3 (monoatomic gas) the previous sytem coincide witht the one of Euler fluid.

69 It si interesting to observe that the maximum characteristic velocity λ of the system in the frame moving with the fluid is independent of D and coincide with the sound velocity of monoatomic gas in contrast with the sound velocity of Euler fluid λ: λ = v k B D + 2 m T, k B λ = v + D m T. (32) According that Euler system is a principal subsystem (Boillat- Ruggeri) and we have automatically satisfied the sub-characteristic conditions : λ λ. (33) If we consider a shock structure solution, we know that the continuous solution degenerate in sub-shocks when the shock velocity s reach the maximum characteristic speed. Therefore smooth shock solutions can be exists only for λ < s < λ. Therefore we have the following behaviour described in the following figures (Taniguchi calculations).

70 Figure: The profiles of the dimensionless velocity ˆv (left) and dynamic pressure ˆΠ for D = 4.0. The unperturbed Mach numbers are M 0 = 1.05.

71 Figure: The profiles of the dimensionless velocity ˆv (left) and dynamic pressure ˆΠ for D = 3.1. The unperturbed Mach numbers are M 0 = 1.05.

72 Figure: The profiles of the dimensionless velocity ˆv (left) and dynamic pressure ˆΠ for D = The unperturbed Mach numbers are M 0 = 1.05.

73 Figure: The profiles of the dimensionless velocity ˆv (left) and dynamic pressure ˆΠ for D = 3.0. The unperturbed Mach numbers are M 0 = 1.05.

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