Molecular Extended Thermodynamics of Rarefied …Molecular Extended Thermodynamics of Rare ed...

Molecular Extended Thermodynamics of RarefiedPolyatomic Gases and Wave Velocities for Increasing

Number of Moments

Tommaso Ruggeri

Mathematical Departement and Research Center of Applied MathematicsUniversity of Bologna

Problems on Kinetic Theory and PDE′s.September 25-27, 2014

Novi Sad, Serbia

Tommaso RuggeriMolecular Extended Thermodynamics of Rarefied Polyatomic Gases and Wave Velocities for Increasing Number of Moments

Modelling non-equilibrium phenomena in which steep gradients and rapid changesoccur represents a challenging task which has been tackled by means of twocomplementary approaches: the continuum approach and the kinetic theory.The continuum approach consists in describing the system by means ofmacroscopic equations (e.g. fluid-dynamic equations) obtained on the basis ofconservation laws and appropriate constitutive equations. As an example,Thermodynamics of Irreversible Processes (TIP), which relies on the assumptionof local thermodynamic equilibrium (LTE), has proven to be a useful and soundtheory characterized by a systematic and comprehensive theoretical structure. Inthis framework, the Navier-Stokes-Fourier (NSF) theory has gained muchpopularity mainly due to its practical usefulness in many applications.Nonetheless, TIP and NSF suffer from serious weaknesses: since the mathematicalstructure of this theories is characterized by a system of differential equations ofparabolic type, an infinite speed is predicted for the propagation of signals – anissue which has been addressed as the paradox of heat conduction.


Moreover, the applicability of the classical macroscopic theory is inherentlyrestricted to processes characterized by small Knudsen numbers (dense gas), andthe transport coefficients associated to the dissipation processes are not providedby the theory except for the sign.On the other hand, the approach based on the kinetic theory, which postulatesthat the state of the gas can be described by a velocity distribution functionwhose evolution is governed by the celebrated Boltzmann equation, is applicableto processes characterized by a large Knudsen number, and transport coefficientsnaturally emerge from the theory itself.


In particular Extended Thermodynamics (ET) which describe a nonequilibriumphenomena beyond the assumption of local equilibrium was presented to fills thegap between the above-mentioned macroscopic and microscopic approaches andshows successful result and full agreement with Kinetic Theory.

Nevertheless the weak point of ET and KT is that the applicable range is limitedto rarefied monatomic gas.

Figure: semiconductor(Copyright c©iTak (international) Limited.) Atmospheric ReentryJAXA)


The Extended Thermodynamics of Rarefied MonoatomicGas

The kinetic theory describes the state of a rarefied gas through the phase densityf (x, t, ξ), where f (x, t, ξ)dξ is the number density of atoms at point x and time tthat have velocities between ξ and ξ + dξ. The phase density obeys theBoltzmann equation

∂f

∂t+ ξi

∂f

∂xi= Q (1)

where Q represents the collisional terms. Most macroscopic thermodynamicquantities are identified as moments of the phase density

Fk1k2···kj =

∫R3

f ξk1ξk2 · · · ξkj dξ, (2)

and due to the Boltzmann equation (1), the moments satisfy an infinity hierarchyof balance laws in which the flux in one equation becomes the density in the nextone:


∂tF + ∂iFi = 0

∂tFk1 + ∂iFik1 = 0

∂tFk1k2 + ∂iFik1k2 = P<k1k2>

∂tFk1k2k3 + ∂iFik1k2k3 = Pk1k2k3

...

∂tFk1k2...kn + ∂iFik1k2...kn = Pk1k2...kn

...

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.


Taking into account that Pkk = 0, the first five equations are conservation lawsand 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 monatomicgases. In fact due to the previous structure we have

3p = 2ρε Π = 0

that implies

γ =cpcV

=5

3,

i.e. monatomic gas.

The Closure Problem

When we cut the hierarchy at the density with tensor of rank n, we have theproblem of closure because the last flux end the production terms are not in thelist of the densities.


The first idea of Rational Extended Thermodynamics Muller and Ruggeri(Springer - Verlag 1993, 1998) was to view the truncated system as aphenomenological system of continuum mechanics and then we consider the newquantities as constitutive functions:

Fk1k2...knkn+1 ≡ Fk1k2...knkn+1 (F ,Fk1 ,Fk1k2 , . . .Fk1k2...kn)

Pk1k2...kj ≡ Pk1k2...kj (F ,Fk1 ,Fk1k2 , . . .Fk1k2...kn) 2 ≤ j ≤ n.

According with the continuum theory, the restrictions on the constitutiveequations come only from universal principles, i.e.: Entropy principle, ObjectivityPrinciple and Causality and Stability (convexity of the entropy).

The most interesting physical cases was the 13 fields theory in classical framework(I.S.-Liu & I. Muller - ARMA 1983) and the 14- fields in the context of relativisticfluids (I.S.-Liu, I. Muller & T. Ruggeri -ANNALS of PHYSICS 1984)


Equilibrium distribution function for polyatomic gases

The 14 moment phenomenological theory for dense gas and in particularpolyatomic ones was recently obtained using the universal principles of ET andpostulating a double hierarchy of equations by T. Arima, S. Taniguchi, T. Ruggeriand M. Sugiyama - Continuum Mech. Thermodyn., (2012).The question is if the macroscopic system have a kinetic counterpart. We will seethat in the case of rarefied gas the previous structure can be explained in clearmanner. We shall, therefore, briefly describe the kinetic model for polyatomicgases and point out the important consequences related to internal energy density.The idea is to consider an additional parameter in the distribution functionf (t, x, ξ, I ) defined on extended domain [0,∞)× R3 × R3 × [0,∞). Its rate ofchange is determined by the Boltzmann equation which has the same form as formonatomic gas but collision integral Q(f ) takes into account the influence ofinternal degrees of freedom through collisional cross section (Bourgat,Desvillettes, Le Tallec and Perthame, see also Borgnakke and Larsen)


Collision invariants for this model form a 5-vector:

ψ(ξ, I ) = m

(1, ξi ,

1

2mξ2 + I

)T

, (3)

which lead to hydrodynamic variables in the form: ρρvi

12ρv 2 + ρε

=

∫R3

∫ ∞0

ψ(ξ, I )f (t, x, ξ, I )ϕ(I ) dI dξ, (4)

where ρ, v and ε are mass density, hydrodynamic velocity and internal energy,respectively. A non-negative measure ϕ(I ) dI is property of the model aimed atrecovering classical caloric equation of state for polyatomic gases in equilibrium.Entropy is defined by the following relation:

h = −k

∫R3

∫ ∞0

f log f ϕ(I ) dI dξ. (5)


We shall introduce the peculiar velocity:

C = ξ − v (6)

and rewrite the Eq. (4) in terms of it. Then: ρ0i

2ρε

=

∫R3

∫ ∞0

m

1Ci

C 2 + 2I/m

f (t, x,C, I )ϕ(I ) dI dC. (7)

Note that the internal energy density can be divided into the translational partρεT and part related to the internal degrees of freedom ρεI :

ρεT =

∫R3

∫ ∞0

1

2mC 2f (t, x,C, I )ϕ(I ) dI dC,

ρεI =

∫R3

∫ ∞0

If (t, x,C, I )ϕ(I ) dI dC. (8)

The former can be related to kinetic temperature in the following way:

εT =3

2

k

mT , (9)

whereas the latter should determine the contribution of internal degrees offreedom to internal energy of a polyatomic gas.


In fact if D is the number of degrees of freedom of a molecule, it can be shownthat ϕ(I ) = Iα leads to appropriate caloric equation in equilibrium provided:

α =D − 5

2. (10)

In fact if the weighting function is chosen to be ϕ(I ) = Iα, internal energy of apolyatomic gas in equilibrium reads:

ε|E =

(5

2+ α

)k

mT , α > −1. (11)

The relation between α and D (10) follows directly from comparison between (11)and well-know caloric equation for polyatomic gases:

ε|E =D

2

k

mT .

Observe that model for a monatomic gas (D = 3) cannot be recovered from theone with continuous internal energy, since the value of parameter α in monatomiccase violates the overall restriction α > −1.


Pavic, Ruggeri and Simic [5] firstly considered the Euler fluid with 5 moments andthey considered the the maximum entropy principle expressed in terms of thefollowing variational problem: determine the velocity distribution functionf (t, x, ξ, I ) such that h→ max., being subjected to the constraints (4). In thisway they were able to determine the following equilibrium distribution functionwhich maximizes the entropy (5) with the constraints (4):

fE =ρ

m q(T )

( m

2πkT

)3/2

exp

− 1

kT

(1

2mC 2 + I

), (12)

where C = ξ − v is the peculiar velocity and

q(T ) =

∫ ∞0

exp

(− I

kT

)ϕ(I ) dI , (13)

that, in the case ϕ(I ) = Iα, becomes

q(T ) = (kT )1+αΓ(1 + α)

where Γ is the gamma function.


Pavic, Ruggeri and Simic [5] secondly considered the case of 14 moments. Thiscase is completely in agreement with the binary hierarchy with the moments: F

Fi1

Fi1i2

=

∫R3

∫ ∞0

m

1ξi1ξi1ξi2

f (t, x, ξ, I )ϕ(I ) dI dξ,

(14)(Gpp

Gppk1

)=

∫R3

∫ ∞0

m

(ξ2 + 2 I

m(ξ2 + 2 I

m

)ξk1

)f (t, x, ξ, I )ϕ(I ) dI dξ.

For the entropy defined by (5), the following variational problem, expressing themaximum entropy principle, can be formulated: determine the velocity distributionfunction f (t, x,C, I ) such that h→ max., being subjected to the constraints (14).The solution of the problem is as follows.


Near the equilibrium state the velocity distribution function, which maximizes theentropy (5) with the constraints (14) and the weighting function ϕ(I ) = Iα, hasthe form:

f = fE

1− ρ

p2qiCi +

ρ

p2

[−S〈ij〉 +

(5

2+ α

)(1 + α)−1Πδij

]CiCj (15)

− 3

2(1 + α)

ρ

p2Π

(1

2C 2 +

I

m

)+

(7

2+ α

)−1ρ2

p3qi

(1

2C 2 +

I

m

)Ci

,

where fE is the equilibrium distribution (12) and q(T ) is the auxiliary function(13). The non-equilibrium distribution (15) reduces to the velocity distributionobtained by Mallinger for gases composed of diatomic molecules (α = 0), and, forany α > −1, the closure gives exactly the same equations obtained before byusing the macroscopic approach and the entropy principle:

[T. Arima, S. Taniguchi, T. Ruggeri and M. Sugiyama - Continuum Mech.Thermodyn., (2012).]


The general hierarchy of moment equations for polyatomicgases

In non-equilibrium motivated by the idea of phenomenological ET, we shallgeneralize the moment equations for polyatomic gases by constructing twoindependent hierarchies. One will be much alike classical “momentum” hierarchyof monatomic gases (F−hierarchy); the other one, “energy” hierarchy, commenceswith the moment related to energy collision invariant and proceeds with standardincrease of the order through multiplication by velocities (G−hierarchy). Theyread:

∂tF + ∂iFi = P, ∂tG + ∂iGi = Q.

[Pavic, Ruggeri & Simic Physica A,(2012)][Arima, Mentrelli & Ruggeri - Annals of Physics (2014)]


Moments, fluxes and productions of F−hierarchy are defined as:

F(t, x) =

∫R3

∫ ∞0

Ψ(ξ)f ϕ(I ) dI dξ,

Fi (t, x) =

∫R3

∫ ∞0

ξiΨ(ξ)f ϕ(I ) dI dξ,

P(t, x) =

∫R3

∫ ∞0

Ψ(ξ)Q(f )ϕ(I ) dI dξ,

with:

Ψ(ξ) = m

1ξi1ξi1ξi2

...ξi1 · · · ξin

...

.


Moments, fluxes and productions of G−hierarchy are defined as:

G(t, x) =

∫R3

∫ ∞0

Θ(ξ, I )f ϕ(I ) dI dξ,

Gi (t, x) =

∫R3

∫ ∞0

ξiΘ(ξ, I )f ϕ(I ) dI dξ,

Q(t, x) =

∫R3

∫ ∞0

Θ(ξ, I )Q(f )ϕ(I ) dI dξ,

with:

Θ(ξ, I ) = m

ξ2 + 2 Im(

ξ2 + 2 Im

)ξk1(

ξ2 + 2 Im

)ξk1ξk2

...(ξ2 + 2 I

m

)ξk1 · · · ξkm

...

,

Note that minimal order of the moment in F−hierarchy is 0, while minimal orderin G−hierarchy is 2.


Polyatomic rarefied gas with many moments

More explicit :

For simplicity, we adopt the following notation:

FA =

F for A = 0

Fk1k2···kA for 1 ≤ A ≤ N,GLLa =

GLL for a = 0

GLLk1k2···ka for 1 ≤ a ≤ M,

Then the system can be rewritten as simple form:

∂tFA + ∂iFiA = PA, ∂tGLLA′ + ∂iGiLLA′ = QLLA′ . (16)Tommaso RuggeriMolecular Extended Thermodynamics of Rarefied Polyatomic Gases and Wave Velocities for Increasing Number of Moments

Maximum Entropy Principle

The variational problem from which the distribution function f(N,M) is obtained isconnected to the functional:

L(N,M) (f ) = −k

∫R3

∫ ∞0

f log f Iα dI dξ+u′A

(FA −m

∫R3

∫ ∞0

ξA f Iα dI dξ

)+

+ v ′A′

(GllA′ −m

∫R3

∫ ∞0

(ξ2 +

2I

m

)ξA′ f IαdIdξ

),

where u′A and v ′a are the Lagrange multipliers and

ξA =

1 for A = 0

ξk1ξk2 · · · ξkA for 1 ≤ A ≤ N,ξA′ =

1 for A′ = 0

ξk1ξk2 · · · ξkA′ for 1 ≤ A′ ≤ M.

The distribution function f(N,M) which maximizes the functional L(N,M) is givenby:

f(N,M) = exp(−1− m

kχ(N,M)

), χ(N,M) = u′AξA +

(ξ2 +

2I

m

)v ′A′ξA′ . (17)


Then, the system may be rewritten as follows:(J0AB J1

AB′

J1A′B J2

A′B′

)∂t

(u′B

v ′B′

)+

(J0iAB J1

iAB′

J1iA′B J2

iA′B′

)∂i

(u′B

v ′B′

)=

(PA

QllA′

),

(18)where

J0AB = −m

k

∫R3

∫ ∞0

f ξAξB Iα dIdξ,

J1AB′ = −m

k

∫R3

∫ ∞0

f ξAξB′

(ξ2 +

2I

m

)Iα dIdξ,

J2A′B′ = −m

k

∫R3

∫ ∞0

f ξA′ξB′

(ξ2 +

2I

m

)2

Iα dIdξ.

(19)

The system (18) is symmetric hyperbolic according with the general theory ofsystems of balance laws with a convex entropy density (Boillat & RuggeriContinuum Mech. Thermodyn. (1997)).


Problem ê Are N and M independent?

The following two theorems give the answer [Arima, Mentrelli and Ruggeri- Annals of

Physics (2014)]:

TheoremThe differential system is Galilean invariant if and only if M ≤ N − 1.

Theorem

If M < N − 1, all characteristic velocities are independent from the internaldegrees of freedom D and coincides with the one of F -hierarchy of monatomicgases with the truncation order N.

êThe relation between N and M for the physically meaningful system

The requirement that the system is Galilean invariant and the characteristicvelocities are function of D require

M = N − 1


Examples:

N=1, M=0 Euler system:

N=2, M=1 14-field system:


Singular limit of monatomic gas as particular case ofpolyatomic one

We introduce

Iα = FLLα − GLLα (0 ≤ α ≤ M − 1(= N − 2)).

where

FLLα = FLLk1k2···kα ,

Then the basic equations can be transformed to the equivalent one:

where Iiα = FiLLα − GiLLα and Rα = Pα − QLLα.Tommaso RuggeriMolecular Extended Thermodynamics of Rarefied Polyatomic Gases and Wave Velocities for Increasing Number of Moments

We define

Πα = limD→3

Iα,

and we can prove that

limD→3

Rα = −ΨαβΠβ .

where Ψαβ are the coefficient depending on the collisional integral. In the case ofBGK model,

Ψαβ =1

τδαβ .

In the limit D → 3, we can obtain the balance equations of Πα:

∂tΠα + JMiaαJM

aβ

−1∂iΠβ = −

(Ψαβ + ∂iJ

MiaαJM

aβ

−1)

Πβ .

with

JMaα =

∫fMξaξαdc,

where fM is the Maxwellian distribution function.Tommaso RuggeriMolecular Extended Thermodynamics of Rarefied Polyatomic Gases and Wave Velocities for Increasing Number of Moments

When we impose the initial data compatible with rarefied monatomic gases:

Πα(0, x) = 0,

we obtain

Πα(t, x) = 0, ∀t

Furthermore we obtain

limD→3

GLLM = limD→3

FLLM


Results [Arima, Taniguchi, Ruggeri & Sugiyama (in preparation)]

Therefore we proved in the limit D → 3 the system of polyatomic gases formed by16 (N + 1)(N + 2)(2N + 3) equations have the same solution of the system ofrarefied monatomic gases formed by 1

6 (N + 1)(N2 + 8N + 6) equations.

Example N = 2: The solutions of 14-field theory for polyatomic gases, whenD → 3, have the same solutions of 13-field theory for monatomic gases. [Arima,Taniguchi, Ruggeri & Sugiyama, Phys. Lett. A. 2013]:


Maximum characteristic velocity

It has been proved that, in the case that D → 3, the solutions of the ET theoryfor rarefied polyatomic gases converge to those of corresponding rarefiedmonatomic gases. Therefore, when D → 3, all the characteristic velocitiesconverge to the one of the monatomic one as in figure and in particular the λmax :

Moreover we have the following property with respect to λmax .

Theorem

When D →∞, λmax coincides with the one of the system of rarefied monatomicgases with truncation order N.


The dependence of the maximum characteristic velocity on D and N:

N=3 , 30-eld system

N=2 , 14-eld system

N=1 , Euler system


Using the property of the principal subsystem [Boillat & Ruggeri (1997)], we canprove that the lower bound of monatomic gases remains also for polyatomic gases.

λ(N)max

cs≥ λ

(N)max |D→∞

cs=λ

(N)max |mon

cs≥

√6

5

(N − 1

2

)s

Therefore, also for the case of polyatomicgases, λmax is unbounded when N →∞.

[Arima, Mentrelli & Ruggeri - Annals of Phys.

2014]


The 14 moments system for polyatomic gases

In the case of 14 moments and polyatomic gas p = kmρT , ε = D

2kmT , the system

become:

ρ + ρ∂vk

∂xk

= 0,

ρvi +∂p

∂xi

+∂Π

∂xi

−∂S〈ij〉∂xj

= 0,

T +2

DkBmρ

(p + Π)∂vk

∂xk

−2

DkBmρ

∂vi

∂xk

S〈ik〉 +2

DkBmρ

∂qk

∂xk

= 0,

S〈ij〉 + S〈ij〉∂vk

∂xk

− 2Π∂v〈i∂xj〉

+ 2∂v〈i∂xk

S〈j〉k〉 −4

D + 2

∂q〈i∂xj〉

− 2p∂v〈i∂xj〉

= −1

τS

S〈ij〉,

Π +5D − 6

3DΠ∂vk

∂xk

−2(D − 3)

3D

∂v〈i∂xk〉

S〈ik〉 +4(D − 3)

3D(D + 2)

∂qk

∂xk

+2(D − 3)

3Dp∂vk

∂xk

= −1

τΠ

Π,

qi +D + 4

D + 2qi

∂vk

∂xk

+2

D + 2qk

∂vk

∂xi

+D + 4

D + 2qk

∂vi

∂xk

+kB

mT∂Π

∂xi

−kB

mT∂S〈ik〉∂xk

+ Π

− kBm

T

ρ

∂ρ

∂xi

+D + 2

2

kB

m

∂T

∂xi

−1

ρ

∂Π

∂xi

+1

ρ

∂S〈ik〉∂xk

−S〈ik〉

− kBm

T

ρ

∂ρ

∂xk

+D + 2

2

kB

m

∂T

∂xk

−1

ρ

∂Π

∂xk

+1

ρ

∂S〈pk〉∂xp

+D + 2

2

(kB

m

)2ρT

∂T

∂xi

= −1

τqqi ,


The 6 Moments case

The most simple case of dissipative polyatomic gas is the one in which we neglectheat conductivity and shear viscosity and we suppose that only the bulk viscosityis not negligible In this case we have the most simple model after Euler in thepresence of dissipation due to the role of the dynamical pressure (Arima,Taniguchi, Ruggeri and Sugiyama Physics Lett. A -2012). The field equations are

ρ+ ρ div v = 0,

ρvi +∂

∂xi(p + Π) = 0,

ρε+ (p + Π)div v = 0,

τ Π +

(ν + τ

5D − 6

3DΠ

)div v = −Π,

(20)

where the bulk viscosity ν ∝ D − 3. When D → 3 (monatomic gas) the previoussystem have the same solution of the Euler fluid provided Π(x, 0) = 0.


The parabolic case τ → 0:

ρ+ ρ div v = 0,

ρvi +∂

∂xi(p + Π) = 0,

ρε+ (p + Π)div v = 0,

ν div v = −Π,

(21)

was studied in same papers, e.g. :

- P. Secchi - Rend. Sem. Padova (1984)- V. Shelukhin - Journal of Differential Equations (2000)- H. Frid & V. Shelukhin - Siam J. Math. Anal. (2000)


Applications of ET14 and ET6

Applications of the ET14 theory to specific problems have been made. In thissection we review briefly some of them:

Dispersion relation for sound in rarefied diatomic gases

Light scattering

Shock structure

Riemann problem


Dispersion relation

The dispersion relation for sound in rarefied diatomic gases; hydrogen, deuteriumand hydrogen deuteride gases basing on the ET14 theory was recently studied indetail in Arima, T., Taniguchi, S., Ruggeri, T. and Sugiyama, M. Cont. Mech.Thermodyn., (2013).The relation was compared with those obtained in experiments and by the NSFtheory. As is expected, the applicable frequency-range of the ET theory wasshown to be much wider than that of the NSF theory. The values of the bulkviscosity and the relaxation times involved in nonequilibrium processes wereevaluated. It was found that the relaxation time related to the dynamic pressurehas a possibility to become much larger than the other relaxation times related tothe shear stress and the heat flux. The isotope effects on sound propagation werealso clarified.


4. An application of the theory to ultrasonic soundsA test of the ET theory for rarefied diatomic gases : Dispersion relation for sound

Normal hydrogen (n-H2) gases:

ááááááá

á

á

áá

óóóóóóó

ó

óóó

óETH293 KLNSFH293 KL

ó

á RhodesH273.5 KL

RhodesH296.8 KL

n -H2

10-4 10-3 10-2 10-11.00

1.04

1.08

1.12

W

v ph

c 0

ççççççççççççççççççç

n -H2

Sluijter et al.H293 KL

ETH293 KL

NSFH293 KL

ç

10-4 10-3 10-2 10-1 100 10110-6

10-4

10-2

100

W

c 0HΤ

SL 0Α

Figure: Dependence of the dimensionless phase velocity vph/c0 (left) and the attenuationfactor c0(τS)0α (right) on the dimensionless frequency Ω(= τSω) for n-H2 gases. Thesquares and triangles in the left figure are the experimental data at T0 = 273.5 and296.8K, respectively, and the circles in the right figure are those at T0 = 293K. The solidand dashed lines are predictions at 293K by the ET and NSF theories, respectively.


Shock wave structure in a rarefied polyatomic gas

The shock wave structure in a rarefied polyatomic gas is, under some conditions,quite different from the shock wave structure in a rarefied monatomic gas due tothe presence of the microscopic internal modes in a polyatomic molecule such asthe rotational and vibrational modes. For examples: 1) The shock wave thicknessin a rarefied monatomic gas is of the order of the mean free path. On the otherhand, owing to the slow relaxation process involving the internal modes, thethickness of a shock wave in a rarefied polyatomic gas is several orders larger thanthe mean free path. 2) As the Mach number increases from unity, the profile ofthe shock wave structure in a polyatomic rarefied gas changes from the nearlysymmetric profile (Type A) to the asymmetric profile (Type B), and then changesfurther to the profile composed of thin and thick layers (Type C)


Schematic profiles of the mass density are shown in Figure 3. Such change of theshock wave profile with the Mach number cannot be observed in a monatomic gas.

x

ρ Type A

x

ρ Type B

x

ρ Type C

∆

Ψ

Figure: Schematic representation of three types of the shock wave structure in a rarefiedpolyatomic gas, where ρ and x are the mass density and the position, respectively. Asthe Mach number increases from unity, the profile of the shock wave structure changesfrom Type A to Type B, and then to Type C that consists of the thin layer ∆ and thethick layer Ψ.


In order to explain the shock wave structure in a rarefied polyatomic gas, therehave been two well-known approaches. One was proposed by Bethe and Teller andthe other is proposed by Gilbarg and Paolucci. Although the Bethe-Teller theorycan describe qualitatively the shock wave structure of Type C, its theoretical basisis not clear enough. The Gilbarg-Paolucci theory, on the other hand, cannotexplain asymmetric shock wave structure (Type B) nor thin layer (Type C).Recently it was shown that the ET14 theory can describe the shock wave structureof all Types A to C in a rarefied polyatomic gas In other words the ET14 theoryhas overcome the difficulties encountered in the previous two approaches. Thisnew approach indicates clearly the usefulness of the ET theory for the analysis ofshock wave phenomena.


Shock structure in ET14

−5 0 5

1

1.5

2

−0.002 0 0.002

1

1.2

1.4

x

ρ^

^−5 0 5

1

1.5

−0.002 0 0.002

1

1.5

x

v^

^

−5 0 5

1

1.1

1.2

−0.002 0 0.002

1

1.2

x

T^

^ −5 0 5

0

0.2

−0.002 0 0.002

0

0.2

x

Π^

^

−5 0 5

−0.05

0

−0.002 0 0.002

−0.05

0

x

σ^

^ −5 0 5

−0.05

0

−0.002 0 0.002

−0.05

0

x

q^

^

S. Taniguchi, T. Arima, T. Ruggeri, and M. SugiyamaPhys. Rev. E (2014), Phys. of Fluids (2014).


Riemann Problem in ET6

−20 −15 −10 −5 0 5 10 15 20

position

1.0

1.2

1.4

1.6

1.8

2.0

mass

densi

tyEuler

−20 −15 −10 −5 0 5 10 15 20

position−0.1

0.0

0.1

0.2

0.3

0.4

0.5

velocity

Euler

−20 −15 −10 −5 0 5 10 15 20

position

300

320

340

360

temperature

Euler

−20 −15 −10 −5 0 5 10 15 20

position

1.0

1.2

1.4

1.6

1.8

2.0

mass

densi

ty

ET6

−20 −15 −10 −5 0 5 10 15 20

position−0.1

0.0

0.1

0.2

0.3

0.4

0.5

velocity

ET6

−20 −15 −10 −5 0 5 10 15 20

position

300

320

340

360

temperature

ET6

−20 −15 −10 −5 0 5 10 15 20

position

−0.04

−0.02

0.00

0.02

0.04

0.06

0.08

0.10

dynam

ic p

ress

ure

ET6

S. Taniguchi (in preparation (2014)).


Bibliography

T. Arima, A. Mentrelli, T. Ruggeri, Molecular extended thermodynamics ofrarefied polyatomic gases and wave velocities for increasing number of moments.Annals of Physics 345 (2014) 111-140

T. Arima, S. Taniguchi, T. Ruggeri and M. Sugiyama, Extended thermodynamicsof dense gases, Continuum Mech. Thermodyn., 24, 271 (2012).

T. Arima, S. Taniguchi, T. Ruggeri and M. Sugiyama, Dispersion relation forsound in rarefied polyatomic gases based on extended thermodynamics,Continuum Mech. Thermodyn., doi:10.1007/s00161-012-0271-8 (2012).

T. Arima, S. Taniguchi, T. Ruggeri, M. Sugiyama, Extended thermodynamics ofreal gases with dynamic pressure: An extension of Meixner’s theory, PhysicsLetters A 376 2799–2803, (2012).

M. Pavic, T. Ruggeri and S. Simic, Maximum entropy principle for rarefiedpolyatomic gases. Physica A, 392 (2012) 1302.

T. Arima, S. Taniguchi, T. Ruggeri and M. Sugiyama, Monatomic rarefied gas asa singular limit of polyatomic gas in extended thermodynamics, Physics Letters A377(2013) 2136.


I. Muller and T. Ruggeri (1998) Rational Extended Thermodynamics, 2nd ed.,Springer Tracts in Natural Philosophy 37, Springer-Verlag, New York.

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I-S. Liu and I. Muller (1983) Extended thermodynamics of classical anddegenerate ideal gases, Arch. Rat. Mech. Anal. 83, 285–332.

C. Borgnakke and P.S. Larsen (1975) Statistical Collision Model for Monte CarloSimulation of Polyatomic Gas Mixture, Journal of Computational Physics, 18,405–420.

J.-F. Bourgat, L. Desvillettes, P. Le Tallec and B. Perthame (1994) Microreversiblecollisions for polyatomic gases, Eur. J. Mech., B/Fluids, 13 (2), 237–254.

L. Desvillettes, R. Monaco and F. Salvarani (2005) A kinetic model allowing toobtain the energy law of polytropic gases in the presence of chemical reactions,European Journal of Mechanics B/Fluids, 24, 219–236.


F. Mallinger (1998) Generalization of the Grad theory to polyatomic gases, INRIA– Research Report, No. 3581.

M.N. Kogan (1967) On the principle of maximum entropy, in Rarefied GasDynamics, Vol. I, 359–368, Academic Press, New York.

W. Dreyer (1987) Maximisation of the entropy in non-equilibrium, J. Phys. A:Math. Gen., 20, 6505–6517.

M.N. Kogan (1969) Rarefied Gas Dynamics, Plenum Press, New York.

I. Muller and T. Ruggeri (1993) Extended Thermodynamics, Springer Tracts inNatural Philosophy 37, Springer-Verlag, New York.

G. Boillat and T. Ruggeri (1997) Moment equations in the kinetic theory of gasesand wave velocities, Continuum Mech. Thermodyn., 9, 205–212.


G. Boillat and T. Ruggeri (1997) Hyperbolic Principal Subsystems: EntropyConvexity and Subcharacteristic Conditions, Arch. Rat. Mech. Anal., 137,305–320.

T. Ruggeri (1989) Galilean invariance and entropy principle for systems of balancelaws. The structure of extended thermodynamics, Continuum Mech. Thermodyn.,1, 3–20.

S.K. Godunov (1961) An interesting class of quasilinear systems, Sov. Math. Dokl.

G. Boillat (1974) Sur l’existence et la recherche d’equations de conservationsupplementaires pour les systemes hyperboliques, C. R. Acad Sci. Paris A 278,909–912.


T. Ruggeri, and A. Strumia (1981) Main field and convex covariant density forquasi-linear hyperbolic systems. Relativistic fluid dynamics, Ann. Inst. H. Poincare.Section A 34, 65–84.

, P. Secchi (1983), Existence theorems for compressible viscous fluid having zeroshear viscosity, Rend. Sem. Padova, 70, 73-102

V. Shelukhin, Vanishing Shear Viscosity in a Free-Boundary Problem for theEquations of Compressible Fluids. Journal of Differential Equations 167, 73–86(2000)

H. Frid & V. Shelukhin, Vanishing shear viscosity in the equations of compressiblefluids for the flows with the cylinder symmetry, Siam J. Math. Anal (2000) Vol.31, No. 5, pp. 11441156.


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