moment equations for polyatomic gases

Acta Appl MathDOI 10.1007/s10440-014-9928-6

Moment Equations for Polyatomic Gases

Milana Pavic-Colic · Srboljub Simic

Received: 25 November 2013 / Accepted: 3 April 2014© Springer Science+Business Media Dordrecht 2014

Abstract The aim of this paper is to analyze the moment equations for polyatomic gaseswhose internal degrees of freedom are modeled by a continuous internal energy function.The closure problem is resolved using the maximum entropy principle. The macroscopicequations are divided in two hierarchies—“momentum” and “energy” one. As an example,the system of 14 moments equations is studied. The main new result is determination ofthe production terms which contain two parameters. They can be adapted to fit the expectedvalues of Prandtl number and/or temperature dependence of the viscosity. The ratios ofrelaxation times are also discussed.

Keywords Kinetic theory of gases · Polyatomic gases · Moment equations · Transportcoefficients

Mathematics Subject Classification 82C40 · 82D05 · 82C05

1 Introduction

Modeling a non-equilibrium processes in polyatomic gases can be performed in a variety ofways. The choice of the model strongly depends on the problem itself, as well as needed

This work was supported by the Ministry of Education, Science and Technological Development,Republic of Serbia, within the project “Mechanics of nonlinear and dissipative systems—contemporarymodels, analysis and applications”, Project No. ON174016.

M. Pavic-ColicDepartment of Mathematics and Informatics, University of Novi Sad, Novi Sad, Serbiae-mail: [email protected]

M. Pavic-ColicCMLA, ENS Cachan, Cachan, France

S. Simic (B)Department of Mechanics, University of Novi Sad, Novi Sad, Serbiae-mail: [email protected]

mailto:[email protected]

mailto:[email protected]

M. Pavic-Colic, S. Simic

level of accuracy. The main difference with respect to the monatomic case appears in mod-eling internal degrees of freedom of the molecule. This can be done using classical, semi-classical, or quantum-mechanical approach [15].

This paper is devoted to the macroscopic modeling of polyatomic gases by means ofthe moment equations. In the monatomic case, they can be derived by three different ap-proaches. The first one is the Grad’s method [11] based upon expansion of velocity distribu-tion function in terms of the Hermite polynomials. Closure is achieved through compatibilityof the moments of polynomial approximation with macroscopic variables. The second oneis the application of maximum entropy principle [3, 4, 7, 10, 16, 20], where the velocitydistribution function comes out as a solution of the variational problem with constraints.Finally, the third approach is developed within macroscopic theory of a non-equilibriumprocesses—extended thermodynamics [21]. It fills the gap between macroscopic level (clas-sical TIP) and mesoscopic level (kinetic theory of gases) and resolves the closure problemusing the entropy principle. In the polyatomic case, the problem was partially treated byseveral approaches [14, 17, 25], but there is no complete picture, yet.

In this study we give a contribution to the modeling of polyatomic gases using the modelwith continuous internal energy. It was firstly developed for the purpose of direct simulations[5], and its basic feature is the presence of a single variable I ≥ 0, which models internaldegrees of freedom as communicable internal energy during the collisions. In such a way,the caloric equation of state for polyatomic gases can be easily recovered [6]. Another con-sequence of the basic assumption is that internal energy of the gas is not anymore equal tothe trace of pressure tensor. Therefore, transfer equations for the moments of distributionfunction cannot be structured into a single hierarchy of the macroscopic equations.

The paper is organized as follows. Section 2 contains description of polyatomic gaseswith continuous internal energy, based upon Borgnakke-Larsen procedure. In Sects. 3 and 4we give a review of the results about moment equations for polyatomic gases, recently givenin [22], obtained by the application of maximum entropy principle. It is shown that momentequations are structured in two hierarchies which are in full correspondence with recentlyderived model for dense gases in extended thermodynamics [1]. The main new result, pre-sented in Sect. 5, is concerned with derivation of production terms which govern the non-equilibrium processes. Starting from the collision integral, they are derived in a linear form.By a direct comparison with the results of extended thermodynamics we will discuss thevalues of transport coefficients, Prandtl number and temperature dependence of viscosity.The paper is closed by a brief observation about the Grad’s method for polyatomic gases.

2 Polyatomic Gases with Continuous Internal Energy

To establish the model for polyatomic gases within kinetic theory we will assume that in-ternal degrees of freedom of the molecules are described by a single non-negative variableI ≥ 0. It determines the energy of all degrees of freedom other than translational, so thatconservation laws of momentum and energy in a binary collision read:

ξ ′ + ξ ′∗ = ξ + ξ ∗,

1

2m|ξ ′|2 + 1

2m|ξ ′

∗|2 + I ′ + I ′∗ = 1

2m|ξ |2 + 1

2m|ξ ∗|2 + I + I∗,

(1)

where |ξ | = (ξiξi)1/2 (i, j, k, . . . = 1,2,3, summation convention is assumed).


Besides, the distribution function f := f (t,x, ξ , I ) that aims at describing polyatomicgas depends on time t > 0, space position x ∈ R

3, microscopic velocity ξ ∈ R3, but also on

microscopic internal energy I ∈R+.The post-collisional quantities are described by the so-called Borgnakke-Larsen proce-

dure, based on a repartition of the kinetic and internal energy [5, 6, 9]. To achieve this goalwe should pass to the center of mass reference frame and rewrite the total energy of incom-ing and outgoing molecules using (1):

ε = 1

4m|ξ − ξ ∗|2 + I + I∗ = 1

4m|ξ ′ − ξ ′

∗|2 + I ′ + I ′∗.

By R ∈ [0,1] we denote a fraction of the total energy of the outgoing molecules attributedto the translational degrees of freedom, and thus obtain:

Rε = 1

4m|ξ ′ − ξ ′

∗|2

(1 − R)ε = I ′ + I ′∗.

(2)

Distribution of internal energy between the two outgoing molecules is parametrized byr ∈ [0,1]:

I ′ = r(1 − R)ε, I ′∗ = (1 − r)(1 − R)ε.

Furthermore, it is assumed that collisions between the molecules are of specular reflectiontype, so we can parameterize (2) by a unit vector ω ∈ S2:

ξ ′ − ξ ′∗ =

√4Rε

mTω

[ξ − ξ ∗|ξ − ξ ∗|

],

where Tω is the symmetry operator with respect to the plane {ω}⊥, i.e.

Tωz = z − 2(ω · z)ω, ∀z ∈ R3.

After straightforward computation we end up with expressions for post-collisional velocitiesin a laboratory reference frame:

ξ ′ = ξ + ξ ∗2

+√

Rε

mTω

[ξ − ξ ∗|ξ − ξ ∗|

], ξ ′

∗ = ξ + ξ ∗2

−√

Rε

mTω

[ξ − ξ ∗|ξ − ξ ∗|

].

It will be useful in the sequel to express peculiar post-collisional velocities with help of thepre-collisional ones, C = ξ − u and C∗ = ξ ∗ − u, u being the hydrodynamic velocity:

C′ = C + C∗2

+√

Rε

mTω

[C − C∗|C − C∗|

], C′

∗ = C + C∗2

−√

Rε

mTω

[C − C∗|C − C∗|

]. (3)

For convenience, we introduce the extra parameters R′ ∈ [0,1] and r ′ ∈ [0,1] in the follow-ing equalities

R′ε = 1

4m|ξ − ξ ∗|2, I = r ′(1 − R′)ε, I∗ = (

1 − r ′)(1 − R′)ε.


The time rate of change of the velocity distribution function in the absence of external forcesis determined by the Boltzmann equation

∂tf + ξj ∂jf = Q(f ),

where symbols ∂t and ∂j denote partial derivatives with respect to time t and space vari-ables xj . In this paper, we consider the model introduced in [9] for the collision integral i.e.for any t ∈ R+ and x ∈ R

3 we write

Q(f )(ξ , I ) =∫

Ω

∫S2

(f ′f ′

∗ − ff∗)B(1 − R)R

12

1

ϕ(I)dω dr dR dI∗ dξ ∗, (4)

where Ω = R3 ×R+ × [0,1] × [0,1] and we have used the standard abbreviations:

f∗ := f (t,x, ξ ∗, I∗), f ′ := f(t,x, ξ ′, I ′), f ′

∗ := f(t,x, ξ ′

∗, I′∗).

Due to the presence of an additional parameter, the collision integral takes into ac-count the influence of internal degrees of freedom through collisional cross sectionB := B(ξ , ξ ∗, I, I∗, r,R,ω) which is restricted to satisfy usual microreversibility assump-tions:

B(ξ , ξ ∗, I, I∗, r,R,ω) = B(ξ ∗, ξ , I∗, I,1 − r,R,ω),

B(ξ , ξ ∗, I, I∗, r,R,ω) = B(ξ ′, ξ ′

∗, I′, I ′

∗, r′,R′,ω

).

(5)

Corresponding collision invariants for this model form a 5-vector:

ψ(ξ , I ) = m

(1, ξi, |ξ |2 + 2

I

m

)T

,

i.e. for vector ψ(ξ , I ) the following holds

∫∫R3×R+

ψ(ξ , I )Q(f )(ξ , I )ϕ(I )dI dξ = 0. (6)

Also, hydrodynamic variables are given in the form:

⎛⎝ ρ

ρui

ρ|u|2 + 2ρε

⎞⎠ =

∫∫R3×R+

ψ(ξ , I )f ϕ(I)dI dξ , (7)

where ρ, u and ε are mass density, hydrodynamic velocity and internal energy, respectively.A non-negative measure ϕ(I)dI is a property of the model introduced with the aim to re-cover classical caloric equation of state for polyatomic gases in equilibrium. One of thepossible choices, which will be used in the sequel, is ϕ(I) = Iα .

By introducing the peculiar velocity C = ξ − u, we can rewrite Eq. (7):

⎛⎝ ρ

0i

2ρε

⎞⎠ =

∫∫R3×R+

m

⎛⎝ 1

Ci

|C|2 + 2I/m

⎞⎠f ϕ(I)dI dC. (8)


Taking into account the structure of energy collision invariant, internal energy density canbe divided into the translational part ρεT and the part related to the internal degrees offreedom ρεI :

ρεT =∫∫

R3×R+

1

2m|C|2f ϕ(I)dI dC,

ρεI =∫∫

R3×R+If ϕ(I)dI dC.

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

εT = 3

2

k

mT,

while εI is expected to determine the contribution of internal degrees of freedom to theinternal energy of a polyatomic gas.

3 Maximum Entropy Principle

The idea of entropy maximization is deeply rooted in statistical mechanics and informationtheory. In the context of kinetic theory it is well known that the Maxwellian distribution formonatomic gases maximizes the physical entropy, i.e. it corresponds to the state of max-imum probability. The idea was revived when Kogan [12, 13] recognized that the Grad’s13 moments distribution also maximizes the entropy. Thus, it became a sort of a principlewhich helps to determine the appropriate velocity distribution function which is compatiblewith the macroscopic quantities, and yields a proper closure for the macroscopic equations.This is well explained in the series of papers [3, 7, 10, 16].

We will proceed with direct application of the maximum entropy principle to polyatomicgases, following the approach recently given in [22]. The entropy is defined by the followingrelation:

h = −k

∫∫R3×R+

f logf ϕ(I)dI dξ . (9)

In general, among all admissible functions f we seek for the one which maximizes theentropy, i.e. h → max, subject to the constraints:

F(N)(t,x) =∫∫

R3×R+Ψ (N)(ξ , I )f ϕ(I)dI dξ , (10)

where F(t,x) ∈RN is the vector of macroscopic observable quantities, i.e. the macroscopic

densities. Once the maximizer f is determined by means of the Lagrange multiplier method,one can derive the set of macroscopic equations which determine the time rate of change ofmacroscopic quantities:

∂t F(N) + ∂j F(N)j = P(N),

where:

F(N)(t,x) =∫∫

R3×R+Ψ (N)(ξ , I )f ϕ(I )dI dξ ,


F(N)j (t,x) =

∫∫R3×R+

ξjΨ(N)(ξ , I )f ϕ(I )dI dξ , (11)

P(N)(t,x) =∫∫

R3×R+Ψ (N)(ξ , I )Q(f )ϕ(I )dI dξ

are the densities, the fluxes and the production terms, respectively, evaluated at the maxi-mizer f .

It is obvious that the choice of constraints (10) decisively influences the structure of max-imizer, and thus has to be made properly. For example, if the densities of mass, momentumand energy are taken as constraints, the maximum entropy principle yields an equilibriumvelocity distribution as a maximizer. On the other hand, if more constraints are taken intoaccount, the exact maximizer does not provide convergent macroscopic densities F(t,x) andhas to be approximated, i.e. expanded in the neighborhood of local equilibrium state. Thisis the point where the maximum entropy principle meets the Grad’s 13 moments approx-imation of the distribution function in the case of monatomic gases (see [13]). For thesereasons we will recall the results about polyatomic gases, given in [22], which determinethe equilibrium velocity distribution and the non-equilibrium 14 moments approximation.

To determine the equilibrium distribution, one may use the constraints (8) in the max-imization problem due to invariance under the Galilean transformations. This yields thefollowing

Theorem 1 The distribution function which maximizes the entropy (9) subject to the con-straints (8) has the form:

f = fE = ρ

mq(T )

(m

2πkT

)3/2

exp

{− 1

kT

(1

2m|C|2 + I

)},

where

q(T ) =∫ ∞

0exp

(− I

kT

)ϕ(I)dI.

This result is in complete agreement with the ones given in [6, 9], where the H -theoremwas used. The proof, given in [22], will be omitted. Using fE one can easily recover standardEuler equations of gas dynamics. Moreover, taking ϕ(I) = Iα the following form of internalenergy of polyatomic gases in equilibrium is obtained:

ε|E =(

5

2+ α

)k

mT, α > −1,

where the overall constraint for the parameter α comes from the structure of auxiliary func-tion q(T ) = (kT )1+αΓ (1 + α), Γ (z) being the Gamma function. By a proper choice of α

one may obtain the correct form of internal energy density for a polyatomic gas.To determine an approximate velocity distribution function akin to the Grad’s 13 mo-

ments one, it is necessary to take the pressure tensor pij and the heat flux qi into the set ofconstraints. At this point the study of polyatomic gases departs from the monatomic case:the internal energy density is not related to the trace of the pressure tensor due to exis-tence of internal degrees of freedom. Therefore, it is natural to split the constraints in two


groups—“momentum” and “energy” ones:

⎛⎝ F

Fi1

Fi1i2

⎞⎠ =

∫∫R3×R+

m

⎛⎝ 1

ξi1

ξi1ξi2

⎞⎠f ϕ(I)dI dξ ,

(Gpp

Gppk1

)=

∫∫R3×R+

m

( |ξ |2 + 2 Im

(|ξ |2 + 2 Im)ξk1

)f ϕ(I)dI dξ ,

(12)

where the following relations hold due to the Galilean invariance [1, 23]:

F = ρ, Fi1 = ρui1 , Fi1i2 = ρui1ui2 + pi1i2 ,

Gpp = 2ρε + ρ|u|, Gppk1 = (2ρε + ρ|u|2)uk1 + 2pik1ui + 2qk1 .

This separation is of substantial importance since it will yield the new hierarchical structureof macroscopic equations, to be discussed in the next section. Actually, the constraints arenot used in the form (12), but for the quantities determined by the peculiar velocity:

⎛⎝ ρ

0i

pij

⎞⎠ =

∫∫R3×R+

m

⎛⎝ 1

Ci

CiCj

⎞⎠f ϕ(I)dI dC,

(ρε

qi

)= 1

2

∫∫R3×R+

m

( |C|2 + 2 Im

(|C|2 + 2 Im)Ci

)f ϕ(I)dI dC.

(13)

If we take into account that the pressure tensor in the polyatomic case has the followingstructure:

pij = (p + Π)δij + p〈ij 〉, (14)

where p〈ij 〉 is its traceless part, and Π the so-called dynamic pressure (which vanishes inthe monatomic case), application of the maximum entropy principle is summarized in thefollowing

Theorem 2 The velocity distribution function which maximizes the entropy (9) subject toconstraints (13), for the choice of weighting function ϕ(I) = Iα , has the form:

f ≈ f14 = fE

{1 − ρ

p2qiCi + ρ

2p2

[p〈ij 〉 +

(5

2+ α

)(1 + α)−1Πδij

]CiCj

− 3

2(1 + α)

ρ

p2Π

(1

2|C|2 + I

m

)+

(7

2+ α

)−1ρ2

p3qi

(1

2|C|2 + I

m

)Ci

}(15)

where equilibrium distribution fE and auxiliary function q(T ) are given in Theorem 1.

The proof is given in [22]. The non-equilibrium distribution function (15) contains 14macroscopic moments and is valid for any polyatomic gas. It generalizes the one provided in[19] for diatomic gases, obtained by a sort of Grad’s procedure. However, it cannot be triv-ially reduced to the Grad’s 13 moments distribution for monatomic gases due to restrictionα > −1.


4 Hierarchical Structure of the Moment Equations

The structure of macroscopic densities (12) indicated that the second order moments are notrelated as in the monatomic gases, i.e. Fii = Gpp , due to the presence of internal energyparameter I . This fact prevents structuring the moment equations into a single hierarchy,but calls for introduction of two coupled hierarchies:

∂tF + ∂j Fj = P, ∂tG + ∂j Gj = Q, (16)

where the densities F and G are the moments of the distribution function:

F(t,x) =∫∫

R3×R+Ψ (ξ)f ϕ(I)dI dξ ,

G(t,x) =∫∫

R3×R+Θ(ξ , I )f ϕ(I)dI dξ ,

for:

Ψ (ξ) = m

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

1ξi1

ξi1ξi2...

ξi1 · · · ξin

...

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

, Θ(ξ , I ) =(

2I

m+ |ξ |2

)Ψ (ξ).

The fluxes Fj and Gj , and the productions P and Q, are defined in a usual way, likein Eq. (11). Equation (16) presents two infinite hierarchies of the moment equations: theF -hierarchy which is much alike the classical “momentum” hierarchy in monatomic gases,and the G-hierarchy—the “energy” one—whose essential ingredient is the energy collisioninvariant.

Moment equations (16) in a certain way replace the Boltzmann equation by an infiniteset of macroscopic equations. To make the analysis feasible, the system is usually truncatedat a certain order:

∂tF(N) + ∂j F(N)j = P(N), ∂tG(M) + ∂j G(M)

j = Q(M),

where N ≥ 0, M ≥ 2 denote the highest order of moments which appear in the hierarchy. Onthe other hand, truncation calls for the solution of the so-called closure problem. Maximumentropy principle is one of the methods to resolve it, and Theorem 1 provides its solutionat an Euler level (N = 1, M = 2), while Theorem 2 gives the 14 moments approximation(N = 2, M = 3). Distribution functions uniquely determine the fluxes which appear in mo-ment equations of the highest order in the hierarchy, and serve to calculate the productionterms once the type of interaction is prescribed by the collision cross section.

Detailed analysis of the moment equations is given in [22]. In the sequel, we will consideronly the 14 moments approximation. Momentum hierarchy for this system reads:

∂tρ + ∂i(ρui) = 0,

∂t (ρui) + ∂j (ρuiuj + pij ) = 0,


∂t (ρuiuj + pij ) + ∂k

{ρuiujuk + uipjk + ujpki + ukpij

+ (7/2 + α)−1(qiδjk + qj δki + qkδij )} = Pij ,

while energy hierarchy has the form:

∂t

(1

2ρ|u|2 + ρε

)+ ∂i

{(1

2ρ|u|2 + ρε

)ui + pijuj + qi

}= 0,

∂t

{(1

2ρ|u|2 + ρε

)ui + pijuj + qi

}+ ∂j

{(1

2ρ|u|2 + ρε

)uiuj

+ uiukpjk + ujukpik + 1

2ρ|u|2pij + uiqj + ujqi

+(

7

2+ α

)−1

(uiqj + ujqi + ukqkδij ) +(

9

2+ α

)p

ρpij − p2

ρδij

}= Qi.

The production terms, which describe the rate of change of momentum and energy fluxes,are defined as:

Pij =∫∫

R3×R+mξiξjQ(f )ϕ(I)dI dξ , (17)

Qi =∫∫

R3×R+

(m

2|ξ |2 + I

)ξiQ(f )ϕ(I)dI dξ , (18)

and will be analyzed in detail in the next section.

5 Production Terms and Transport Coefficients

Explicit calculation of production terms is always tedious, even for an approximate form ofdistribution function like f14. It is therefore common practice to make further approxima-tions and determine production terms as a linear form of non-convective macroscopic fluxes.In this section the appropriate approximations for source terms that correspond to the dis-tribution function (15) will be determined. Computation itself will be omitted. Instead, wewill explain the main steps.

Introduction of the peculiar velocity C = ξ − u in conjunction with the collision invari-ants (6) lead to the following form of the production terms (17)–(18):

Pij =∫∫

R3×R+mCiCjQ(f )ϕ(I)dI dξ , (19)

Qi = ukPki +∫∫

R3×R+

(m

2|C|2 + I

)CiQ(f )ϕ(I)dI dξ . (20)

We consider 14 moments approximation of the source terms, obtained by plugging (15)into (19) and (20). Moreover, as non-equilibrium effects are supposed small, we linearizeproducts of the distribution functions appearing in the collision integral (4) with respectto the non-equilibrium quantities manifested in our setting—traceless part of the pressure


tensor p〈k�〉, dynamic pressure Π and heat flux qn i.e. we are led to:

f ′14f

′14∗ − f14f14∗

≈ fEfE∗{

ρ

2p2

(p〈k�〉 +

(5

2+ α

)(1 + α)−1Πδk�

)

× [C ′

kC′� + C ′

∗kC′∗� − CkC� − C∗kC∗�

]

+(

7

2+ α

)−1ρ2

mp3qn

[(m

2

∣∣C′∣∣2 + I ′)

C ′n +

(m

2

∣∣C′∗∣∣2 + I ′

∗

)C ′

∗n

−(

m

2|C|2 + I

)Cn −

(m

2|C∗|2 + I∗

)C∗n

]}. (21)

In addition, we consider the polyatomic version of the modified variable hard sphere modelfor the cross section, which reads:

B(ξ , ξ ∗, I, I∗, r,R,ω) = K2Rs |C − C∗|2s

∣∣∣∣ω · C − C∗|C − C∗|

∣∣∣∣,where K is an appropriate dimensional constant and the parameter s satisfies the overallrestriction s > − 3

2 . One may easily verify that this model satisfies assumptions (5).Computation of such production terms, denoted by P 14

ij and Q14i , starts with an analy-

sis of parity of their ingredients. From (3) we can see that both C′ and C′∗ are odd func-tions of (C,C∗) for ω fixed i.e. C′(−C,−C∗,ω) = −C′(C,C∗,ω) and C′∗(−C,−C∗,ω) =−C′∗(C,C∗,ω). Consequently, (21) consists of two parts: one part that is even and the an-other one that is odd in (C,C∗). Likewise, the cross section is even function of (C,C∗).This consideration leads us to the conclusion that only even part of (21) is of importance forthe calculation of the production term P 14

ij , whereas only odd part of (21) contributes to theproduction term Q14

i . Therefore, if we introduce the following notation:

Pijk� =∫∫

R3×R+

∫Ω

∫S2

mCiCjfE(C + u, I )fE(C∗ + u, I∗)

× (C ′

kC′� + C ′

∗kC′∗� − CkC� − C∗kC∗�

)

× K2(1 − R)Rs+ 12 |C − C∗|2s

∣∣∣∣ω · C − C∗|C − C∗|

∣∣∣∣dω dr dR dI∗ dC∗ dI dC,

Qin =∫∫

R3×R+

∫Ω

∫S2

(m

2|C|2 + I

)CifE(C + u, I )fE(C∗ + u, I∗)

×((

m

2

∣∣C′∣∣2 + I ′)

C ′n +

(m

2

∣∣C′∗∣∣2 + I ′

∗

)C ′

∗n

−(

m

2|C|2 + I

)Cn −

(m

2|C∗|2 + I∗

)C∗n

)

× K2(1 − R)Rs+ 12 |C − C∗|2s

∣∣∣∣ω · C − C∗|C − C∗|

∣∣∣∣dω dr dR dI∗ dC∗ dI dC,


it can be written:

P 14ij = ρ

2p2

[p〈k�〉 +

(5

2+ α

)(1 + α)−1Πδk�

]Pijk�,

Q14i = ukP

14ki +

(7

2+ α

)−1ρ2

mp3qnQin.

(22)

Note that Pijkl vanishes unless indices are equal by pairs, and because of the symmetry itmay be represented in the form:

Pijk� = P1δij δk� +P2(δikδj� + δi�δjk). (23)

Combining (22) and (23), in conjunction with (14) and symmetry of the pressure tensor, wecan extract some more information:

P 14ij = ρ

2p2

(2p〈ij 〉P2 + 1

3δij

(5

2+ α

)(1 + α)−1ΠPrrtt

).

Term P2 can be determined from the system of equations obtained from the representa-tion (23):

Prrtt = 9P1 + 6P2, Prtrt = 3P1 + 12P2.

Therefore, (22) reduces to

P 14ij = ρ

2p2

(p〈ij 〉

3Prtrt −Prrtt

15+ 1

3δij

(5

2+ α

)(1 + α)−1ΠPrrtt

).

From this point, pure calculation is required. On the other hand, one can see that Qin van-ishes unless i = n. Therefore, it can be written

Q14i = ukP

14ki +

(7

2+ α

)−1ρ2

mp3qi

1

3Qrr .

Long but straightforward calculation finally leads to the following:

Theorem 3 Linearized production terms in the 14 moments approximation read:

P 14ij = −22s+4ρ(kT )2

15mq(T )2

√π

(kT

m

)s

Γ

[s + 3

2

]

× K

(p〈ij 〉 + 20

(2s + 5)(2s + 7)

(α + 5

2

)(α + 1)−1Πδij

), (24)

Q14i = ukP

14ik −

(7

2+ α

)−1

× K22s+5(s(2s + 15) + 30)

9(2s + 5)(2s + 7)

ρ(kT )2

mq(T )2

√π

(kT

m

)s

Γ

[s + 3

2

]qi. (25)

Production terms (24)–(25) contain two real parameters, α and s. The former is aimedat modeling internal degrees of freedom, while the latter models the interaction between the


molecules. Apart from the global restriction s > −3/2, it can be chosen in a way which fitsthe compatibility of the model with some other macroscopic properties.

It is in the spirit of extended thermodynamics, as well as the theory of hyperbolic systemsof balance laws, to recast production terms in the following form:

P 14ij = 1

τs

p〈ij 〉 + 1

τΠ

Πδij , Q14i = ukP

14ik − 1

τq

qi,

where τs , τΠ and τq are appropriate relaxation times. They estimate the rate of decay ofnon-equilibrium quantities. It is easy to recognize from (24)–(25) that:

1

τs

= K22s+4

15

ρ(kT )2

mq(T )2

√π

(kT

m

)s

Γ

[s + 3

2

]

1

τΠ

= K

(α + 5

2

)(α + 1)−1 22s+6

3(2s + 5)(2s + 7)

ρ(kT )2

mq(T )2

√π

(kT

m

)s

Γ

[s + 3

2

]

1

τq

= K

(7

2+ α

)−1 22s+5(s(2s + 15) + 30)

9(2s + 5)(2s + 7)

ρ(kT )2

mq(T )2

√π

(kT

m

)s

Γ

[s + 3

2

]

On the other hand, it was shown [1] that relaxation times can be related to the transportcoefficients—shear viscosity μ, bulk viscosity ν, and heat conductivity κ :

μ = pτS, ν = 4(1 + α)

3(5 + 2α)pτΠ, κ =

(α + 7

2

)p2

ρTτq,

so that one obtains:

μ = 15

22s+4

ms

√πKΓ [s + 3

2 ] (kT )−s−1q(T )2,

ν = (α + 1)2

(2α + 5)2

(2s + 5)(2s + 7)

22s+3

ms

√πKΓ [s + 3

2 ] (kT )−s−1q(T )2, (26)

κ =(

α + 7

2

)2 9(2s + 5)(2s + 7)

22s+5(s(2s + 15) + 30)

k

m

ms

√πKΓ [s + 3

2 ] (kT )−s−1q(T ).

Although we arrived at (26) by an ad-hoc procedure, in contrast to the standard asymptoticmethods like Chapman-Enskog one, it gives us first impression about transport coefficientsand their dependence on the collision model.

In [22] results similar to (26) were obtained in the special case s = 0. Generalized form ofthe transport coefficients leaves the possibility to adapt the collision cross section and matchcertain macroscopic quantities. For example, we can choose s to obtain the appropriatevalue of Prandtl number, Pr = cpμ/κ . Theoretically obtained value for Prandtl number forpolyatomic gases is given by Eucken’s relation, Pr = 4γ /(9γ − 5), while from (26) oneobtains:

Pr =(

α + 7

2

)−1 10(s(2s + 15) + 30)

3(2s + 5)(2s + 7).

Matching these two relations, with the aid of restriction s > −3/2, the obtained values ofparameter s are presented in Table 1.


Table 1 Prandtl number andvalues of the parameter s Gas γ α Pr s

Diatomic 7/5 0 28/38 0.678

Three-atomic 4/3 1/2 16/21 −0.311

Since we have only one free parameter in the model, we can adapt it to match only onemacroscopic quantity, while for the others we may expect just better or worse matching. Oneof these important properties is temperature dependence of viscosity, for which it is easy todetermine from (26)1 that μ ∝ T 2α+s−1. For the values from Table 1 one can determinethat one obtains μ ∝ T 0.322 for diatomic gases, and μ ∝ T 2.311 for three-atomic gases, bothresults being unsatisfactory when compared with experiment.

However, a different strategy can be applied: first to adapt s to match the temperaturedependence, and then to calculate the corresponding Prandtl number. According to [8], forCO one has μ ∝ T 0.734, which gives s = 0.262 and yields Pr = 0.781, which is in satisfac-tory agreement with the theoretical value for diatomic gases Pr = 0.737 from Table 1. Forthree-atomic gases the results are a bit worse.

6 Towards the Grad’s Method for Polyatomic Gases

It was noted at the beginning that the Grad’s method, along with the maximum entropyprinciple and the extended thermodynamics, can be applied to resolve the closure problemfor a truncated system of the moment equations. Equivalence of these methods has alreadybeen demonstrated for monatomic gases, while there is lack of completeness of such resultsin the polyatomic case. The only attempt in this direction, within the framework of themodel with continuous internal energy parameter, was given in [19]. It was based upon ideaproposed already in [6], to extend the velocity space in order to include internal degrees offreedom, and to apply the standard Grad’s procedure to such a modified Boltzmann equation.

It is our aim in future studies to develop a Grad-like procedure for polyatomic gaseswith continuous internal energy, without extending the velocity space. Our intension is alsoto remove the arbitrariness in the choice of macroscopic moments, which is present in themonatomic case. In our opinion, this could strengthen the idea of two hierarchies proposedin the papers [1, 22].

7 Conclusion

In this paper we gave a review of recent results concerned with the moment equations forpolyatomic gases, derived by the application of maximum entropy principle. We adoptedthe model with continuous internal energy parameter, and showed that it naturally leadsto the set of macroscopic equations structured in two hierarchies—a momentum and anenergy one. We also generalized the previous results about production terms in 14 momentsapproximation, which now contain a single free parameter from the collision cross section.This provided the possibility to match certain macroscopic quantities, like Prandtl numberor temperature dependence of viscosity.

The results presented in this study are a part of a more general project aimed at modelingand analysis of polyatomic gases. Apart from the idea to develop a counterpart for the Grad’smethod in polyatomic gases, it is our aim to apply it to the mixtures of polyatomic gases,especially to the so-called multi-temperature models, which became the topic of intensivecontemporary research [2, 18, 24].


References

1. Arima, T., Taniguchi, S., Ruggeri, T., Sugiyama, M.: Extended thermodynamics of dense gases. Contin.Mech. Thermodyn. 24(4-6), 271–292 (2012)

2. Bisi, M., Martaló, G., Spiga, G.: Multi-temperature hydrodynamic limit from kinetic theory in a mixtureof rarefied gases. Acta Appl. Math. 122, 37–51 (2012)

3. Boillat, G., Ruggeri, T.: Hyperbolic principal subsystems: entropy convexity and subcharacteristic con-ditions. Arch. Ration. Mech. Anal. 137, 305–320 (1997)

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

5. Borgnakke, C., Larsen, P.S.: Statistical collision model for Monte Carlo simulation of polyatomic gasmixture. J. Comput. Phys. 18, 405–420 (1975)

6. Bourgat, J.-F., Desvillettes, L., Le Tallec, P., Perthame, B.: Microreversible collisions for polyatomicgases. Eur. J. Mech. B, Fluids 13(2), 237–254 (1994)

7. Brini, F., Ruggeri, T.: Entropy principle for the moment systems of degree α associated to the Boltzmannequation. Critical derivatives and non controllable boundary data. Contin. Mech. Thermodyn. 14, 165–189 (2002)

8. Chapman, S., Cowling, T.G.: The Mathematical Theory of Non-Uniform Gases, 3rd edn. CambridgeUniversity Press, Cambridge (1990)

9. Desvillettes, L., Monaco, R., Salvarani, F.: A kinetic model allowing to obtain the energy law of poly-tropic gases in the presence of chemical reactions. Eur. J. Mech. B, Fluids 24, 219–236 (2005)

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

11. Grad, H.: On the kinetic theory of rarefied gases. Commun. Pure Appl. Math. 2, 331–407 (1949)12. Kogan, M.N.: On the principle of maximum entropy. In: Rarefied Gas Dynamics, Vol. I, pp. 359–368.

Academic Press, New York (1967)13. Kogan, M.N.: Rarefied Gas Dynamics. Plenum, New York (1969)14. Kremer, G.M.: Extended thermodynamics of non-ideal gases. Physica A 144, 156–178 (1987)15. Kremer, G.M.: An Introduction to the Boltzmann Equation and Transport Processes in Gases. Springer,

Berlin (2010)16. Levermore, C.D.: Moment closure hierarchies for kinetic theories. J. Stat. Phys. 83(5/6), 1021–1065

(1996)17. Liu, I.-S.: Extended thermodynamics of fluids and virial equations of state. Arch. Ration. Mech. Anal.

88, 1–23 (1985)18. Madjarevic, D., Simic, S.: Shock structure in helium-argon mixture—a comparison of hyperbolic multi-

temperature model with experiment. Europhys. Lett. 102, 44002 (2013)19. Mallinger, F.: Generalization of the Grad theory to polyatomic gases. INRIA Research Report, No. 3581

(1998)20. Müller, I., Ruggeri, T.: Extended Thermodynamics. Springer Tracts in Natural Philosophy, vol. 37.

Springer, New York (1993)21. Müller, I., Ruggeri, T.: Rational Extended Thermodynamics, 2nd edn. Springer Tracts in Natural Philos-

ophy, vol. 37. Springer, New York (1998)22. Pavic, M., Ruggeri, T., Simic, S.: Maximum entropy principle for rarefied polyatomic gases. Physica A

392, 1302–1317 (2013)23. Ruggeri, T.: Galilean invariance and entropy principle for systems of balance laws. The structure of

extended thermodynamics. Contin. Mech. Thermodyn. 1, 3–20 (1989)24. Ruggeri, T., Simic, S.: On the hyperbolic system of a mixture of Eulerian fluids: a comparison between

single- and multi-temperature models. Math. Methods Appl. Sci. 30, 827–849 (2007)25. Wang Chang, C.S., Uhlenbeck, G.E., de Boer, J.: The heat conductivity and viscosity of polyatomic

gases. In: Studies in Statistical Mechanics, vol. II, pp. 243–268. North-Holland, Amsterdam (1964)

moment equations for polyatomic gases

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