Mathematical modelling and analysis of polyatomic gases ... · Mathematical modelling and analysis of ... physics the object of kinetic theory is the ... modelling and analysis of

Mathematical modelling and analysis ofpolyatomic gases and mixtures

in the context ofkinetic theory of gases and fluid mechanics

Presented by Milana Pavic

University of Novi Sad Ecole Normale Superieure de Cachan

Jury Members Examiners: Laurent Boudin, invitedKlemens FellnerBerenice GrecMaria GroppiMarko NedeljkovTommaso RuggeriFrancesco Salvarani

Supervisors: Laurent DesvillettesSrboljub Simic

*with the support of the bilateral project between France and Serbia CNRS/MSTD no25794and the project noON174016 of the Ministry of Education and Science of Serbia

We deal with...

Mixture of monatomic gases

Polyatomic gas

Mixture of polyatomic gases

Milana Pavic

Mathematical modelling and analysis of polyatomic gases and mixtures in the context of kinetic theory and fluid mechanics

Contents

Introduction: presentation of models and results

Part 1. Presentation of models

Chapter 1. Description of the kinetic models used in this thesisDescription of monatomic gasesDescription of mixtures of monatomic gasesDescription of polyatomic gasesDescription of mixtures of polyatomic gases

Part 2. Presentation of the results of the thesis

Chapter 2. Maximum entropy principle for polyatomic gases

Chapter 3. Multivelocity and multitemperature models ofEulerian polyatomic gases

Chapter 4. Diffusion asymptotics of a kinetic modelfor the mixtures of monatomic gases

Milana Pavic


Introduction: kinetic theory

a part of nonequilibrium statistical physics

the object of kinetic theory is the modelling of a gas throughthe distribution function

distribution function f (t, x , ξ) ≥ 0– defined on the phase space R+ × RN × Ω, Ω ⊆ Rdim(ξ)

– for identical molecules f (t, x , ξ) dx dξ determines thenumber of molecules in an elementary volume dx dξ centeredaround (x , ξ)

if the distribution function changes due to interactionsbetween particles, then the evolution of the distributionfunction is governed by the Boltzmann equation:

∂t f + v · ∇x f = Q(f , f )

ξ = (v , . . .)

Milana Pavic


Assumptions

(i) particles interact via binary collisions (the gas should be diluteenough so that the effect of interactions involving more thantwo particles can be neglected)

(ii) collisions are assumed localized both in space and time

(iii) collisions are microreversible

(iv) collisions involve only uncorrelated particles (molecular chaosassumption)

Milana Pavic


Assumptions ⇒ structure of Q(f , f )

Q(f , f ) =

∫Ω

∫SN−1

(f ′f ′∗ − ff∗

)B dω dξ

(i) particles interact via binary collisions (the gas should be diluteenough so that the effect of interactions involving more thantwo particles can be neglected)⇒ quadratic form of the collisional operator Q(f , f )

(ii) collisions are assumed localized both in space and time⇒ variables t and x appear as parameters

(iii) collisions are microreversible⇒ microreversibility assumptions should be made on B

(iv) collisions involve only uncorrelated particles (molecular chaosassumption)⇒ appearance of the products f ′f ′∗ and ff∗

Milana Pavic


Binary elastic collision between moleculesof monatomic, monocomponent gas

v ′

v ′∗

v∗

v

Conservation law of momentum

mv ′ + mv ′∗ = mv + mv∗

Conservation law of kinetic energy

m

2v ′2 +

m

2v ′2∗ =

m

2v 2 +

m

2v 2∗

Milana Pavic


Mixture of monatomic gases

miv′ + mjv

′∗ = miv + mjv∗

mi

2v ′2 +

mj

2v ′2∗ =

mi

2v2 +

mj

2v2∗

Polyatomic gas

mv ′ + mv ′∗ = mv + mv∗m

2v ′2 +

m

2v ′2∗ + I ′ + I ′∗ =

m

2v2 +

m

2v2∗ + I + I∗

Bourgat, Desvillettes, Le Tallec, Perthame, 1994

Desvillettes, Monaco, Salvarani, 2005

Mixture of polyatomic gases

miv′ + mjv

′∗ = miv + mjv∗

mi

2v ′2 +

mj

2v ′2∗ + I ′ + I ′∗ =

mi

2v2 +

mj

2v2∗ + I + I∗

Milana Pavic


Diffusion asymptotics of a kinetic model forthe mixtures of monoatomic gases

Laurent Boudin, Berenice Grec, Milana Pavic, and Francesco Salvarani.

Diffusion asymptotics of a kinetic model for gaseous mixtures.

Kinetic and Related Models, 6(1): 137-157, 2013.

Milana Pavic


Literature overview

derivation of macroscopic equations starting from kinetic theorybased on asymptotic expansions

Chapman, Cowling, 1990,Bardos, Golse, Levermore, 1989, 1991, 1993,Golse, Saint-Raymond, 2004

study of gaseous mixtures (much more intricate)

Sirovich, 1962, Groppi, Spiga, 1999, Rossani, Spiga, 1999,Desvillettes, Monaco, Salvarani, 2005, Bisi, Desvillettes, 2006, 2014

diffusive scaling in the monatomic and mono-species case

Grad, 1963

Milana Pavic


Model

each species Ai of the mixture is described by fi := fi (t, x , v) ≥ 0

Boltzmann equation ∂t fi + v · ∇x fi =∑j=1

Qij(fi , fj)

collision operator associated to species Ai and Aj , 1 ≤ i , j ≤ `

Qij(f , g)(v) =

∫∫R3×S2

[f (v ′)g(v ′∗)− f (v)g(v∗)

]Bij(ω, v−v∗)dωdv∗

pre-collisional velocities

v ′ =miv + mjv∗mi + mj

+mj

mi + mjTω(v − v∗)

v ′∗ =miv + mjv∗mi + mj

− mi

mi + mjTω(v − v∗),

ω ∈ S2 is arbitrary, and Tωz = z − 2(ω · z)ω, ∀z ∈ R3

Milana Pavic


Statement of the problem

diffusion scaling of BE : ε ∂t fεi + v · ∇x f

εi = 1

ε

∑`j=1 Qij(f

εi , f

εj )

formal development : f εi = f 0i + εf 1

i + ε2f 2i + . . .

after integration w. r. to v ...∂tni +∇x · Ni = 0, where

Ni (t, x)=

∫R3

v gi (t, x , v)Mi (v)1/2dv

1 ≤ i ≤ `

identification of thesame order terms:

. determines f 0i :

f εi (t, x , v) = ni (t, x)Mi (v)+εMi (v)1/2gi (t, x , v)+...

∀t ≥ 0, ∀x , v ∈ R3

. existence of g = (g1, . . . , g`)with respect to a given (n1, . . . , n`)?. we work in L2 in the variable v &

‖g‖2L2 =

∑j=1

‖gj‖2L2 =

∑j=1

∫R3

gj(t, x , v)2dv

(K − ν Id) g =(M

1/2i (v · ∇xni )

)1≤i≤`

ε−1

ε0

ε1

Milana Pavic


The main result

Theorem. Suppose that the cross sections (Bij)1≤i ,j≤` are positive

functions satisfying assumption:

Bij(ω,V ) ≤ a | sin θ| | cos θ|(|V |+ 1

|V |1−δ).

If we assume that∑i=1

ni (t, x) does not depend on x ,

then, for any t, x , there exists g(t, x , ·) ∈ L2(R3v )` satisfying

(K − ν Id) g =(M

1/2i (v · ∇xni )

)1≤i≤`

.

Proposition. The operator K is compact from L2(R3v )` to L2(R3

v )`.

Milana Pavic


Sketch of the proof of Theorem

(K − ν Id) g = Lg =(M

1/2i (v · ∇xni )

)1≤i≤`

(?)

. we here assume that the compactness of the operator K is known

. idea: since K is compact, we can apply the Fredholm alternativeto the operator K − ν Id =: L

Step 1 – Study of ker L Step 2 – Computation of L∗ Step 3 – Conclusion

kerL 6= 0Fredholm alternative statesthat (?) has a solution iff(M

1/2i (v · ∇xni )

)1≤i≤`∈(kerL∗)⊥

∀ t > 0, x ∈ R3

kerL∗=SpanM

1/2i ,miM

1/2i vj ,

mi

2 M1/2i v2

,

j = 1, 2, 3

∑i=1

3∑k=1

∂ni∂xk

∫R3

vkmivkvj

mivkv2/2

Mi (v)dv = 0,

j = 1, 2, 3

X by parity & assumption of theorem

Milana Pavic


Sketch of the proof of Proposition

Lg = (K − ν Id) g ; we want to prove compactness of K on L2(R3)`

(Lg)i (v) ∼∑j

∫∫ω,v∗

Bij(ω, v − v∗) e− 1

4miv

2e−

12mjv

2∗

(e

14miv′2gi (v

′) + e14mjv′2∗ gj(v

′∗)− e

14miv

2gi (v)︸︷︷︸

gives νId

− e14mjv

2∗gj(v∗)

).

An operator J is compact if it satisfiesa uniform decay at infinity

‖J g‖L2(B(0,R)c ) ≤ σ(R) ‖g‖L2(R3), with σ(R)→ 0,R → +∞

an equiintegrability property

‖(τw − Id)J g‖L2(R3) ≤ %(w)‖g‖L2(R3), with %(w)→ 0,w → 0

where τwJ g (v) = J g (v + w), ∀v ,w ∈ R3.Milana Pavic


(Kg)i (v) ∼∑j

∫∫R×S2

Bij(ω, v − v∗) e− 1

4miv

2e−

12mjv

2∗

(e

14miv′2gi (v

′)︸︷︷︸gives K4g

+ e14mjv′2∗ gj(v

′∗)︸︷︷︸

depending on mi ,mj

− e14mjv

2∗gj(v∗)︸︷︷︸

gives K1g

)dω dv∗

v ′, v ′∗ are functions of v , v∗, ω

K1g : easy, using properties of BijK4g : monatomic case treated by Grad in 1963 can be extended

to the mixtures

for mi = mj : using symmetry properties of a collision betweenmolecules of same masses, reduces to K4g

for mi 6= mj : this symmetry is broken! We propose a newapproach that allows to reduce to K1g .

Milana Pavic


(Kg)i (v) ∼∑j=1

∫∫R3×S2

Bij(ω, v − v∗) e− 1

4miv

2e−

12mjv

2∗

(e

14miv′2gi (v



′∗)︸︷︷︸

mi=mj gives K3g

− e14mjv

2∗gj(v∗)︸︷︷︸

gives K1g

)dω dv∗

mi 6=mj gives K2g

(K3g)i (v) ∼∑j=1

s.t.mi=mj

∫∫R3×S2

Bij(ω, v − v∗) e− 1

4miv

2− 12mjv

2∗+ 1

4mjv′2∗ gj(v

′∗) dω dv∗

=∑j=1

s.t.mi=mj

∫∫R3×S2

Bij(ω, v − v∗) e− 1

4miv

2− 12mjv

2∗+ 1

4mjv′2gj(v

′) dω dv∗

(K2g)i (v) ∼∑j=1

s.t.mi 6=mj

∫∫R3×S2

Bij(ω, v − v∗) e− 1

4miv

2− 12mjv

2∗+ 1

4mjv′2∗ gj(v

′∗)dω dv∗

Milana Pavic


(Kg)i (v) ∼∑j=1

∫∫R3×S2

Bij(ω, v − v∗) e− 1

4miv

2e−

12mjv

2∗

(e

14miv′2gi (v



′∗)︸︷︷︸

mi=mj gives K3g

− e14mjv

2∗gj(v∗)︸︷︷︸

gives K1g

)dω dv∗

mi 6=mj gives K2g

(K3g)i (v) ∼∑j=1

s.t.mi=mj

∫∫R3×S2

Bij(ω, v − v∗) e− 1

4miv

2− 12mjv

2∗+ 1

4mjv′2∗ gj(v

′∗) dω dv∗

=∑j=1

s.t.mi=mj

∫∫R3×S2

Bij(ω, v − v∗) e− 1

4miv

2− 12mjv

2∗+ 1

4mjv′2gj(v

′) dω dv∗

(K2g)i (v) ∼∑j=1

s.t.mi 6=mj

∫∫R3×S2

Bij(ω, v − v∗) e− 1

4miv

2− 12mjv

2∗+ 1

4mjv′2∗ gj(v

′∗)dω dv∗

Milana Pavic


(Kg)i (v) ∼∑j=1

∫∫R3×S2

Bij(ω, v − v∗) e− 1

4miv

2e−

12mjv

2∗

(e

14miv′2gi (v



′∗)︸︷︷︸

mi=mj gives K3g

− e14mjv

2∗gj(v∗)︸︷︷︸

gives K1g

)dω dv∗

mi 6=mj gives K2g

(K3g)i (v) ∼∑j=1

s.t.mi=mj

∫∫R3×S2

Bij(ω, v − v∗) e− 1

4miv

2− 12mjv

2∗+ 1

4mjv′2∗ gj(v

′∗) dω dv∗

=∑j=1

s.t.mi=mj

∫∫R3×S2

Bij(ω, v − v∗) e− 1

4miv

2− 12mjv

2∗+ 1

4mjv′2gj(v

′) dω dv∗

(K2g)i (v) ∼∑j=1

s.t.mi 6=mj

∫∫R3×S2

Bij(ω, v − v∗) e− 1

4miv

2− 12mjv

2∗+ 1

4mjv′2∗︸︷︷︸

=− 14mjv2∗− 1

4miv ′2

gj(v′∗)dω dv∗

Milana Pavic


(K2g)i (v) ∼∑j=1

s.t.mi 6=mj

∫∫R3×S2

Bij(ω, v − v∗) e− 1

4mjv

2∗− 1

4miv′2gj(v

′∗) dω dv∗

Lemma. There exists b > 0 s.t. for any i , j satisfying mi 6= mj ,

miv′2 + mjv

2∗ ≥ b

(miv

2 + mjv′2∗)

for any v , v∗ ∈ R3 and v ′, v ′∗ given by collisional rules.

Proof. We define A = I3− 2mimi+mj

ω ωT ⇒ det A =(mj−mi )(mi+mj )

. Then√mi v ′

√mj v∗

=

mi+mj

miI3−

mj

miA−1 −

√mj

mi

(I3−A−1

)√mj

mi

(I3−A−1

)A−1

︸︷︷︸

A

√mi v

√mj v ′∗

& b = λmin(ATA)

(K2g)i (v) .∑j=1

s.t.mi 6=mj

∫∫R3×S2

Bij(ω, v − v∗) e− b

4miv

2− b4mjv′2∗ gj(v

′∗)dω dv∗.

Milana Pavic


(K2g)i (v) ∼∑j=1

s.t.mi 6=mj

∫∫R3×S2

Bij(ω, v − v∗) e− 1

4mjv

2∗− 1

4miv′2gj(v

′∗) dω dv∗

Lemma. There exists b > 0 s.t. for any i , j satisfying mi 6= mj ,

miv′2 + mjv

2∗ ≥ b

(miv

2 + mjv′2∗)

for any v , v∗ ∈ R3 and v ′, v ′∗ given by collisional rules.

Proof. We define A = I3− 2mimi+mj

ω ωT ⇒ det A =(mj−mi )(mi+mj )

. Then√mi v ′

√mj v∗

=

mi+mj

miI3−

mj

miA−1 −

√mj

mi

(I3−A−1

)√mj

mi

(I3−A−1

)A−1

︸︷︷︸

A

√mi v

√mj v ′∗

& b = λmin(ATA)

(K2g)i (v) .∑j=1

s.t.mi 6=mj

∫∫R3×S2

˜Bij(ω, v − v ′∗) e− b

4miv

2− b4mjv′2∗ gj(v

′∗)dω dv ′∗.

Milana Pavic


Maximum entropy principlefor polyatomic gases

Milana Pavic, Tommaso Ruggeri and Srboljub Simic.

Maximum entropy principle for rarefied polyatomic gases.

Physica A, 392(6): 1302–1317, 2013.

Milana Pavic


Literature overview – monatomic case

Grad, 1949

If f := f (t, x , v) is a solution to the Boltzmann equation∫v

/∂t f +

3∑j=1

∂xj vj f = Q(f , f )/vi1vi2 . . . vin

then the following moment equations hold:

∂t

F (0)

F(1)i1

F(2)i1 i2

.

.

.

F(n)i1 i2 i3...in

.

.

.

+3∑

j=1

∂xj

F(1)j

F(2)i1 j

F(3)i1 i2 j

.

.

.

F(n+1)i1 i2 i3...in j

.

.

.

=

P(0)

P(1)i1

P(2)i1 i2

.

.

.

P(n)i1 i2 i3...in

.

.

.

,

F(n)i1 i2...in

=∫R3 vi1vi2 ...vin f dv , P

(n)i1 i2...in

=∫R3 vi1vi2 ...vin Q(f ,f ) dv .

(i) the hierarchical structure: flux in one equation becomes densityin the next equation

Milana Pavic



collision invariants 1, v , 12 |v |

2 for which

P(0)

P(1)i1

12

∑3k=1 P

(2)kk

= 0

physical interpretation of momentsF (0)

F(1)i1

12

3∑k=1

F(2)kk

=

∫R3

m

1

vi112 |v |

2

f dv =

ρ

ρui112ρ |u|

2 + ρe

F(2)i1i2

=

∫R3

mvi1vi2 f dv = ρui1ui2 + pi1i2

(ii) ρ e = 12

3∑k=1

pkk and therefore energy density = 12

3∑k=1

F(2)kk

Milana Pavic



1

2

3∑i=1

(∂t

F (0)

F(1)i1

F(2)i1i2...

F(n)i1i2...in

...

+

3∑j=1

∂xj

F(1)j

F(2)i1j

F(3)i1i2j...

F(n+1)i1i2...inj

...

=

P(0)

P(1)i1

P(2)i1i2...

P(n)i1i2...in

...

)

∂tρ+∇x · (ρ u) = 0,

∂t (ρui ) +3∑

j=1∂xj

(ρuiuj + pij

)= 0,

∂t

(m2 |u|

2 + ρ e)

+3∑

j=1∂xj

((m2 |u|

2 + ρ e)uj +

3∑i=1

pijui + qj

)= 0.

Milana Pavic



1

2

3∑i=1

(∂t

F (0)

F(1)i1

F(2)i1i2...

F(n)i1i2...in

...

+

3∑j=1

∂xj

F(1)j

F(2)i1j

F(3)i1i2j...

F(n+1)i1i2...inj

...

=

P(0)

P(1)i1

P(2)i1i2...

P(n)i1i2...in

...

)

∂tρ+∇x · (ρ u) = 0,

∂t (ρui1) +3∑

j=1∂xj

(ρui1uj + pi1j

)= 0,

∂t

(m2 |u|

2 + ρ e)

+3∑

j=1∂xj

((m2 |u|

2 + ρ e)uj +

3∑i=1

pijui + qj

)= 0.

Milana Pavic



1

2

3∑k=1

(∂t

F (0)

F(1)i1

F(2)kk...

F(n)i1i2...in

...

+

3∑j=1

∂xj

F(1)j

F(2)i1j

F(3)kkj...

F(n+1)i1i2...inj

...

=

P(0)

P(1)i1

P(2)kk...

P(n)i1i2...in

...

)

∂tρ+∇x · (ρ u) = 0,

∂t (ρui1) +3∑

j=1∂xj

(ρui1uj + pi1j

)= 0,

∂t

(12ρ |u|

2 + ρ e)

+3∑

j=1∂xj

((12ρ |u|

2 + ρ e)uj +

3∑i=1

pijui + qj

)= 0.

Milana Pavic


∂tρ+∇x · (ρ u) = 0,

∂t (ρui1) +3∑

j=1∂xj (ρui1uj + pi1j) = 0,

∂t

(12ρ |u|

2 + ρ e)

+3∑

j=1∂xj

((12ρ |u|

2 + ρ e)uj +

3∑i=1

pijui + qj

)= 0.

monatomic gas:3∑

k=1

pkk = 3p = 2 ρ e

rarefied polyatomic gas or dense gas:3∑

k=1

pkk = 3 (p + Π) 6= 2ρ e

Milana Pavic


Literature overview – Extended Thermodynamics of dense gases

Arima, Taniguchi, Ruggeri, Sugiyama, 2012

two independent hierarchies:

∂tF +3∑

j=1

∂xjFj = 0

∂tFi1 +3∑

j=1

∂xjFi1j = 0

∂tFi1i2 +3∑

j=1

∂jFi1i2j = Pi1i2 ∂tG +3∑

j=1

∂xjGj = 0

∂tGk1 +3∑

j=1

∂xjGk1j = Qk1

F - mass density, Fi1 - momentum density, Fi1i2 - momentum flux;G - twice the total energy density, Gk1 - twice the total energy flux;Fi1i2j , Gppk1j - fluxes of Fi1i2 , Gppk1 , respectively;

Pi1i2 , Qppk1 - productions with respect to Fi1i2 , Gppk1 , respectively.

Milana Pavic


Literature overview – kinetic model for polyatomic gases

Bourgat, Desvillettes, Le Tallec, Perthame, 1994Desvillettes, Monaco, Salvarani, 2005

introduces one parameter I ∈ R+ that captures all phenomenarelated to peculiarity of polyatomic gases

f := f (t, x , v , I ):

∂t f + v · ∇x f = Q(f , f )

introduces ϕ(I ) aiming to recover a good equation of state

Milana Pavic



collision invariants: m, mv , m2 |v |

2 + I

macroscopic quantities

ρ =

∫∫R3×R+

m f ϕ(I )dIdv

ρui1 =

∫∫R3×R+

mvi1 f ϕ(I )dIdv

ρui1ui2 + pi1i2 =

∫∫R3×R+

mvi1vi2 f ϕ(I )dIdv

12ρ |u|

2 + ρe = 12ρ |u|

2 + ρeT + ρeI =

∫∫R3×R+

(m2 |v |

2 + I)f ϕ(I )dIdv

ρeT = 12

3∑k=1

pkk

Milana Pavic



collision invariants: m, mv , m2 |v |

2 + I

macroscopic quantities

ρ =

∫∫R3×R+

m f ϕ(I )dIdv

ρui1 =

∫∫R3×R+

mvi1 f ϕ(I )dIdv

ρui1ui2 + pi1i2 =

∫∫R3×R+

mvi1vi2 f ϕ(I )dIdv

12ρ |u|

2 + ρe = 12ρ |u|

2 + ρeT + ρeI =

∫∫R3×R+

(m2 |v |

2 + I)f ϕ(I )dIdv

ρeT = 12

3∑k=1

pkk

Milana Pavic


”Momentum – like”

F (0)

F(1)i1

F(2)i1 i2

F(3)i1 i2 i3...

F(n)i1···in...

=∫∫

m

1

vi1

vi1vi2

vi1vi2vi3...

vi1 ···vin...

f ϕ(I )dIdv

∂tF(n)i1···in +

3∑j=1

∂xjF(n+1)i1···in j = P

(n)i1···in

Milana Pavic


”Momentum – like”

F (0)

F(1)i1

F(2)i1 i2

F(3)i1 i2 i3...

F(n)i1···in...

=∫∫

m

1

vi1

vi1vi2

vi1vi2vi3...

vi1 ···vin...

f ϕ(I )dIdv

∂tF(n)i1···in +

3∑j=1

∂xjF(n+1)i1···in j = P

(n)i1···in

”Energy – like”

G (2)

G(3)k1...

G(m)k1···im−2

...

=∫∫

(m2 v2+I)

1

vk1

...vk1···vkm−2

...

f ϕ(I )dIdv

∂tG(m)k1···km−2

+3∑

j=1

∂xjG(m+1)k1···km−2 j

= Q(m)k1···km−2

(i) the hierarchical structure is conserved,albeit separately for the two hierarchies

(ii) conservation laws can be recovered

Milana Pavic


Literature overview – Maximum entropy principle for monatomic gases

Kogan, 1969, Dreyer, 1987, Levermore, 1996, Boillat, Ruggeri, 1997

1. Truncation of moment equations

∂tF + ∂xjFj = P

∂tFi1 + ∂xjFi1 j = Pi1

∂tFi1i2 + ∂xjFi1i2 j = Pi1i2

...

∂tFi1i2i3i4...iN−1+ ∂xjFi1i2i3i4...iN−1 j = Pi1i2i3i4...iN−1

∂tFi1i2i3i4...iN + ∂xjFi1i2i3i4...iN j = Pi1i2i3i4...iN

∂tFi1i2i3i4...iN+1+ ∂xjFi1i2i3i4...iN+1 j = Pi1i2i3i4...iN+1

...

Milana Pavic




1. Truncation of moment equations

∂tF + ∂xjFj = P



...




...

Milana Pavic




2. Closure problem

∂tF + ∂xjFj = P



...




...

Milana Pavic




2. Maximum entropy principle

entropy becomes the objective function

maxf

h = −k∫R3

f log f dv

truncated moments become constraints

subject to F (N)=∫∫

m

1

vi1

vi1vi2...

vi1 ···viN

f dv

Milana Pavic


Maximum entropy principle for polyatomic gases

truncated moment equations

∂tF(N)i1···iN +

3∑j=1

∂xjF(N+1)i1···iN j = P

(N)i1···iN ∂tG

(M)k1···kM−2

+3∑

j=1

∂xjG(M+1)k1···kM−2 j

= Q(M)k1···kM−2

entropy becomes the objective function

maxf

h = −k∫R3

∫ ∞0

f log f ϕ(I ) dI dv

truncated moments become constraints

subject to

F (N)=∫∫

m

1

vi1

vi1vi2...

vi1 ···viN

f ϕ(I )dIdv G (M)=

∫∫ (12mv2+I

)

1

vk1

vk1k2

...vk1···vkM−2

f ϕ(I )dIdv

Milana Pavic


N = 1 M = 2

the solution to the variational problem is the equilibriumdistribution function fE

the corresponding equations are Euler equations

N = 2 M = 3

the solution to the variational problem is

f (t,x ,c+u,I )≈f14(t,x ,c+u,I )=fE (t,x ,c+u,I )

1− ρ

p2 q·c

+ ρ

2p2

∑3i,j=1[p〈ij〉+( 5

2+α)(1+α)−1Πδij ]cicj

− 32(1+α)

ρ

mp2 Π(m2 |c|2+I)+( 7

2+α)

−1 ρ2

mp3 q·(m2 |c|2+I)c

the corresponding equations are...

Milana Pavic


∂tρ+3∑

i=1

∂xi (ρui ) = 0,

∂t(ρui ) +3∑

j=1

∂xj (ρuiuj + pij) = 0,

∂t (ρuiuj + pij)

+3∑

k=1

∂xk

ρuiujuk + uipjk + ujpki + ukpij +

(α +

7

2

)−1

(qiδjk + qjδki + qkδij)

= −K 22s+4ρ (k T )2

15m ζ0(T )2

√π

(k T

m

)s

Γ

[s +

3

2

](p〈ij〉 +

20

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

(α +

5

2

)(α + 1)−1Πδij

)

∂t

(12ρ |u|

2 + ρe)

+3∑

i=1

∂xi

(12ρ |u|

2 + ρe)ui +

3∑j=1

pijuj + qi

= 0,

∂t

(12ρ |u|

2 + ρe)ui +

3∑j=1

pijuj + qi

+

3∑j=1

∂xj

(12ρ |u|

2 + ρe)uiuj +

3∑k=1

(uiukpjk + ujukpik) + 12ρ |u|

2 pij

+

(α +

9

2

)(α +

7

2

)−1

(qiuj + qjui ) +

(α +

7

2

)−1

u · q δij +

(α +

9

2

)p

ρpij −

p2

ρδij

= −K√π

(k T

m

)s

Γ

[s +

3

2

]ρ (k T )2

m ζ0(T )2

×

(22s+4

15

(3∑

k=1

ukp〈ik〉 +20

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

(α +

5

2

)(α + 1)−1Πui

)

+

(7

2+ α

)−1 22s+5 (s (2s + 15) + 30)

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

), for B(v , v∗, I , I∗, r ,R, ω) = K2Rs |v − v∗|2s

∣∣∣∣ω · v − v∗|v − v∗|

∣∣∣∣ .Milana Pavic


Multivelocity and multitemperature modelsof Eulerian polyatomic gases

Laurent Desvillettes, Milana Pavic and Srboljub Simic

Milana Pavic


de Groot, Mazur, 1962, Truesdell, 1969, Muller, Ruggeri, 1998

we consider a mixture of ` Euler fluids A1, . . . ,A`

each species Ai of the mixture obey the following balancelaws:

∂tρi +∇x · (ρi ui ) = 0,

∂t (ρi ui ) +∇x (ρi ui ⊗ ui + pi Id) = Ni ,

∂t

(12ρi |ui |

2 + ρiei

)+∇x ·

(12ρi |ui |

2 + ρiei

)ui + piui

= Ei .

summation yields the conservation laws for mixture

∂tρ+∇x · (ρ u) = 0,

∂t (ρ u) +∇x (ρ u ⊗ u + p Id) = 0,

∂t

(12ρ |u|

2 + ρe)

+∇x ·(

12ρ |u|

2 + ρe)u + pu

= 0.

Milana Pavic


Production terms obtained by

extended thermodynamics

Ruggeri, Simic, 2007

Ni = −`−1∑j=1

αij(w)

(ujTj− u`

T`− u

(1

Tj− 1

T`

)),

Ei = u · Ni −`−1∑j=1

βij(w)

(− 1

Tj+

1

T`

).

kinetic theory

Ni =∑j=1j 6=i

∫∫RN×R+

mi v Qbij (fEi

, fEj)(v , I )ϕi (I )dv dI ,

Ei =∑j=1j 6=i

∫∫RN×R+

(mi2 |v |

2 + I)Qb

ij (fEi, fEj

)(v , I )ϕi (I )dv dI .

Milana Pavic


Calculation of production terms

Main steps:

choice of the cross section

Bij(v , v∗, I , I∗, r ,R, ω) = 2N−1KRsij |v − v∗|2sij∣∣∣∣ω · v − v∗|v − v∗|

∣∣∣∣N−2

passage to the center of mass reference frame

passage to the σ−notation:

ω 7→ z − 2(ω · z)ω =: σ, for z ∈ R3

Milana Pavic


Ni = −∑j=1j 6=i

(ui − uj)Kµij ni njk2 Ti Tj

ζ0i (Ti )ζ0j (Tj)

2∣∣SN−1

∣∣ ∣∣SN−2∣∣

(N + 2sij) (N + 2sij + 2)

×(

2 k Ti

mi+

2 k Tj

mj

)sij

π1−N

2 e−(

2 k Timi

+2 k Tjmj

)−1

|ui−uj |2Γ

[N − 1

2

]Γ

[sij +

N

2+ 1

]× 1F1

(sij +

N

2+ 1;

N

2+ 1;

(2 k Ti

mi+

2 k Tj

mj

)−1

|ui − uj |2),

Ei =∑j=1j 6=i

(mi

2 k Ti+

mj

2 k Tj

)−1( mi

2 k Tiui +

mj

2 k Tjuj

)· Nij

+∑j=1j 6=i

Kninjk2 Ti Tj

ζ0i (Ti )ζ0j (Tj)

2∣∣SN−1

∣∣ ∣∣SN−2∣∣

(N + 2sij + 2)×(

2 k Ti

mi+

2 k Tj

mj

)sij

π1−N

2 e−(

2 k Timi

+2 k Tjmj

)−1

|ui−uj |2Γ

[N − 1

2

]Γ

[N + 2sij

2

]

×(

mj k Ti + mi k Tj

mi + mj

)(µij

k Tj − k Ti

mikTj + mjkTi+

mi −mj

(mi + mj)

1

(N + 2sij + 4)

)× 1F1

(N + 2sij + 2

2;N

2;

(2 k Ti

mi+

2 k Tj

mj

)−1

|ui − uj |2)

+

(mj −mi

2 (mi + mj)

1

(N + 2sij + 4)(k Ti + k Tj) +

1

2 (N + 2sij)(k Tj − k Ti )

)× 1F1

(N + 2sij

2;N

2;

(2 k Ti

mi+

2 k Tj

mj

)−1

|ui − uj |2)

.

Bisi, Martalo, Spiga, 2011, 2012

Milana Pavic


Comparison

not possible in the general, because of a completely differentstructure


Ni = −`−1∑j=1

αij(w)

(ujTj− u`

T`− u

(1

Tj− 1

T`

)),

Ei = u · Ni −`−1∑j=1

βij(w)

(− 1

Tj+

1

T`

).

kinetic theory

Ni =∑j=1j 6=i

∫∫RN×R+

mi v Qbij (fEi

, fEj)(v , I )ϕi (I ) dv dI ,

Ei =∑j=1j 6=i

∫∫RN×R+

(mi2 |v |

2 + I)Qb

ij (fEi, fEj

)(v , I )ϕi (I ) dv dI .

Milana Pavic


Comparison

not possible in the general, because of a completely differentstructure


Ni = −s−1∑j=1j 6=i

αij(w)

(ujTj− ui

Ti− u

(1

Tj− 1

Ti

))+

s−1∑j=1

αij(w)

( usTs− ui

Ti− u

(1

Ts− 1

Ti

)),

Ei = u · Ni −s−1∑j=1j 6=i

βij(w)

(− 1

Tj+

1

Ti

)+

s−1∑j=1

βij(w)

(− 1

Ts+

1

Ti

).

kinetic theory

Ni =∑j=1j 6=i

∫∫RN×R+

mi v Qbij (fEi

, fEj)(v , I )ϕi (I ) dv dI ,

Ei =∑j=1j 6=i

∫∫RN×R+

(mi2 |v |

2 + I)Qb

ij (fEi, fEj

)(v , I )ϕi (I ) dv dI .

Milana Pavic


Comparison

the relation is established in the limit

ui → u, Ti → T , ∀i1 ≤ i , j ≤ s − 1 and i 6= j

αij (w0) = −Kµ

1−sijij ni nj

2sij 64π1/2

3(2sij + 3)(2sij + 5)

Γ[sij + 52

]

Γ[αi + 1]Γ[αj + 1](kT )−(αi +αj )+sij T ,

βij (w0) = −Kµ

−sijij ni nj

2sij 16π1/2

(2sij + 5)

Γ[sij + 32

]

Γ[αi + 1]Γ[αj + 1](kT )−(αi +αj )+sij kT 2

×(mi − mj )

2(2sij + 3) + (mi + mj )2(2sij + 7) + 2(2sij + 3)(2sij + 7)mimj

(mi + mj )2(2sij + 3)(2sij + 7)

,

i = j

αii (w0) = Kµ

1−sisis ni ns

2sis 64π1/2

3(2sis + 3)(2sis + 5)

Γ[sis + 52

]

Γ[αi + 1]Γ[αs + 1](kT )−(αi +αs )+sis T

+

s−1∑j=1j 6=i

Kµ1−sijij ni nj

2sij 64π

3(2sij + 3)(2sij + 5)

Γ[sij + 2]

Γ[αi + 1]Γ[αj + 1](kT )−(αi +αj )+sij T ,

βii (w0) = Kµ

−sisis ni ns

2sis 16π1/2

(2sis + 5)

Γ[sis + 32

]

Γ[αi + 1]Γ[αs + 1](kT )−(αi +αs )+sis kT 2

×(mi − ms )2(2sis + 3) + (mi + ms )2(2sis + 7) + 2(2sis + 3)(2sis + 7)mims

(mi + ms )2(2sis + 3)(2sis + 7)

+

s−1∑j=1j 6=i

Kµ−sijij ni nj

2sij 16π1/2

(2sij + 5)

Γ[sij + 32

]

Γ[αi + 1]Γ[αj + 1](kT )−(αi +αj )+sij kT 2

×(mi − mj )

2(2sij + 3) + (mi + mj )2(2sij + 7) + 2(2sij + 3)(2sij + 7)mimj

(mi + mj )2(2sij + 3)(2sij + 7)

,

Milana Pavic


In summary...

Mixture of monatomic gases- existence of a first order perturbation of Hilbertexpansion in diffusion scaling of the BE,i.e. solution g of the equation

(K − ν Id) g =(M

1/2i (v · ∇xni )

)1≤i≤`

Polyatomic gas- 14 moments system via MEP(determination of 14 moments non-equilibriumdistribution function)- one parameter in production terms that can be fitted

Mixture of polyatomic gases- calculation of production terms for Eulerian fluids- comparison with the results of ET

Milana Pavic


Thank you for your attention!

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