Recent advances in kinetic theory for mixtures of polyatomic gases · M. Bisi, Parma Kinetic theory...

Recent advances in kinetic theoryfor mixtures of polyatomic gases

Marzia Bisi

Parma University, Italy

Conference “Problems on Kinetic Theory and PDE’s”

Novi Sad (Serbia), September 25–27, 2014

M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 1 / 25

Summary

Kinetic Boltzmann model for (inert or reactive) polyatomic gasmixtures


Summary


BGK relaxation model (joint work with M.J. Caceres)


Summary



Hydrodynamic limit leading to incompressible Navier–Stokesequations (based on a joint work with L. Desvillettes)


Summary



Hydrodynamic limit leading to incompressible Navier–Stokesequations (based on a joint work with L. Desvillettes)

Work in progress and open problems


Kinetic Boltzmann approach to polyatomic gases

Motivation: It is well known that gas mixtures involved in physicalapplications are usually composed also of polyatomic species (forinstance in simple dissociation and recombination processes, or inthe evolution of powders in the atmosphere)




∗ In kinetic approaches, each gas is endowed with a suitableinternal energy variable to mimic non–translational degrees offreedom

Groppi, Spiga, J. Math. Chem. (1999): discrete internal energy levels

Desvillettes, Monaco, Salvarani, Europ. J. Mech. B/Fluids (2005):continuous internal energy variable







Our physical frame:

Mixture of M polyatomic gases Gs , s = 1, . . . ,M







Our physical frame:

Mixture of M polyatomic gases Gs , s = 1, . . . ,M

Each gas Gs is considered as a mixture of Q monatomiccomponents A i , i = s, s + M, s + 2M, s + M(Q − 1),each one characterized by a different internal energy E i


In the frame of each species, energies are monotonicallyincreasing with their index:

∀ i, j ≡ s , i < j ⇒ E i < E j



∀ i, j ≡ s , i < j ⇒ E i < E j

Besides classical elastic scattering, particles may undergoalso inelastic transitions in which they change their internalenergy level

A i + A j⇋ Ah + Ak h ≡ i, k ≡ j



∀ i, j ≡ s , i < j ⇒ E i < E j


A i + A j⇋ Ah + Ak h ≡ i, k ≡ j

Boltzmann equations for distribution functions of single components

∂f i

∂t+ v · ∇xf i =

∑

(j, h, k )∈Di

"K ijhk

i [f ](v,w, n′)dwdn′ 1≤ i≤QM

K ijhki [f ] = Θ

(

g2 −2∆Ehk

ij

µij

)

gσhkij (g, n · n′)

[

fh(

vhkij

)

fk(

whkij

)

− f i(v)f j(w)]



∀ i, j ≡ s , i < j ⇒ E i < E j


A i + A j⇋ Ah + Ak h ≡ i, k ≡ j

Boltzmann equations for distribution functions of single components

∂f i

∂t+ v · ∇xf i =

∑

(j, h, k )∈Di

"K ijhk

i [f ](v,w, n′)dwdn′ 1≤ i≤QM

K ijhki [f ] = Θ

(

g2 −2∆Ehk

ij

µij

)


[

fh(

vhkij

)

fk(

whkij

)

− f i(v)f j(w)]

Here the set Di includes all possible collisions,vhk

ij , whkij are post–collision velocities, g = v −w = g n,

Θ is the Heaviside function providing a threshold for all endothermicinteractions in which ∆Ehk

ij = Eh + Ek − E i − E j > 0


Collision invariants:number density of each gas Ns =

∑

i≡s

ni , s = 1, . . . ,M,

global velocity u, global energy32

nKT +QM∑

i=1

E ini



∑

i≡s

ni , s = 1, . . . ,M,


nKT +QM∑

i=1

E ini

Collision equilibria of the Boltzmann equations:

f iM(v) = ni

(

ms

2πKT

)3/2

exp

[

− ms

2KT|v − u|2

]

∀i ≡ s, ∀s = 1, . . . ,M,



∑

i≡s

ni , s = 1, . . . ,M,


nKT +QM∑

i=1

E ini


f iM(v) = ni

(

ms

2πKT

)3/2

exp

[

− ms

2KT|v − u|2

]

∀i ≡ s, ∀s = 1, . . . ,M,

with equilibrium number densities related by the constraints

ni = Ns ψ(E i ,T) i ≡ swhere

ψ(E i ,T) =exp

(

−E i−Es

KT

)

∑

i≡s exp(

−E i−Es

KT

) =exp

(

−E i−Es

KT

)

Zs(T)



∑

i≡s

ni , s = 1, . . . ,M,


nKT +QM∑

i=1

E ini


f iM(v) = ni

(

ms

2πKT

)3/2

exp

[

− ms

2KT|v − u|2

]

∀i ≡ s, ∀s = 1, . . . ,M,

with equilibrium number densities related by the constraints

ni = Ns ψ(E i ,T) i ≡ swhere

ψ(E i ,T) =exp

(

−E i−Es

KT

)

∑

i≡s exp(

−E i−Es

KT

) =exp

(

−E i−Es

KT

)

Zs(T)

Remark: ψ(E i ,T) represents the fraction of particles Gs (s ≡ i) belonging to the

component A i in any equilibrium configuration; for any i, j with i ≡ j and i < j, we

have ψ(E i ,T) > ψ(E j , T)


Generalization to chemically reactive frames (4 species)

Besides elastic scattering and inelastic transitions (with transfer ofinternal energy), particles may undergo the binary and reversiblechemical reaction G1 + G2 ⇋ G3 + G4, hence, for singlecomponents,

A i + A j⇋ Ah + Ak i ≡ 1, j ≡ 2, h ≡ 3, k ≡ 4




A i + A j⇋ Ah + Ak i ≡ 1, j ≡ 2, h ≡ 3, k ≡ 4

Basic properties:

For reactive encounters

K ijhki [f ] = Θ

(

g2 −2∆Ehk

ij

µij

)


[(

µij

µhk

)3fh

(

vhkij

)

fk(

whkij

)

− f i(v)f j(w)]




A i + A j⇋ Ah + Ak i ≡ 1, j ≡ 2, h ≡ 3, k ≡ 4

Basic properties:


K ijhki [f ] = Θ

(

g2 −2∆Ehk

ij

µij

)


[(

µij

µhk

)3fh

(

vhkij

)

fk(

whkij

)

− f i(v)f j(w)]

Collision invariants: N1 + N3, N1 + N4, N2 + N4,global momentum, and total energy




A i + A j⇋ Ah + Ak i ≡ 1, j ≡ 2, h ≡ 3, k ≡ 4

Basic properties:


K ijhki [f ] = Θ

(

g2 −2∆Ehk

ij

µij

)


[(

µij

µhk

)3fh

(

vhkij

)

fk(

whkij

)

− f i(v)f j(w)]

Collision invariants: N1 + N3, N1 + N4, N2 + N4,global momentum, and total energy

Collision equilibria: Maxwellian distributions f i = f iM(ni ,u,T)

with ni = Ns ψ(E i ,T) plus the mass action law of chemistry

N1N2

N3N4=

(

µ12

µ34

)3/2 Z1(T)Z2(T)

Z3(T)Z4(T)e

∆E3412

KT ∆E3412 = E3 + E4 − E1 − E2 > 0


BGK approximation for polyatomic INERT mixtures

∂f i

∂t+ v · ∇xf i = νi(Mi − f i) i = 1, . . . ,QM



∂f i

∂t+ v · ∇xf i = νi(Mi − f i) i = 1, . . . ,QM

where attractorsMi =Mi(ni , u, T) take the form

Mi(v) = ni(

mi

2πKT

)3/2

exp

[

− mi

2KT|v − u|2

]



∂f i

∂t+ v · ∇xf i = νi(Mi − f i) i = 1, . . . ,QM


Mi(v) = ni(

mi

2πKT

)3/2

exp

[

− mi

2KT|v − u|2

]

with densities ni bound together as

ni = Ns ψ(E i , T)



∂f i

∂t+ v · ∇xf i = νi(Mi − f i) i = 1, . . . ,QM


Mi(v) = ni(

mi

2πKT

)3/2

exp

[

− mi

2KT|v − u|2

]


ni = Ns ψ(E i , T)

Key points:Only one relaxation operator for each component[Andries, Aoki, Perthame, J. Stat. Phys. (2002)]



∂f i

∂t+ v · ∇xf i = νi(Mi − f i) i = 1, . . . ,QM


Mi(v) = ni(

mi

2πKT

)3/2

exp

[

− mi

2KT|v − u|2

]


ni = Ns ψ(E i , T)

Key points:Only one relaxation operator for each component[Andries, Aoki, Perthame, J. Stat. Phys. (2002)]AttractorsMi fulfill the equilibrium conditions[Bisi, Groppi, Spiga, Proceedings RGD26 (2009)]



∂f i

∂t+ v · ∇xf i = νi(Mi − f i) i = 1, . . . ,QM


Mi(v) = ni(

mi

2πKT

)3/2

exp

[

− mi

2KT|v − u|2

]


ni = Ns ψ(E i , T)

Key points:Only one relaxation operator for each component[Andries, Aoki, Perthame, J. Stat. Phys. (2002)]AttractorsMi fulfill the equilibrium conditions[Bisi, Groppi, Spiga, Proceedings RGD26 (2009)]Ns , u, T are M + 4 independent free parameters to beproperly determined as functions of the actual macroscopicfields ni, ui , T i


Strategy: we impose that the BGK model preserves the correctcollision invariants



∑

i≡s

νi∫

(Mi − f i)dv = 0 s = 1, . . . ,M (a)

M∑

s=1

∑

i≡s

νi∫

msv(Mi − f i)dv = 0 (b)

M∑

s=1

∑

i≡s

νi∫ (

12

ms |v|2 + E i)

(Mi − f i)dv = 0 (c)



∑

i≡s

νi∫

(Mi − f i)dv = 0 s = 1, . . . ,M (a)

M∑

s=1

∑

i≡s

νi∫


M∑

s=1

∑

i≡s

νi∫ (

12

ms |v|2 + E i)

(Mi − f i)dv = 0 (c)

• For any s = 1, . . . ,M, condition (a) provides∑

i≡s

νini =∑

i≡s

νini

from which, bearing in mind the constraint ni = Ns ψ(E i , T) we get

Ns =

∑

i≡s

νini

/

∑

i≡s

νiψ(E i , T)



∑

i≡s

νi∫

(Mi − f i)dv = 0 s = 1, . . . ,M (a)

M∑

s=1

∑

i≡s

νi∫


M∑

s=1

∑

i≡s

νi∫ (

12

ms |v|2 + E i)

(Mi − f i)dv = 0 (c)

• For any s = 1, . . . ,M, condition (a) provides∑

i≡s

νini =∑

i≡s

νini

from which, bearing in mind the constraint ni = Ns ψ(E i , T) we get

Ns =

∑

i≡s

νini

/

∑

i≡s

νiψ(E i , T)

• Momentum conservation (b) yields u =

M∑

s=1

ms∑

i≡s

νiniui

/

M∑

s=1

ms∑

i≡s

νini


• Energy conservation (c) provides

M∑

s=1

∑

i≡s

νi[

12

ms ni |u|2 +32

niKT + E ini − 12

msni |ui |2 − 32

niKT i − E ini]

= 0


• Energy conservation (c) provides

M∑

s=1

∑

i≡s

νi[

12

ms ni |u|2 +32

niKT + E ini − 12

msni |ui |2 − 32

niKT i − E ini]

= 0

that, recalling the explicit expressions for ni, u, may be written as atranscendental equation for T :

F(T) = Λ where F(T) =M∑

s=1

∑

j≡s

νjnj

32

KT +

∑

i≡s νiE iψ(E i , T)

∑

j≡s νjψ(E j , T)

and Λ is a known explicit function of the actual macroscopic fieldsthat turns out to fulfill Λ ≥ ∑M

s=1

(

∑

i≡s νini

)

Es


Lemma: For any Λ, the equation F(T) = Λ has a unique positivesolution



Steps of the proof:by direct computations, we check that F(T) is a monotonicallyincreasing function of its argument;



Steps of the proof:by direct computations, we check that F(T) is a monotonicallyincreasing function of its argument;recalling that, for each gas Gs , energy levels are such thatEs < Es+M < Es+2M < · · · < Es+QM, we get

∑


∑


=νsEs +

∑

i≡si,sνiE i exp

(

− E i−Es

KT

)

νs +∑

i≡si,sνi exp

(

− E i−Es

KT

) ≥ mini≡s

E i = Es

therefore

limT→0

F(T) =M∑

s=1

∑

i≡s

νini

Es , limT→+∞

F(T) = +∞



Steps of the proof:by direct computations, we check that F(T) is a monotonicallyincreasing function of its argument;recalling that, for each gas Gs , energy levels are such thatEs < Es+M < Es+2M < · · · < Es+QM, we get

∑


∑


=νsEs +

∑

i≡si,sνiE i exp

(

− E i−Es

KT

)

νs +∑

i≡si,sνi exp

(

− E i−Es

KT

) ≥ mini≡s

E i = Es

therefore

limT→0

F(T) =M∑

s=1

∑

i≡s

νini

Es , limT→+∞

F(T) = +∞

⇒ Since for T > 0 the function F(T) monotonically increasesfrom the minimum admissible value for Λ to +∞, existence anduniqueness of solution to F(T) = Λ is guaranteed

•M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 10 / 25

Remarks and basic properties

We have thus proved that the proposed BGK model is welldefined, in the sense that all auxiliary parameters are uniquelydetermined in terms of the actual fields




Collision equilibria:

f i(v) =Mi(v) ∀v ∈ R3 ⇒ ni = ni , ui = u, T i = T

hence also the actual number densities fulfill the constraintsni = Ns ψ(E i ,T) (i ≡ s)




Collision equilibria:

f i(v) =Mi(v) ∀v ∈ R3 ⇒ ni = ni , ui = u, T i = T

hence also the actual number densities fulfill the constraintsni = Ns ψ(E i ,T) (i ≡ s)

It may be proved that the usual H–functional

H =M∑

s=1

∑

i≡s

∫

f i log f i dv

is a Lyapunov functional even for the BGK kinetic model


BGK approximation for polyatomic REACTING mixtures(4 species)

∂f i

∂t+ v · ∇xf i = νi(Mi − f i) i = 1, . . . , 4Q

with parameters of the Maxwellian attractorsMi =Mi(ni , u, T)bound together as

ni = Ns ψ(E i , T) ,N1N2

N3N4=

(

µ12

µ34

)3/2 Z1(T)Z2(T)

Z3(T)Z4(T)e

∆E3412

KT (a)



∂f i

∂t+ v · ∇xf i = νi(Mi − f i) i = 1, . . . , 4Q



N3N4=

(

µ12

µ34

)3/2 Z1(T)Z2(T)

Z3(T)Z4(T)e

∆E3412

KT (a)

⇒ 7 independent free parameters (u, T , three among Ns) to bedetermined imposing the preservation of correct collision invariants



∂f i

∂t+ v · ∇xf i = νi(Mi − f i) i = 1, . . . , 4Q



N3N4=

(

µ12

µ34

)3/2 Z1(T)Z2(T)

Z3(T)Z4(T)e

∆E3412

KT (a)

⇒ 7 independent free parameters (u, T , three among Ns) to bedetermined imposing the preservation of correct collision invariants

Main difference with respect to the inert mixture:the constraint coming from energy conservation and mass actionlaw (a) are two transcendental equations for the unknowns (N1, T)


Incompressible hydrodynamic limit of the Boltzmannequations (four reacting species)

ε ∂t fiε + v · ∇xf i

ε =1ε

4Q∑

j=1

Q ij(f iε, f

jε) + ε Ji i = 1, . . . , 4Q

where Q ij(f iε, f

jε) is the classical elastic operator and Ji takes into

account all non–conservative collisions (inelastic transitions andchemical reactions)


Incompressible hydrodynamic limit of the Boltzmannequations (four reacting species)

ε ∂t fiε + v · ∇xf i

ε =1ε

4Q∑

j=1

Q ij(f iε, f

jε) + ε Ji i = 1, . . . , 4Q

where Q ij(f iε, f

jε) is the classical elastic operator and Ji takes into

account all non–conservative collisions (inelastic transitions andchemical reactions)

As in [Bardos, Golse, Levermore, J. Stat. Phys. (1991)] and in[Bisi, Desvillettes, ESAIM - Math. Model. Numer. Anal. (2014)], welook for solutions in the form

f iε = ρi Mi

(1,0,1)(1 + ε giε)

(perturbations of collision equilibria)


Here Mi(1,0,1)

are absolute normalized Maxwellians

Mi(v) =

(

mi

2π

)3/2

e−mi2 v2

and ρi > 0 are constants (without loss of generality ρ =∑4Q

i=1 = 1)such that ρi Mi

(1,0,1)are equilibria even of the non–conservative

operators Ji


Here Mi(1,0,1)

are absolute normalized Maxwellians

Mi(v) =

(

mi

2π

)3/2

e−mi2 v2

and ρi > 0 are constants (without loss of generality ρ =∑4Q

i=1 = 1)such that ρi Mi

(1,0,1)are equilibria even of the non–conservative

operators Ji

In other words, if we denote

Ns =∑

i≡s

ρi , Zs =∑

i≡s

e−(Ei−Es), s = 1, . . . , 4 ,

for any i ≡ s the constant ρi has to be related to Ns and Zs as

ρi =Ns

Zs e−(Ei−Es)

and global number densities have to fulfill the mass action law

N1N2

Z1Z2=

(

µ12

µ34

)3/2

e∆E3412

N3N4

Z3Z4


Incompressible Navier–Stokes equations

By inserting the ansatz f iε = ρi Mi

(1,0,1)(1 + εgi

ε) into the scaledBoltzmann equations we get

4Q∑

j=1

ρiρj[

Q ij(giεM

i ,Mj) + Q ij(Mi , gjεM

j)]

= O(ε)




(1,0,1)(1 + εgi


4Q∑

j=1

ρiρj[

Q ij(giεM


j)]

= O(ε)

hencegiε(v) = αi + mi v · u +

(

12

mi v2 − 32

)

T + O(ε)




(1,0,1)(1 + εgi


4Q∑

j=1

ρiρj[

Q ij(giεM


j)]

= O(ε)


(

12

mi v2 − 32

)

T + O(ε)

The parameters αi, u, T (depending on t and x) are perturbationsof number densities, velocity and temperature∫

f iε(v) dv = ρi(1 + ε αi) + O(ε2)

∫

v f iε(v) dv = ε ρi u + O(ε2)

mi∫

v2f iε(v) dv = 3 ρi + ε 3 ρi(αi + T) + O(ε2)




(1,0,1)(1 + εgi


4Q∑

j=1

ρiρj[

Q ij(giεM


j)]

= O(ε)


(

12

mi v2 − 32

)

T + O(ε)

The parameters αi, u, T (depending on t and x) are perturbationsof number densities, velocity and temperature∫

f iε(v) dv = ρi(1 + ε αi) + O(ε2)

∫

v f iε(v) dv = ε ρi u + O(ε2)

mi∫

v2f iε(v) dv = 3 ρi + ε 3 ρi(αi + T) + O(ε2)

We look for evolution equations for αi (i = 1, . . . , 4Q), u, T


We consider interactions of Maxwell molecules type: gσij(g, χ) = ϑij(χ)


We consider interactions of Maxwell molecules type: gσij(g, χ) = ϑij(χ)and define

κij = 2π∫ π

0ϑij(χ)(1−cos χ) sin χ dχ νij = 2π

∫ π

0ϑij(χ)(1−cos2 χ) sin χ dχ



κij = 2π∫ π


∫ π


We formally get that parameters αi (i = 1, . . . , 4Q), u, T satisfythe following Navier–Stokes system for polyatomic mixtures



κij = 2π∫ π


∫ π



Incompressibility condition:

∇x · u = 0



κij = 2π∫ π


∫ π



Incompressibility condition:

∇x · u = 0

Boussinesq identity:

∇x

4Q∑

i=1

(

ρi αi)

+ T

= 0

[Such constraints follow from conservation of total number densityand of global momentum, respectively]


Convection-diffusion equations for the densities of thecomponents (main difference with respect to the singlespecies frame):

∂t

[

∑

j,i

ρjµij κij(αi − αj)

]

+ u · ∇x

[

∑

j,i

ρjµijκij(αi − αj)

]

= ∆x

[

∑

j,i

ρj(αi − αj)

]

+

(

∑

j,i

ρj

ρiµij κij

) ∫

Ji(1) dv −

∑

j,i

µij κij∫

Jj(1)

dv i = 1, . . . , 4Q − 1 ,

where µij = mi mj

mi+mj is the reduced mass and Ji(1)

is the O(ε) part of

the operator Ji (contributions will be made explicit for Maxwellmolecules)


Convection-diffusion equations for the densities of thecomponents (main difference with respect to the singlespecies frame):

∂t

[

∑

j,i

ρjµij κij(αi − αj)

]

+ u · ∇x

[

∑

j,i

ρjµijκij(αi − αj)

]

= ∆x

[

∑

j,i

ρj(αi − αj)

]

+

(

∑

j,i

ρj

ρiµij κij

) ∫

Ji(1) dv −

∑

j,i

µij κij∫

Jj(1)

dv i = 1, . . . , 4Q − 1 ,

where µij = mi mj

mi+mj is the reduced mass and Ji(1)

is the O(ε) part of

the operator Ji (contributions will be made explicit for Maxwellmolecules)

More precisely, if D iin and D i

ch denote the sets of all inelastictransitions and chemical reactions involving particles A i , we have

Ji(1)

=∑

(j,h,k )∈D iin∪D i

ch

{

ρhρk[

Jiijhk+(gh

εMh ,Mk ) + Jiijhk+(Mh , gk

εMk )]

− ρiρj[

Jiijhk−(g

iεM

i ,Mj) + Jiijhk−(M

i , gjεM

j)]}


Convection-diffusion equation for the momentum

∂tu + u · ∇xu + ∇xp = d1 ∆xu



∂tu + u · ∇xu + ∇xp = d1 ∆xu

Convection-diffusion equation for the temperature

∂tT + u · ∇xT = d2 ∆xT +4Q∑

i=1

∫ (

15

miv2 − 1

)

Ji(1) dv

where diffusion coefficients d1, d2 are the unique solutions ofsuitable linear systems and depend on masses and on averagedcollision frequencies κij , νij



∂tu + u · ∇xu + ∇xp = d1 ∆xu


∂tT + u · ∇xT = d2 ∆xT +4Q∑

i=1

∫ (

15

miv2 − 1

)

Ji(1) dv


Remarks:The final system is not strongly coupled, evolution equationfor u could be solved separately



∂tu + u · ∇xu + ∇xp = d1 ∆xu


∂tT + u · ∇xT = d2 ∆xT +4Q∑

i=1

∫ (

15

miv2 − 1

)

Ji(1) dv


Remarks:The final system is not strongly coupled, evolution equationfor u could be solved separately

Velocities or temperatures specific to each species wouldappear only if we considered higher orders in expansions ofdistributions, or if we took as dominant operator (of order 1/ε)only Q ii(f i

ε, fiε)


In the one-species case concentration α is completely knownfrom the equation for T , while for a mixture 4Q − 1 additionalindependent evolution equations for αi are needed



For a mixture of only two monatomic species, the additionalequation is simply provided by the difference of the two kineticequations




Computation of diffusion coefficients d1 and d2 and the proofthat they are strictly positive is not a trivial extension of theone-species case




Computation of diffusion coefficients d1 and d2 and the proofthat they are strictly positive is not a trivial extension of theone-species case

Contributions due to the polyatomic nature of gases (and tochemical reactions) affect equations for number densities andglobal temperature


Some steps of the derivation

Equations for concentrations (i = 1, . . . , 4Q)

ε ∂t

∫

(giεM

i)(v) dv + ∇x ·∫

v (giεM

i)(v) dv = ε1

ρi

∫

Ji(1) dv

We need a closure for the streaming terms to O(ε) accuracy




ε ∂t

∫

(giεM

i)(v) dv + ∇x ·∫

v (giεM

i)(v) dv = ε1

ρi

∫

Ji(1) dv


We resort to momentum equations of each component

ε2 ρi∂t

∫

v(giεM

i)(v) dv + ε ρi∇x ·∫

v ⊗ v(giεM

i)(v) dv =

=∑

j,i

{

ρiρj∫

v[

Q ij(giεM


j)]

dv

+ ε ρiρj∫

v Q ij(giεM

i , gjεM

j)dv}

+ ε2∫

v Ji(1) dv




ε ∂t

∫

(giεM

i)(v) dv + ∇x ·∫

v (giεM

i)(v) dv = ε1

ρi

∫

Ji(1) dv


We resort to momentum equations of each component

ε2 ρi∂t

∫

v(giεM

i)(v) dv + ε ρi∇x ·∫

v ⊗ v(giεM

i)(v) dv =

=∑

j,i

{

ρiρj∫

v[

Q ij(giεM


j)]

dv

+ ε ρiρj∫

v Q ij(giεM

i , gjεM

j)dv}

+ ε2∫

v Ji(1) dv

We find an explicit relation between the terms in red


∑

j,i

ρiρj∫

v[

Q ij(giεM


j)]

dv

= −

∑

j,i

ρjµij κij

ρi

mi

∫

v (giεM

i)(v) dv +ρi

mi

∑

j,i

ρjµij κij∫

v (gjεM

j)(v) dv

hence we can “insert” the i–th momentum equation into a suitablelinear combination of number densities equations, and we evaluatethen each term recalling thatgiε(v) = αi + mi v · u +

(

12 mi v2 − 3

2

)

T + O(ε)


∑

j,i

ρiρj∫

v[

Q ij(giεM


j)]

dv

= −

∑

j,i

ρjµij κij

ρi

mi

∫

v (giεM

i)(v) dv +ρi

mi

∑

j,i

ρjµij κij∫

v (gjεM

j)(v) dv

hence we can “insert” the i–th momentum equation into a suitablelinear combination of number densities equations, and we evaluatethen each term recalling thatgiε(v) = αi + mi v · u +

(

12 mi v2 − 3

2

)

T + O(ε)

∗ Analogously for closure of momentum and temperature equations:

integrals of streaming terms ∇x ·

4Q∑

i=1

ρi∫

Bin(v)(gi

εMi)(v) dv

with Bi1(v) = mi

(

v ⊗ v − 13

v2I)

, Bi2(v) =

(

12

miv2 − 52

)

v

are proportional to4Q∑

i,j=1

ρiρj θin

∫

Bin(v)

[

Q ij(giεM

i ,Mj)+Q ij(Mi , gjεM

j)]

dv (n = 1, 2)


Inelastic collision contributions∫

Ji(1)dv =

∑


ch

K iijhk

where K iijhk represents the net production of particles of species A i

due to the interaction A i + A j ⇄ Ah + Ak



Ji(1)dv =

∑


ch

K iijhk



Obvious symmetry property: K iijhk = K j

ijhk = −Khijhk = −Kk

ijhk



Ji(1)dv =

∑


ch

K iijhk





ijhk

We adopt a Maxwell molecule assumption for any option (i, j, h, k)corresponding to an endothermic direct interaction (i.e. ∆Ehk

ij > 0):

νhkij =

∫

gσhkij (g, χ) dn′



Ji(1)dv =

∑


ch

K iijhk





ijhk


ij > 0):

νhkij =

∫


The relation for the cross section of the reverse exothermicinteraction σij

hk follows from the microreversibility condition



Ji(1)dv =

∑


ch

K iijhk





ijhk


ij > 0):

νhkij =

∫


The relation for the cross section of the reverse exothermicinteraction σij

hk follows from the microreversibility condition

⇒∫

Ji(1)dv =

∑

(j,h,k )∈D iEn

K iijhk −

∑

(j,h,k )∈D iEx

Khhkij


Recalling that giε(v) = αi + mi v · u +

(

12 mi v2 − 3

2

)

T + O(ε) and

the assumptions on the leading order number densities ρi we get∫

Ji(1)dv =

2√π

∑


ch

Λhkij

[

αh + αk − αi − αj − T∆Ehkij

]

Γ

(

32,∣

∣

∣

∣

∆Ehkij

∣

∣

∣

∣

)

where

Λhkij =

νhkij ρ

iρj if (j, h, k) ∈ D iEn

νijhkρ

hρk if (j, h, k) ∈ D iEx



(

12 mi v2 − 3

2

)

T + O(ε) and


Ji(1)dv =

2√π

∑


ch

Λhkij

[


]

Γ

(

32,∣

∣

∣

∣

∆Ehkij

∣

∣

∣

∣

)

where

Λhkij =

νhkij ρ


νijhkρ


The content of the square brackets is the linearization (i.e.,the O(ε) terms) of the mass action law for global distributionfunctions (f i

ε, fjε, fh

ε , fkε )



(

12 mi v2 − 3

2

)

T + O(ε) and


Ji(1)dv =

2√π

∑


ch

Λhkij

[


]

Γ

(

32,∣

∣

∣

∣

∆Ehkij

∣

∣

∣

∣

)

where

Λhkij =

νhkij ρ


νijhkρ


The content of the square brackets is the linearization (i.e.,the O(ε) terms) of the mass action law for global distributionfunctions (f i

ε, fjε, fh

ε , fkε )

Suitable combinations of∫

Ji(1)dv complete the derivation of

incompressible Navier–Stokes equations for number densitiesand global temperature



Work in progress (joint with S. Brull): formal compressiblehydrodynamic limit (at Navier–Stokes level) for polyatomic gasmixtures, owing to the Chapman–Enskog method




Open problem: Fredholm alternative for the linearizedBoltzmann operator for polyatomic gases (with discrete orcontinuous internal energy)




Open problem: Fredholm alternative for the linearizedBoltzmann operator for polyatomic gases (with discrete orcontinuous internal energy)

Open problem: more appropriate descriptions for chains ofchemical reactions occurring in physical applications


Thank you for your attention


Recent advances in kinetic theory for mixtures of polyatomic gases · M. Bisi, Parma Kinetic theory...

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Transcript of Recent advances in kinetic theory for mixtures of polyatomic gases · M. Bisi, Parma Kinetic theory...