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

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

gσhkij (g, n · n′)

− 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

− 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 =

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

global velocity u, global energy32

nKT +QM∑

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

nKT +QM∑

Collision equilibria of the Boltzmann equations:

f iM(v) = ni

− ms

2KT|v − u|2

∀i ≡ s, ∀s = 1, . . . ,M,

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

nKT +QM∑

f iM(v) = ni

− 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

i≡s exp(

−E i−Es

) =exp

−E i−Es

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

nKT +QM∑

f iM(v) = ni

− 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

i≡s exp(

−E i−Es

) =exp

−E i−Es

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

Basic properties:

For reactive encounters

K ijhki [f ] = Θ

g2 −2∆Ehk

− 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

− 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

− 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

)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

2KT|v − u|2

∂f i

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

Mi(v) = ni(

− 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

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

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

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

νi∫

(Mi − f i)dv = 0 s = 1, . . . ,M (a)

νi∫

msv(Mi − f i)dv = 0 (b)

νi∫ (

ms |v|2 + E i)

(Mi − f i)dv = 0 (c)

νi∫

(Mi − f i)dv = 0 s = 1, . . . ,M (a)

νi∫

νi∫ (

ms |v|2 + E i)

(Mi − f i)dv = 0 (c)

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

νini =∑

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

νiψ(E i , T)

νi∫

(Mi − f i)dv = 0 s = 1, . . . ,M (a)

νi∫

νi∫ (

ms |v|2 + E i)

(Mi − f i)dv = 0 (c)

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

νini =∑

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

νiψ(E i , T)

• Momentum conservation (b) yields u =

νiniui

• Energy conservation (c) provides

ms ni |u|2 +32

niKT + E ini − 12

msni |ui |2 − 32

niKT i − E ini]

• Energy conservation (c) provides

ms ni |u|2 +32

niKT + E ini − 12

msni |ui |2 − 32

niKT i − E ini]

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

F(T) = Λ where F(T) =M∑

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

i≡s νini

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

νs +∑

i≡si,sνi exp

− E i−Es

) ≥ mini≡s

E i = Es

therefore

limT→0

F(T) =M∑

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

νs +∑

i≡si,sνi exp

− E i−Es

) ≥ mini≡s

E i = Es

therefore

limT→0

F(T) =M∑

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∑

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

)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

)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

)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

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

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) =

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) =

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 =∑

ρi , Zs =∑

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

e∆E3412

Incompressible Navier–Stokes equations

By inserting the ansatz f iε = ρi Mi

(1,0,1)(1 + εgi

ε) into the scaledBoltzmann equations we get

ρiρj[

Q ij(giεM

i ,Mj) + Q ij(Mi , gjεM

= O(ε)

(1,0,1)(1 + εgi

ρiρj[

Q ij(giεM

= O(ε)

hencegiε(v) = αi + mi v · u +

mi v2 − 32

T + O(ε)

(1,0,1)(1 + εgi

ρiρj[

Q ij(giεM

= O(ε)

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)

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

(1,0,1)(1 + εgi

ρiρj[

Q ij(giεM

= O(ε)

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)

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:

ρi αi)

[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):

ρjµij κij(αi − αj)

+ u · ∇x

ρjµijκij(αi − αj)

= ∆x

ρj(αi − αj)

ρiµij κij

Ji(1) dv −

µij κij∫

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):

ρjµij κij(αi − αj)

+ u · ∇x

ρjµijκij(αi − αj)

= ∆x

ρj(αi − αj)

ρiµij κij

Ji(1) dv −

µij κij∫

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

(j,h,k )∈D iin∪D i

ρhρk[

Jiijhk+(gh

εMh ,Mk ) + Jiijhk+(Mh , gk

εMk )]

− ρiρj[

Jiijhk−(g

i ,Mj) + Jiijhk−(M

i , gjεM

Convection-diffusion equation for the momentum

∂tu + u · ∇xu + ∇xp = d1 ∆xu

Convection-diffusion equation for the temperature

∂tT + u · ∇xT = d2 ∆xT +4Q∑

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∑

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∑

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

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

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

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 =

ρiρj∫

Q ij(giεM

+ ε ρiρj∫

v Q ij(giεM

i , gjεM

+ ε2∫

v Ji(1) dv

ε ∂t

(giεM

i)(v) dv + ∇x ·∫

v (giεM

i)(v) dv = ε1

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 =

ρiρj∫

Q ij(giεM

+ ε ρiρj∫

v Q ij(giεM

i , gjεM

+ ε2∫

v Ji(1) dv

We find an explicit relation between the terms in red

ρiρj∫

Q ij(giεM

ρjµij κij

v (giεM

i)(v) dv +ρi

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

T + O(ε)

ρiρj∫

Q ij(giεM

ρjµij κij

v (giεM

i)(v) dv +ρi

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

T + O(ε)

∗ Analogously for closure of momentum and temperature equations:

integrals of streaming terms ∇x ·

ρi∫

Bin(v)(gi

εMi)(v) dv

with Bi1(v) = mi

v ⊗ v − 13

, Bi2(v) =

miv2 − 52

are proportional to4Q∑

ρiρj θin

Bin(v)

Q ij(giεM

i ,Mj)+Q ij(Mi , gjεM

dv (n = 1, 2)

Inelastic collision contributions∫

Ji(1)dv =

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 =

K iijhk

Obvious symmetry property: K iijhk = K j

ijhk = −Khijhk = −Kk

Ji(1)dv =

K iijhk

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 =

K iijhk

ij > 0):

νhkij =

The relation for the cross section of the reverse exothermicinteraction σij

hk follows from the microreversibility condition

Ji(1)dv =

K iijhk

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

T + O(ε) and

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

Ji(1)dv =

2√π

Λhkij

αh + αk − αi − αj − T∆Ehkij

32,∣

∆Ehkij

Λhkij =

νhkij ρ

iρj if (j, h, k) ∈ D iEn

νijhkρ

hρk if (j, h, k) ∈ D iEx

12 mi v2 − 3

T + O(ε) and

Ji(1)dv =

2√π

Λhkij

32,∣

∆Ehkij

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

T + O(ε) and

Ji(1)dv =

2√π

Λhkij

32,∣

∆Ehkij

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

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

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FLASH CARDS: Polyatomic Names

Quasi-chemical approximation for polyatomic - arXiv · Quasi-chemical approximation for polyatomic mixtures M. V. Dávila, ... E-mail: maradav@unsl.edu.ar; mpasi@unsl.edu.ar April

Discrete Velocity Models for Polyatomic Molecules Without ...The construction of normal DVMs for single species as well as for binary mixtures has been well studied, see for example

Lecture 4: Polyatomic Spectra - Princeton University · Lecture 4: Polyatomic Spectra Ammonia molecule 1. From diatomic to polyatomic 2. Classification of polyatomic molecules 3

Lewis Diagrams for Polyatomic Ions

Diatomic and polyatomic molecules

HEAT RADIATION OF BURNING HYDROGEN/AIR MIXTURES …conference.ing.unipi.it/ichs2007/fileadmin/user_upload/... · 2016. 12. 2. · Non-linear polyatomic molecules ... published in

Mass and Thermal Diffusion in Binary Polyatomic Gas Mixtures · Mass and Thermal Diffusion in Binary Polyatomic Gas Mixtures D. OMEIRI Laboratoire de Recherche sur la Physico-Chimie

Polyatomic Molecular Orbital Theory

Polyatomic Ions & Chemical Nomenclature

Naming Compounds Having Polyatomic Ions

KINETIC THEORY FOR GRANULAR MIXTURES AT MODERATE DENSITIES: SOME APPLICATIONS

Chemical kinetic modeling development and validation ... · Project Summary. Develop a chemical kinetic mechanism for Supercritical Carbon Dioxide (sCO. 2) Mixtures. Validate the

Physics of polyatomic gases in non- equilibrium based on ...acml.gnu.ac.kr/download/Conference/RGD31-Myong-180727.pdf · 2nd-order Boltzmann-type kinetic theory: Boltzmann-based gas

Ionic Compounds – Polyatomic Ions (Metal + Polyatomic Ion)

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

Common polyatomic ions

Polyatomic Molecular Orbital Theoryprushan/IC-articles/Polyatomic... · 2007-09-27 · 1 Polyatomic Molecular Orbital Theory Transformational properties of atomic orbitals Atomic

KINETIC MONTE CARLO SIMULATION OF …372183/s4214339...KINETIC MONTE CARLO SIMULATION OF MIXTURES: PHASE EQUILIBRIA SHILIANG JOHNATHAN TAN BEng (Chem) A thesis submitted for the degree

The Maxwell-Stefan di usion limit for a kinetic model of ... · The Maxwell-Stefan di usion limit for a kinetic model of mixtures Laurent Boudin, B er enice Grec, Francesco Salvarani