recent advances in kinetic theory for mixtures of polyatomic gases · m. bisi, parma kinetic theory...
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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
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Summary
Kinetic Boltzmann model for (inert or reactive) polyatomic gasmixtures
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Summary
Kinetic Boltzmann model for (inert or reactive) polyatomic gasmixtures
BGK relaxation model (joint work with M.J. Caceres)
M. Bisi, Parma Kinetic theory for polyatomic gas mixtures 2 / 25
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Summary
Kinetic Boltzmann model for (inert or reactive) polyatomic gasmixtures
BGK relaxation model (joint work with M.J. Caceres)
Hydrodynamic limit leading to incompressible Navier–Stokesequations (based on a joint work with L. Desvillettes)
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Summary
Kinetic Boltzmann model for (inert or reactive) polyatomic gasmixtures
BGK relaxation model (joint work with M.J. Caceres)
Hydrodynamic limit leading to incompressible Navier–Stokesequations (based on a joint work with L. Desvillettes)
Work in progress and open problems
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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)
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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
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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
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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
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
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In the frame of each species, energies are monotonicallyincreasing with their index:
∀ i, j ≡ s , i < j ⇒ E i < E j
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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
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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
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)]
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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
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)]
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
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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
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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
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,
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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
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,
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)
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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
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,
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)
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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
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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
ij
µij
)
gσhkij (g, n · n′)
[(
µij
µhk
)3fh
(
vhkij
)
fk(
whkij
)
− f i(v)f j(w)]
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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
ij
µij
)
gσhkij (g, n · n′)
[(
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
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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
ij
µij
)
gσhkij (g, n · n′)
[(
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
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BGK approximation for polyatomic INERT mixtures
∂f i
∂t+ v · ∇xf i = νi(Mi − f i) i = 1, . . . ,QM
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BGK approximation for polyatomic INERT mixtures
∂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
]
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BGK approximation for polyatomic INERT mixtures
∂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
]
with densities ni bound together as
ni = Ns ψ(E i , T)
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BGK approximation for polyatomic INERT mixtures
∂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
]
with densities ni bound together as
ni = Ns ψ(E i , T)
Key points:Only one relaxation operator for each component[Andries, Aoki, Perthame, J. Stat. Phys. (2002)]
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BGK approximation for polyatomic INERT mixtures
∂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
]
with densities ni bound together as
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)]
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BGK approximation for polyatomic INERT mixtures
∂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
]
with densities ni bound together as
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
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Strategy: we impose that the BGK model preserves the correctcollision invariants
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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)
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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)
• 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)
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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)
• 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
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• 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
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• 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
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Lemma: For any Λ, the equation F(T) = Λ has a unique positivesolution
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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;
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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;recalling that, for each gas Gs , energy levels are such thatEs < Es+M < Es+2M < · · · < Es+QM, we get
∑
i≡s νiE iψ(E i , T)
∑
j≡s νjψ(E j , T)
=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) = +∞
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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;recalling that, for each gas Gs , energy levels are such thatEs < Es+M < Es+2M < · · · < Es+QM, we get
∑
i≡s νiE iψ(E i , T)
∑
j≡s νjψ(E j , T)
=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
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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
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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)
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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)
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
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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)
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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)
⇒ 7 independent free parameters (u, T , three among Ns) to bedetermined imposing the preservation of correct collision invariants
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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)
⇒ 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)
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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)
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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)
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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
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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
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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)
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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)
hencegiε(v) = αi + mi v · u +
(
12
mi v2 − 32
)
T + O(ε)
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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)
hencegiε(v) = αi + mi v · u +
(
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)
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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)
hencegiε(v) = αi + mi v · u +
(
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
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We consider interactions of Maxwell molecules type: gσij(g, χ) = ϑij(χ)
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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
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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
We formally get that parameters αi (i = 1, . . . , 4Q), u, T satisfythe following Navier–Stokes system for polyatomic mixtures
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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
We formally get that parameters αi (i = 1, . . . , 4Q), u, T satisfythe following Navier–Stokes system for polyatomic mixtures
Incompressibility condition:
∇x · u = 0
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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
We formally get that parameters αi (i = 1, . . . , 4Q), u, T satisfythe following Navier–Stokes system for polyatomic mixtures
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]
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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)
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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)]}
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Convection-diffusion equation for the momentum
∂tu + u · ∇xu + ∇xp = d1 ∆xu
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Convection-diffusion equation for the momentum
∂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
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Convection-diffusion equation for the momentum
∂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
Remarks:The final system is not strongly coupled, evolution equationfor u could be solved separately
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Convection-diffusion equation for the momentum
∂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
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)
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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
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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
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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
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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
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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
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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
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
i ,Mj) + Q ij(Mi , gjεM
j)]
dv
+ ε ρiρj∫
v Q ij(giεM
i , gjεM
j)dv}
+ ε2∫
v Ji(1) dv
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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
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
i ,Mj) + Q ij(Mi , gjεM
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
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∑
j,i
ρiρj∫
v[
Q ij(giεM
i ,Mj) + Q ij(Mi , gjεM
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)
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∑
j,i
ρiρj∫
v[
Q ij(giεM
i ,Mj) + Q ij(Mi , gjεM
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)
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Inelastic collision contributions∫
Ji(1)dv =
∑
(j,h,k )∈D iin∪D i
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
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Inelastic collision contributions∫
Ji(1)dv =
∑
(j,h,k )∈D iin∪D i
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
Obvious symmetry property: K iijhk = K j
ijhk = −Khijhk = −Kk
ijhk
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Inelastic collision contributions∫
Ji(1)dv =
∑
(j,h,k )∈D iin∪D i
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
Obvious symmetry property: K iijhk = K j
ijhk = −Khijhk = −Kk
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′
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Inelastic collision contributions∫
Ji(1)dv =
∑
(j,h,k )∈D iin∪D i
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
Obvious symmetry property: K iijhk = K j
ijhk = −Khijhk = −Kk
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′
The relation for the cross section of the reverse exothermicinteraction σij
hk follows from the microreversibility condition
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Inelastic collision contributions∫
Ji(1)dv =
∑
(j,h,k )∈D iin∪D i
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
Obvious symmetry property: K iijhk = K j
ijhk = −Khijhk = −Kk
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′
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
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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
∑
(j,h,k )∈D iin∪D i
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
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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
∑
(j,h,k )∈D iin∪D i
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
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 )
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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
∑
(j,h,k )∈D iin∪D i
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
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
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Work in progress and open problems
Work in progress (joint with S. Brull): formal compressiblehydrodynamic limit (at Navier–Stokes level) for polyatomic gasmixtures, owing to the Chapman–Enskog method
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Work in progress and open problems
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)
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Work in progress and open problems
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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Thank you for your attention
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