Discussion on kinetic and asymptotic diffusion modelsfor gaseous mixtures: Fick vs. Maxwell-Stefan

Bérénice Grec1

in collaboration with L. Boudin and V. Pavan

1MAP5 – Université Paris Descartes, France

Franco-italian meeting on kinetic theory and singular parabolic equationsUniv. Paris Diderot, March 16th, 2017

Diffusion models for mixtures: Fick and Maxwell-Stefan

I Mixture of p ≥ 2 speciesI ni : number of molecules of species iI Ni : flux of species i

I n =

n1...

np

, N =

N1...

Np

I mass conservation: ∂tn +∇ · N = 0

(Generalized) Fick’s law:

N = −F (n)∇x n

Maxwell-Stefan’s equations:

−∇x n = S(n)N

Properties of the matrices F and SI F (n) and S(n) are not invertible (rank p − 1).I Well-posedness of the Cauchy problem [Giovangigli, Bothe, Jüngel & Stelzer]I Use of Moore-Penrose pseudo-inverse: structural similarity

Bérénice Grec Asymptotic diffusion models and kinetic theory for mixtures 2/15

Fick vs. Maxwell-Stefan

Formal analogy of the two systems,but Fick’s and Maxwell-Stefan’s are not obtained in the same way!

Obtention of Fick’s lawI Thermodynamical considerations (entropy minimization)I Thermodynamics of irreversible processes approach

Obtention of Maxwell-Stefan’s equationsI Mechanical considerations on forcesI Extended thermodynamics approach

IdeaUse kinetic theory and study the diffusion asymptotics

Bérénice Grec Asymptotic diffusion models and kinetic theory for mixtures 3/15

Kinetic framework

I Distribution function fi (t, x , v), t ≥ 0, x ∈ Ω, v ∈ R3.

I Number of molecules ni =∫

fidv .

I Mixture of ideal gases without chemical reaction

I Collisions: (mi , v ′), (mj , v ′∗) 7→ (mi , v), (mj , v∗), obtained by the conservationof momentum and kinetic energy

v ′ = 1mi + mj

(miv + mjv∗ + mj |v − v∗|σ)

v ′∗ = 1mi + mj

(miv + mjv∗ −mi |v − v∗|σ)

with mi : mass of species i and σ ∈ S2.

Bérénice Grec Asymptotic diffusion models and kinetic theory for mixtures 4/15

Collision operators

I Collision operator

Qij(f , g)(v) =∫

v∗∈R3

∫σ∈S2

Bij(v , v∗, σ) [f ′g ′∗ − f g∗] dσ dv∗

I Weak form:∫Qij(f , g)ψdv = −1

2

∫∫∫Bij [f ′g ′∗ − f g∗] [ψ′ − ψ] dσ dv∗ dv

I Conservation properties: ∫R3

Qij(f , g)(v) dv = 0∫R3

Qij(f , g)(v)(

mi v12mi |v |2

)dv +

∫R3

Qji (g , f )(v)(

mj v12mj |v |2

)dv = 0

Bérénice Grec Asymptotic diffusion models and kinetic theory for mixtures 5/15

Kinetic model for mixtures

I Diffusive scaling : ε ∼ Kn ∼ Ma

∂t fi + v · ∇x fi =∑

jQij(fi , fj), ∀i

I Study of the equilibrium (H-theorem) [Desvillettes, Monaco, Salvarani]

Mi (v) =( mi2πkT

)3/2exp

(−mi |v |22kT

)I Well-posedness of the Cauchy problem [Briant, Daus]I Asymptotic analysis: introduce the linearized operator L, whose i-th

component is

[Lg ]i =M−1/2i

∑j

ninj(Qij(Mi ,M1/2

j gj) + Qji (M1/2i gi ,Mj)

).

I Compactness of the operator [Boudin, Grec, Pavić, Salvarani]I Spectral gap [Daus, Jüngel, Mouhot, Zamponi]

Bérénice Grec Asymptotic diffusion models and kinetic theory for mixtures 6/15

Kinetic model for mixtures

I Diffusive scaling : ε ∼ Kn ∼ Ma

ε∂t fi + v · ∇x fi = 1ε

∑j

Qij(fi , fj), ∀i

I Study of the equilibrium (H-theorem) [Desvillettes, Monaco, Salvarani]

Mi (v) =( mi2πkT

)3/2exp

(−mi |v |22kT

)I Well-posedness of the Cauchy problem [Briant, Daus]I Asymptotic analysis: introduce the linearized operator L, whose i-th

component is

[Lg ]i =M−1/2i

∑j

ninj(Qij(Mi ,M1/2

j gj) + Qji (M1/2i gi ,Mj)

).

I Compactness of the operator [Boudin, Grec, Pavić, Salvarani]I Spectral gap [Daus, Jüngel, Mouhot, Zamponi]

Bérénice Grec Asymptotic diffusion models and kinetic theory for mixtures 6/15

Fick’s and Maxwell-Stefan’s equations are not obtained in the same wayfrom the Boltzmann equation!

Perturbative method (Fick)I Based on Chapman-Enskog expansion and the use of L−1 [Bardos, Golse,

Levermore]

Moment method (Maxwell-Stefan)I Based on the ansatz (below) and the use of L [Levermore], [Müller, Ruggieri]

Ansatz

f εi (t, x , v) = nεi (t, x)( mi2πkT

)3/2exp

(− mi2kT |v − εu

εi (t, x)|2

)thus

∫v

f εi dv = nεi and∫

vvf εi dv = εnεi uεi .

Maxwell-Stefan’s equations can be written

−∇x ni = 1∑ni

∑j 6=i

njni (ui − uj)Dij

.

Bérénice Grec Asymptotic diffusion models and kinetic theory for mixtures 7/15

ε∂t f εi + v · ∇x f εi = 1ε

∑j

Qij(f εi , f εj ), ∀i

I Mass conservationMoment of order 0:

ε∂

∂t

(∫R3

f εi (v) dv)

+∇x ·(∫

R3v f εi (v) dv

)= 0,

where the collision operator Qij(fi , fj) vanishes by invariance.

∂tnεi +∇x · (nεi uεi ) = 0.

I Momentum equationMoment of order 1:

ε∂

∂t

∫R3

v f εi (v) dv +∫R3

v (v · ∇x f εi (v)) dv = 1ε

∑j 6=i

∫R3

v Qij(f εi , f εj )(v) dv

where the mono-species collision term vanishes by invariance.Bérénice Grec Asymptotic diffusion models and kinetic theory for mixtures 8/15

Computation of the divergence term

ε∂

∂t

∫R3

v f εi (v) dv +∫R3

v (v · ∇x f εi (v)) dv = 1ε

∑j 6=i

∫R3

v Qij(f εi , f εj )(v) dv

I Use of the Ansatz, translation in v

∂

∂x

(∫R

v2 f εi (v) dv)∝ ∂

∂x nεi∫R

(v + εuiε)2e−mi |v |2/2kT dv

Bérénice Grec Asymptotic diffusion models and kinetic theory for mixtures 9/15

Computation of the divergence term

ε∂

∂t

∫R3

v f εi (v) dv +∫R3

v (v · ∇x f εi (v)) dv = 1ε

∑j 6=i

∫R3

v Qij(f εi , f εj )(v) dv

I Use of the Ansatz, translation in v + parity argument

∂

∂x

(∫R

v2 f εi (v) dv)∝ ∂

∂x nεi∫R

(v2 + ε2(ui

ε)2)

e−mi |v |2/2kT dv

I In terms of macroscopic quantities∂

∂x

(∫R

v2 f εi (v) dv)

= kTmi

∂nεi∂x + ε2

∂

∂x

(nεi (ui

ε)2)

Bérénice Grec Asymptotic diffusion models and kinetic theory for mixtures 9/15

Galilean transformations

I Galilean invariance:

For any translation w ∈ R3, and any rotation Θ ∈ O+(R3):

Bij(v + w , v∗ + w , σ) = Bij(v , v∗, σ) = Bij(Θv ,Θv∗,Θσ), ∀v , v∗, σ

Galilean transformations for the collision rules

Let πij(v , v∗, σ) := v ′ = 1mi + mj

(miv + mjv∗ + mj |v − v∗|σ).

For any translation w ∈ R3, and any rotation Θ ∈ O+(R3):πij(v + w , v∗ + w , σ) = πij(v , v∗, σ) + wπij(Θv ,Θv∗,Θσ) = Θπij(v , v∗, σ)

The same holds for v ′∗.

Bérénice Grec Asymptotic diffusion models and kinetic theory for mixtures 10/15

Computation of the collision term

ε∂

∂t

∫R3

v f εi (v) dv +∫R3

v (v · ∇x f εi (v)) dv = 1ε

∑j 6=i

∫R3

v Qij(f εi , f εj )(v) dv

Let Γεij(εui , εuj) :=∫R3

v Qij(f εi , f εj )(v) dv

∝ nεi nεj∫∫∫v ,v∗,σ

Bij(v , v∗, σ)(

e−mi (v′−εui )2

2kT −mj (v′∗−εuj )2

2kT − e−mi (v−εui )2

2kT −mj (v∗−εuj )2

2kT

)(v ′ − v)

Find Dij such that limε→0

Γεij(εui , εuj)ε

= Dijninj(ui − uj).

I Invariance property: from v ′ = πij(v , v∗, σ) (and the same for v ′∗)

Γεij(εui + w , εuj + w) = Γεij(εui , εuj), Γεij(εΘui , εΘuj) = ΘΓεij(εui , εuj).

I LetHε

ij (u) :=Γεij(εu, 0)

ε, where u := ui − uj .

Bérénice Grec Asymptotic diffusion models and kinetic theory for mixtures 11/15

I Expand

Γεij(εu, 0) ∝ nεi nεj∫∫∫

Bij

(exp

[− mi2kT (v ′2 − 2εv ′ · u + ε2u2)

]exp

[− mj2kT v ′2∗

]− exp

[− mi2kT (v2 − 2εv · u + ε2u2)

]exp

[− mj2kT v∗2

] )(v − v ′)

by the conservation of kinetic energy.I Expansion in ε of the exponentials:

∝ nεi nεj∫∫∫

Bij exp[− mi2kT v2 − mj

2kT v∗2]

(v − v ′)

I Matrix form of the operator Lij(u) = limε→0

Γεijε , involving (v ′ − v)(v ′ − v)ᵀ

I Since Lij commutes with any rotation, Schur’s lemma ensures that Lij is anhomothecy:

Lij(ui − uj) = Dijninj(ui − uj)I Explicit form of the coefficients from kinetic quantities (+ positivity,

symmetry)Bérénice Grec Asymptotic diffusion models and kinetic theory for mixtures 12/15

I Expand

Γεij(εu, 0) ∝ nεi nεj∫∫∫

Bij

(exp

[− mi2kT (v2 − 2εv ′ · u + ε2u2)

]exp

[− mj2kT v2

∗

]− exp

[− mi2kT (v2 − 2εv · u + ε2u2)

]exp

[− mj2kT v∗2

] )(v − v ′)

by the conservation of kinetic energy.I Expansion in ε of the exponentials:

∝ nεi nεj∫∫∫

Bij exp[− mi2kT v2 − mj

2kT v∗2]

(v − v ′)

I Matrix form of the operator Lij(u) = limε→0

Γεijε , involving (v ′ − v)(v ′ − v)ᵀ

I Since Lij commutes with any rotation, Schur’s lemma ensures that Lij is anhomothecy:

Lij(ui − uj) = Dijninj(ui − uj)I Explicit form of the coefficients from kinetic quantities (+ positivity,

symmetry)Bérénice Grec Asymptotic diffusion models and kinetic theory for mixtures 12/15

I Expand

Γεij(εu, 0) ∝ nεi nεj∫∫∫

Bij

(exp

[− mi2kT (v2 − 2εv ′ · u + ε2u2)

]exp

[− mj2kT v2

∗

]− exp

[− mi2kT (v2 − 2εv · u + ε2u2)

]exp

[− mj2kT v∗2

] )(v − v ′)

by the conservation of kinetic energy.I Expansion in ε of the exponentials:

∝ nεi nεj∫∫∫

Bij exp[− mi2kT v2 − mj

2kT v∗2]

(1 + ε

mikT v ′ · u − 1− εmi

kT v · u + O(ε2))

(v − v ′)

I Matrix form of the operator Lij(u) = limε→0

Γεijε , involving (v ′ − v)(v ′ − v)ᵀ

I Since Lij commutes with any rotation, Schur’s lemma ensures that Lij is anhomothecy:

Lij(ui − uj) = Dijninj(ui − uj)I Explicit form of the coefficients from kinetic quantities (+ positivity,

symmetry)Bérénice Grec Asymptotic diffusion models and kinetic theory for mixtures 12/15

I Expand

Γεij(εu, 0) ∝ nεi nεj∫∫∫

Bij

(exp

[− mi2kT (v2 − 2εv ′ · u + ε2u2)

]exp

[− mj2kT v2

∗

]− exp

[− mi2kT (v2 − 2εv · u + ε2u2)

]exp

[− mj2kT v∗2

] )(v − v ′)

by the conservation of kinetic energy.I Expansion in ε of the exponentials:

∝ nεi nεj∫∫∫

Bij exp[− mi2kT v2 − mj

2kT v∗2]

(ε

mikT (v ′ − v) · u + O(ε2)

)(v − v ′)

I Matrix form of the operator Lij(u) = limε→0

Γεijε , involving (v ′ − v)(v ′ − v)ᵀ

I Since Lij commutes with any rotation, Schur’s lemma ensures that Lij is anhomothecy:

Lij(ui − uj) = Dijninj(ui − uj)I Explicit form of the coefficients from kinetic quantities (+ positivity,

symmetry)Bérénice Grec Asymptotic diffusion models and kinetic theory for mixtures 12/15

I Expand

Γεij(εu, 0) ∝ nεi nεj∫∫∫

Bij

(exp

[− mi2kT (v2 − 2εv ′ · u + ε2u2)

]exp

[− mj2kT v2

∗

]− exp

[− mi2kT (v2 − 2εv · u + ε2u2)

]exp

[− mj2kT v∗2

] )(v − v ′)

by the conservation of kinetic energy.I Expansion in ε of the exponentials:

∝ nεi nεj∫∫∫

Bij exp[− mi2kT v2 − mj

2kT v∗2]

(ε

mikT (v ′ − v) · u + O(ε2)

)(v − v ′)

I Matrix form of the operator Lij(u) = limε→0

Γεijε , involving (v ′ − v)(v ′ − v)ᵀ

I Since Lij commutes with any rotation, Schur’s lemma ensures that Lij is anhomothecy:

Lij(ui − uj) = Dijninj(ui − uj)I Explicit form of the coefficients from kinetic quantities (+ positivity,

symmetry)Bérénice Grec Asymptotic diffusion models and kinetic theory for mixtures 12/15

I Expand

Γεij(εu, 0) ∝ nεi nεj∫∫∫

Bij

(exp

[− mi2kT (v2 − 2εv ′ · u + ε2u2)

]exp

[− mj2kT v2

∗

]− exp

[− mi2kT (v2 − 2εv · u + ε2u2)

]exp

[− mj2kT v∗2

] )(v − v ′)

by the conservation of kinetic energy.I Expansion in ε of the exponentials:

∝ nεi nεj∫∫∫

Bij exp[− mi2kT v2 − mj

2kT v∗2]

(ε

mikT (v ′ − v) · u + O(ε2)

)(v − v ′)

I Matrix form of the operator Lij(u) = limε→0

Γεijε , involving (v ′ − v)(v ′ − v)ᵀ

I Since Lij commutes with any rotation, Schur’s lemma ensures that Lij is anhomothecy:

Lij(ui − uj) = Dijninj(ui − uj)I Explicit form of the coefficients from kinetic quantities (+ positivity,

symmetry)Bérénice Grec Asymptotic diffusion models and kinetic theory for mixtures 12/15

Asymptotics

Formal limitni (t, x) = lim

ε→0nεi (t, x), Ni (t, x) = lim

ε→0nεi (t, x)uεi (t, x)

I Mass conservation

∂tni +∇x · Ni = 0

I Momentum equation

ε2mikT

(∂t(nεi uεi ) +∇x · (nεi uεi ⊗ uεi )

)+∇x nεi =

∑j 6=i

nεi nεj uεj − nεj nεi uεi∆ij

−∇x ni =∑j 6=i

njNi − niNj∆ij

.

No closure relation at the limit! The closure relation can be obtained e.g. inthe isothermal setting with equimolar diffusion.

Bérénice Grec Asymptotic diffusion models and kinetic theory for mixtures 13/15

Link with the linearized operator

I We obtained Dij ∝∫∫∫

Bij(v , v∗, σ) exp[− mi

2kT v2 − mj2kT v2

∗]

(v ′ − v)2.

I Link with the linearized Boltzmann operator (dimension 3)

Dij ∝ 〈miC (k)i ,L(mjC (k)

j )〉, for any k = 1, 2, 3,

where C (k)i = (0, · · · , 0, v (k)M1/2

i , 0, · · · , 0)ᵀ.I Matrix form of Maxwell-Stefan’s system: ∇x n = − 1

kT AU, withA ∝ (Dijninj)i,j and U = (u1, · · · , up)ᵀ.

Self-adjointness + non-positivity of L ↔ A symmetric + semi-definite negative

I Diffusion coefficients for Fick’s law

Dij ∝ 〈L−1(C (k)i − αi Γ(k)),C (k)

j 〉, for any k = 1, 2, 3,

where Γ(k) =∑

i miC (k)i and αi Γ(k) the projection of C (k)

i on KerL.

Bérénice Grec Asymptotic diffusion models and kinetic theory for mixtures 14/15

Conclusion and prospects

I Asymptotics of Boltzmann equation for mixtures towards Maxwell-StefanI Explicit form of the diffusion coefficients w.r.t. the cross sectionsI Also true for polyatomic gasesI Same computations with the moment method

I Fick’s law models a gas in the continuous regimeI Maxwell-Stefan’s equations can be invoked when moderate rarefaction occurs

I Maxwell-Stefan’s equations model moderately rarefied mixturesI Apply the formalism of hyperbolic conservation laws with stiff relaxation terms

[Chen, Levermore, Liu]I Obtain an explicit form of Fick’s coefficients from Maxwell-Stefan’s for low

Knudsen numbers (ongoing work with L. Boudin, V. Pavan)I Compare experimental and analytical relaxation times

I Rigorous convergence towards both Fick’s and Maxwell-Stefan’s equations(ongoing works with A. Bondesan, L. Boudin, M. Briant, E. Daus)

Bérénice Grec Asymptotic diffusion models and kinetic theory for mixtures 15/15

Thank you for your attention!

Bérénice Grec Asymptotic diffusion models and kinetic theory for mixtures 15/15

Moment method

I Entropy associated to the Boltzmann equation for mixtures and its associatedconjugate convex function

η(y) =∑

iyi ln(yi )− yi , η∗(z) =

∑i

exp(zi )

I Tensorial moment basis Ni , Ci (v), E (v)

N1 =

10... , Np =

...01

, C1(v) =

m1v0...

, Cp(v) =

...0

mpv

, E (v) = 12

m1v2

...mpv2

I Even for not so small Knudsen numbers, we obtain from the entropy and the

moment basis the form of

fi (v) = ni

( mi2πkT

)3/2exp

(mi (v − ui )2

2πkT

),

and its moments give, with the same computations, the macroscopicequations.

Bérénice Grec Asymptotic diffusion models and kinetic theory for mixtures 15/15

Closure relation

I Moment of order 2

ε∂

∂t

(mi

∫v

|v |2 f εi)

+∇x ·(

mi

∫v

|v |2 v f εi)

= 1ε

∑j 6=i

mi

∫v

|v |2 Qij(f εi , f εj )

I Ansatz ⇒ mi

∫v|v |2f εi dv = 3kTnεi and mi

∫v|v |2v f εi dv = 5kTεnεi uεi

I Sum over i : 3∂t∑

ini + 5∇x ·

∑i

Ni = 0

I Along with mass conservation, it follows ∂t∑

ini = 0 et ∇ ·

∑i

Ni = 0

I If∑

inini is initially uniform in space, then

∑i

ni is constant

Bérénice Grec Asymptotic diffusion models and kinetic theory for mixtures 15/15