Diffusion models for mixtures using a stiff dissipative hyperbolic formalism Bérénice Grec1 in collaboration with L. Boudin, V. Pavan 1
MAP5 – Université Paris Descartes, France
Workshop on kinetic and fluid Partial Differential Equations March 7th, 2018
Outline of the talk
1
Gaseous mixtures: macroscopic models
2
Stiff dissipative hyperbolic formalism
3
Entropy and equilibrium
4
Local equilibrium approximation
5
Conclusion and prospects
12 :29
Bérénice Grec
Stiff dissipative hyperbolic formalism for diffusion for mixtures
1/16
Diffusion models for mixtures: Fick/Maxwell-Stefan I
Mixture of p ≥ 2 species
I
ρi : mass density of species i
I
Ni = ρi ui : momentum of species i ρ1 N1 .. .. ρ = . , N = .
I
ρp I
Fick law: N = −F (ρ)∇x ρ Maxwell-Stefan equations:
Np
−∇x ρ = S(ρ)N
Mass conservation: ∂t ρ + ∇ · N = 0
Properties of matrices F and S I
F (ρ) and S(ρ) are not invertible (rank p − 1).
I
Cauchy problem for Maxwell-Stefan’s equations [Giovangigli, Bothe, Jüngel & Stelzer]
I
Using Moore-Penrose pseudo-inverse: structural similarity
12 :29
Bérénice Grec
Stiff dissipative hyperbolic formalism for diffusion for mixtures
2/16
Fick vs. Maxwell-Stefan (macroscopic point of view) Formal analogy of the two systems, but Fick and Maxwell-Stefan are not obtained in the same way
Obtention of the Fick law I
Thermodynamics of irreversible processes (entropy decay) [Onsager]
I
Thermodynamical considerations on fluxes, written as linear combinations of potential gradients
I
Stems from mass equations
Obtention of the Maxwell-Stefan equations I
Mechanical considerations on forces (equilibrium of pressure and friction forces)
I
Assumption: different species have different macroscopic velocities on macroscopic time scales
I
Stems from momentum equations
12 :29
Bérénice Grec
Stiff dissipative hyperbolic formalism for diffusion for mixtures
3/16
Fick vs. Maxwell-Stefan (kinetic point of view) Formal analogy of the two systems, but Fick and Maxwell-Stefan are not obtained in the same way
Perturbative method (Fick) I
Based on the Chapman-Enskog expansion [Bardos, Golse, Levermore], [Bisi, Desvillettes]
Moment method (Maxwell-Stefan) I
Based on the ansatz that the distribution functions are at local Maxwellian states [Levermore], [Müller, Ruggieri]
The Maxwell-Stefan equations can be written X ρi ρj (uj − ui ) 1 , − ∇x ρ i = mi Dij j6=i
mi : molecular mass of species i, Dij > 0 symmetric. 12 :29
Bérénice Grec
Stiff dissipative hyperbolic formalism for diffusion for mixtures
4/16
Outline of the talk
1
Gaseous mixtures: macroscopic models
2
Stiff dissipative hyperbolic formalism
3
Entropy and equilibrium
4
Local equilibrium approximation
5
Conclusion and prospects
12 :29
Bérénice Grec
Stiff dissipative hyperbolic formalism for diffusion for mixtures
4/16
Stiff dissipative model for mixtures For any species i with velocity u i , we write mass and momentum conservation ∂t ρi + ∇x · (ρi u i ) = 0, 1 ∂t (ρi u i ) + ∇x · (ρi u i ⊗ u i + Pi (ρi )) + Ri = 0 ε I I
Ideal gas law for the partial pressure Pi (ρi ) = ρi kB T /mi Relaxation term: friction force exerted by the mixture on species i Ri =
X
aij ρi ρj (u j − u i ) =
j6=i
X
αij (u j − u i ) =
j6=i
p X
αij u j .
j=1
Using the formalism of [Chen, Levermore, Liu, CPAM, ’94] Obtain a reduced system involving the aligned velocity u when ε remains small P ( ∇x Pj p ∂t ρi + ∇x · (ρi u) = ε∇x · ` , ij j=1 ρj ∂t (ρu) + ∇x · (ρu ⊗ u) + ∇x P = 0, where P = 12 :29
P
j
Pj (ρj ) is the total pressure, and (`ij ) are real constants. Bérénice Grec
Stiff dissipative hyperbolic formalism for diffusion for mixtures
5/16
Stiff dissipative model for mixtures For any species i with velocity u i , we write mass and momentum conservation ∂t ρi + ∇x · (ρi u i ) = 0, 1 ∂t (ρi u i ) + ∇x · (ρi u i ⊗ u i + Pi (ρi )) + Ri = 0 ε I I
Ideal gas law for the partial pressure Pi (ρi ) = ρi kB T /mi Relaxation term: friction force exerted by the mixture on species i Ri =
X
aij ρi ρj (u j − u i ) =
j6=i
X
αij (u j − u i ) =
j6=i
p X
αij u j .
j=1
Using the formalism of [Chen, Levermore, Liu, CPAM, ’94] This approach has been used in previous papers for other systems: I
[Kawashima], [Yong], [Kawashima,Yong], ...
I
[Giovangigli, Matuszewski], [Giovangigli, Yong], ...
12 :29
Bérénice Grec
Stiff dissipative hyperbolic formalism for diffusion for mixtures
5/16
Vectorial expression of the system 1 ∂t W + ∇x · F(W) + R(W) = 0 ε I
(∗)
Unknown W = [W1 , . . . , Wp , W p+1 , · · · , W 2p ] ∈ (R∗+ )p × Rdp , with Wi = ρi , W p+i = ρi u i
the k-th column of F(W) and R(W) are given by W p+1 · e k 0p×1 .. . p W · e X 2p k W p+j α1j W j , R(W) = j=1 W ⊗ W Fk (W) = . p+1 p+1 + P (W )I e . 1 1 d k . W 1 . .. p X W p+j . αpj W 2p ⊗ W 2p W j j=1 + Pp (Wp )Id e k Wp
I
12 :29
Bérénice Grec
Stiff dissipative hyperbolic formalism for diffusion for mixtures
6/16
Aim of the work I
Adapt the formalism by Chen, Levermore and Liu in the gaseous mixture framework
I
Limiting behavior for small ε Derive an approximation of the local equilibrium and its first-order correction
I
I
Build a relevant entropy which ensures...
I
... the hyperbolicity of the local equilibrium approximation...
I
... and the dissipativity of its first-order correction
Analogy with the kinetic theory I
relaxation time ε ←→ mean free path
I
unknown W ←→ distribution function f
I
friction relaxation term R ←→ collision operator Q (with dissipativity and collisional invariant properties)
I
local equilibria ←→ Maxwellian functions
12 :29
Bérénice Grec
Stiff dissipative hyperbolic formalism for diffusion for mixtures
7/16
Outline of the talk
1
Gaseous mixtures: macroscopic models
2
Stiff dissipative hyperbolic formalism
3
Entropy and equilibrium
4
Local equilibrium approximation
5
Conclusion and prospects
12 :29
Bérénice Grec
Stiff dissipative hyperbolic formalism for diffusion for mixtures
7/16
Building the entropy 1 ∂t W + ∇x · F(W) + R(W) = 0 ε
(∗)
Entropy for Equation (∗) The function η defined by 2
1 W p+i kB T η(W) = + Ei (Wi ), with Ei (Wi ) = 2 Wi mi
Wi − Wi Wi ln Wi0
is a strictly convex entropy for Equation (∗), i.e. I
∇2W η(W)∇W Fk (W) is symmetric for any k
I
∇W η(W) · R(W) ≥ 0
I
∇2W η(W) is a positive definite quadratic form Choice of the internal energy Ei00 (Wi ) = Pi0 (Wi )/Wi Nonnegativity of the matrix A = (αij )
12 :29
Bérénice Grec
Stiff dissipative hyperbolic formalism for diffusion for mixtures
8/16
Conserved quantities The matrix
0p×dp
Ip
Q= 0d×p
(p+d)×(p+dp) ∈R Id , · · · , Id
satisfies QR(W) = 0(p+d)×1 . It allows to define " w = QW = W1 , · · · , Wp ,
p X
#| W |p+j
∈ Rp+d
j=1
the p + d independent conserved quantities. | Conversely, to any w = w1 , · · · , wp , w |p+1 , we associate, via the equilibrium function E, the equilibrium Weq = E(w) such that R(Weq ) = 0, where | p X wp w1 | | E(w) = w1 , · · · , wp , w ,··· , w , with σ(w) = wi . σ(w) p+1 σ(w) p+1 i=1 12 :29
Bérénice Grec
Stiff dissipative hyperbolic formalism for diffusion for mixtures
9/16
Characterization of the local equilibria Equilibrium function | p X w1 | wp E(w) = w1 , · · · , wp , w p+1 , · · · , w |p+1 , with σ(w) = wi . σ(w) σ(w) i=1 Characterization of the equilibrium I I
I
R(Weq ) = 0 there exists u ∈ Rd such that Weq = [W1 , · · · , Wp , W1 u | , · · · , Wp u | ]| this corresponds to saying that all species velocities are aligned there exists v ∈ Rp+d such that ∇W η(Weq ) = v| Q
Use of the Legendre-Fenchel transform of η I
there exists v ∈ Rp+d such that Weq = ∇V η ∗ (v| Q)
In terms of the physical variables, |
E(w) = [ρ1 , · · · , ρp , ρ1 u | , · · · , ρp u | ] . Define P(w) = ∇w E(w)Q, which is a projection matrix. 12 :29
Bérénice Grec
Stiff dissipative hyperbolic formalism for diffusion for mixtures
10/16
Outline of the talk
1
Gaseous mixtures: macroscopic models
2
Stiff dissipative hyperbolic formalism
3
Entropy and equilibrium
4
Local equilibrium approximation
5
Conclusion and prospects
12 :29
Bérénice Grec
Stiff dissipative hyperbolic formalism for diffusion for mixtures
10/16
Look for an expansion W = M[w] (with w = QW), such that ∂t w + ∇x · QF(M[w]) ' 0. Then 1 1 ∂t W + ∇x · F(W) + R(W) ' (I − ∇w M[w]Q)∇x · F(M[w]) + R(M[w]). ε ε Satisfying equation (∗) leads to cancelling the RHS. Introduce the formal expansion W = M[w] = E(w) + εM(1) [w] + · · · Linearizing, order 1 in ε becomes (I − ∇w E(w)Q)∇x · F(E(w)) + ∇W R(E(w))M(1) [w] = 0 Provided that the inversion of ∇W R(E(w)) is possible in some sense, −1
M(1) [w] = − (∇W R(E(w)))
(I − ∇w E(w)Q)∇x · F(E(w)) | {z } =P(w)
and the equation on w becomes, with f(w) = QF(E(w)) h i ∂t w + ∇x · f(w) + ε∇x · Q∇W F(E(w))M(1) [w] = 0. 12 :29
Bérénice Grec
Stiff dissipative hyperbolic formalism for diffusion for mixtures
11/16
Local equilibrium approximation Formal expansion Wε = E(w) + εM(1) [w] + · · ·
Chen, Levermore and Liu’s computations The first-order correction is given by M(1) [w] = −B [Ip+dp − P(w)] ∇x · F(E(w)), with P(w) = ∇w E(w)Q. The system (∗) becomes ∂t w + ∇x · f(w) = ε∇x · g(w), where fk (w) = QFk (E(w)), gk (w) = Q∇W Fk (E(w))B [Ip+dp − P(w)] ∇x · F(E(w)), where B is the pseudo-inverse of ∇W R(E(w)) such that im B = ker Q. 12 :29
Bérénice Grec
Stiff dissipative hyperbolic formalism for diffusion for mixtures
12/16
Explicit computations Expression of the equilibrium, definition of P(w) = ∇w E(w)Q 0p×1 ∇x P1 (ρ1 ) − ρ1 ∇x P ρ (Ip+dp − P(w))∇x · F(E(w)) = , .. . ρp ∇x Pp (ρp ) − ∇x P ρ P P where ρ = i ρi and P = i Pi (ρi ). Computation of X = B(Ip+dp − P(w))∇x · F(E(w)): I
B pseudo-inverse of ∇W R(E(w)) such that im B = ker Q
I
ker Q =
I
X ∈ im B = ker Q thus Xi = 0 and
I
im(∇W R(E(w))) = ker Q =⇒ (Ip+dp − P(w))∇x · F(E(w)) ∈ im(∇W R(E(w)))
I
∇W R(E(w))X = (Ip+dp − P(w))∇x · F(E(w)) p X ρi X p+j = ∇x Pi (ρi ) − ∇x P αij ρj ρ
0, · · · , 0, X |p+1 , · · · , X |2p
| Pp ,
Pp j=1
i=1
X p+i = 0d×1
X p+j = 0
Pseudo-invert A = (αij )
j=1
12 :29
Bérénice Grec
Stiff dissipative hyperbolic formalism for diffusion for mixtures
13/16
Explicit computations Expression of the equilibrium, definition of P(w) = ∇w E(w)Q 0p×1 ∇x P1 (ρ1 ) − ρ1 ∇x P ρ (Ip+dp − P(w))∇x · F(E(w)) = , .. . ρp ∇x Pp (ρp ) − ∇x P ρ P P where ρ = i ρi and P = i Pi (ρi ). Computation of X = B(Ip+dp − P(w))∇x · F(E(w)): I
B pseudo-inverse of ∇W R(E(w)) such that im B = ker Q
I
ker Q =
I
X ∈ im B = ker Q thus Xi = 0 and
I
im(∇W R(E(w))) = ker Q =⇒ (Ip+dp − P(w))∇x · F(E(w)) ∈ im(∇W R(E(w)))
I
∇W R(E(w))X = (Ip+dp − P(w))∇x · F(E(w)) p X X p+j ρi αij = ∇x Pi (ρi ) − ∇x P ρj ρ
0, · · · , 0, X |p+1 , · · · , X |2p
| Pp ,
Pp j=1
i=1
X p+i = 0d×1
X p+j = 0
Pseudo-invert A = (αij )
j=1
12 :29
Bérénice Grec
Stiff dissipative hyperbolic formalism for diffusion for mixtures
13/16
Explicit computations Expression of the equilibrium, definition of P(w) = ∇w E(w)Q 0p×1 ∇x P1 (ρ1 ) − ρ1 ∇x P ρ (Ip+dp − P(w))∇x · F(E(w)) = , .. . ρp ∇x Pp (ρp ) − ∇x P ρ P P where ρ = i ρi and P = i Pi (ρi ). Computation of X = B(Ip+dp − P(w))∇x · F(E(w)): I
B pseudo-inverse of ∇W R(E(w)) such that im B = ker Q
I
ker Q =
I
X ∈ im B = ker Q thus Xi = 0 and
I
im(∇W R(E(w))) = ker Q =⇒ (Ip+dp − P(w))∇x · F(E(w)) ∈ im(∇W R(E(w)))
I
∇W R(E(w))X = (Ip+dp − P(w))∇x · F(E(w)) p X X p+j ρi αij = ∇x Pi (ρi ) − ∇x P ρj ρ
0, · · · , 0, X |p+1 , · · · , X |2p
| Pp ,
Pp j=1
i=1
X p+i = 0d×1
X p+j = 0
Pseudo-invert A = (αij )
j=1
12 :29
Bérénice Grec
Stiff dissipative hyperbolic formalism for diffusion for mixtures
13/16
Pseudo-inversion of A I
Let 1 = (1, · · · , 1)|
I
ker A = Span 1 and im A = (Span 1)⊥
I
Let r = [ρ1 , · · · , ρp ] , which is not orthogonal to 1
I
Decompositions
|
Rp = (Span 1) ⊕ (Span r)⊥ = (Span 1)⊥ ⊕ (Span r) I
Existence of a unique pseudo-inverse L = (λij )1≤i,j≤p of A with im L = (Span r)⊥ and ker L = Span r
I
L is symmetric
I
Since A = (αij ) = (ρi ρj aij ), we have that L = (λij ) =
12 :29
Bérénice Grec
1 ρi ρj `ij
Stiff dissipative hyperbolic formalism for diffusion for mixtures
14/16
End of the computations p X j=1 I
αij
X p+j = ρj
ρi ∇x Pi (ρi ) − ∇x P ρ
|
r = [ρ1 , · · · , ρp ] spans ker L, i.e.
P
X p+i = ρi
j
λij ρj = 0
p X
λij ∇x Pj (ρj )
j=1
P Expression of the equilibrium, and the fact that i X p+i = 0 allow to compute gk (w) = Q∇W Fk (E(w))X p p X `1j X ∂xk Pj (ρj ) λ1j ∂xk Pj (ρj ) ρ1 j=1 j=1 ρj .. .. . . gk (w) = = p p X X `pj ∂xk Pj (ρj ) λpj ∂xk Pj (ρj ) ρp j=1 j=1 ρj 0d×1 12 :29
Bérénice Grec
0d×1 Stiff dissipative hyperbolic formalism for diffusion for mixtures
15/16
End of the computations p X X p+i ρj = λij ∇x Pj (ρj ) − ∇x P ρi ρ j=1 I
|
r = [ρ1 , · · · , ρp ] spans ker L, i.e.
P
X p+i = ρi
j
λij ρj = 0
p X
λij ∇x Pj (ρj )
j=1
P Expression of the equilibrium, and the fact that i X p+i = 0 allow to compute gk (w) = Q∇W Fk (E(w))X p p X `1j X ∂xk Pj (ρj ) λ1j ∂xk Pj (ρj ) ρ1 j=1 j=1 ρj .. .. . . gk (w) = = p p X X `pj ∂xk Pj (ρj ) λpj ∂xk Pj (ρj ) ρp j=1 j=1 ρj 0d×1 12 :29
Bérénice Grec
0d×1 Stiff dissipative hyperbolic formalism for diffusion for mixtures
15/16
End of the computations p X X p+i ρj = λij ∇x Pj (ρj ) − ∇x P ρi ρ j=1 I
|
r = [ρ1 , · · · , ρp ] spans ker L, i.e.
P
X p+i = ρi
j
λij ρj = 0
p X
λij ∇x Pj (ρj )
j=1
P Expression of the equilibrium, and the fact that i X p+i = 0 allow to compute gk (w) = Q∇W Fk (E(w))X p p X `1j X λ1j ∂xk Pj (ρj ) ∂xk Pj (ρj ) ρ1 j=1 j=1 ρj .. .. . . gk (w) = = p p X X `pj λpj ∂xk Pj (ρj ) ∂xk Pj (ρj ) ρp j=1 j=1 ρj 0d×1 12 :29
Bérénice Grec
0d×1 Stiff dissipative hyperbolic formalism for diffusion for mixtures
15/16
Outline of the talk
1
Gaseous mixtures: macroscopic models
2
Stiff dissipative hyperbolic formalism
3
Entropy and equilibrium
4
Local equilibrium approximation
5
Conclusion and prospects
12 :29
Bérénice Grec
Stiff dissipative hyperbolic formalism for diffusion for mixtures
15/16
Conclusion and prospects Reduced system involving the bulk velocity u for small ε (
∂t ρi + ∇x · (ρi u) = ε∇x ·
P
∇x Pj p j=1 `ij ρj
,
∂t (ρu) + ∇x · (ρu ⊗ u) + ∇x P = 0. I I
Hyperbolicity & dissipativity Diffusion correction term of Fick’s type (on the equation of mass conservation)
I
No viscosity term on the momentum equation (convective diffusive fluxes)
I
Maxwell-Stefan can describe a moderate rarefied regime more than Fick
Prospects I
Obtain an explicit form of Fick’s coefficients from the Maxwell-Stefan’s ones
I
Compare the experimental and theoretical relaxation times (experiments being designed at IUSTI)
I
Taking into account the non isothermal effects
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Bérénice Grec
Stiff dissipative hyperbolic formalism for diffusion for mixtures
16/16
Thank you for your attention!
12 :29
Bérénice Grec
Stiff dissipative hyperbolic formalism for diffusion for mixtures
16/16
Pseudo-inverses Proposition Let A ∈ Rp×p .Let S and T be two subspaces of Rp such that Rp = ker A ⊕ S and Rp = im A ⊕ T . Then there exists a unique matrix B such that: 1
ABA = A,
2
BAB = B,
3
ker B = T and im B = S.
This matrix B is then called the pseudo-inverse of A with prescribed range S and null space T .
Corollary Let Y ∈ im A. Then there exists a unique X ∈ im B such that AX = Y, it is given by X = BY.
Symmetric case Consider a symmetric matrix A ∈ Rp×p , and a subspace N such that Rp = im A ⊕ N. Then the only symmetric pseudo-inverse of A is the one with prescribed range N ⊥ and null space N. 12 :29
Bérénice Grec
Stiff dissipative hyperbolic formalism for diffusion for mixtures
16/16
Legendre-Fenchel transform We introduce the following domain V = V ∈ Rp+dp | V = ∇W η(W) for some W ∈ (R∗+ )p × Rdp . The Legendre-Fenchel transform η ∗ of η is the convex function satisfying η(W) + η ∗ (V) = V · W. We can compute η ∗ (V) = V · W − η(W) =
p X kB T i=1
mi
Wi0 exp
mi kB T
1 Vi + V 2p+i . 2
Let φ∗ : Rp+d → R, v 7→ η ∗ (v| Q). Denote by φ the Legendre-Fenchel transform of φ∗ , we compute p X wi 1 w 2p+1 kB T wi ln − 1 + . φ(w) = mi 2 σ(w) Wi0 i=1 Then, following [Chen, Levermore, Liu] E(w) = ∇V η ∗ (∇w φ(w)| Q). 12 :29
Bérénice Grec
Stiff dissipative hyperbolic formalism for diffusion for mixtures
16/16
