Some mathematical properties of a barotropic multiphase flow model

We study a model for compressible multiphase ﬂows involving N non miscible phases where N is arbitrary. This model boils down to the Baer-Nunziato model when N = 2 . For the barotropic version of model, and for more general equations of state, we prove the weak hyperbolicity property, the convexity of the natural phasic entropies, and the existence of a symmetric form.


Introduction
The modeling and numerical simulation of multiphase flows is a relevant approach for a detailed investigation of some patterns occurring in many industrial sectors. In the nuclear industry for instance, some accidental configurations involve three phase flows such as the steam explosion, a phenomenon consisting in violent boiling or flashing of water into steam, occurring when the water is in contact with hot molten metal particles of "corium": a liquid mixture of nuclear fuel, fission products, control rods, structural materials, etc.. resulting from a core meltdown. We refer the reader to [3,12] and the references therein in order to have a better understanding of that phenomenon.
The modeling and numerical simulation of the steam explosion is an open topic up to now. Since the sudden increase of vapor concentration results in huge pressure waves including shock and rarefaction waves, compressible multiphase flow models with unique jump conditions and for which the initial-value problem is well posed are mandatory. Some modeling efforts have been provided in this direction in [10,9,5,14]. The N -phase flow models developed therein 1 consist in an extension to N ≥ 3 phases of the well-known Baer-Nunziato two phase flow model [1]. They consist in N sets of partial differential equations (PDEs) accounting for the evolution of phase fraction, density, velocity and energy of each phase. As in the Baer-Nunziato model, the PDEs are composed of a hyperbolic first order convective part consisting in N Euler-like systems coupled through non-conservative terms and zero-th source terms accounting for pressure, velocity and temperature relaxation phenomena between the phases. It is worth noting that the latter models are quite similar to the classical two phase flow models in [4,2,7].
In [6], two crucial properties have been proven for a class of two phase flow models containing the Baer-Nunziato model, namely, the convexity of the natural entropy associated with the system, and the existence of a symmetric form. As recalled in that paper, such properties are well understood for systems of conservation laws since Godunov [8] and Mock [13], but remain an open question for non conservative and non strictly hyperbolic models such as those considered here.
In the present paper, we prove the convexity of the entropy and the existence of a symmetric form for a multiphase flow model with N -where N is arbitrarily large -phases. We restrict the study to the case where the interfacial velocity coincides with one of the phasic material velocities. We consider two versions of the model. Firstly, the model with a barotropic pressure law, introduced in [10], and secondly, a similar model with a more general equation of state.

The barotropic multiphase flow model
We consider the following system of partial differential equations (PDEs) introduced in [10] for the modeling of the evolution of N distinct compressible phases in a one dimensional space: for k = 1, .., N , x ∈ R and t > 0: The model consists in N coupled Euler-type systems. The quantities α k , ρ k and u k represent the mean statistical fraction, the mean density and the mean velocity in phase k (for k = 1, .., N ). The quantity p k is the pressure in phase k. We assume barotropic pressure laws for each phase so that the pressure p k is a given function of the density p k : ρ k → p k (ρ k ) with the classical assumption that p ′ k (ρ k ) > 0. The mean statistical fractions and the mean densities are positive and the following saturation constraint holds everywhere at every time: Thus, among the N equations (1a), N − 1 are independent and the main unknown U is expected to belong to the physical space: such that 0 < α 2 , .., α N < 1 and α k ρ k > 0 for all k = 1, .., N .
Following [10], we make the following choice for the closure laws of the socalled interface pressures P kl (U ): Observing that the saturation constraint gives N l=1,l =k ∂ x α l = −∂ x α k for all k = 1, .., N the momentum equations (1c) can be simplified as follows:

Eigenstructure of the system
The following result characterizes the wave structure of system (1): is weakly hyperbolic on Ω U : it admits the following 3N − 1 real eigenvalues: . The corresponding right eigenvectors are linearly independent if, and only if, The characteristic field associated with σ 1 (U ), .., σ N −1 (U ) is linearly degenerate while the characteristic fields associated with σ N −1+k (U ) and σ 2N −1+k (U ) for k = 1, .., N are genuinely non-linear. When (6) fails, the system is said to be resonant.
Proof. In the following, we denote p k and c k instead of p k (ρ k ) and c k (ρ k ) for k = 1, ..N in order to ease the notations. Choosing the variable U = (α 2 , .., α N , u 1 , p 1 , .., u N , p N ) T , the smooth solutions of system (1) satisfy the following equivalent system: where A (U) is the block matrix: 3 Defining M k = (u k − u 1 )/c k the Mach number of phase k relatively to phase 1 for k = 2, .., N , the matrices A, B 1 , .., B N and C 1 , .., C N are given as follows.
where δ p,q is the Kronecker symbol: for p, q ∈ N, δ p,q = 1 if p = q and δ p,q = 0 otherwise. Since A is diagonal and C k is R-diagonalizable with eigenvalues u k − c k and u k + c k , the matrix A (U) admits the eigenvalues u 1 (with multiplicity N − 1), u k − c k and u k + c k for k = 1, .., N . In addition, A (U) is Rdiagonalizable provided that the corresponding right eigenvectors span R 3N −1 .
The right eigenvectors are the columns of the following block matrix: where A ′ , B ′ 1 , .., B ′ N and C ′ 1 , .., C ′ N are matrices defined by: for k = 2, .., N , The first N − 1 columns are the eigenvectors associated with the eigenvalue u 1 . For k = 1, .., N , the N + 2(k − 1) -th and N + (2k − 1) -th columns are the eigenvectors associated with u k − c k and u k + c k respectively. We can see that R(U) is invertible if and only if M k = 1 for all k = 2, .., N i.e. if and only if inequations Hence, the field associated with the eigenvalue u 1 is linearly degenerated. Now we observe that all the acoustic fields are genuinely non linear since for all k = 1, .., N : Proposition 2.2. The the linearly degenerated field σ 1 (U ) = .. = σ N −1 (U ) = u 1 admits the following 2N independent Riemann invariants: The computation is tedious but straightforward.

Mathematical Entropy
An important consequence of the closure law (3) for the interface pressures P kl (U ) is the existence of an additional conservation law for the smooth solutions of (1). Defining the specific internal energy of phase k, e k by e ′ k (ρ k ) = p k (ρ k )/ρ 2 k and the specific total energy of phase k by E k = u 2 k /2 + e k (ρ k ), the smooth solutions of (1) satisfy the following identities: Summing for k = 1, .., N ,the smooth solutions of (1) are seen to satisfy the following additional conversation equation which expresses the conservation of the total mixture energy : As regards the non-smooth weak solutions of (1), one has to add a so-called entropy criterion in order to select the relevant physical solutions. For this purpose, we prove the following result.
is a non strictly convex function of U . Consequently, the total mixture energy, defined by is also a non strictly convex function of U . In the light of (10), the total mixture energy is a mathematical entropy of system (1).
The monophasic mathematical entropy of phase k is given by: Without loss of generality, we can rearrange the components of U and assume that: has the following block-diagonal structure for k = 2, .., N : Let be given (a, b T ) T ∈ R 3×1 with a ∈ R and b ∈ R 2×1 . Then, we easily see 6 that: is a positive matrix by the strict convexity of the monophasic mathematical entropy S k , the right hand side is positive, which yields the positivity of the matrix S ′′ k (U k ) and hence the (non-strict) convexity of (α k ρ k E k )(U ) for k = 2, .., N .
Thus, the Hessian matrix (α 1 ρ 1 E 1 ) ′′ (U ) has the following structure: Defining A 1 , B 1 and C 1 as in (11), the matrices A 1 and B 1 are given by: Let be given with a k ∈ R for k = 2, .., N and b k ∈ R 2×1 for all k = 1, .., N . An easy computation gives: We easily check that Hence, Since S ′′ 1 (V 1 ) is a positive matrix by the strict convexity of the monophasic mathematical entropy S 1 , the right hand side is positive, which yields the positivity of the matrix (α The convexity of the total mixture energy is a direct consequence of the convexity of all the fractional specific energies and we have: Thus, the total mixture energy in non strictly convex.

Symmetrizability
Definition 2.1. The system (1) is said to be symmetrizable if there exists a , and a symmetric matrix Q(U) such that the smooth solutions of (1) satisfy: Since the total mixture energy defined in the previous section is not strictly convex, we cannot use it to prove the symmetrizability of system (1) by multiplication by its hessian matrix. However we can find a suitable positive definite matrix P(U) which symmetrizes the system. Theorem 2.4. System (1) is symmetrizable as long as the non resonance condition (6) holds.
Proof. Let us define U = (α 2 , .., α N , u 1 , p 1 , .., u N , p N ) T . The smooth solutions of system (1) satisfy where the matrix A (U) is given in (7). Let us seek for a symmetric positive definite matrix P(U) that symmetrizes the system. We seek for P(U) in the form: We can easily see that the matrix P k C k is symmetric for all k = 1, .., N . A necessary and sufficient condition for Q(U) to be symmetric is: for all k = 1, .., N, The matrix C T k − u 1 I 2 is a 2 × 2 matrix the determinant of which is c 2 k (M 2 k − 1) where M k = (u k − u 1 )/c k is the relative Mach number of phase k. Hence, the matrices C T k − u 1 I 2 are invertible if and only if the non resonance condition (6) holds. Assuming (6), the matrix D k is therefore given by: An easy computation shows that the matrix (C T k − u 1 I 2 ) −1 P k is symmetric and we get that D T k B k = B T k (C T k − u 1 I 2 ) −1 P k B k is also symmetric. Thus, condition (6) is a necessary and sufficient condition for matrix Q(U) to be symmetric. The matrix P(U) is clearly symmetric. Therefore, it remains to prove that there exists θ > 0 such that P(U) is positive definite. Let x = (a T , b T 1 , .., b T n ) T ∈ R (3N −1)×1 \{0} with a ∈ R (N −1)×1 and for k = 1, .., N , b k ∈ R 2×1 . We have: by the Cauch-Schwarz inequality. The right hand side of this inequality is a polynomial of degree 2 in |a| and its second discriminant ∆ ′ is given by: again by the Cauch-Schwarz inequality. Since D k D T k is symmetric and P k is symmetric positive definite, there exists an invertible 2 × 2 matrix Q k which simultaneously diagonalizes these two matrices. More precisely, we have Q T k P k Q k = Hence, choosing θ larger than the two the eigenvalues of N δ k for all k = 1, .., N (observe that these eigenvalues only depend on U and not on the vector x), we get that ∆ ′ < 0 and therefore x T P(U)x > 0 for all x ∈ R (3N −1)×1 \{0}.

The multiphase flow model with energies
We still consider the evolution of N distinct compressible phases in a one dimensional space. We now consider the following multiphase flow model where the evolution of the phasic energies is now governed by additional PDEs: for k = 1, .., N , x ∈ R and t > 0: The saturation constraint is still valid: and the main unknown U is expected to belong to the physical space: Defining e k := E k − u 2 k /2 the specific internal energy of phase k, the pressure p k = p k (ρ k , e k ) is now given by an equation of state (e.o.s.) as a function defined for all positive ρ k and all positive e k We assume that, taken separately, all the phases follow the second principle of thermodynamics so that for each phase k = 1, .., N , there exists a positive integrating factor T k (ρ k , e k ) and a strictly convex function s k (ρ k , e k ), called the (mathematical) specific entropy of phase k such that: Finally, the closure laws for the interface pressures P kl (U ) are given by: for k = 1, P 1l (U ) = p l (ρ l , e l ), for l = 2, .., N for k = 1, P kl (U ) = p k (ρ k , e k ), for l = 1, .., N, l = k. (15) Observing that the saturation constraint gives N l=1,l =k ∂ x α l = −∂ x α k for all k = 1, .., N the momentum equations (1c) can be simplified as follows: In the same way, the energy equations (12d) can be simplified as follows:

Eigenstructure of the system
The following result characterizes the wave structure of system (12): Theorem 3.1. System (12) admits the following 4N − 1 eigenvalues: . If c k (ρ k , e k ) 2 > 0, then system (12) is weakly hyperbolic on Ω U in the following sense: all the eigenvalues are real and the corresponding right eigenvectors are linearly independent if, and only if, The characteristic fields associated with σ 1 (U ), .., σ N −1 (U ) and σ 2N −1+k (U ) = u k for k = 1, .., N are linearly degenerate while the characteristic fields associated with σ N −1+k (U ) and σ 3N −1+k (U ) for k = 1, .., N are genuinely non-linear. When (20) fails, the system is said to be resonant.
Remark 3.1. The condition c k (ρ k , e k ) 2 > 0 is a classical condition that ensures the hyperbolicity for monophasic flows. In general, assuming U ∈ Ω U is not sufficient to guarantee that c k (ρ k , e k ) 2 > 0. For the stiffened gas e.o.s. for instance, where the pressure is given by where γ k > 1 and p ∞,k ≥ 0 are two constants, a classical calculation yields ρ k c k (ρ k , e k ) 2 = γ k (γ k − 1)(ρ k e k − p ∞,k ). Hence, the hyperbolicity of the system requires a more restrictive condition than simply the positivity of the internal energy which reads : ρ k e k > p ∞,k .
Proof. We choose the variable U = (α 2 , .., α N , u 1 , p 1 , s 1 , .., u N , p N , s N ) T . We denote p k and c k instead of p k (ρ k , e k ) and c k (ρ k , e k ) for k = 1, ..N in order to ease the notations. The smooth solutions of system (1) satisfy the following equivalent system (see Section 3.2 for the entropy equations on s k for k = 1, .., N ): where A (U) is the block matrix: Defining M k = (u k − u 1 )/c k the Mach number of phase k relatively to phase 1 for k = 2, .., N , the matrices A, B 1 , .., B N and C 1 , .., C N are given as follows.
A = diag(u 1 , .., u 1 ) ∈ R (N −1)×(N −1) Since A is diagonal and C k is R-diagonalizable if c 2 k > 0, with eigenvalues u k −c k , u k and u k + c k , the matrix A (U) admits the eigenvalues u 1 (with multiplicity N ), u k − c k and u k + c k for k = 1, .., N and u k for k = 2, .., N . In addition, A (U) is R-diagonalizable provided that the corresponding right eigenvectors span R 4N −1 . The right eigenvectors are the columns of the following block 12 matrix: ., B ′ N and C ′ 1 , .., C ′ N are matrices defined by: we can see that the N -th component of R j (U) is zero. This implies that for all 1 ≤ j ≤ N − 1 and for j = N + 2, R j (U) · ∇ U (u 1 ) = 0. Hence, the field associated with the eigenvalue u 1 is linearly degenerated. In the same way, since the N + 2(k − 1) -th component of R N +2k (U) is zero, the field associated with the eigenvalue u k is linearly degenerated. Now we observe that all the acoustic fields are genuinely non linear since for all k = 1, .., N : Proposition 3.2. The the linearly degenerated field σ 1 (U ) = .. = σ N −1 (U ) = σ 2N (U ) = u 1 admits the following 3N − 1 independent Riemann invariants:

Mathematical Entropy
A consequence of the second law of thermodynamics (14) and the closure laws (15) is the following convection equations satisfied by the specific phasic entropies: We have the following result. is a non strictly convex function of U . Consequently, the total mixture entropy, defined by N k=1 α k ρ k s k (U ) is also a non strictly convex function of U . In the light of (22), the fractional specific entropies are mathematical entropies of system (12).

Symmetrizability
We have the following symmetrisability result for system (12). where θ ∈ R + , I N −1 is the (N − 1) × (N − 1) identity matrix and for k = 1, .., N , and D k is the 3 × (N − 1) matrix given by: and a necessary and sufficient condition for the 3 × 3 matrix C T k − u 1 I 3 to be invertible is the non resonance condition (20). As in the proof of Theorem 2.4, we can show that Q(U) = P(U)A (U) is symmetric and that P(U) is a symmetric positive definite matrix provided that θ is large enough.

Conclusion
For both the barotropic and non barotropic multiphase flow models described in (1) and (12), we have proven the weak hyperbolicity, the existence of convex mathematical entropies as well as the existence of a symmetric form. This last property is valid only far from resonance, i.e. as long as the considered models remain in their domain of hyperbolicity. These properties have been obtained for any admissible phasic equations of state (increasing phasic pressure laws for the barotropic system, and for the system with energies, equations of state abiding by the second law of thermodynamics). What is more, the proven properties can be extended to the two and three dimensional versions of theses models thanks to their frame invariance.
An important consequence of the symmetrisability and Kato's theorem on quasi-linear symmetric systems ( [11]) is that, far from resonance, there exists a unique local-in-time smooth solution to the Cauchy problem. The blow-up in finite time still holds, but with the additional restriction due to the non resonance conditions (6) and (20).