Entropy supplementary conservation law for non-linear systems of PDEs with non-conservative terms: application to the modelling and analysis of complex fluid flows using computer algebra
EENTROPY SUPPLEMENTARY CONSERVATION LAW FORNON-LINEAR SYSTEMS OF PDES WITH NON-CONSERVATIVETERMS: APPLICATION TO THE MODELLING AND ANALYSIS OFCOMPLEX FLUID FLOWS USING COMPUTER ALGEBRA ∗ PIERRE CORDESSE † AND
MARC MASSOT ‡ Abstract.
In the present contribution, we investigate first-order nonlinear systems of partialdifferential equations which are constituted of two parts: a system of conservation laws and non-conservative first order terms. Whereas the theory of first-order systems of conservation laws is wellestablished and the conditions for the existence of supplementary conservation laws, and more specif-ically of an entropy supplementary conservation law for smooth solutions, well known, there exists sofar no general extension to obtain such supplementary conservation laws when non-conservative termsare present. We propose a framework in order to extend the existing theory and show that the presenceof non-conservative terms somewhat complexifies the problem since numerous combinations of the con-servative and non-conservative terms can lead to a supplementary conservation law. We then identifya restricted framework in order to design and analyze physical models of complex fluid flows by meansof computer algebra and thus obtain the entire ensemble of possible combination of conservative andnon-conservative terms with the objective of obtaining specifically an entropy supplementary conserva-tion law. The theory as well as developed computer algebra tool are then applied to a Baer-Nunziatotwo-phase flow model and to a multicomponent plasma fluid model. The first one is a first-order fluidmodel, with non-conservative terms impacting on the linearly degenerate field and requires a closuresince there is no way to derive interfacial quantities from averaging principles and we need guidancein order to close the pressure and velocity of the interface and the thermodynamics of the mixture.The second one involves first order terms for the heavy species coupled to second order terms for theelectrons, the non-conservative terms impact the genuinely nonlinear fields and the model can be rig-orously derived from kinetic theory. We show how the theory allows to recover the whole spectrum ofclosures obtained so far in the literature for the two-phase flow system as well as conditions when oneaims at extending the thermodynamics and also applies to the plasma case, where we recover the usualentropy supplementary equation, thus assessing the effectiveness and scope of the proposed theory.
Keywords.
Nonlinear PDEs with non-conservative terms, supplementary conservation law, en-tropy, computer algebra, two-phase flow, Baer-Nunziato model, multicomponent plasma fluid model
AMS subject classifications.
1. Introduction
First-order nonlinear systems of partial differential equations and more specificallysystems of conservation laws have been the subject of a vast literature since the secondhalf of the twentieth century because they are ubiquitous in mathematical modellingof fluid flows and are used extensively for numerical simulation in applications andindustrial context [1, 2]. Such systems of equation can either be rigorously derived fromkinetic theory of gases through various expansion techniques [3, 4], or can be derivedusing rational thermodynamics and fluid mechanics including stationary action principle(SAP) [5, 6, 7]. As far as Euler or Navier-Stokes equations are concerned for a gaseous ∗ Received date, and accepted date (The correct dates will be entered by the editor). † ONERA, DMPE, 8 Chemin de la Huni`ere, 91120 Palaiseau, France, and CMAP, Ecole polytech-nique, Route de Saclay 91128 Palaiseau Cedex, France, ([email protected]) ‡ CMAP, Ecole polytechnique, Route de Saclay, 91128 Palaiseau Cedex, France,([email protected]) 1 a r X i v : . [ c s . S C ] N ov Entropy conservation law for nonlinear systems of PDEs with non-conservative terms flow field, the outcome of both approaches are similar and the mathematical propertiesof these systems have been thoroughly investigated for the past decades.An interesting related problem is the quest for supplementary conservation laws.Noether’s theorem [8] leads, within the framework of SAP, to the derivation of supple-mentary conservation laws based on symmetry transformations of the variational prob-lem under investigation . Examples of such derivations on two-phase flow modellingcan be found in [9, 10]. However, to the authors knowledge, no symmetry transforma-tions have been identified yielding a conservative law on the entropy of the system. Infact, SAP does not allow to reach a closed system of equations, and one has to pro-vide a closure for the entropy (see [11] for example). A specific type of supplementaryconservation equation for smooth solution is especially important, namely the entropyequation , derived through the theory developed in [12, 13] for systems of conservationlaws. Such systems of PDEs are hyperbolic at any point where a locally convex entropyfunction exists [14], and when they are equipped with a strictly convex entropy, theycan be symmetrized [13] [15] and thus are hyperbolic. These properties have been atthe heart of the mathematical theory of existence and uniqueness of smooth solutions[16] [17], but they are also a corner stone for the study of weak solutions for which thework of [18] proves the well-posedness of Cauchy problem for one-dimensional systems.Nonetheless, for a number of applications, where reduced-order fluid models haveto be used for tractable mathematical modelling and numerical simulations, be it inthe industry or in other disciplines, micro-macro kinetic-theory-like approaches as wellas rational thermodynamics and SAP approaches often lead to system of conservationlaws involving non-conservative terms . Among the large spectrum of applications, wefocus on two types of models, which exemplify the two approaches: 1- two phase flowsmodels which rely on a hierarchy of diffuse interface models among which stands theBaer-Nunziato [19] model used when full disequilibrium of the phases must be taken intoaccount. Since this model is derived through rational thermodynamics, the macroscopicset of equations can not be derived from physics at small scale of interface dynamicsand thus require closure of interfacial pressure and velocity, 2- multicomponent fluidmodelling of plasmas flows out of thermal equilibrium, where the equations can bederived rigorously from kinetic theory using a multi-scale Chapman-Enskog expansionmixing a hyperbolic scaling for the heavy species and a parabolic scaling for the electrons[20]. Concerning the thermodynamics, whereas for the first model it has to be postulatedand requires assumptions, it can be obtained from kinetic theory in the second model. Inboth cases, the models involve non-conservative terms, but these terms do not act on thesame fields; linearly degenerate field is impacted for the two-phase flow model, whereasit acts on the genuinely nonlinear fields in the second [21]. Whereas hyperbolicitydepends on the closure and is not guaranteed for the first class of models [22], thesecond is naturally hyperbolic [20] and also involves second-order terms and eventuallysource terms [23].Thus, the presence of non-conservative terms encompasses several situations andrequires a general theoretical framework. While Noether’s theorem can still appliedto obtain some supplementary conservation laws, it does not permit to exhibit all ofthem and especially not an entropy supplementary conservation law. A unifying theory Among the most well-known symmetry transformations, the time translation yields the conserva-tion of the total energy of the system if the associated Lagrangian is invariant to time-shift and thespace translation yields the conservation of the total momentum of the system if the Lagrangian isinvariant to space-shiftORDESSE, P. and MASSOT, M. non-conservative first order terms .We emphasize how the presence of non-conservative terms somewhat complexifies the problem since numerous combinations ofthe conservative and non-conservative terms can lead to supplementary conservationlaws. We then identify a restricted framework in order to design and analyze physicalmodels of complex fluid flows by means of computer algebra and thus obtain the entireensemble of possible combination of conservative and non-conservative terms to obtainan entropy supplementary conservation law. The proposed theoretical approach is thenapplied to the two systems identified so far for their diversity of behaviour. Even if thewhole theory is valid for any supplementary conservation law, we focus on obtainingan entropy supplementary conservation law. For the two-phase flow model, assuminga thermodynamics of non-miscible phases, we derive conditions to obtain an entropysupplementary conservative equation together with a compatible thermodynamics andclosures for the non-conservative terms. Interestingly enough, all the closures proposedso far in the literature are recovered [19, 26, 27, 28, 29]. The strength of the for-malism lies also in the capacity to derive such conditions for some level of mixing ofthe phases. By introducing a mixing term in the definition of the entropy, the newtheory brings out constraints on the form of the added mixing term. We recover notonly the closure proposed to account for a configuration energy as in the context ofdeflagration-to-detonation [19] or in [30], but we also rigorously find new closures lead-ing to a conservative system of equations . We also prove that the theory encompassesthe plasma case, where we recover the usual entropy supplementary equation assessingthe effectiveness and scope of the proposed theory.The paper is organized as follows. The extension of the theory for system of con-servation laws to first-order nonlinear systems of partial differential equations includingnon-conservative terms, as well as the framework to apply the theory by means of com-puter algebra are introduced in Section 2. These results are then applied first to theBaer-Nunziato model in Section 3 and then to the plasma model in Section 4 to obtainan entropy supplementary conservation law compatible with the model closure. Notations:
Let a ∈ R p , b ∈ R p , B ∈ R p × p , C ∈ R p × p , D ∈ R p × p × p be a p -componentline first-order tensor, a p -component column first-order tensor, two p -square second-order tensor and a third-order tensor respectively. We introduce the following notations: • a B is a line first-order tensor in R p whose i component are defined by( a B ) i = (cid:88) j =1 ,p a j B j,i , (1.1) Such closure is similar to the one used in [31, 32] which led to a controversy [27, 33, 34]
Entropy conservation law for nonlinear systems of PDEs with non-conservative terms • B b is a column first-order tensor in R p whose i component is defined by( B b ) i = (cid:88) j =1 ,p B i,j b j , (1.2) • B × C is p -square second-order tensor whose ( i,j ) component is defined by( B × C ) i,j = (cid:88) k =1 ,p B i,k C k,j , (1.3) • a ⊗ D is a p -square second-order tensor whose ( i,j ) component is defined by( a ⊗ D ) ( i,j ) = (cid:88) k =1 ,p a k × D k,i,j . (1.4)Hereafter, we will name zero- first- and second-order tensors by scalar, vector andmatrix respectively and for convenience we will use vector and matrix representationsof functions. Moreover, given a scalar function S , the partial differentiation of S bya column vector a , ∂ a S is a line vector in R p . Finally, · denotes the Euclidean scalarproduct in R p .
2. Supplementary conservation law
First we recall the theory of the existence of a supplementary conservative equationfor first-order nonlinear systems of conservation laws. Second, this notion is extended tosystems containing first order non-conservative terms. Third, we introduce a frameworkto apply this new theory to design and analyze physical models using computer algebra.A one-dimensional framework is adopted from now on, x ∈ R , in order to simplifythe derivation. Nonetheless, the results can easily be extended to the multi-dimensionalapproach as presented in [35] for systems of conservation laws. The homogeneous form ofa first-order nonlinear system of p conservation laws writes ∂ t u + ∂ x f ( u ) = , (2.1)where u ∈ Ω ⊂ R p denotes the conservative variables with Ω an open convex of R p and f : u ∈ Ω (cid:55)→ R p the conservative fluxes. Focusing on smooth solution of the system (2.1),its quasi-linear form is given by ∂ t u + ∂ u f ( u ) ∂ x u = . (2.2) Theorem 2.1.
Let H : u ∈ Ω (cid:55)→ R be a scalar function, not necessarily convex. Thefollowing statements are equivalent: ( C ) System (2.1) admits a supplementary conservative equation ∂ t H ( u ) + ∂ x G ( u ) = , (2.3) where u ∈ R p is a smooth solution of System (2.1) and G : u ∈ Ω (cid:55)→ R is a scalarfunction. ORDESSE, P. and MASSOT, M. C ) There exists a scalar function G : u ∈ Ω (cid:55)→ R such that ∂ u H ( u ) ∂ u f ( u ) = ∂ u G ( u ) . (2.4)( C ) ∂ uu H ( u ) × ∂ u f ( u ) is a p -square symmetric matrix.Proof . The proofs of the theorem can be found in the literature. We would like torecall how the last statement is obtained. Assuming ( C ), differentiating Equation (2.4)leads to ∂ uu H ( u ) × ∂ u f ( u ) + ∂ u H ( u ) ⊗ ∂ uu f ( u ) = ∂ uu G ( u ) , (2.5)where ∂ u H ( u ) ⊗ ∂ uu f ( u ) is a p -square matrix defined as (cid:80) i ∂ u i H ( u ) ∂ uu f i ( u ) which isa linear combination of Hessian matrices and hence symmetric. Moreover, the RHS ofEquation (2.5) ∂ uu G ( u ) is symmetric. Therefore ∂ uu H ( u ) × ∂ u f ( u ) is symmetric.Theorem 2.1 applies for any type of supplementary conservative equations andother formulations of Theorem 2.1 can be found in the literature [15, 35, 36]. Let us nowconsider the homogeneous form of a first-order nonlinear system of partial differentialequations constituted of two parts: conservations laws and first-order non-conservativeterms. Its quasi-linear form can be written as ∂ t u + [ ∂ u f ( u ) + N ( u )] ∂ x u = , (2.6)where u ∈ Ω ⊂ R p is a smooth solution with Ω an open convex of R p , f : u ∈ Ω (cid:55)→ R p the conservative fluxes, N : u ∈ Ω (cid:55)→ R p × p the p -square matrix containing the first-ordernon-conservative terms.In the following we extend the theory introduced in Section 2.1 to system (2.6).Given a scalar function H : u ∈ Ω (cid:55)→ R , multiplying system (2.6) by the line vector ∂ u H ( u )yields ∂ t H + ∂ u H ( u )[ ∂ u f ( u ) + N ( u )] ∂ x u = 0 . (2.7)Compared to Equation (2.3), the presence of the non-conservative terms in Equa-tion (2.7) complexifies the question of the existence of a supplementary conservativeequation. Therefore we propose to decompose in a specific way the conservative andnon-conservative terms in Definition 2.1. Definition 2.1.
Given a scalar function H : u ∈ Ω (cid:55)→ R and a first-order nonlinear non-conservative system (2.6) , let us define the four p -square matrices, C ( u ) , Z ( u ) , C ( u ) and Z ( u ) in R p × p such that ∂ u f ( u ) = C ( u ) + Z ( u ) , (2.8) N ( u ) = C ( u ) + Z ( u ) , (2.9) with the condition ∂ u H ( u )[ Z ( u ) + Z ( u )] = . (2.10)In light of Definition 2.1, Theorem 2.1 can be extended as follows: Theorem 2.2.
Let H : u ∈ Ω (cid:55)→ R be a scalar function, not necessarily convex. Given afirst-order nonlinear system of non-conservation laws (2.6) , if we introduce the decom-position as in Definition 2.1, then the following statements are equivalent: Entropy conservation law for nonlinear systems of PDEs with non-conservative terms ( C ) System (2.6) admits a supplementary conservative equation ∂ t H ( u ) + ∂ x G ( u ) = 0 , (2.11) where u ∈ R p is a smooth solution of System (2.6) and G : u ∈ Ω (cid:55)→ R is a scalarfunction.( C ) There exists a scalar function G : u ∈ Ω (cid:55)→ R such that ∂ u H ( u )[ C ( u ) + C ( u )] = ∂ u G ( u ) . (2.12) ( C ) ∂ uu H ( u ) × [ C ( u ) + C ( u )] + ∂ u H ( u ) ⊗ ∂ u [ C ( u ) + C ( u )] is a p -square symmet-ric matrix.Proof . Rewriting Equation (2.7) using the decomposition of the conservative andnon-conservative terms as ∂ t H ( u ) + ∂ u H ( u )[ C ( u ) + C ( u )] ∂ x u = − ∂ u H ( u )[ Z ( u ) + Z ( u )] ∂ x u (2.13)outlines the result. Remark 2.1.
Theorem 2.2 applies for any type of supplementary conservative equa-tions. The usual symmetry condition on which relies the existence of a supplementaryconservation equation is strongly modified when non-conservation terms are present.From Theorem 2.1 to Theorem 2.2 the condition ∂ uu H ( u ) × ∂ u f ( u ) symmetric , is modified into ∂ uu H ( u ) × [ C ( u ) + C ( u )] + ∂ u H ( u ) ⊗ ∂ u [ C ( u ) + C ( u )] symmetric . In the context of systems of conservation laws, an interesting algebraic approach is pro-posed in [37] based on the reinterpretation of the symmetric Condition ( C ) in The-orem 2.1 as a Frobenuis problem. Nevertheless, when dealing with additional non-conservative terms, the above new symmetry condition prevents us from applying ef-ficiently such an approach. Remark 2.2.
In Definition 2.1, the condition (2.10) implies that the conservative andnon-conservative terms depend only on the variables u , and not on their gradient. Someauthors have allowed the matrices Z k to depend also on the gradients of the variables u , then a more general condition for the decomposition can be written ∂ u H ( u )[ Z ( u ,∂ x u ) + Z ( u ,∂ x u )] ∂ x u ≤ . (2.14) In Section 3, we will see that such a condition has been chosen to close the Baer-Nunziato model [29]. However, since it changes the mathematical nature of the PDEunder investigation, we will not include it in our study.
From a modelling perspective, System (2.6) under consideration is not necessaryclosed. Therefore, the following corollary yields conditions on the model to obtain asupplementary conservative equation once we have postulated the thermodynamics.
Corollary 2.1.
Let H : u ∈ Ω (cid:55)→ R be a scalar function, not necessarily convex. Given afirst-order nonlinear system of non-conservation laws (2.6) where f : u ∈ Ω (cid:55)→ R p and N : u ∈ Ω (cid:55)→ R p × p are unknown functions to be modelled. If we introduce the decompositionas in Definition 2.1, then System (2.6) admits a supplementary conservative equation ∂ t H ( u ) + ∂ x G ( u ) = 0 , (2.15) where u ∈ Ω ⊂ R p is a smooth solution of System (2.6) and G : u ∈ Ω (cid:55)→ R a scalar func-tion, if and only if the following conditions hold ORDESSE, P. and MASSOT, M. ( C ) ∂ uu H ( u ) × [ C ( u ) + C ( u )] + ∂ u H ( u ) ⊗ ∂ u [ C ( u ) + C ( u )] is a p -square symmet-ric matrix.( C ) ∂ u H ( u )[ Z ( u ) + Z ( u )] = . We would like to apply the theory on first-order nonlinear non-conservative systemsintroduced in Section 2.2 to physical models such as the Baer-Nunziato model and theplasma model in order to design and analyze them. We recall that our prior interest is toobtain an entropy supplementary conservation law. However, the difficulty is manifold: − The combination of the non-conservative terms and conservative terms proposedin Definition 2.1 to build a supplementary conservative equation is not uniqueand thus many degrees of freedom exist in defining the matrices C k and Z k . − When the model is derived trough rational thermodynamics, terms in the sys-tem of equations might need closure and the thermodynamics has to be postu-lated. Therefore, the matrices C k and Z k can contain unknowns related to thesystem and the definition of H . − The calculations needed to derive a supplementary conservative equation areheavy and choice-based. Any change of C k and Z k that respects Definition 2.1,or any new postulated thermodynamics would require to derive again all theequations, and eventually a very limited range of possibilities would be exam-ined.These difficulties to apply the theory and examine all the possibilities makes computeralgebra very appealing since it allows symbolic operations to be implemented and thuscan derive equations systematically and quasi-instantaneously for any combinations ofconservative and non-conservative terms as well as model closure and H definition.Furthermore, the generic level handled by computer algebra is not unlimited andtherefore Definition 2.1 requires further assumptions to circumscribe the number ofdegrees of freedom that can be accounted for.Even if the theory proposed hereinbefore is valid to obtain any kind of supplemen-tary conservation laws, we are mainly interested in obtaining an entropy supplementaryconservation law. We thus need to define the notions of entropy and entropic variables in the following two definitions. Definition 2.2. H : u ∈ Ω (cid:55)→ R is said to be an entropy of the system (2.6) if H ( u ) is aconvex scalar function of the variables u which fulfills Theorem 2.1. The supplementaryconservative equation (2.3) is then named the entropy equation and G : u ∈ Ω (cid:55)→ R is theassociated entropy flux. Definition 2.3.
Let H : u ∈ Ω (cid:55)→ R be a scalar function, not necessarily convex. Givena first-order nonlinear conservative system (2.1) , let us define the entropic variables v : u ∈ Ω (cid:55)→ R p such that v ( u ) = ( ∂ u H ( u )) t . (2.16)The entropic variables have been studied in [17] in order to obtain symmetric andnormal forms of the system of equation and used in the framework of gaseous mixtures, Entropy conservation law for nonlinear systems of PDEs with non-conservative terms where the mathematical entropy H is usually defined as the opposite of a physicalentropy density per unit volume of the system [17]. Definition 2.4.
Given a scalar function H : u ∈ Ω (cid:55)→ R , a first-order nonlinear non-conservative system (2.6) , and the four p -square matrices C ( u ) , Z ( u ) , C ( u ) and Z ( u ) in R p × p defined in Definition 2.1, we introduce the unknown line vector t : u ∈ Ω (cid:55)→ R p such that ∂ u H ( u )[ C ( u ) + C ( u )] = ∂ u H ( u ) ∂ u f ( u ) + t ( u ) , (2.17) ∂ u H ( u )[ Z ( u ) + Z ( u )] = ∂ u H ( u ) N ( u ) − t ( u ) . (2.18) The condition of Equation (2.10) rewrites into ∂ u H ( u ) N ( u ) − t ( u ) = . (2.19) Remark 2.3.
Since Definition 2.4 is a projection of the matrix equations of Defini-tion 2.1 on the vector ∂ u H ( u ) , it may be interesting to introduce an unknown matrix T ( u ) ∈ R p × p associated to the unknown line vector t ( u ) such that t ( u ) = ∂ u H ( u ) T ( u ) . (2.20) Thus, Definition 2.4 can be formulated as follows C ( u ) + C ( u ) = ∂ u f ( u ) + T ( u ) , (2.21) Z ( u ) + Z ( u ) = N ( u ) − T ( u ) , (2.22) with the condition ∂ u H ( u )[ N ( u ) − T ( u )] = . (2.23)The unknown functional line vector t ( u ) ∈ R p represents the transfer of non-conservative terms to the conservative terms. In the degenerate case where t = , C k receives all the conservative terms and Z k all the non-conservative terms. Con-dition (2.19) forces all the non-conservative terms to vanish and System (2.6) is fullyconservative, hence the theory of conservative system can be applied.Definition 2.4 being more restrictive than Definition 2.1, computer algebra is nowapplicable to analyze the properties of a first-order nonlinear non-conservative systemleading to a reformulation of Theorem 2.2. Theorem 2.3.
Let H : u ∈ Ω (cid:55)→ R be a scalar function, not necessarily convex. Con-sider a first-order nonlinear system of non-conservation laws (2.6) . If we introduce thedecomposition as in Definition 2.4, then the following statements are equivalent: ( C ) System (2.6) admits a supplementary conservative equation ∂ t H ( u ) + ∂ x G ( u ) = 0 , (2.24) where u ∈ R p is a smooth solution of System (2.6) and G : u ∈ Ω (cid:55)→ R is a scalarfunction. ORDESSE, P. and MASSOT, M. C ) There exists a scalar function G : u ∈ Ω (cid:55)→ R such that ∂ u H ( u ) ∂ u f ( u ) + t ( u ) = ∂ u G ( u ) . (2.25)( C ) ∂ uu H ( u ) × ∂ u f ( u ) + ∂ u t ( u ) is a p -square symmetric matrix.Proof . Injecting Definition 2.4 into Theorem 2.2 leads to these results.When H is the entropy of the system, Theorem 2.3 provides equations that relatethe thermodynamics of the model through H , the model itself with possible terms to beclosed in f ( u ) and N ( u ), and the unknown line vector t ( u ). Combined with the Defi-nition 2.4, Theorem 2.3 brings out conditions on the model to obtain a supplementaryconservative equation given a postulated thermodynamics and it leads to the followingcorollary. Corollary 2.2.
Consider a first-order nonlinear system of non-conservation laws (2.6) where u ∈ Ω ⊂ R p is a smooth solution with Ω an open convex of R p but f : u ∈ Ω (cid:55)→ R p and N : u ∈ Ω (cid:55)→ R p × p are unknown functions to be modelled. Let H : u ∈ Ω (cid:55)→ R be a scalar function, not necessarily convex. If we introduce the decomposition as inDefinition 2.4, then System (2.6) admits a supplementary conservative equation ∂ t H ( u ) + ∂ x G ( u ) = 0 , (2.26) where G : u ∈ Ω (cid:55)→ R is a scalar function if and only if the following conditions hold ( C ) ∂ uu H ( u ) × ∂ u f ( u ) + ∂ u t ( u ) is symmetric. ( C ) ∂ u H ( u ) N ( u ) − t ( u ) = . Remark 2.4.
The previous framework can be extended to the multi-dimensional case ina straightforward manner. If the original system is isotropic, such as for the applicationswe have in mind, then the previous conditions will be the same in the various directions.In the framework of more general non-isotropic systems, which satisfy Galilean androtational invariances for example, we will obtain different conditions and we have tocheck that the decomposition we perform in the various directions satisfies some com-patibility relations so that the obtained conservation law satisfies the original invarianceproperties of the system.
Corollary 2.2 draws the methodology we have implementedin the Maple TM computer algebra software in order to obtain an entropy supplementaryconservation law. Our methodology is the following:( Step
1) We define the thermodynamics by postulating - if need be - an entropy function H : u ∈ Ω (cid:55)→ R .( Step
2) We then use Condition ( C ) and ( C ) of Corollary 2.2 to ensure the existenceof an entropy flux G : u ∈ Ω (cid:55)→ R and solve (cid:40) ∂ uu H ( u ) × ∂ u f ( u ) + ∂ u t ( u ) symmetric ,∂ u H ( u ) N ( u ) − t ( u ) = . (2.27)In System (2.27), t ( u ) is systematically an unknown, f ( u ), N ( u ) as well as H ( u ) can include unknown terms for which the variable dependency is specified.Maple TM generates then an exhaustive solution for t ( u ) and constraints on allthe other unknown terms. Maple is a trademark of Waterloo Maple Inc. Entropy conservation law for nonlinear systems of PDEs with non-conservative terms ( Step
3) From that, the software derives the admissible entropy flux G : u ∈ Ω (cid:55)→ R whichgives then the supplementary conservative equation.
3. Application to the Baer-Nunziato model3.1. Context and presentation of the model.
The Baer-Nunziato modelhas been derived through rational thermodynamics in [19] and describes a two-phaseflow out of equilibrium. Extended by the work of [38] thanks to the introduction ofinterfacial quantities, the homogeneous form of the Baer-Nunziato model is ∂ t u + [ ∂ u f ( u ) + N ( u )] ∂ x u = ,∂ u f ( u ) = ∂ u f ( u )
00 0 ∂ u f ( u ) , N ( u ) = v I , (3.1)where the column vector u ∈ R is defined by u T = (cid:0) α , u T , u T (cid:1) , u Tk = ( α k ρ k , α k ρ k v k ,α k ρ k E k ). The conservative flux f : u ∈ Ω (cid:55)→ R reads f ( u ) T = (0 , f ( u ) T , f ( u ) T ) with f k ( u k ) T = ( α k ρ k v k , α k ( ρ k v k + p k ) , α k ( ρ k E k + p k ) v k ). N : u ∈ Ω (cid:55)→ R × is the matrixcontaining the non-conservative terms with n ( u ) T = − n ( u ) T = (0 , − p I , − p I v I ). Then, α k is the volume fraction of phase k ∈ [1 , ρ k the partial density, v k the phase velocity, p k the phase pressure, E k = (cid:15) k + v k / (cid:15) k the internalenergy, v I the interfacial velocity and p I the interfacial pressure.Two levels of ingredients are still missing for this model. First, the macroscopic setof equations includes the interface dynamics through the interfacial terms v I and p I andthus needs closure on these terms. Second the thermodynamics has to be postulated.The mathematical properties of the model have been studied by [22, 30, 39] amongothers and many closure have been proposed for the interfacial terms based on wave-typeconsiderations and the entropy inequality.Regarding the thermodynamics, for non-miscible phases, the entropy H ( u ) is com-monly defined by Equation (3.2) as in [28, 30], H ( u ) = − (cid:88) k =1 , α k ρ k s k , (3.2)with s k = s k ( ρ k ,p k ) the phase entropy which takes for the Ideal Gas equation of statethe form s k = c v,k ln (cid:18) p k ρ γ k k (cid:19) , (3.3)with c v,k the heat capacity, p k the pressure, ρ k the density and γ k the isentropic coeffi-cient of phase k .If we were to account for partial miscibility between the two phases, we would haveto add a mixing term to the definition of the non-miscible entropy. The mixing termcould take the form proposed in [22], so that the entropy rewrites H = − (cid:88) k =1 , α k ρ k [ s k ( ρ k ,p k ) − ψ k ( α k )] , (3.4) ORDESSE, P. and MASSOT, M. ψ k , k = [1 , ψ k ( α k ) = ψ k (cid:48) ( α k (cid:48) ) . (3.5)In this section, we apply to the Baer-Nunziato model the framework introducedin Section 2 by means of computer algebra. We will firstly assume the phases arenon-miscible and derive an entropy supplementary conservative equation along withconditions on the interfacial terms. All the closures proposed in the literature will berecovered. Secondly, we will also apply the methodology in the case of a thermodynam-ics with partial miscibility and derive an entropy supplementary conservative equationtogether with conditions on both the interfacial terms and the mixing terms of theentropy. Not only all the closures proposed in the literature are recovered but alsonew ones and we also propose explicit formulations of the mixing terms and show thatdepending on their expression, the condition expressed in [22] is not necessary. We start without any condition on( v I ,p I ). We need initially to fix a decomposition of ∂ u f ( u ) and N ( u ) including a certaindegree of freedom as explained in Section 2.3.Given an entropy H : u ∈ Ω (cid:55)→ R of System (3.1), by expressing the entropic variablesas v ( u ) T = (cid:0) v α , v T , v T (cid:1) , we use the decomposition proposed in Definition (2.4). Sincewe do not want to generate other non-conservative terms, we choose to define the linevector t : u ∈ Ω (cid:55)→ R p by t ( u ) = ( t α ( u ) , , ) where t α : u ∈ Ω (cid:55)→ R is the unknown scalarfunction a priori of all the variables u . We obtain the following decompositions( ∂ u H [ C + C ]) T = t α ( u ) v · ∂ u f ( u ) v · ∂ u f ( u ) , (3.6a)( ∂ u H [ Z + Z ]) T = − t α ( u ) + v α v I + (cid:80) k =1 , v k · n k . (3.6b) t α allows fractions of the non-conservative terms to feed the matrix C k .Given this decomposition, we use the methodology proposed in Section 2.4. ( Step
2) will be split here into two sub-steps.(
Step .a ) Condition ( C ) on the symmetry of the matrix ∂ uu H ( u ) × ∂ u f ( u ) + ∂ u t ( u ) en-sures the existence of an entropy flux G ( u ). It will determine t ( u ).( Step .b ) Knowing t ( u ), Condition ( C ), ∂ u H ( u ) N ( u ) − t ( u ) = , will return an equationlinking ( v I ,p I ) and also ψ k when miscibility is accounted for. We start applying our method (
Step H as in Equation (3.2). The thermodynamics is entirely known and weuse the Ideal Gas EOS. The entropic variables v are then v = v α v v with v α = p T − p T and v k = 1 T k g k − v k / v k − , (3.7)2 Entropy conservation law for nonlinear systems of PDEs with non-conservative terms with g k the Gibbs free energy, g k = (cid:15) k + p k /ρ k − T k s k . We now apply the conditions todetermine t α ( u ) and derive the equation that links the interfacial quantities v I and p I . Theorem 3.1.
Consider System (3.1) . If the mixture entropy is defined as H = − (cid:80) k =1 , α k ρ k s k then with the decomposition proposed in Equations (3.6) ∂ uu H ( u ) × ∂ u f ( u ) + ∂ u t ( u ) symmetric ⇔ t α ( u ) = F ( α ) + p T u − p T u , (3.8) with F a strictly convex arbitrary function depending on the volume fraction α . As aconsequence the condition on ∂ u H ( u )[ Z ( u ) + Z ( u )] gives ∂ u H ( u )[ Z ( u ) + Z ( u )]= ⇔ − F ( α ) + (cid:88) k =1 , ( − k T k ( p I − p k )( v k − v I )= 0 . (3.9) Proof . The function t α is found relying on symbolic computation and it holds as aproof.As explained in ( Step .a ), Equation (3.8) guarantees the existence of an entropyflux G associated with the mixture entropy H chosen as in Equation (3.2) by definingthe unknown function t α ( u ).Then as described in ( Step .b ), Equation (3.9) relates the interfacial terms ( v I ,p I ).By choosing F ( α ) = 0, the condition on ∂ u H × [ Z + Z ] writes (cid:88) k =1 , T k ( p k − p I )( v I − v k ) = 0 . (3.10)So now, to obtain a closed model along with a supplementary conservative equation,we can postulate an interfacial velocity v I and derive the corresponding p I . We willlimit ourselves to defining v I such that the field associated to v I is linearly degenerate.In that case, the only admissible interfacial velocities are v I = βu + (1 − β ) u with β ∈ [0 , ,α ρ /ρ ] [30], [28]. We will focus on the particular case where F ( α ) = 0. We obtainthe following results: − If v I = v k , then Equation (3.10) returns p I = p k (cid:48) . ( v k ,p k (cid:48) ) is the closure proposedfirst by [19], [26], [27], in the context of deflagration-to-detonation. − If v I = βu + (1 − β ) u with β = α ρ /ρ , then Equation (3.10) returns p I = µp +(1 − µ ) p with µ ( β ) = (1 − β ) T / ( βT + (1 − β ) T ). It is the closure found in [28]among others.We see that first these closures are a specific case where F ( α ) is chosen to be zero inEquation (3.9). Second, one could have chosen another interfacial velocity v I and itwould have led to another interfacial pressure p I compatible with an entropy pair. Remark 3.1.
If we had used the extended condition expressed in Equation (2.14) , thenthe condition on ∂ u H [ Z + Z ] would be (cid:88) k =1 , T k [ p k − p I ( u ,∂ x u )][ v I ( u ,∂ x u ) − v k ] ∂ x α k ≤ ORDESSE, P. and MASSOT, M. ⇔ − (cid:88) k =1 , T k Z k ( Z + Z ) [ p k (cid:48) − p k + sgn ( ∂ x α )( u k (cid:48) − v k ) Z k (cid:48) ] ≤ , (3.12) where Z k is defined by Z k = ρ k a k with the phase sound speed a k = ∂p k /∂ρ k | s k . FromEquation (3.11) , one sees that the dependency on ∂ x u reduces to ∂ x α otherwise someterms would not be signable. Then closures such as the one found through DiscreteElement Method (DEM) [29] are obtained v I = Z u + Z u Z + Z + sgn ( ∂ x α ) p − p Z + Z , (3.13) p I = Z p + Z p Z + Z + sgn ( ∂ x α ) Z Z Z + Z ( u − u ) . (3.14) Now, let us add a degree of freedomin the thermodynamics by introducing mixing terms in the definition of the entropy H as in Equation (3.4) to account for partial miscibility of the phases. The added terms, ψ k , functions of the volume fraction α k only, are to be determined.The entropic variables v are v = (cid:80) k =1 , ( − k +1 p k T k (cid:20) − α k r k ψ (cid:48) k ( α k ) (cid:21) v v with v k = 1 T k g k − v k / v k − (3.15) Theorem 3.2.
Consider System (3.1) . If the mixture entropy is defined as H = − (cid:80) k =1 , α k ρ k [ s k − ψ k ( α k )] with ψ k , k = [1 , , two strictly convex arbitrary functions de-pending on the volume fraction, then with the decomposition proposed in Equations (3.6) ,we have ∂ uu H × ∂ u f + ∂ u t symmetric ⇔ t α ( u ) = F ( α ) + p T u (cid:20) − α r ψ (cid:48) ( α ) (cid:21) − p T u (cid:20) − α r ψ (cid:48) ( α ) (cid:21) (3.16) with F a strictly convex arbitrary function depending on the volume fraction. As aconsequence the condition on ∂ u H [ Z + Z ] gives = ∂ u H ( u )[ Z ( u ) + Z ( u )] ⇔ − F ( α ) + (cid:88) k =1 , ( − k +1 α k ρ k ψ (cid:48) k ( α k )( u k − v I )+ (cid:88) k =1 , ( − k T k ( p I − p k )( v k − v I ) (3.17)Again, Equation (3.16) guarantees the existence of an entropy flux G ( u ) conditioningthe function t α ( u ) ( Step .a ). The interfacial quantities ( v I ,p I ) and ψ k are linked byEquation (3.17) ( Step .b ).The difference with the previous case for immiscible phases is that there are twosupplementary unknowns ψ k , k = 1 ,
2. We thus are free to either postulate first aninterfacial velocity v I and then derive the corresponding p I and ψ k or postulate first thefunctions ψ k and see what choices we have for the interfacial terms. In the following weinvestigate the two approaches.4 Entropy conservation law for nonlinear systems of PDEs with non-conservative terms
Let us postulate v I and limit ourselves to the case F ( α ) = 0. We will again seek alinearly degenerate field for v I . In such case, the results in Table 3.1 are obtained. Table 3.1 . Admissible thermodynamics and model closures obtained by postulating v I v I p I ( ψ k ,ψ k (cid:48) )Case 1 v k p k (cid:48) ( ψ k , βu + (1 − β ) u β = α ρ /ρ µp + (1 − µ ) p µ ( β ) = (1 − β ) T βT +(1 − β ) T ψ k ( α k ) = ψ k (cid:48) ( α k (cid:48) )In Case 1 of Table 3.1, ψ k can be interpreted as a configuration energy of phase k as in [19], [26] [27], in the context of deflagration-to-detonation. It is a term definingan interaction of one phase with itself only. More importantly, Equation (3.17) showsthat it is not possible to include a configuration energy for each phase when choosingthe closure ( v I ,p I ) = ( v k ,p k (cid:48) ).In Case 2 of Table 3.1, the condition on the mixing term introduced in Equa-tion (3.5) by [22] is recovered and the closures are the one stated in [30]. However, thecondition on the mixing terms imposes a constraint on the volume fraction and thus onthe flow topology. Since mixing of the phases should be able to occur disregarding theflow topology, these terms fail to introduce free mixing among the phases. Since Case1 and Case 2 of Table 3.1 do not allow the phases to mix, let us choose first thethermodynamics of the system and induce the admissible interfacial terms.It has been shown that the mixing entropy of an ideal compressible binary mixtureis of the form (cid:80) k =1 , α k ln( α k ). Therefore, we choose to define the functions ψ k by ψ k ( α k ) = r k ln( α k ). In this case, the entropy writes H = − (cid:88) k =1 , α k ρ k [ s k − r k ln( α k )] , (3.18)with r k the specific gas constant of phase k , we now account for quasi-miscibility betweenthe phases.The condition on t α degenerates, t α = F ( α ) and the condition on ∂ u H [ Z + Z ] isnow − F ( α ) + p I (cid:18) u − v I T − u − v I T (cid:19) = 0 . (3.19)It is no more possible to obtain the classic definition on v I and p I . In the case F ( α ) = 0two choices are possible to verify Equation (3.19) and summarized in Table 3.2.Case 3 of Table 3.2 proposes a temperature-based averaged velocity for v I , whichdoes not seem to be physically reasonable. In Case 4, the interfacial pressure must vanishfor the system to admit an entropy supplementary conservation equation and the Baer-Nunziato model becomes a conservative system if one assumes the field associated to v I ORDESSE, P. and MASSOT, M. Table 3.2 . Admissible thermodynamics and model closures obtained by postulating ψ k v I p I Case 3 βu + (1 − β ) u with β = T / ( T − T ) no constraintCase 4 no constraint 0to be linearly degenerate. One knows how much it simplifies the problem in terms ofnumerical implementation. This result can be interpreted as an incompatibility betweenthe existence of a mixing process in the thermodynamics of the mixture and an interfacialpressure, that stays meaningful as long as there is an interface between the two phases. When the thermodynamics accountsfor mixing (Case 4 Table 3.2), the existence of an entropy supplementary conservativeequation is incompatible with the interfacial pressure, and thus the nozzling terms p I ∂ x α k vanish.In separated two-phase flows, these terms are known to be necessary to preserveuniformity in velocity and pressure of the flow during its temporal evolution [34] andare usually compared to the terms obtained in a single gas with a variable section [40].Whereas these arguments seem valid for separated two-phase flows, one may questionthe role these terms play in a dispersed phase flows.Taking the particular case p I = 0 and p = 0 in the Baer-Nunziato model seems tolead to a system of equations similar to one that would describe a flow of incompressiblesuspended particles, where 1 would denote the carrier phase and 2 the dispersed phase.Doing so, one recovers not only the Marble model [41], which proposes a pressurelessgas dynamic equations for the particle phase, valid in the limit where α < − , butalso the model obtained by Sainsaulieu [42] in the asymptotic limit where the volumefraction of the particles α →
4. Application to the plasma model
The multicomponent fluid modellingof plasma flows out of thermal equilibrium has been derived rigorously from kinetictheory using a multi-scale Chapman-Enskog expansion mixing a hyperbolic scaling forthe heavy species with a parabolic scaling for the electrons [20]. The system takes theform ∂ t u + [ ∂ u f ( u ) + N ( u )] ∂ x u = ∂ x ( D ( u ) ∂ x u ) , (4.1)6 Entropy conservation law for nonlinear systems of PDEs with non-conservative terms with ∂ u f ( u ) = κ/ − v (2 − κ ) v κ κ/ v − h tot ρ h ) v h tot ρ h − κv (1 + κ ) v − ρ e ρ h v ρ e ρ h v − ρ e (cid:15) e ρ h v ρ e (cid:15) e ρ h v , (4.2) N ( u ) = − ρ e (cid:15) e ρ h κv ρ e (cid:15) e ρ h κ , (4.3) D ( u ) = − λκ(cid:15) e ρ e λκ(cid:15) e ρ e + γD DκT e − λκ(cid:15) e ρ e λκ(cid:15) e ρ e + γD , (4.4)where the column vector u ∈ R is defined by u T = ( ρ h ,ρ h v,E,ρ e ,ρ e (cid:15) e ) with ρ h is thedensity of the heavy particles, v the hydrodynamic velocity, E the total energy definedby E = ρ h v / ρ h (cid:15) h + ρ e (cid:15) e , (cid:15) h the internal energy of the heavy particles, ρ e the densityof the electrons, (cid:15) e the internal energy of the electrons, h tot the total enthalpy defined by h tot = E + p with p = p h + p e , T e the temperature of the electrons, the constant κ definedby κ = γ − γ the isentropic coefficient, p h is the pressure of the heavy particlesand p e is the pressure of the electrons. In the diffusive terms, λ is the electron thermalconductivity, D the electron diffusion coefficient.Concerning the thermodynamics, it can be obtained from kinetic theory. Theelectrons and the heavy particles thermodynamics are defined by an ideal gas equationof state, and they share both the same isentropic coefficient: p h = κρ h (cid:15) h , p e = κρ e (cid:15) e where p h is the pressure of the heavy particles and p e is the pressure of the electrons, r is the constant of the gas r = c v κ with c v the calorific heat at constant volume, themodel being adimensionalized r = c v ( γ −
1) = 1.The model is naturally hyperbolic [20] and also involves second-order terms andeventually source terms [23]. Here we considered the homogeneous form.In this section, we would like to derive the usual entropy supplementary conservativeequation found by [20] and show that it is unique, to attest the effectiveness of the theory.
We need to proceed to the decomposition of the conser-vative and non conservative terms of System (4.1). We restrict ourselves again to the de-composition proposed in Definition (2.4) and we add a degree of liberty to each non-nullnon-conservative components by defining t : u ∈ Ω (cid:55)→ R as t ( u ) T = ( t ( u ) ,t ( u ) , , , ∂ u H ( u )[ C ( u ) + C ( u )]) T = v ( u ) · ∂ u f ( u ) + t ( u ) t ( u )000 , (4.5) ORDESSE, P. and MASSOT, M. ∂ u H ( u )[ Z ( u ) + Z ( u )]) T = − t ( u ) − ρ e ρ h (cid:16) − T e T h (cid:17) v − t ( u ) + ρ e ρ h (cid:16) − T e T h (cid:17) . (4.6)The unknown scalar functions t k ( u ) give the possibility to fractions of the non-conservative terms to be given to the matrix C k . The entropy H : u ∈ Ω (cid:55)→ R for two perfect gases isdefined as H = − ρ h s h − ρ e s e , (4.7)with the partial entropies defined by s h = c v ln (cid:18) p h κρ t h (cid:19) , s e = c v ln (cid:18) p e κρ t e (cid:19) . (4.8)This entropy includes mixing between the electrons and the heavy particles. Thus, westart applying our method ( Step
1) by postulating H as in Equation (4.7). The entropicvariables v are then v = T h (cid:0) g h − v / (cid:1) T h v − T h T e g e T h − T e , (4.9)with g k the Gibbs free energy, g k = (cid:15) k + p k /ρ k − T k s k . Remark 4.1.
In the fourth component of the entropic variable, the kinetic energy of theelectrons has vanished. This is due to the low-Mach assumption made for the electrons.
We now apply the conditions to determine t k ( u ). Theorem 4.1.
Consider System (4.1) . If the mixture entropy is defined as H = − ρ h s h − ρ e s e , then with the decomposition proposed in Equations (4.5) , we have ∂ uu H ( u ) × ∂ u f ( u ) + ∂ u t ( u ) symmetric ⇔ t ( u ) = ρ e ρ h (cid:18) − T e T h (cid:19) v and t ( u ) = − ρ e ρ h (cid:18) − T e T h (cid:19) , (4.10) and the condition on ∂ u H ( u )[ Z ( u ) + Z ( u )] is ∂ u H ( u )[ Z ( u ) + Z ( u )] = (0 , , , , . (4.11)8 Entropy conservation law for nonlinear systems of PDEs with non-conservative terms
Proof . Using Maple TM , we find ∂ uu H ( u ) × ∂ u f ( u ) + ∂ u t ( u ) symmetric ⇔ t ( u ) = ρ e ρ h (cid:18) − T e T h (cid:19) v + (cid:90) [ − v∂ v F ( ρ h ,v ) + ρ h ∂ ρ h F ( ρ h ,v )] dv + F ( ρ h )and t ( u ) = − ρ e ρ h (cid:18) − T e T h (cid:19) + F ( ρ h ,v ) , with F , F two arbitrary functions and the condition on ∂ u H ( u )[ Z ( u ) + Z ( u )] is( ∂ u H ( u )[ Z ( u ) + Z ( u )]) T = − (cid:82) [ − v∂ v F ( ρ h ,v ) + ρ h ∂ ρ h F ( ρ h ,v )] dv − F ( ρ h ) − F ( ρ h ,v )000 = . One sees that the last equation imposes first F = 0 and thus F = 0. Reinjecting theseterms into the first equation gives the result.As explained in ( Step .a ), the Equation (4.10) guarantees the existence of anentropy flux G : u ∈ Ω (cid:55)→ R associated with the entropy H defined in Equation (4.7) bysolving the unknown functions t ( u ) and t ( u ).Therefore, for the entropy H defined in Equation (4.7), there is a unique decom-position which ensures the existence of a supplementary conservative equation which isgiven by ( ∂ u H [ C + C ]) T = v T · ∂ u f ( u ) + ρ e ρ h (cid:16) − T e T h (cid:17) v ρ e ρ h (cid:16) − T e T h (cid:17) , (4.12) ∂ u H [ Z + Z ] = . (4.13)It leads to the following entropy flux couple H = − ρ h s h − ρ e s e , (4.14) G = − ( ρ h s h + ρ e s e ) v. (4.15)The theory recovers the supplementary conservative equation already found in the lit-erature from the kinetic theory [20].
5. Conclusion
In the present contribution, we have proposed a theoretical frame-work for the derivation of supplementary conservation laws for systems of partial dif-ferential equation including first-order non-conservative terms - commonly encounteredin modeling of complex flows - thus extending the standard approach for systems ofconservation laws. Since our main objective is deriving an entropy supplementary con-servation law, we have used this framework to make a first step to extend the theory ofGodunov-Mock to such non-conservative systems.
EFERENCES
Acknowledgment.
The authors would like to acknowledge the support of aCNES/ONERA PhD Grant for P. Cordesse and the help of M. Th´eron (CNES). Theywould like to express their special thanks to F. Coquel, S. Kokh, V. Giovangigli andA. Murrone for their invaluable help and numerous pieces of advice during the writing ofthe paper. We also would like to thank discussions with J.M. H´erard, which promptedthis research path. Part of this work was conducted during the Summer Program 2018at NASA Ames Research Center and the support and help of Nagi N. Mansour is alsogratefully acknowledged.
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