Closure conditions for non-equilibrium multi-component models

Transcription

1 Closure conditions for non-equilibrium multi-component models Siegfried Müller, Maren Hante 2 and Pascal Richter 3 Preprint No. 44 December 204 Key words: multi-component flows, entropy, relaxation, phase transition, closure conditions, hyperbolicity, subcharacteristic property, Galilean invariance AMS Subject Classifications: 76T30, 76T0, 74A5, 35L60, 82C26 Institut für Geometrie und Pratische Mathemati RWTH Aachen Templergraben 55, D Aachen, Germany Institut für Geometrie und Pratische Mathemati, RWTH Aachen University, Templergraben 55, D Aachen, Germany address: mueller@igpm.rwth-aachen.de 2 Institut für Analysis und Numeri, Otto-von-Guerice-Universität Magdeburg, PSF 420, D-3906 Magdeburg, Germany address: maren.hante@ovgu.de 3 Center for Computational Engineering Sciences, RWTH Aachen University, Schinelstr. 2, D Aachen, Germany address: richter@mathcces.rwth-aachen.de The third author acnowledges funding by the Friedrich-Naumann-Stiftung für die Freiheit.

2 Closure conditions for non-equilibrium multi-component models Siegfried Müller Maren Hante 2 Pascal Richter 3 Institut für Geometrie und Pratische Mathemati, RWTH Aachen University, Templergraben 55, D Aachen, Germany address: mueller@igpm.rwth-aachen.de 2 Institut für Analysis und Numeri, Otto-von-Guerice-Universität Magdeburg, PSF 420, D-3906 Magdeburg, Germany address: maren.hante@ovgu.de 3 Center for Computational Engineering Sciences, RWTH Aachen University, Schinelstr. 2, D Aachen, Germany address: richter@mathcces.rwth-aachen.de. Abstract. A class of non-equilibrium models for compressible multi-component fluids in multi-dimensions is investigated taing into account viscosity and heat conduction. These models are subject to the choice of interfacial pressures and interfacial velocity as well as relaxation terms for velocity, pressure, temperature and chemical potentials. Sufficient conditions are derived for these quantities that ensure meaningful physical properties such as a non-negative entropy production, thermodynamical stability, Galilean invariance as well as mathematical properties such as hyperbolicity, subcharacteristic property and existence of an entropy-entropy flux pair. For the relaxation of chemical potentials a two-component and a three-component model for vapor-water and gas-water-vapor, respectively, is considered. Keywords. multi-component flows, entropy, relaxation, phase transition, closure conditions, hyperbolicity, subcharacteristic property, Galilean invariance. Math. classification. 76T30, 76T0, 74A5, 35L60, 82C26.. Introduction Flows of compressible multi-component fluids, where the single components may be in the liquid or the gas phase, respectively, have a wide range of applications. Difficulties in the modeling result from the interaction of the fluids, especially from the exchange of mass and energy across the phase interfaces. So the treatment of the phase interfaces is in the focus of the modeling. In the literature several models are available that are distinguished in sharp interface and diffuse interface models. A detailed survey of these models can be found in Zein [26]. Here our interest is on multi-component fluids derived from an ensemble averaging procedure of Drew [5]. A comprehensive introduction to these models can be found in the classical boo of Drew and Passman [6]. Baer and Nunziato [3] proposed a two-phase model for detonation waves in granular explosives. This model is a full non-equilibrium model, which means, each component has its own pressure, velocity and temperature and is governed by its own set of fluid equations. It was modified and generalized by several authors. For instance, Saurel and Abgrall [24] also included relaxation terms for the pressure and the velocities of the components. By instantaneous relaxation procedures equilibrium values for the pressure and the velocity can be found. Using further relaxation procedures to drive the temperatures and the Gibbs free energies chemical potentials into equilibrium mass transfer between the phases can be modeled, see Abgrall et al. [25, 22] or Zein et al. [27]. The third author acnowledges funding by the Friedrich-Naumann-Stiftung für die Freiheit.

3 S. Müller, M. Hante, & P. Richter There are simplified models available in the literature that can be derived from the above general model by assuming zero relaxation times, see [5]. Typically these are classified by the number of equations in case of two phases in one space dimension. For instance, a six-equation model with a single velocity is derived by assuming a zero velocity relaxation time. Assuming zero relaxation time for both the velocity and the pressure a five-equation model with mechanical equilibrium, i.e., single velocity and single pressure, is deduced in the asymptotic limit. The four-equation model has a single velocity, single pressure and also single temperature coinciding with the single-fluid reactive Euler equations. While the three-equation model is the system of Euler equations. It has single velocity, pressure, temperature, and also single Gibbs free energy, i.e., it is in full equilibrium. A detailed discussion of these models is beyond the scope of this wor. For this purpose the interested reader is referred to [26] and the references cited therein. Typically reduced models suffer from some short-comings. For instance, conservation of energy might be violated or the system looses its hyperbolicity. Therefore we prefer a full non-equilibrium model taing into account viscosity and heat conduction. For this purpose we consider a general class of non-equilibrium models that is a generalization of the three-phase model investigated by Hèrard, see Remar 7 in [4]. For instance, the Saurel-Abgrall approach [24] fits into this class. Characteristic for models based on ensemble averaging is the problem to close the set of equations, i.e., find appropriate interfacial pressures and interfacial velocity as well as relaxation terms for velocity, pressure, temperature and chemical potentials. Since the closing procedure is not unique, there is some freedom left for modeling. However, a reasonable model that is acceptable from a physical point of view has to be consistent with the fundamental principles of thermodynamics, e.g., the second law of thermodynamics. Besides this there are also constraints from a mathematical point of view that are related to existence and uniqueness of solutions to the model, e.g., the existence of entropy-entropy flux pairs. When it comes to the numerical solution additional properties are helpful for the design of appropriate schemes, e.g., the hyperbolicity of the transport operator or the sub-characteristic condition. The objective of this paper is to derive constraints for the closing terms such that the aforementioned physical, analytical and numerical properties hold for the nonequilibrium multi-component model. Similar investigations have been performed in case of two-phase models [2, 0, 26, 23] and three-phase models [4]. Here we do not confine ourselves to two and three phases but an arbitrary number of components. Drew and Passmann [6] consider multi-component fluids from a physical point of view but do not investigate analytical and numerical properties of the models. The paper is organized as follows. In Section 2 we introduce the non-equilibrium multi-component model and derive the model for the mixture as well the model at equilibrium. Then we rewrite these models in terms of primitive quantities in Section 3. Neglecting viscosity and heat conduction some mathematical properties of the models are investigated. In particular, we verify hyperbolicity and the sub-characteristic condition, see Section 4. Furthermore, a physical meaningful model should be Galilean invariant. This is investigated in Section 5. In Section 6 we are concerned with the entropies corresponding to the non-equilibrium model and the mixture model. From the 2nd law of thermodynamics we derive constraints for the definition of the interfacial velocity and pressures. By means of the physical entropy we define in Section 7 a convex entropy function and a compatible entropy flux that form an entropy-entropy flux pair. In particular, the compatibility conditions coincide with constraints for the interfacial pressures and the interfacial velocity derived from thermodynamic principles. In Section 8 we introduce the relaxation terms for mechanical and thermal relaxation as well as relaxation of chemical potentials. In particular, we verify that they are in agreement with the 2nd law of thermodynamics. 2

4 CLOSURE CONDITIONS FOR NON-EQUILIBRIUM MULTI-COMPONENT MODELS 2. Mathematical model First of all, we describe the full non-equilibrium model and then derive from this the mixture model and the equilibrium model. 2.. Non-equilibrium model The multi-component flow is described by a non-equilibrium model where all components are present in each point of the space-time continuum. Each component =,..., K has density ρ, velocity v and pressure p, The amount of each component is determined by its volume fraction α. The volume fractions are related by the saturation constraint α =, α 0,. 2. In analogy to the three-phase model of Hèrard [4] the fluid equations for each component can be written as t α ρ + α ρ v = S αρ,, 2.2 t α ρ v + α ρ v v T + α p I = 2.3 P,l α l + α T + S αρv,, l= t α ρ E + α ρ v E + p /ρ = 2.4 P,l V I α l + α v T q + S αρe,, l= taing into account viscosity and heat conduction via the stress tensor T and the heat flux q, but neglecting effects due to surface tension and gravity. In our notation E = e +v 2 /2 is the total specific energy with e the specific internal energy of component. There may be other contributions to be accounted for, see [6], p. 68 ff and 44 ff. In particular, the term P,l accounts for different pressures at the phase interface. Without loss of generality we may assume P, = Otherwise we replace P,l by P,l P, due to the saturation condition 2.. The interfacial velocity is denoted by V I. Obviously, the equations cannot be written in conservative form. Finally, the fluid equations are supplemented by an equation of state p = p ρ, e resp. e = e ρ, p 2.6 for each of the components. The evolution of the volume fractions is characterized by the non-conservative equation t α + V I α = S α,, =,..., K. 2.7 Due to the saturation condition 2. we only need K equations. Without loss of generality we express α K by the other volume fractions, i.e., K α K = K α, α K = K α, S α,k = S α,

5 S. Müller, M. Hante, & P. Richter The source terms S α,, S ρ,, S ρv, and S ρe, on the right-hand sides of 2.2, 2.3, 2.4 and 2.7 describe the relaxation process due to mass, momentum, energy transfer and volume fraction between the different components corresponding to the relaxation of velocity, pressure, temperature and chemical potentials, ξ {v, p, T, µ}, i.e., S α, := ξ S ξ α,, S αρ, := ξ S ξ αρ,, S αρv, := ξ S ξ αρv,, S αρe, := ξ S ξ αρe,. 2.9 These depend on the specific components at hand that will be discussed in Section 8. So far, the model is not yet closed. For this purpose, we have to find closing conditions for the pressures P,l, the interfacial velocity V I and the relaxation terms S α,, S αρ,, S αρv, and S αρe,. In the following sections we will derive appropriate constraints. However, these will not specify a unique model but some options are still remaining for the choice of the interfacial velocity, the relaxation terms, the stress tensor and the heat conduction Mixture model From the non-equilibrium model we can derive the equations for the mixture. For this purpose we introduce the mixture quantities p := α p, ρ := α ρ, v := ρ α ρ v, 2.0 for pressure, density and velocity. Accordingly, we define the specific internal energy, the specific total energy and the specific total enthalpy of the mixture as e := α ρ e, E := ρ ρ α ρ E, H := ρ α ρ H = E + p ρ 2. with H := E + p /ρ the total enthalpy of component. The stress tensor and the heat flux of the mixture are determined by T := α T, q := α q. 2.2 In order to ensure conservation of mass, momentum and energy of the mixture the relaxation terms 2.9 have to satisfy the conservation constraints S ξ α, = 0, K S ξ αρ, = 0, K S ξ αρv, = 0, K S ξ αρe, = for each relaxation type ξ {v, p, T, µ}. In addition, we need that the interfacial pressures satisfy P l := P,l P = const l =,..., K. 2.4, l 4

6 CLOSURE CONDITIONS FOR NON-EQUILIBRIUM MULTI-COMPONENT MODELS Then by summation of the single-component fluid equations 2.2, 2.3, 2.4 and employing the saturation constraint 2. as well as the conservation constraints 2.3 and 2.4 we obtain t ρ + ρ v = 0, 2.5 t ρ v + ρ v v T + p I = T 2.6 K α ρ v v v v, T t ρ E + ρ v E + p/ρ = v T q 2.7 K K α T v v α ρ H H v v. We note that there are contributions corresponding to the slip between the mixture velocity v and the velocities of the components v. In the multi-component model of Drew and Passman these terms are added to the mixture stress tensor and the mixture heat flux, see [6], p In contrast to the non-equilibrium model, the mixture model is in conservative form if and only if the conditions 2.3 and 2.4 hold Equilibrium model If the relaxation processes are much faster than the transport and dissipation effects, then the fluid can be considered to be at equilibrium. This state is characterized by vanishing relaxation terms, i.e., S α, = 0, S αρ, = 0, S αρv, = 0, S αρe, = At equilibrium the velocities, pressures and temperatures coincide, i.e., v =... = v K = v, p =... = p K = p, T =... = T K = T, 2.9 and the chemical potentials of reacting components are equal. In particular, for the interfacial pressures and interfacial velocity it holds P,l = p, l, V I = v Then the mixture model 2.5, 2.6 and 2.7 reduces to the the equilibrium model t ρ + ρ v = 0, 2.2 t ρ v + ρ v v T + p I = T, 2.22 t ρ E + ρ v E + p/ρ = v T q Note that by definition 2.0 and 2.2 the mixture pressure p, the mixture stress tensor T and the mixture heat flux q depend on the volume fractions α. These are determined by the algebraic conditions 2.8 and Primitive variables For the verification of some physical and mathematical properties it will be helpful to rewrite the systems of equations for the non-equilibrium, mixture and the equilibrium model in terms of primitive quantities 5

7 S. Müller, M. Hante, & P. Richter 3.. Non-equilibrium model By means of the system 2.2, 2.3, 2.4 and 2.7 we derive evolution equations for the density ρ, the velocity v and the pressure p for each component. Inserting the evolution equation for the volume fraction 2.7 into the continuity equation 2.2 we obtain t ρ + ρ α v V I α + v ρ + ρ v = α S αρ, ρ S α,. 3. From the momentum equation 2.3 we deduce with 2.2 Cauchy s equation of motion t v + v v + p + p α = P,l α l + α T + ρ α ρ α ρ l= α ρ S αρv, S αρ, v. 3.2 Here the gradient of the velocity is defined as v = v,,..., v,d T. Then we immediately obtain the evolution equation for the inetic energy u := v 2 /2 p t u + v v v + v p + v α = 3.3 ρ α ρ v P,l α l + α T + v S αρv, S αρ, v. α ρ α ρ l= Since the total energy is composed of the internal energy and the inetic energy, we derive the evolution equation for the internal energy e = E u from the energy equation 2.4, where we employ 2.2 and 3.3. Finally we obtain with the relaxation term t e + v e = α ρ d l= ρ i,l= P,l v V I α l p ρ v v,l x i T l,i α ρ α q + α ρ S e, S e, := S αρe, v S αρv, + S αρ, u e. 3.5 Next we derive the evolution equation for the pressure p. For this purpose we first note that for any equation of state 2.6 the following relation holds dp = p / ρ dρ + p / e de. 3.6 By means of the continuity equation 3. and the energy equation 3.4 we then derive from 3.6 t p + with the relaxation term ρ C,l 2 α v V I α l + v p + ρ c 2 v = 3.7 l=,l p / e d v,l T l,i α q ρ x i α ρ + S p, α ρ i,l= S p, := ρ p / ρ S αρ, ρ S α, + p / e S e,

8 CLOSURE CONDITIONS FOR NON-EQUILIBRIUM MULTI-COMPONENT MODELS Here the interfacial sound speed and the phase sound speed are defined as C 2,l := p / e P,l /ρ 2 + p / ρ, c 2 := p / ρ + p /ρ 2 p / e Mixture model Similar to the non-equilibrium model we derive evolution equations for the mixture quantities ρ, v, e and p defined by 2.0 and 2. from the evolution equations 2.5, 2.6 and 2.7. First of all, we determine Cauchy s equation of motion from the momentum equation 2.6 where we use the continuity equation 2.5 and the constraint 2.4: t v + v v + ρ p = ρ T ρ P l α l K ρ α ρ v vv v T. 3.0 l= Since definition 2. of the mixture energy e implies that ρe = K α ρ e, we obtain by 3.4 and 2.2 the evolution equation for the internal energy t e + v e + ρ ρ α p v = ρ α ρ e v v + ρ d i,l= P,l v V I α l 3. l, v,l x i α T l,i ρ q ρ v S αρv, S αρ, u. Finally we determine the evolution equation for the mixture pressure p. Applying the time derivative to the definition 2.0 of p and using 2.7 and 3.7 we obtain t p + v p + ρc 2 v = 3.2 K C,l 2 v V I α l α v p p + v v p ρ l=,l α ρ c 2 v v + p / e α ρ + p S α, + ρ S p,. Here the sound speed of the mixture is defined as 3.3. Equilibrium model c 2 := ρ d v,l T i,l α q x i i,l= α ρ c In case of the equilibrium model the evolution equations for the primitive variables can be directly determined from those of the mixture model where we mae use of the equilibrium assumptions 2.8, 2.9 and 2.20 and the saturation condition 2.. Then Cauchy s equation of motion reads t v + v v + ρ p = ρ T

9 S. Müller, M. Hante, & P. Richter The energy equation reduces to t e + v e + p ρ v = ρ Finally the pressure equation becomes d i,l= v l T l,i q. 3.5 x i ρ t p + v p + ρc 2 v = 3.6 d p / e α ρ v l T i,l α q x. i i,l= 4. Mathematical properties: hyperbolicity and sub-characteristic condition Neglecting viscosity and heat conduction as well as relaxation processes in the fluid equations introduced in Section 2 the models reduce to first order systems describing transport effects only. Therefore these systems should be hyperbolic. This ensures that all wave speeds are finite and the system may be locally decoupled. From a mathematical point of view, this property is helpful in the construction of Riemann solvers. Therefore we determine the eigenvalues and eigenvectors corresponding to the non-equilibrium model. From a numerical point of view the relation between the eigenvalues of the non-equilibrium and the equilibrium model are of special interest. 4.. Non-equilibrium model Starting point are the evolution equations for the primitive variables 2.7, 3., 3.2 and 3.7. The corresponding first order system then reads t α + t ρ + t v + t p + d V I,i α x i = 0 4. i= d ρ v,i V I,i α ρ v,i + v,i + ρ = 0, α i= x i x i x i 4.2 d v α l v,i + P,l p e i,d + p e i,d = 0, x i= i α ρ x i ρ x i l=,l 4.3 d p ρ v,i + C,l 2 x i α v,i V I,i α l + ρ c 2 v,i = 0, x i x i 4.4 i= l=,l where e i,d R d denotes the unit vector in the ith coordinate direction. In order to characterize hyperbolicity of this system we consider its projection onto normal direction ξ := x n for arbitrary unit direction n R d. Introducing the vector of primitive variables w = α,..., α K, w T,..., w T K T, w = ρ, v T, p T 4.5 the projected system can be written in quasi-conservative form as t w + B n w w ξ =

10 CLOSURE CONDITIONS FOR NON-EQUILIBRIUM MULTI-COMPONENT MODELS The matrix B n is determined by the bloc matrix V I,n I K d A,n B,n B n := B i n i =. i= A K,n... B K,n, 4.7 with the blocs defined as ρ α v,n V I,n e T,K T K δ,k v,n ρ n T 0 A,n := nβ T, B,n := 0 d v,n I d ρ n. 4.8 ρ α v,n V I,n γ T 0 ρ c 2 nt v,n Here e,k is the th unit vector in R K. In particular, we mae the convention e K,K = 0. Furthermore, I d and I K are the unit matrices in R d d and R K K, respectively, and 0 d and K are vectors in R d and R K with value 0 or, respectively. The vectors β and γ are defined by its components l =,..., K as β,l := α ρ P,l p δ,l P,K p δ,k, 4.9 γ,l := C 2,l δ,l C 2,K δ,k 4.0 with δ,l the Kronecer symbol. The normal components of the velocities and the interfacial velocity are defined as v,n := v n, V I,n := V n. 4. Obviously, the eigenvalues of the matrix 4.7 can now be easily computed where we employ the bloc structure: K detb n λ I = detv n,i I K λ I K detb,n λ I d+2 = Since the matrices B,n coincide with those in case of a single-phase fluid, we then compute Hence we obtain the following eigenvalues: detb,n λ I d+2 = v,n λ d v,n λ 2 c λ I, = V I,n, =,..., K 4.4 λ,i = v,n, =,..., K, i =,..., d 4.5 λ,± = v,n ± c, =,..., K. 4.6 Motivated by the bloc structure of the matrix B n we mae the following ansatz for computing the corresponding left and right eigenvectors with blocs R 0 I,n R I,n R,n R n :=. R K I,n... R K,n R 0 I,n := κ 0 I K, L 0 I,n R K K, L 0 I,n, L L I,n L,n n :=. L K I,n... L K,n R I,n, L I,n R d+2 K, R,n, L,n R d+2 d+2, =,..., K

11 S. Müller, M. Hante, & P. Richter Here the matrix R n and L n are composed of the right and left eigenvectors in its columns and rows, respectively. To determine the blocs R,n we note that λ,i and λ,± are also eigenvalues of the matrix B,n and the eigenvalue problem for B n decouples into eigenvalue problems for the matrices B,n corresponding to a single component. In a first step, we therefore compute the eigenvectors to these sub-problems where we first determine an orthonormal basis {n, t,..., t d } of R d such that t i t j = δ i,j and t i n = 0. Then the right and left eigenvectors to the eigenvalues 4.5 and 4.6 are r,d =, 0 T d, 0T, r,i = 0, t T i, 0 T, i =,..., d, r,± =, ±c /ρ n T, c 2 T, 4.8 l,d =, 0 T d, c 2 T, l,i = 0, t T i, 0 T, i =,..., d, l,± = 0.5c 2, ±c ρ n T, T. 4.9 Thus there exists an eigenvalue decomposition of the matrix B,n, i.e., L,n B,n R,n = Λ,n, 4.20 where L,n and R,n are defined by the left and right eigenvectors and Λ,n is a diagonal matrix with eigenvalues on the diagonal R,n := c /ρ n t... t d 0 d c /ρ n, 4.2 c c 2 L,n := c2 T 0 2c 2 c ρ n 2c 2 t... 2c 2 t d 0 d c ρ n, Λ,n := v,n c 0 T d 0 0 d v,n I d 0 d T d v,n + c To calculate the eigenvectors to the multiple eigenvalue λ I,i we employ the nowledge of the matrices R,n. According to the bloc structure of the matrix R n the matrix of corresponding right eigenvectors needs to satisfy B n λ I,i IR n = 0 B,n λ I,i I d+2 R I,n = A,n R 0 I,n = κ 0 A,n, =,..., K. Assuming that the eigenvalue λ I,i does not coincide with one of the eigenvalues λ,i and λ,±, then B,n λ I,i I d+2 is regular and there exists a unique solution for R I,n. For its representation we introduce κ 0 := K α l σ, κ := l= K l=,l with δ n := v,n V I,n and σ := δ n2 c 2. Then we obtain α l σ, =,..., K R 0 I,n := κ 0 I K, 4.24 δ R n 2 ρ γ T + α ρ c 2 βt I,n := κ nα β T + γt δn 4.25 σ e T i,k δ,k T K δ,k + α β T + γt Since α 0, according to 2., these matrices are regular, i.e., the columns are linearly independent, if and only if σ 0 =,..., K,

12 CLOSURE CONDITIONS FOR NON-EQUILIBRIUM MULTI-COMPONENT MODELS holds. This condition is referred to as the non-resonance condition, see [4] in case of a two-velocitytwo-pressure model in two-phase flows. Thus the corresponding left eigenvectors are determined by the rows of the inverse of R n. Since L n R n = I, the blocs turn out to be L,n = R,n, =,..., K, 4.27 L 0 I,n = R 0 I,n = κ 0 I K, 4.28 L K I,n = L,n R K I,nR 0 I,n = κ 0 L,n R K I,n, =,..., K, 4.29 and we obtain for the right eigenvectors r I,i := κ 0 r I,0 T, r I,i T,..., r K I,i T T, i =,..., K with vectors r I,i = x,i, y T,i, z,i T and their components z,i = ρ δ n 2 γ,i + α ρ c 2 β,i /α σ, 4.3 y,i = n α β,i + γ,i δ n /α σ, 4.32 x,i = σ δ,i δ,k δ,k + α β,i + γ,i ρ α σ These are well-defined also in case of α = 0 or σ = 0. Similar to 4.30 the left eigenvectors are then given by with and l I,i := l I,0 T, l I,i T,..., l K I,i T T, i =,..., K l 0 I,i = K α l σ l e i,k, l I,i = l= 2c 2 α σ a i, 0,..., 0, bi T a i := c ρ α β,i + γ,i δ n + σ δ,i δ,k δ,k α β,i γ,i, b i := 2c 2 ρ δ n 2 γ,i + α c 2 β,i 2σ δ,i δ,k δ,k + 2α β,i + γ,i. After having determined the eigenvalues and the corresponding linearly independent right and left eigenvectors we finally end up with the eigenvalue decomposition of the matrix B n L n B n R n = Λ n 4.35 with the bloc-diagonal matrix Λ n = diagλ 0,n, Λ,n,..., Λ K,n and Λ 0,n := V I,n I K. To verify this decomposition we mae use of the identity B,n R I,n = R I,n Λ 0,n A,n R 0 I,n. To conclude the investigation on the hyperbolicity we summarize the findings in the following theorem. Theorem 4.. Hyperbolicity Let the interfacial pressures satisfy the conditions 2.5 and 2.4. Let the interfacial velocity V I not coincide with one of the eigenvalues λ,i and λ,± of B n. Then the first order system 4., 4.2, 4.3 and 4.4 is hyperbolic, i.e., i the eigenvalues of the matrix 4.7 are all real but not necessarily distinct and ii there exists a system of linearly independent left and right eigenvectors with 4.35 if and only if the non-resonance condition 4.26 is satisfied. This theorem holds true also for the non-equilibrium model 2.2, 2.3, 2.4 and 2.7 neglecting viscosity and heat conduction as well as relaxation processes, because eigenvalues are invariant under a regular, bijective change of variables and the corresponding eigenvectors can be determined by scaling of the original eigenvectors by the Jacobian of the transformation and its inverse, respectively. Finally we want to remar that the eigenvectors coincide with the one given in [26, 2] in case of a 7-equation model in one space dimension K = 2, d =.

13 S. Müller, M. Hante, & P. Richter 4.2. Equilibrium model Similar to the non-equilibrium case we can determine the eigenvalues of the equilibrium model. Starting point are the evolution equations 2.2, 2.22 and The corresponding first order system then reads d ρ t ρ + v i + ρ v i = 0, 4.36 x i= i x i d v t v + v i + x i= i ρ e p i = 0, 4.37 x i d p t p + v i + ρ c 2 v i = 0, 4.38 x i x i i= Again we consider its projection onto normal direction ξ := x n for arbitrary unit direction n R d that can be written in quasi-conservative form t w + B n w w ξ = with the vector of primitive variables w = ρ, v T, p T and matrix v n ρ n T 0 B n := 0 d v n I d ρ n ρ c 2 n T v n The normal component of the velocity is defined as v n := v n. 4.4 The eigenvalues of B n are then characterized by the roots of the characteristic polynomial Hence we obtain the following eigenvalues: detb n λ I d+2 = V n λ d v n λ 2 c 2. λ ± = v n ± c, λ i = v n, i =,..., d With regard to a stable numerical discretization of the non-equilibrium model in the limit of vanishing relaxation terms, the so-called sub-characteristic condition has to hold true. This condition was originally introduced by Liu [8]. For this purpose we evaluate the eigenvalues 4.4, 4.5 and 4.6 with respect to an equilibrium state, i.e.,2.8, 2.9 and 2.20 hold, λ I, = V I,n = v I n = v n = v n, =,..., K λ,i = v,n = v n = v n = v n, =,..., K, i =,..., d λ,± = v,n ± c = v n ± c, =,..., K. Here the bar indicates evaluation with respect to an equilibrium state. Then by definition 3.3 and 2.0 of the mixture sound speed and the mixture density we conclude from the positivity of the densities ρ and the volume fractions α A straight forward estimation gives c [min,...,k c, max,...,k c ] =: [c min, c max ]. min,...,k λ,+ λ + max,...,k λ,+, min,...,k λ, λ max,...,k λ,. 2

14 CLOSURE CONDITIONS FOR NON-EQUILIBRIUM MULTI-COMPONENT MODELS In the second estimate we use that c [ max,...,k c, min,...,k c, ] = [min,...,k c, max,...,k c ]. This immediately implies that the following theorem holds true. Theorem 4.2. Sub-characteristic condition Let α, ρ and c, =,..., K, be non-negative. Then the eigenvalues 4.4, 4.5, 4.6 and 4.43, 4.43, 4.43 of the non-equilibrium model and the equilibrium model, respectively, evaluated with respect to an equilibrium state, i.e., 2.8, 2.9 and 2.20 hold, satisfy the sub-characteristic condition λ i min [ λ ± min min,...,k λ I,, max min,...,k i=,...,d λ,±, λ,±,...,k,...,k ]. λ,i, max max,...,k λ I,, max,...,k i=,...,d λ,i = {v n }, Note that similar results have been proven recently by Flatten and Lund [9] for a hierarchy of two-phase relaxation models. 5. Frame invariance and objectivity Since the results of an experiment should be independent of the observer s position in the Euclidean space, a physical meaningful model should reflect this behavior. This property is referred to as frame indifference and objectivity in the literature, cf. [6], p. 3 ff, and [7]: performing the general Euclidean change of frame t = t + a, x = x 0t + Qtx x 0, 5. with constant values a and x 0 and Q an orthogonal matrix, i.e., then a scalar f, a vector u and a tensor T are called objective, if QQ T = QQ T = I, 5.2 f t, x = ft, x, u t, x = Qtut, x, T t, x = QtT t, xq T t. 5.3 From the orthogonality of Q we conclude A := QQ T = Q Q T = A T, 5.4 i.e., A is a sew-symmetric matrix and it holds A ii = 0, A ij = A ji. Then under the general Euclidean change of frame 5. the velocities v := ẋ and v := ẋ of the different frames are lined via v ẋ 0 = Qv ẋ 0 + Ax x 0 equiv. v ẋ 0 = Q T v ẋ 0 Ax x Thus, the velocity and the acceleration are not objective under a general Euclidean change of frame. For the rate of deformation D and the rotation tensor W defined as D := xv + xv T = D T, 2 W := xv xv T = W T we conclude from 5. D := xv + xv T = QDQ T, 2 W := xv xv T = QW Q T + A T, i.e., the rate of deformation is objective but the rotation tensor is not objective under a general Euclidean change of frame. 3

15 S. Müller, M. Hante, & P. Richter 5.. General Euclidean change of frame In order to rewrite the fluid equations 2.2, 2.3, 2.4 and 2.7 in terms of the general Euclidean change of frame we need some rules of calculus that will be summarized below. First of all, we determine from 5. the Jacobian of the transformation t, x t, x = 0 T d ẋ 0 + Qx x 0 Q, Then we introduce the change of variables t, x t, x = 0 T d ẋ 0 + Q T x x 0 QT ẋ 0 Q T. 5.8 ft, x = ft a, x 0 + Q T tx x 0t, 5.9 ut, x = ut a, x 0 + Q T tx x 0t, 5.0 T t, x = T t a, x 0 + Q T tx x 0t. 5. for a scalar f, a vector u and a tensor T. Next we consider the derivatives of these variable transformations. For a scalar we derive from 5.8 and 5.8 f = f t t + x f ẋ 0 + Ax x 0, 5.2 xf = Q T x f. 5.3 Applying these equations componentwise we obtain for a vector u = u t t + x u ẋ 0 + Ax x 0, 5.4 x u = x Qu, 5.5 ui xu = x u Q. 5.6 x j ij In particular, if u is an objective vector, i.e., u = Q T u according to 5.3, then we have u u = Q T t t Au + x u Qẋ 0 + Ax x 0, 5.7 xu = Q T x u Q. 5.8 Finally, we consider a tensor matrix T with t i and t i the ith column and row of T, respectively. Then we obtain T = T t t + x t ẋ 0 + Ax x 0,..., x t dẋ 0 + Ax x 0, 5.9 x T x t,..., x t d T = x T Q T For T = fi the divergence reads x fi = Q T x fi. 5.2 In case of an objective tensor, i.e., T = Q T T Q according to 5.3, we conclude from x t m = Q T n x t nq nm T t T = Q T t Q AT Q + T Q Q n x t x 0 + Ax x 0,..., Q nd x t d x 0 + Ax x 0, x T = Q T x T. n

16 CLOSURE CONDITIONS FOR NON-EQUILIBRIUM MULTI-COMPONENT MODELS 5.2. Galilean transformation It is well-nown that the fluid equations for a single phase are not invariant under a general Euclidean change of frame. For instance, Coriolis forces enter in case of a time-dependent rotation. However, the fluid equations are invariant under a Galilean transformation where we choose ẋ 0 = 0, Q = 0 equiv. Q = const, x 0t = c 0 + c t, c, c 2 = const 5.24 in 5.. For this transformation the velocity vector is still not objective but the acceleration and the rotation tensor are objective, because A = 0. In the following we will confine ourselves to a Galilean transformation and derive constraints for the source terms S α,, S αρ,, S αρv, and S αρe,. First of all, we derive from the evolution equation 2.7 of the volume fractions using 5.2 and 5.3 and assuming that the volume fractions are objective, i.e., α = α, t α + V I x α = S α,, =,..., K, 5.25 where the interfacial velocity and the source term are given as V I := ẋ 0 + QV I + Qx x 0 = ẋ 0 + QV I = c + QV I, 5.26 S α, := S α, Next we consider the evolution of mass. Assuming that the mass is objective, i.e., ρ = ρ, and using 5.2 and 5.5 we derive from 2.3 with source term t α ρ + x α ρ v = S α ρ,, 5.28 S α ρ, := S αρ, Note that 5.25 and 5.28 hold true for a general Euclidean change of frame 5.. The transformation of the momentum equation 2.3 is cumbersome. It significantly simplifies in case of a Galilean transformation. Starting from 2.3 one has to employ 5.4, 5.5, 5.5, 5.3, 5.23 and incorporate Assuming that the pressures p and P,l and the stress tensors T are objective, i.e., p = p, P,l = P,l and T = QT Q T, we obtain K t α ρ v + x α ρ v v T + α p I = P,l x α l + x α T +S α ρ v,, 5.30 with source term l= S α ρ v, := S αρ, ẋ 0 + Q S αρv,. 5.3 Finally we apply the Galilean transformation to the energy equation 2.4. Since the velocity is not an objective vector, the inetic energy in the Galilean frame becomes e in, = e in, + ẋ 0 v 2 ẋ 0 2, e in, := 2 v using 5.5. Thus the total energy and the total enthalpy are E := e + 2 v2 = E ẋ 0 v + 2 ẋ 0 2, E := e + 2 v 2, 5.33 H := E + p ρ = H ẋ 0 v + 2 ẋ 0 2, H := E + p ρ, 5.34 where we also assume objectivity of the internal energy, i.e., e = e. Again, after some tedious wor of calculus using 5.2, 5.5, 5.3, 5.5 and incorporating 5.28, 5.30, the energy equation 2.4 5

17 S. Müller, M. Hante, & P. Richter in the Galilean frame becomes t α ρ E + x α ρ v E + p /ρ = 5.35 P,l V I x α l + x α v T q + S α ρ E,, l= with source term Sα ρ E, := S αρe, + QS αρv, + S αρ, ẋ 0 x 0 2 ẋ 0 2 S αρ, Here we again assume objectivity of the pressure P,l and the heat flux q, i.e., P,l q = Qq. Thus we have proven the following = P,l and Theorem 5.. Galilean Invariance Let the following assumptions hold true α, ρ, p, e and P,l are objective scalars, 2 q, T are objective vectors and tensors, respectively, 3 all material parameters, e.g., µ, λ, are objective, 4 the source terms 5.27, 5.29, 5.3, 5.36 are invariant under a Galilean transformation, i.e., S α, = S α,, S αρ, = S αρ,, S αρv, = S αρv,, S αρe, = S αρe, Then the non-equilibrium model 2.2, 2.3, 2.4 and 2.7 with velocity v = ẋ 0 + Qv, interfacial velocity 5.26 and total energy 5.33 is Galilean invariant. This also holds true for the mixture model and the equilibrium model because they are derived from the non-equilibrium model by summation. 6. Thermodynamical properties: 2nd law of thermodynamics From a physical point of view, a model is admissible if it is in agreement with the principles of thermodynamics. For this purpose we first derive the entropy law for the non-equilibrium model. Then we determine the entropy production terms of the mixture. To be consistent with the 2nd law of thermodynamics we have to chec the sign of the entropy production terms. This will provide us with admissibility criteria for the interfacial pressures and velocity as well as the relaxation terms. 6.. Entropy In order to investigate thermodynamical properties of the non-equilibrium model 2.2, 2.3, 2.4 and 2.7, we assume that the entropy of each component satisfies de = T ds p dτ, 6. where τ := /ρ is the specific volume of component. Thus the pressure and the temperature are the partial derivatives of e τ, s that are assumed to be positive, i.e., p τ, s = e τ τ, s 0, T τ, s = e s τ, s 0. 6

18 CLOSURE CONDITIONS FOR NON-EQUILIBRIUM MULTI-COMPONENT MODELS Furthermore, to ensure thermodynamical stability we assume that the Hessian of e function with respect to τ and s, i.e., is a convex 2 e 2 τ τ, s 0, 2 e 2 s τ, s 0, Finally, the third law of thermodynamics implies 2 e 2 τ, s 2 e 2 2 e τ 2 τ, s τ, s. s τ s τ 0, s 0. Assuming that p and T are strictly positive, then e becomes a monotone function in τ and s and we may change variables, i.e., s = s τ, e satisfying with partial derivatives s τ τ, e = p T > 0, T ds = de + p dτ 6.2 s e τ, e = T > It is well-nown that s = s τ, e is a concave function, i.e., the Hessian is negative-definite 2 s 2 s 2 s 2 τ, e 0, τ 2 τ, e 0, e 2 τ, e 2 s 2 2 e τ 2 τ, e τ, e, 6.4 e τ e if and only if e τ, s is a convex function, i.e., thermodynamic stability holds Entropy equation In order to derive the entropy equation we rewrite 6.2 as T ds = de p ρ 2 dρ. 6.5 By means of the evolution equations 3. and 3.4 for the density and the internal energy we then deduce the entropy law t s + v s = K P,l v V I α l p v V I α α ρ T l= 6.6 d v,l +α T l,i α q x + S s, i with the relaxation term S s, := S e, p S αρ, + p S α,. 6.7 ρ For the volume specific entropy we then obtain together with 2.2 i,l= t α ρ s + α ρ s v + α q T = K P,l v V I α l 6.8 T with the relaxation term d v,l +p v V I α + α T l,i α q x i T T + S αρs, i,l= l= S αρs, := T S s, + s S αρ,

19 S. Müller, M. Hante, & P. Richter Introducing the entropy of the components of the mixture ρs := α ρ s 6.0 we finally obtain with 2.2 the entropy law of the mixture K K t ρs + α ρ s v + α q T = α Σ + α + Π + S αρs,, 6. where the production terms are defined as Π := K P,l v V I α l + p v V I α, 6.2 T l= Σ := T d i,l= v,l x i T l,i, 6.3 := T 2 q T. 6.4 Note that the total entropy of a homogeneous mixture is determined by the sum of ρs and the nonnegative mixture entropy, [9], p We discuss the mixture entropy in the context of the relaxation terms for chemical potentials for a three-component mixture, see Section Entropy production According to the 2nd law of thermodynamics the production terms 6.2, 6.3 and 6.4 have to be non-negative. In the subsequent sections we will derive sufficient conditions that ensure thermodynamical compatibility Entropy production due to viscosity and heat conduction In order to verify the physically admissible sign of the entropy production term Σ we have to specify the viscous stress tensor T for each component. For an isotropic Newtonian fluid the stress tensor reads T = µ v + v T 2 3 v I. 6.5 Thus the components of this symmetric tensor are T l,i = µ v,l x i + v,i x l 2 3 Then we compute for the entropy production term 6.3 d v,j δ l,i = T i,l. x j j= T µ Σ = 2 d i v,l i= l= x i + v 2,i + 2 x l 3 Γ 8

20 CLOSURE CONDITIONS FOR NON-EQUILIBRIUM MULTI-COMPONENT MODELS with Γ := = d 2 v,i d v,j v,i 2 x i= i x j x i j=,j i v, 2 2 x, d = v, 2 v,2 2 v, x + x 2 + x v 2,2 x 2, d = 2. v, x v 2,2 v,2 x 2 + x 2 v 2,3 v, x 3 + x v 2,3 x 3, d = 3 Obviously, the following theorem holds true. Theorem 6.. Entropy production due to viscosity The viscous stress tensors are determined by 6.5. Let the temperatures T and the viscosity coefficients µ, =,..., K, be non-negative. Then the production terms 6.3 are non-negative. In addition, because of the saturation condition 2., the entropy production due to viscosity K α Σ is non-negative. The heat fluxes are modeled by Fourier s law of heat conduction for a fluid with isotropic material property q = λ T. 6.6 Then the entropy production term 6.4 reads From this we directly conclude = T 2 λ T T. Theorem 6.2. Entropy production due to heat conduction The heat fluxes are determined by 6.6. Let the temperatures T and the heat conduction coefficients λ, =,..., K, be non-negative. Then the production terms 6.4 are non-negative. In addition, because of the saturation condition 2., the entropy production due to heat conduction K α is non-negative Interfacial velocity and pressure To investigate the admissibility of the production terms Π we first mae use of the assumptions 2.5 and 2.4 for the interfacial pressures. Then these terms become Π = T K l=,l P,l p v V I α l. Obviously, we cannot control the sign of Π. According to the conservation constraints 2.4 all P,l, l, are coupled. Therefore we determine the interfacial pressures P,l and the interfacial velocity V I such that the sum Π := K Π vanishes. For this purpose we substitute α K by the other gradients using 2.8. According to the saturation condition 2. the gradients α, =,..., K, are linearly independent. Thus rearranging the terms in Π with respect to the K gradients of α the coefficients in front of these gradients must be zero when Π vanishes. This yields the following K conditions K, l P,l p P,K p v V I + P K,l p K v K V I = T T K 9

21 S. Müller, M. Hante, & P. Richter for l =,..., K. Next we assume that the interfacial velocity is a convex combination of the single component velocities v, i.e., V I = β v, β [0, ], β =. 6.8 This is motivated by Gallouët et al. [0] and Hèrard [4] for a two-phase and a three-phase model, respectively. Then we may rewrite the velocity differences in 6.7 as v K V I = i= j= i β j v i v i+ + K i= j=i+ β j v i v i+. Rearranging 6.7 in terms of the linearly independent differences v i v i+, i =,..., K, we obtain the following conditions i K T P,l p δ l, P,K p =i+ or, equivalently, i δ l, T K =i+ i j=i+ β j i P,l p δ l, P,K p + P K,l p K T T K T δ l, T p δ l, j=i+ j=i+ i j= β j β j β j + P,l P,l + K =i+ i T K =i+ T K j=i+ i j= β j β j P,K i p δ l, p K T T K P,K j= β j T K j= β j = 0 i β j P K,l = 6.20 for l, i =,..., K. This gives K 2 equations for K K unnowns P,l,, l =,..., K, l. Thus we need additional K equations to ensure uniqueness for fixed parameters β,..., β K. These equations are determined by the conservation constraints 2.4 that read These equations are equivalent to, l, l P,l = P I = const, l =,..., K. 6.2 K P,l P,K = 0, l =,..., K, 6.22 Then 6.20 and 6.22 form a linear system for the interfacial pressures. For a two-component model, i.e., K = 2, the solution is given by P,2 = P 2, = p β 2 /T + p 2 β /T 2 β 2 /T + β /T 2 = P I j= 20

22 CLOSURE CONDITIONS FOR NON-EQUILIBRIUM MULTI-COMPONENT MODELS For a three-component model Hérard [4] proved that 6.20 and 6.22 has a unique solution that is given by and, thus, P,2 = β p 2 T + β 2 p T 2 + β 3 p T 3 β T + β 2 T 2 + β 3 T 3, P 2, = β p 2 T + β 2 p T 2 + β 3 p 2 T 3 β T + β 2 T 2 + β 3 T 3, P,3 = β p 3 T + β 2 p T 2 + β 3 p T 3 β T + β 2 T 2 + β 3 T 3, P 3, = β p 3 T + β 2 p 3 T 2 + β 3 p T 3 β T + β 2 T 2 + β 3 T 3, 6.24 P 2,3 = β p 2 T + β 2 p 3 T 2 + β 3 p T 3 β T + β 2 T 2 + β 3 T 3, P 3,2 = β p 3 T + β 2 p 3 T 2 + β 3 p 2 T 3 β T + β 2 T 2 + β 3 T 3 P I = β T p 2 + p 3 + β 2 T 2 p + p 3 + β 3 T 3 p + p β T + β 2 T 2 + β 3 T 3 As a consequence, we observe that in general P,l = P l, only holds at pressure equilibrium where P I = 2p. Whether there exists a unique solution to 6.20 and 6.22 for K 4 and arbitrary convex combination 6.8 for the interfacial velocity V I is still open. However, for the multi-component model the following result can be proven. Theorem 6.3. Entropy production due to interfacial states Let the assumptions 2.5 and 2.4 hold true. Then the production term Π = K Π vanishes provided that the interfacial pressures P,l are a solution to the linear system 6.20 and the interfacial velocity V I satisfies 6.8. In particular, if we choose β =, β 2 =..., β K = 0, then the interfacial pressures P,l = p, = 2,..., K, l =,..., K P,K = p K, P,K = p, = 2,..., K, P K,l = p K, l =,..., K, P,l = P l,k = p l, l = 2,..., K are a solution of Thus the interfacial pressure and the interfacial velocity are P I = p, V I = v =2 Note that for K = 2 and K = 3 the interfacial states 6.26 coincide with those given in [0] and [4], respectively. In [0] another alternative is given for a two-phase model that has been proven in [23] to cancel the term Π. Since by means of the linear system 6.20 the interfacial pressures P,l depend on the convex combination 6.8 for the interfacial velocity V I there might be other options to choose the interfacial pressures, i.e., the 2nd law of thermodynamics does not uniquely characterize the interfacial pressure and interfacial velocity. In Section 8.3 we discuss another choice for the interfacial velocity Entropy production due to relaxation According to the entropy law 6.8 of a single component the entropy production due to the relaxation processes is determined by S αρs, = T p S α, + u g S αρ, v S αρv, + S αρe,, 6.27 where we plug 6.7 and 3.5 into 6.9. Here the Gibbs free energy of component is defined as g = e + p /ρ T s

23 S. Müller, M. Hante, & P. Richter In addition to the conservation constraints 2.3 the relaxation terms 2.9 have to satisfy S ρs, 0, =,..., K or S αρs, to ensure that the mixture is consistent with the 2nd law of thermodynamics. 7. Entropy-entropy flux pairs From a mathematical point of view, the concept of entropy-entropy flux pairs, cf. [], has been introduced to characterize a unique wea solution of an initial boundary value problem of inhomogeneous conservation laws that in quasi-conservative form reads t u + d A i u xi u = Su, i= A i u := f i u 7. u where u : R + Ω D R m with Ω R d, f i : D R m, i =,..., d and S : D R m, denote the vector of m conserved quantities, the fluxes in the ith coordinate direction, i =,..., d, and the source function, respectively. Motivated by thermodynamics, the entropy inequality d t Uu + xi F i u i= has to hold in a wea sense for any convex function U : D R and functions F i : D R, i =,..., d, referred to as entropy and entropy flux, that satisfy the compatibility conditions u Uu T A i u = u F i u T, i =,..., d. 7.3 Due to these conditions we infer for smooth solutions of 7. the entropy equation d t Uu + xi F i u = u Uu T Su, 7.4 i= Obviously, the entropy inequality 7.2 holds if and only if the entropy production is negative, i.e., u Uu T Su Motivated by the entropy equation 6. a candidate for an entropy-entropy flux pair for our nonequilibrium model 2.2, 2.3, 2.4 and 2.7 neglecting viscosity and heat conduction is Uu := α ρ s = ρs, F i u := α ρ s v,i, i =,..., d. 7.6 It remains to verify the convexity of U and the compatibility conditions Convexity of entropy function In order to verify that U is a convex function of the quantities u := α, α u T,..., α Ku T K T with α := α,..., α K T and u := ρ, ρ v T, ρ E T we extend the proof in [23], Appendix A, for a two-phase model to our K-component model. The ey idea is to employ the fact that U u := ρ s is a convex function of the quantities u of component. In [2], p. 99 ff, it is proven that the entropy U is a convex function of u, if e is a convex function of τ, s. 22

24 CLOSURE CONDITIONS FOR NON-EQUILIBRIUM MULTI-COMPONENT MODELS To verify the convexity of U we need to prove that the Hessian is positive semi-definite. First of all, we note that by 2. α α l = δ,l δ,k, u α l = α u δ,l δ,k, u α l u l = α δ,l I d+2 holds for =,..., K, l =,..., K. Then it follows for the gradient of U: U α l u = U l u l U K u K U l u l u l u l + U K u K u K u K, The Hessian of U is determined by the second order derivatives U α l u l u = U l u l u l U u = δ,l u T 2 U l l α α l α l 2 u l u l + u T 2 U K K u l α K 2 u K u K,, l =,..., K, u K 2 U u α α l u l = δ,l δ K,l 2 U l α l 2 u l u l, l =,..., K, =,..., K, u l 2 U u α u α l u l = 2 U l δ,l α l 2 u l,, l =,..., K. u l For a compact representation of the Hessian we introduce the notation 2 U U α,α := u R K K, α α l U α u,α u := U α u,α := l,,...,k 2 U u R d+2 d+2, α u α u 2 U 2 U u,..., u α u α α u α K According to the above second order derivatives these are determined by U α,α = u T α KU Ku K K + diag u T K α U u R d+2 K.,...,K, 7.8 U α u,α u = U α, 7.9 U α u,α = δ,l δ K, U α u = U T α,α u, 7.0 l=,...,k where U denotes the Hessian of the entropy U = U u of component. Then the Hessian can be represented as bloc-matrix U α,α U α,α u... U α,α u K U U α u,α U α u,α u u = U αk u K,α U α u K,α K u K To verify positive semi-definiteness of the Hessian we introduce the vector x = a T, b T,..., b T K T with a R K and b R d+2, =,..., K. Then we obtain by the bloc-structure 7. of the Hessian x T U x = a T U α,α a + a T U α,α u b + b T U α u,αa + U α u,α u b

25 S. Müller, M. Hante, & P. Richter By means of 7.8, 7.9 and 7.0 we determine a T U α,α a = K K au K T U α Kau K + a u T U K α a u, a := a l, K a T U α,α u b = δ,l δ,k b T α U a lu, l= K b T U α,α u a = b T U α u,α l a l = δ,k b T α KU K au K δ,k b T K α U a u, l= b T U α u,α u b = α b T U b. Incorporating this into 7.2 we finally conclude after some calculus with x T U x = K α b a u T U b a u + α K b K au K T U K b K au K 0, because the Hessians U are assumed to be positive semi-definite. Note that for x 0 we cannot ensure x T U x to be positive even if U is strictly convex because all the terms b a u, =,..., K, and b K au K may vanish at the same time. Thus we have proven the following Theorem 7.. Convexity of entropy function Let e be a convex function of τ, s, =,..., K. Then the entropy U is a convex function of u, i.e, the Hessian of U is positive semi-definite. l= 7.2. Compatibility conditions In order to verify the compatibility conditions 7.3 it is preferable to compute the derivatives in terms of primitive variables instead of conserved variables for reasons of simplicity. Therefore we first consider how an entropy-entropy flux pair transforms under a change of variables u = uw with regular Jacobian ui T w := w. w j i,j=,...,d Under this transformation the system of conservation laws 7. becomes a quasi-conservative system with Introducing t w + d B i w xi w = Sw 7.3 i= B i w := T w A i uw T w, Sw := T w Suw. we conclude by a straight-forward calculation W w := Uuw, G i w := F i uw, i =,..., d, w W w = T w T u Uuw, w G i w T = u Uuw T A i uw T w and, hence, w W w T B i w = w G i w T,

26 CLOSURE CONDITIONS FOR NON-EQUILIBRIUM MULTI-COMPONENT MODELS i.e., U, F i satisfies the compatibility conditions 7.3 if and only if W, G i satisfies the compatibility conditions 7.4. However, the function W must not be convex although U is a convex function. This directly follows from the Hessian of W ww W w = T T w uu Uuw T w + m l= U u l uw ww u l w. If the variable transformation is nonlinear, then the second term on the right-hand side must not be positive-definite. Neglecting viscosity and heat conduction our non-equilibrium model 2.2, 2.3, 2.4 and 2.7 can be written in quasi-conservative form 7. for u = α T, u T,..., ut K T with α = α,..., α K T and u = α ρ, v T, E T, =,..., K. Note that the notation slightly differs from the one in Section 7.. Obviously, the non-equilibrium model cannot be written in conservative form, i.e., the corresponding matrices A i are not Jacobians of fluxes f i. However, this does not affect the above discussion on entropy-entropy flux pairs. This non-equilibrium model is rewritten in terms of w = α T, w T,..., wt K T with the primitive variables w = ρ, v T, e T. The Jacobian of the transformation is given by T w = I K D C. D K... C K, C := α D := ρ 0 T d 0 v ρ I d 0 d e + 2 v2 ρ v T ρ v e + 2 v2, e,k K δ,k T, where e,k is the th unit vector in R K, K the unity vector in R K, 0 d the zero vector in R d and I d the unity matrix in R d d. Obviously, the transformation is regular, if and only if the densities ρ, =,..., K are positive. The quasi-conservative system corresponding to this transformation is given by 2.7, 2.2, 3.2 and 3.4 where we neglect viscosity and heat conduction. The matrices B i in 7.3 are given by the bloc-matrices B i w = V I,i I K A,i B,i. A,K... B K,i, B,i := α A,i := ρ v,i ρ e T i,d 0 p p ρ ρ e i,d v,i I d ρ e e i,d, p 0 ρ e T i,d v,i ρ α v,i V I,i e,k K δ,k T e i,d β T γ T, with β,l = γ,l = P M α ρ,l p δ,l P,K M p δ,k, P E α ρ,l V I,i P,l M v,i δ,l P,K E V I,i P,K M v,i δ,k. Then W and G i are given by W w = α ρ s /ρ, e, G i w = α ρ s /ρ, e v,i 25