Abstract
A sixequation Baer–Nunziato model at pressure equilibrium for two ideal gases is derived from a full nonequilibrium model by applying an asymptotic pressure expansion. Conditions on the interfacial pressure are provided that ensure hyperbolicity of the reduced model. Closure conditions for the relaxation terms are given that ensure consistency of the model with the second law of thermodynamics.
Introduction
Compressible twofluid flows where the fluid is a mixture of two components possibly at different phase have a wide range of applications, for instance, a mixture of reacting gases or a mixture of a liquid and a gas. In the literature, several approaches for the modeling of such flows are available. Our interest is in socalled Baer–Nunziatotype models. These can be derived by the ensemble averaging procedure of Drew [4]. Note that ensemble averaging differs from volume averaging, time averaging or statistical averaging. It accounts for the uncertainty in the exact location of particular constituents at a particular time and, thus, allows the interpretation of flow phenomena in terms of repeatability of flows. A comprehensive introduction to these models can be found in the classical book of Drew and Passman [5].
Originally, Baer and Nunziato [1] proposed a twofluid model for detonation waves in granular explosives. This model is a full nonequilibrium model where each component has its own pressure, velocity and temperature and is governed by its own set of fluid equations. In contrast to other twofluid models separating the fluids, here the fluids are assumed to coexist in each location due to an averaging procedure. The presence of more than one component in the mixture is modeled by the concept of volume fractions \(\alpha _k\) describing the ratio of volume of component k to the total volume in a ball in the limit when the volume of the ball tends to zero, see [5]. It was modified and generalized by several authors. For instance, Saurel and Abgrall [18] 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 into equilibrium, the mass transfer between the two fluids can be modeled, see Abgrall et al. [16, 20] or Zein et al. [30].
According to [14, 19, 20] mechanical relaxation, thermal relaxation and relaxation of the chemical potentials proceed on different time scales. Therefore, reduced models have been derived assuming zero relaxation times for some of the nonequilibrium quantities relaxing much faster than the remaining quantities, see [10]. Thus, the stiffness inherent in the nonequilibrium model is avoided that allows for a much faster numerical simulation of the reduced model. Typically, the reduced models are classified by the number of equations in case of two fluids in one space dimension. Usually reduced models suffer from some shortcomings. For instance, conservation of energy might be violated or the system loses its hyperbolicity. A detailed discussion of these models is beyond the scope of this work. For this purpose, the interested reader is referred to [29] and the references cited therein.
In [6, 12], a hierarchy of twofluid models derived from the Baer–Nunziato model [1] is investigated where mechanical, thermal and chemical relaxation is assumed to proceed in different order. This hierarchy splits into two branches of models distinguished by the assumption of dynamical velocity equilibrium and local velocity equilibrium where either both fluids have the same velocity or the fluids have different velocities that coincide for a particular state.
Since Baer–Nunziatotype models are averaged models, the resulting balance laws are underdetermined. To compensate for the loss in information closure conditions have to be imposed additionally. Therefore, there exists a rich literature on Baer–Nunziatotype models, see, for instance, [3, 7,8,9, 13, 17] investigating their properties, in particular, hyperbolicity and thermodynamical consistency. These properties are essential from an analytical, numerical and physical point of view. For instance, hyperbolicity is needed in the analysis of the Riemann problem and the construction of (approximate) Riemann solvers whereas a nonnegative entropy production ensures consistency with the second law of thermodynamics and also might ensure wellposedness so not yet mathematically rigorous verified. The analysis helps to identify physically admissible closure conditions for the interfacial pressure and the interfacial velocity that occur as model parameters and cannot be closed due to the averaging procedure.
We are particularly interested in a model where both fluids are assumed to have the same pressure, while other quantities may be in nonequilibrium. This model may be applied whenever the pressure relaxation time is much faster than the relaxation terms for the velocities, temperatures and chemical potentials. Such a model of Baer–Nunziato type with highest level of detail consists of six balance equations where each fluid has its own density and its own velocity. Further variables may be the common pressure and the volume fraction of one of the fluids.
In the literature, it is often claimed that the socalled classical sixequation model, which denotes the sixequation pressureequilibrium Baer–Nunziato model, is illposed, see, for instance, [15, 21].
The classical sixequation model is discussed in detail in the review article of Stewart and Wendroff [21]. There it is derived by some averaging procedure. This model differs from the model one obtains by taking the asymptotic limit for the pressure relaxation procedure in the full Baer–Nunziato model as we will discuss below. Moreover, source terms corresponding to relaxation processes for velocity, temperature and chemical potentials are missing. In fact, this model is illposed. It may have complex eigenvalues if the velocities are in nonequilibrium.
In the literature, several results can be found how to enforce hyperbolicity in the model. For instance, Toumi and Raymond, cf. [24, 25], consider a twofluid model for mass, momentum and total enthalpy without source terms but nonconservative product in the momentum equation with equal pressures. They claim that by choosing a particular interfacial pressure hyperbolicity can be achieved. Unfortunately, a proof as well as a physical explanation is missing.
In [22], Tiselj and Petelin introduce a virtual mass term that contains derivatives of the velocities to enforce their model to be hyperbolic. However, this model is not asymptotically correct, i.e., it cannot be derived from the full nonequilibrium model by a Chapman–Enskoglike asymptotic expansion. Moreover, also here a physical motivation is missing.
Another model is discussed by Toro [23] who investigates ignition and combustion of reactive solid particles in an expanding combustion chamber. The system also consists of six equations, two balances of mass and two balances of momentum for both the solid and the gas. In addition, an energy balance equation for the gas is considered while an evolution equation for the number of solid particles is used. The model exhibits relaxation terms and is not hyperbolic as well.
Nowadays, pressureequilibrium Baer–Nunziatotype models seem not to be considered anymore in the literature. Probably, it is common sense that models of this type without modifications are not hyperbolic. In this work, we will show that providing the correct asymptotic limit from a Chapman–Enskoglike asymptotic expansion and using an appropriate closure for the interfacial velocity and the interfacial pressure the Baer–Nunziato model leads to a novel hyperbolic pressureequilibrium model. To the best of our knowledge, it has not yet been discussed in the literature. In particular, up to now this model was not derived. This is surprising because a similar asymptotic procedure has been performed in [10] to derive asymptotically correct models at velocity equilibrium and velocity–pressure equilibrium.
Our main objective is to verify that the pressureequilibrium Baer–Nunziato model is not generally illposed. We exemplify this by means of two ideal gases. At first glance, this might be considered a contradiction to the underlying immiscibility assumption of Baer–Nunziato models because gases are known to mix perfectly at equilibrium, i.e., they are miscible. We will comment on this in more detail below. Note that using general equations of state makes the analysis much more cumbersome without any appreciable benefit. Moreover, since we are using specific equations of state, we are able to derive explicit expressions for the relaxation terms as well as for the constraints on the closure terms.
The paper is organized as follows. First of all, in Sect. 2 we derive the pressureequilibrium Baer–Nunziato model starting from a full nonequilibrium Baer–Nunziato model by investigating the limit of an asymptotic pressure expansion. In Sect. 3, we investigate hyperbolicity of the resulting sixequationmodel. In particular, we derive constraints on the closure of the interfacial pressure ensuring hyperbolicity. Several choices for the interfacial pressure and their influence on the hyperbolicity regime. Furthermore, in Sect. 4 we derive constraints on the relaxation terms that ensure nonnegativity of the entropy production, i.e., the model is in agreement with the second law of thermodynamics. We conclude with a summary of our findings in Sect. 5.
Derivation of the pressureequilibrium model
To derive the pressureequilibrium Baer–Nunziato model, we start with the full nonequilibrium model. Since this model is known to be Galilean invariant, see, for instance, [13], the flux function is the same for all directions \({{{\varvec{\omega }}} }\) when expanding the velocity vector with respect to a particular direction \({{{\varvec{\omega }}} }\) and its normal directions. For the investigation of hyperbolicity, it is thus sufficient to consider only a quasionedimensional flow, for instance, in the xdirection. Since the evolution equations for all the \(d1\) normal momentum components (with d denoting the spatial dimension) correspond to linearly degenerated fields we may confine ourselves only to the genuinely onedimensional case as is typically done in the literature. Therefore, we consider the original model introduced by Baer and Nunziato in 1986, see [1], which is given by
Here \(\alpha _k,\rho _k,v_k,p_k,E_k\) denote the volume fractions, the densities, the velocities, the pressures and the specific total energies of the two components \(k=1,2\). The volume fractions satisfy the saturation condition
The specific total energies \(E_k\) are related to the specific internal energies \(e_k\) via
The pressures \(p_k\) as well as the temperatures \(T_k\) are related to the densities and specific internal energies by the equation of state. In the following, we will consider the ideal gas law which is given by the relations
The quantities \(c_{v,k}\) and \(\gamma _k\) are material parameters, the specific heat capacity at constant volume and the adiabatic exponent, respectively. Moreover, the interfacial velocity \(V_\mathrm{I}\) and the interfacial pressure \(P_\mathrm{I}\) are model parameters. For more details on this, we refer to [13].
The expressions \(\mathcal{C}, \mathcal{M}, \mathcal{F}, \mathcal{E}\) describe the exchange of mass, momentum and energy between the components. Here, \(\mathcal{C}=\mathcal{C}(\mu _2\mu _1)\) is a function of the difference of the chemical potentials \(\mu _k\) of the components, see [10]. Furthermore, we have
with the nonnegative relaxation parameters \(\theta , \nu \) and \(\mathcal{H}\) for pressure, velocity and temperature, respectively. The quantities \({\mathbb E}\) and \(\varrho \) are additional model parameters. Choosing \(P_\mathrm{I}=p_2, V_\mathrm{I}=v_1, \varrho =\rho _1\) and \({\mathbb E}=E_1\) the model coincides with the model given in [10].
For the derivation of the pressureequilibrium model we perform a Chapman–Enskoglike analysis where we proceed as follows: Based on the balance equations for total energy, we determine balance equations for the internal energies as an intermediate step and afterward balance equations for the pressures and the pressure difference. Using an asymptotic expansion for the pressures where the pressure relaxation parameter \(\theta \) tends to infinity, while the pressures \(p_1\) and \(p_2\) tend to the equilibrium pressure p, we find an expression for \(\mathcal{F}\) which has to be applied in the energy balance equations (2.1c) as well as in the transport equation (2.1d). As a consequence, one of these three equations becomes redundant and we conclude with the sixequation pressureequilibrium model.
First we derive
with
From the equation of state (2.3), we then deduce
Finally, we obtain a balance equation for the pressure difference that is given by
For the asymptotic expansion, we use the following notation
From this, we infer
Inserting these expressions into Eq. (2.8a) and ordering the terms by their orders of \(\theta \) leads to the following relation for the pressure difference \(\varDelta p:=p_1p_2\)
with
The relaxation parameter \(\theta \) is assumed to be large. Thus, elimination of the leading order terms corresponding to \(\theta ^{1}\) and \(\theta ^0\) in (2.11) provides us with conditions on the pressures \(p_k^0\) and \(\mathcal{F}^1\). From the highestorder terms, we obtain
This implies
where p and P are referred to as the equilibrium pressure and the equilibrium interfacial pressure, respectively. Note that \(P=p\) if \(P_\mathrm{I}\) is chosen as a convex combination of \(p_1\) and \(p_2\); otherwise, also different equilibrium states are possible.
From the next order terms in (2.11), we then conclude
The last Eq. (2.15f) has to be inserted into system (2.1). This yields seven equations. The balances of mass and momentum are independent. One of the remaining three equations is redundant. Accordingly, there are several possibilities to select six equations for the pressureequilibrium Baer–Nunziato model. Of course, all admissible choices are equivalent due to the correct asymptotic limit.
We prefer to use two energy balance equations due to the symmetry of the resulting system and the similarity to models discussed in the literature, see, for instance, [21]:
The system (2.16) needs to be closed by appropriate models for p, P and \(V_\mathrm{I}\). The equilibrium pressure is determined by the mixture pressure that for an ideal gas reads
Note that in the system (2.16) the volume fraction \(\alpha _1\) is a dependent variable because for an ideal gas it holds \(p_1=p\) at pressure equilibrium. Then, we infer from (2.3)
satisfying the balance law for the volume fraction
Here, we employ the balances for internal energy
The closure of the equilibrium interfacial pressure and the interfacial velocity will be discussed in the subsequent sections.
Remark 1
(Other Baer–Nunziatotype models at pressure equilibrium) When comparing the 6equation model (2.16) with other wellknown models at pressure equilibrium, cf. [21, 24, 25, 28], we note that besides the relaxation terms on the righthand side the term \((D_1^0+D_2^0)/G^0\) is missing in the energy equation (2.16c). Thus, these models are not asymptotically correct derived from the nonequilibrium model (2.1), i.e., they are not the limit of an Chapman–Enskoglike asymptotic expansion. Therefore, the system (2.16) closed by (2.17) and (2.18) is a new model.
Remark 2
(Immiscibility and miscibility) The Baer–Nunziato model describes the multicomponent flow of immiscible fluids. Thus, at equilibrium each component fills a different portion of the accessible volume separated by interfaces. For our analysis, we deliberately have chosen a mixture of ideal gases. This seems to contradict the model since gases are known to mix perfectly, i.e., at equilibrium it must hold \(p=p_1+p_2\) instead of \(p=p_1=p_2\). To our opinion, this contradiction can be resolved by reinterpreting the notions of pressures, densities and volume fractions. We recall that in the equilibrium system (2.16) it holds \(e_k=c_{v,k}T_k\) and \(p=\rho _ke_k(\gamma _k1) \). Introducing the notations \(\hat{\rho }_k:=\alpha _k\rho _k\) and \(\hat{p}_k:=\hat{\rho }_ke_k(\gamma _k1)=\alpha _k p\) the equilibrium system (2.16) can equivalently be rewritten as
where in the equations of state for p and \(\hat{p}_k\) we use the same material parameters. For the pressure of the mixture \(p_{mix}\), we have
whereas the volume fractions \(\alpha _k\) satisfy
Obviously, the miscible and the immiscible notation of the model are equivalent.
We prefer to use the notation of system (2.16) because this notation is more convenient for several reasons: (i) comparability to the full model, (ii) implementation and (iii) extension to the general case of nonideal fluids.
Hyperbolicity
To investigate hyperbolicity of the sixequation model (2.16) at pressure equilibrium closed by (2.17) and (2.19), we first rewrite the system in terms of the primitive variables volume fraction \(\alpha _1\), equilibrium pressure p, velocities \(v_k\) and densities \(\rho _k\). Let
denote the conserved quantities of the system (2.16). Then, the primitive variables can be obtained as follows.
From (2.1), we derive the evolution equations for the primitive variables:
with relaxation terms
Here, we use the notation
Then, we can rewrite \(G^0\) as
In compact form, the system of primitive variables can be written in quasiconservative form as
with \({{{\varvec{w}}}}:=(\alpha _1,p,v_1,v_2,\rho _1,\rho _2)^\mathrm{T}\) and \({{{\varvec{S}}}}:=(\mathcal{A},\mathcal{P},\mathcal{V}_1,\mathcal{V}_2,\mathcal{R}_1,\mathcal{R}_2)^\mathrm{T}\) the vector of primitive variables and relaxation terms, respectively. The matrix \({{{\varvec{J}}}}\) is represented by a \(2\times 2\)block matrix
where the principal part \({{{\varvec{J}}}}_4\) can be split into two parts separating terms depending and not depending on \(\varDelta \)
with
Introducing the convex parameters
the matrix entries can be written as
Obviously, the matrix is well defined if and only if
This is a first constraint on the choice of the interfacial pressure at pressure equilibrium.
From the \(2\times 2\)block matrix (3.5), we conclude for the corresponding characteristic polynomial
where the characteristic polynomial of \({{{\varvec{J}}}}_4\) reads
with
and the singlefluid sound speeds \(c_k\) defined by Eq. (3.4d).
Obviously, the velocities \(v_1\) and \(v_2\) are eigenvalues of \({{{\varvec{J}}}}\). In the following, we investigate the roots of the characteristic polynomial \(\chi \). For this purpose, we introduce the transformation
also applied in [21]. Then, the characteristic polynomial can be written in the form
with coefficients
and linear combinations of the squares of the singlefluid sound speeds
We emphasize that the sign of the terms A and D is independent of the enumeration of the fluids, whereas the terms B and C flip sign.
To ensure hyperbolicity of our model, we have to investigate the roots of the polynomial \(P_4\). In the following, we will derive constraints on \(f(\varDelta )\) and, equivalently, \(\varDelta \) that ensure the existence of real roots. This problem has already been considered in the context of ideal gases for \(f(\varDelta )=0\) or, equivalently, \(\varDelta =0\), see [15, 28]. We start with a characterization for polynomials of the particular type (3.9) that can be found in [26].
Proposition 1
(Real roots of \(P_4\); Viher [26]) Let be
Then, \(P_4\) has only real roots if and only if
This theorem does not distinguish between simple and multiple roots. The latter are characterized in [27].
In the following, we derive sufficient and necessary conditions on \(f(\varDelta )\) or, equivalently, \(\varDelta \) for which the conditions (3.13) hold. First of all, we verify that
is a necessary condition on \(f(\varDelta )\) to ensure the existence of real roots of the characteristic polynomial (3.9) for all admissible physical states
Besides condition (3.7), this is another constraint on the choice of the interfacial pressure P at pressure equilibrium.
Theorem 1
(Necessary condition) To ensure the existence of real roots of the characteristic polynomial (3.9) for all physical admissible states in \(\mathcal{D}\), the condition (3.14) is necessary.
Proof
We consider each of the conditions in (3.13) separately:
Condition (3.13a) This condition only holds if r defined by (3.12) is not positive. We now derive a constraint on \(f(\varDelta )\) that ensures the correct sign of r for all physical admissible states in \(\mathcal{D}\). For this purpose, we consider the term \(3r\) as a quadratic polynomial in \(f\equiv f(\varDelta )\) depending on the parameter \(y\equiv \delta ^2\):
where we suppress the dependency on the other physical quantities \(\alpha _1\), \(\rho _k\), \(v_k\) and p. We now verify for which f this polynomial is nonnegative for all velocity differences \(\delta ^2\).
Since \(D>0\) according to (3.10), we may factorize the polynomial
with
Herein, the discriminant can be written as
Case 1a Obviously, if
the discriminant is negative and, thus, \(G(f;\delta ^2)\) must be positive because \(D^2> 0\), i.e., in this case there is no constraint on \(f(\varDelta )\).
Case 1b On the other hand, if \(0\le \delta ^2\le \hat{\delta }_{crit}^2\), i.e., \(3 c_1^2 c_2^2 \le AD\), then the roots \(\hat{f}_\pm \) are real and we conclude from the factorization (3.15) that \(G(f;\delta ^2)\) is nonnegative if
By definition of \(\hat{f}_\pm \) and the positivity of the singlefluid sound speeds we check that
Note that for \(0< \delta ^2 = A/4 < \hat{\delta }_{crit}^2\) the minimum and the maximum are attained, i.e., \(\hat{f}_=0\) and \(\hat{f}_+=12 c_1^2 c_2^2\).
To summarize the findings of the above investigations for the different cases, we conclude that G is nonnegative for all physical states if the condition (3.14) holds.
Condition (3.13b) For the investigation of this condition, we consider the term \(a_2^24 a_0\) as a quadratic polynomial in \(f\equiv f(\varDelta )\) depending on the parameter \(y\equiv \delta ^2\):
where we suppress the dependency on the other physical quantities \(\alpha _1\), \(\rho _k\), \(v_k\) and p. We now verify for which f this polynomial is nonnegative for all velocity differences \(\delta ^2\).
Since \(D>0\) according to (3.10), we may factorize the polynomial
with
Herein, the discriminant can be factorized by
Since the product AD is positive due to the positivity of the singlefluid sound speeds and (3.11), the roots \(y_\pm \) are real numbers.
Case 1 Obviously, the discriminant g becomes negative if \(y_< y < y_+\). Since \(y\equiv \delta ^2\), the admissible regime reduces to
Thus, the polynomial \(F(f;\delta ^2)\) must be positive because \(D^2> 0\), i.e., in this case there is no constraint on \(f(\varDelta )\).
Case 2 On the other hand, if \(\delta ^2\ge \delta _{crit,+}^2\) or \(\delta ^2 \le \max (0,\delta ^2_{crit,})\) , then the discriminant g is nonnegative and the roots \(f_\pm \) are real. According to the factorization (3.19) the polynomial \(F(f;\delta ^2)\) is nonnegative if and only if
If \(\delta ^2\ge \delta _{crit,+}^2\), then we conclude from the definition of the roots \(f_\pm \):
For the other option, we first note that the interval \(\delta ^2 \le \max (0,\delta ^2_{crit,})\) is nonempty if and only if \(AD \le c_1^2 c_2^2\). Since by definition (3.11) of A and D it holds \(AD \ge c_1^2 c_2^2\), the only choice is \(AD = c_1^2 c_2^2\). This is possible only for a pure fluid, i.e., \(\alpha _1=1\) or \(\alpha _2=1\). Then, the roots coincide, i.e., \(f_ = f_+ = c_1^2 c_2^2/D^2\). Thus, for this case the polynomial \(F(f;\delta ^2)\) is nonnegative, i.e., no constraint is imposed on \(f(\varDelta )\).
Condition (3.13c) Obviously, this condition holds by positivity of A and D due to the positivity of the singlefluid sound speeds. \(\square \)
To derive sufficient conditions on \(f(\varDelta )\) in the nonequilibrium case, the sign of \(D_1\) needs to be further investigated. For this purpose, we first note that by rescaling of \(D_1\) we may equivalently consider the sign of \({\tilde{D}}_1:={\tilde{q}}^2  4 {\tilde{r}}^3\) with \({\tilde{r}}:=  3r = 12 a_0 + a_2^2\) and \({\tilde{q}}:=  27q = 12 a_2^3 + 27 a_1^2 72 a_0 a_2\). To investigate the sign of \({\tilde{D}}_1\), we split this term into two parts
with
and coefficients
and
Here, we have applied the relations
that hold by (3.11). The derivation of this particular splitting is tedious work collecting appropriate terms. It is motivated by the observation that for local singlefluid the sign of \({\tilde{D}}_1\) can be easily checked, see Remark 4.
In case of a genuine twofluid flow, the representation (3.21) of \({\tilde{D}}_1\) is too complex to explicitly determine the roots. The best we may hope for is to find another constraint on \(f(\varDelta )\) that ensures the existence of real roots of the characteristic polynomial.
Theorem 2
(Sufficient condition in case of \(f(\varDelta )\not \equiv 0\)) Let the necessary condition (3.14) hold true for all admissible states in \(\mathcal{D}\). If either \(\varDelta \) is independent of \(\delta \) or in case of \(\varDelta =\varDelta (\delta )\), there exists \(\delta \) such that
then the roots of the characteristic polynomial (3.9) are real for all \(\delta \in {\mathbb R}\) if and only if either the flow locally degenerates to a singlefluid flow, i.e.,
or
Proof
We now factorize the term \({\tilde{D}}_1^{s}\) as follows
Since \(f(\varDelta )\le 0\), this term is nonpositive and vanishes if and only if \(f(\varDelta )=0\) or \(\delta ^2\) coincides with \(y_\pm \), i.e., \(\delta ^2=y_\pm \). Note that whenever \(\varDelta \) depends on \(\delta \) the latter requires to solve a nonlinear problem to determine \(\delta ^2\). That is why we need assumption (3.25). Obviously, \(y_+>0\) is always real, whereas \(y_\) may become negative. We now plug \(\delta ^2=y_+\) into \({\tilde{D}}_1^t\) and check the sign of this term. For this purpose, we first note that
For \(\delta ^2=y_+\), we then obtain
This term vanishes if and only if \(f(\varDelta )=1\) or \(\beta _1 \beta _2 = 0\), i.e., the flow degenerates locally to a singlefluid flow. Otherwise, this term is positive and we conclude for \(\delta ^2=y_+\)
and, thus, by Proposition 1 our characteristic polynomial (3.9) has complex roots and the flow model is not hyperbolic. From this, we finally conclude that the model has only real roots for all \(\delta \), if (3.26) or (3.27) hold true. \(\square \)
Remark 3
(Local velocity equilibrium) For states at local velocity equilibrium, i.e., \(\delta =0\), we immediately conclude from (3.21), (3.22) and (3.23)
Obviously, \({\tilde{D}}_1\) is nonpositive if and only if the condition (3.14) holds. Thus, this condition is both necessary and sufficient to ensure the existence of real roots due to Proposition 1.
The above result can be extended to weak velocity nonequilibrium, i.e., \(\delta  \ll 1\), provided that \(\varDelta \ne 0\) and \(f(\varDelta )<0\) for \(\delta =0\). If \(\varDelta \) is either independent of \(\delta \) or depends smoothly on \(\delta \), then \(\varDelta \ne 0\) and \(f(\varDelta )< 0\) in a neighborhood of local velocity equilibrium. Thus, it holds \({\tilde{D}}_1< 0\) for sufficiently small \(\delta \).
Remark 4
(Local singlefluid flow) If the flow locally degenerates to a singlefluid flow, i.e., \(\alpha _1\alpha _2=0\) or, equivalently, \(\beta _1 \beta _2=0\), then the roots of characteristic polynomial (3.9) are real if and only if the condition (3.14) holds for all admissible states in \(\mathcal{D}\). In particular, there exist one real double root and two single real roots if
This can be concluded from the simplification of (3.21)
and applying Theorem 1, Proposition 1 and the characterization of multiple roots in [27].
Combining Theorems 1 and 2 we conclude with the main result.
Theorem 3
(Global hyperbolicity) Let \(\varDelta \), and, equivalently, P be independent of \(\delta \). Then, the pressureequilibrium system (3.2) is hyperbolic for all admissible states in \(\mathcal{D}\) if and only if the interfacial pressure at pressure equilibrium vanishes, i.e., \(P=0\).
Note that there might exist \(P=P(\delta )\) such that for all \(\delta \) the condition is (3.25) is not satisfied and the system (3.2) is globally hyperbolic.
We will now discuss three particular choices for \(\varDelta \) and, equivalently, P, and their influence on the hyperbolicity.
Remark 5
(Special case: \(\varDelta = 0\), i.e., \(P=p\)) For this particular choice \(f(\varDelta )\) vanishes and, thus, the term \({\tilde{D}}_1^s\equiv 0\) vanishes according to (3.22) and for \({\tilde{D}}_1^t\) we derive from (3.22) and (3.24)
with
Then, we conclude from (3.21) that \({\tilde{D}}_1\) is positive if and only if
Therefore, the roots of the characteristic polynomial (3.9) are real if and only if
i.e., the pressureequilibrium system (3.2) is hyperbolic for admissible states in \(\mathcal{D}\) either at local velocity equilibrium (\(v_1=v_2\)) or at sufficiently strong velocity nonequilibrium (\(v_1v_2\ge \delta _0\). This confirms Wendroff’s Winequality [15, 28].
Remark 6
(Special case: \(\varDelta = \varDelta _B\), i.e., \(P=p+\varDelta _B\)) Kumbaro et al. [11] suggested the following nontrivial choice
The parameter \({\overline{\theta }}\) was introduced by Kumbaro et al. [11], whereas in Bestion’s original work \({\overline{\theta }}=1\), cf. [2]. Obviously, the condition (3.14) that is necessary to ensure the existence of real roots due to Theorem 1 is satisfied if it holds
i.e., for \(\delta ^2\) exceeding the bound on the righthand side the characteristic polynomial (3.9) has complex roots. On the other hand, for \(\delta ^2\) close to zero we may expand \(f(\varDelta _B)\) in powers of \(\delta ^2\) using Taylor expansion as
Applying this expansion in (3.22), we obtain
Then, the term \({\tilde{D}}_1\) defined by (3.21) is nonpositive, if
provided that \(\delta ^2\) is sufficiently small. The equality follows by the definitions of \(c_i^2\) and \(\beta _i\). In [11], it is mentioned that “we find, using a perturbation method Toumi [24], that the model is hyperbolic provided that the interface pressure coefficient \({\overline{\theta }}\) is greater than a minimum value \({\overline{\theta }}_0=1\). However, there is no guarantee that this value will lead to a hyperbolic system for any flow.” In fact, we have verified this statement by the above observations.
Indeed, the above result can be extended to
where \({\overline{M}}\) may depend on the physical state except for the velocities. Performing a similar perturbation analysis, we obtain
For \(\delta \) sufficiently small, we then conclude that \({\tilde{D}}_1\) is nonpositive if and only if
Obviously, for \(q>2\) the term \({\tilde{D}}_1\) is positive.
In summary, choosing the interfacial pressure at pressure equilibrium as \(P=p+\varDelta _B\), the hyperbolicity region for the pressure equilibrium system (3.2) is extended near to local velocity equilibrium in comparison with the choice \(P=p\) in Remark 5.
Remark 7
(Special case: \(\varDelta = p\), i.e., \(P=0\)) Since in this case we have \(f(\varDelta )=1\), we conclude from Theorem 2 that the characteristic polynomial (3.9) has only real roots independent of \(\delta \). In general, these roots are all distinct. To see this, we once more consider \({\tilde{D}}_1\). Here, we obtain for the coefficients (3.23)
and we may rewrite (3.21) as
Obviously, \({\tilde{D}}_1\) is nonpositive and it only vanishes if the nonresonance condition holds, i.e.,
where the characteristic polynomial has one double root and two single roots according to Viher, cf. [27]. Since it is reasonable to assume that there is some symmetry in the distribution of the roots, we can determine the roots by the factorization of polynomial (3.9)
with \(x_1^\pm := \delta \pm c_1\) and \(x_2^\pm := \delta \pm c_2\). By the transformation (3.8), the roots of the polynomial \(\chi _{B}(\lambda )\) and, thus, the eigenvalues of the matrix B are determined by
Note that these eigenvalues are the same as in case of the full nonequilibrium Baer–Nunziato model, cf. [13]. In the nonresonance case, we have four distinct real roots and, thus, there exists a basis of eigenvectors. Hence, the model is hyperbolic and we can explicitly determine left and right eigenvectors as well as Riemann invariants and check whether the corresponding fields are genuinely nonlinear or linearly degenerated.
Finally, we would like to point out that for this particular choice of \(\varDelta \), i.e., \(P=0\), the pressureequilibrium Baer–Nunziato model (2.16) is conservative, i.e., all nonconservative products vanish. Therefore, this choice seems to be optimal. However, there are problems when verifying that the entropy production is nonnegative, see Sect. 4.
Remark 8
(Baer–Nunziato model at full nonequilibrium) Although the above investigation has been performed for the Baer–Nunziato model at pressure equilibrium, the results have a direct consequence for the full nonequilibrium model. Typically, the models are closed by interfacial pressure \(P_\mathrm{I}\) and interfacial velocity \(V_\mathrm{I}\) where \(P_\mathrm{I}\) does not tend to zero when approaching pressure equilibrium. For instance, Baer and Nunziato [1] use
In [18], Saurel and Abgrall suggest
Furthermore, in [20] Saurel et al. apply
Another approach chooses a convex combination of the velocities for the interfacial velocity and then a unique interfacial pressure is derived such that the entropy production resulting from the interfacial pressure and velocity vanishes, cf. [7, 9, 13]
Second 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 briefly summarize the entropy law for the nonequilibrium model and discuss the entropy production terms providing us with admissibility criteria for the interfacial pressure and the interfacial velocity as well as the relaxation terms. Applying the pressure asymptotic to the entropy law, we then derive admissibility criteria for the pressureequilibrium model.
Nonequilibrium model
In order to investigate thermodynamical properties of the nonequilibrium model (2.1), we assume that the entropy \(s_k=s_k(\tau _k,e_k)\) of each component satisfies
with partial derivatives
Furthermore, to ensure thermodynamic stability we assume that the Hessian of \(s_k\) is negativedefinite, i.e.,
By means of the evolution equations for the masses (2.1a), the volume fractions (2.1d), (2.2) and the internal energies (2.5), we obtain the evolution equations for the volume specific entropies
with the entropy production due to the interfacial pressure and the interfacial velocity as well as the relaxation process
with the chemical potentials
Introducing the entropy of the components of the mixture
we finally obtain the entropy law of the mixture
Pressureequilibrium model
To derive the entropy equation at pressure equilibrium, we apply the pressure expansion (2.9) in (4.4). Since the entropies \(s_k\) and the temperatures \(T_k\) depend on the specific volume \(\tau _k\) and the internal energy \(e_k\), we perform a change of variables according to (2.3)
Then, by the asymptotic expansion (2.9) we conclude in the limit \(\theta \rightarrow \infty \) with
where the production terms (4.5) corresponding to the interfacial pressure and the interfacial velocity as well as the relaxation process tend to
Here, \({\mathbb E}_{0k}^0\) is determined by (2.12e). In particular, we note that due to the asymptotic expansions (2.9) and (2.10)
because it holds \(\mathcal{F}^0=0\) in the limit \(\theta \rightarrow \infty \). For the mixture, we then obtain
To investigate the sign of the entropy production \(\mathcal{S}^0\) we first split the term \(E_p^0\) into the contributions of the mechanical, thermal and chemical relaxation processes, i.e.,
Then, we can split the entropy production term similarly
Here, it is important to note that the term \(S_{D}^0\) only depends on spatial derivatives of the volume fraction, the velocities and the pressure but is independent of the relaxation processes.
In the following investigations, we will now neglect relaxation due to chemical reactions, i.e., \(\mathcal{C}=0\). This case becomes significant for frozen flows. Then, the entropy production reduces to
To enforce nonnegativity of the entropy production, we have to impose constraints on the mechanical and thermal relaxation terms \(\mathcal{M}\) and \(\mathcal{H}\) as well as the interfacial velocity \(V_\mathrm{I}\). For this purpose, we draw the following conclusions from (4.14).

1.
In case of local thermal equilibrium, i.e., \({\overline{T}}_1^0={\overline{T}}_2^0={\overline{T}}\), the entropy production reduces to
$$\begin{aligned} \mathcal{S}^0 = \frac{1}{{\overline{T}}} \left( v_1  v_2 \right) \left( (Pp)\frac{\partial \,\alpha _1}{\partial \,x}\mathcal{M}\right) . \end{aligned}$$Obviously, in this case the interfacial velocity does not enter. The sign of the entropy production can be controlled by the term
$$\begin{aligned} \left( v_2  v_1 \right) \mathcal{M}\end{aligned}$$using a sufficiently large velocity relaxation parameter \(\nu \), see (2.4a).

2.
Also in case of thermal nonequilibrium the sign of term \(S_{D}^0\) can be controlled by a sufficiently large velocity relaxation parameter \(\nu \), if the interfacial velocity \(V_\mathrm{I}\) is chosen as a convex combination of the single fluid velocities \(v_1\) and \(v_2\), i.e., \(V_\mathrm{I} = {\overline{\beta }}_1\,v_1+{\overline{\beta }}_2\, v_2\), \({\overline{\beta }}_1+{\overline{\beta }}_2=1\), \({\overline{\beta }}_1,{\overline{\beta }}_2\in [0,1]\). Then, we have
$$\begin{aligned} \left( {\overline{T}}_2^0 v_1  {\overline{T}}_1^0 v_2 + V_\mathrm{I} \left( {\overline{T}}_1^0{\overline{T}}_2^0 \right) \right) = ({\overline{\beta }}_1\,{\overline{T}}_1^0 + {\overline{\beta }}_2\,{\overline{T}}_2^0)\,(v_1v_2). \end{aligned}$$ 
3.
To ensure at least hyperbolicity in the neighborhood of local velocity equilibrium the pressure difference \(\varDelta =Pp\) must be nonpositive. Since \({\overline{\kappa }}\) and \(G^0\) are positive, the entropy production due to thermal relaxation \(S_{\mathcal{H}}^0\, \mathcal{H}\) is nonnegative due to \(\mathcal{H}\ge 0\), see (2.4b). This implies that for sufficiently large temperature relaxation parameter \(\mathcal{H}\) the third term \(S_{\mathcal{H}}^0\, \mathcal{H}\) dominates the second term \(S_{\mathcal{M}}^0\, \mathcal{M}\).
The above observations are summarized in the following
Theorem 4
(Entropy production) Let the interfacial velocity \(V_\mathrm{I}\) be a convex combination of the singlefluid velocities \(v_1\) and \(v_2\). The flow is assumed to be chemically frozen, i.e., \(\mathcal{C}=0\). Then, the entropy production is nonnegative if the velocity relaxation parameter \(\nu \) and the temperature relaxation parameter \(\mathcal{H}\) are sufficiently large.
This theorem applies to both a small pressure difference \(\varDelta =Pp\) choosing for instance the interfacial pressure \(P=p\) or \(P= p+ \varDelta _B\) near to local velocity equilibrium according to Remark 5 and 6 , respectively, and strong pressure difference when choosing \(P=0\), see Remark 7.
Remark 9
(Interfacial pressure) From the above observations, we conclude that at pressure equilibrium the entropy production is always nonnegative if the flow is at local velocity and local temperature equilibrium. Otherwise the local flow must locally tend to velocity and temperature equilibrium. The faster the local relaxation must be, the larger the pressure difference \(\varDelta =Pp\).
Remark 10
(Interfacial velocity) It is worthwhile mentioning that the second law of thermodynamics does not provide a closure for the interfacial velocity. In the literature, the choice of a convex combination is frequently used.
In the literature, several closures for the interfacial pressure and the interfacial velocity have been discussed in the context of the full nonequilibrium Baer–Nunziato model. In the following, we check for some of these closures whether they are also admissible at pressure equilibrium.
Remark 11
(Baer–Nunziato model at full nonequilibrium) In [7, 9, 13], a convex combination of the velocities for the interfacial velocity is considered and then a unique interfacial pressure is derived such that the entropy production due to \({\overline{\varPi } }_1+{\overline{\varPi } }_2\) vanishes, see Equation (8). At pressure equilibrium \(p_1=p_2=p\), these equations are determined by
Then, the entropy production terms read
From this, we conclude for the mechanical relaxation process that \(\mathcal{M}\) must have the same sign as the velocity difference \(v_1v_2\) and the thermal relaxation process \(\mathcal{H}\) must be nonnegative. This is the case, see (2.4a) and (2.4b). Note that the closure proposed by Baer and Nunziato [1], see Eq. (3.33), fits into this class of closures when choosing \({\overline{\beta }}_1=0\) and \({\overline{\beta }}_2=1\).
The closure of Saurel and Abgrall [18], see Equation (3.34), at pressure equilibrium reads
results in similar entropy production terms
The conclusion for the mechanical and thermal relaxation process is the same as before.
Furthermore, in [20] Saurel et al. introduce a closure, see Eq. (8), that at pressure equilibrium reads
For this closure, we derive
While the first term is nonnegative, we cannot control the sign of the second term except at local velocity equilibrium. We emphasize that the second term arises due the asymptotic derivation of the pressure equilibrium model. It is not present in the full nonequilibrium model for which the closure was originally derived. Thus, we conclude that not all closures derived at nonequilibrium are admissible for the equilibrium model.
Conclusion
We have derived a sixequation model at pressure equilibrium where we confine ourselves to two ideal gases. For this purpose, we have incorporated an asymptotic expansion for the singlefluid pressures and the interfacial pressure into the full nonequilibrium model. Considering the asymptotic limit yields the reduced model. In contrast to wellknown sixequation models, an additional term occurs verifying that these models are not asymptotically correct.
Furthermore, we do not insist that the interfacial pressure tends to the equilibrium pressure when the singlefluid pressures are approaching equilibrium. Numerous available closures for the interfacial pressure satisfy this modeling assumption although there is no physical evidence for this assumption. Here, we derive constraints on the equilibrium interfacial pressure that ensure hyperbolicity of the pressureequilibrium model.
Finally, we have presented constraints on the closures for the interfacial velocity as well as the mechanical and thermal relaxation terms in case of a chemically frozen flow.
So far, we considered only the case of two ideal gases as is common practice in the literature on pressureequilibrium models. Currently, we are extending the investigations to nonideal fluids. Preliminary results show that choosing the equilibrium interfacial pressure smaller than the equilibrium pressure ensures hyperbolicity at least in a local neighborhood of local velocity equilibrium.
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Hantke, M., Müller, S. & Grabowsky, L. News on Baer–Nunziatotype model at pressure equilibrium. Continuum Mech. Thermodyn. (2020). https://doi.org/10.1007/s00161020009563
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Keywords
 Twophase flows
 Hyperbolicity
 Entropy
 Relaxation
 Closure conditions