Philippe Helluy, Jean-Marc Hérard and Nicolas Seguin Editors
A HIERARCHY OF NON-EQUILIBRIUM
TWO-PHASE FLOW MODELS
Gaute Linga
1,2and Tore Flåtten
3Abstract. We review and extend a hierarchy of relaxation models for two-phase flow. The models are derived from the non-equilibrium Baer–Nunziato model, which is endowed with relaxation source terms to drive it towards equilibrium. The source terms cause transfer of volume, heat, mass and momentum due to differences between the phases in pressure, temperature, chemical potential and velocity, respectively. In the context of two-phase flow models, thesubcharacteristic conditionimplies that the sound speed of an equilibrium system can never exceed that of the relaxation system. Here, previous work by Flåtten and Lund [Math. Models Methods Appl. Sci., 21 (12), 2011, 2379–2407] and Lund [SIAM J. Appl. Math. 72, 2012, 1713–1741] is extended to encompass two-fluid models, i.e. models with separately governed velocities for the two phases. Each remaining model in the hierarchy is derived, and analytical expressions for the sound speeds are presented. Given only physically fundamental assumptions, the subcharacteristic condition is shown to be satisfied in the entire hierarchy, either in a weak or in a strong sense.
KeywordsTwo-phase flow, relaxation systems, subcharacteristic condition
Classification 2010 MCS.76T10, 35L60
1.
Introduction
The concurrent flow of two fluid phases occurs in a wide range of industrially relevant settings, including in nuclear reactors [11], petroleum production [1,9], heat exchangers [53], cavitating flows [58], and within carbon capture, transport and storage (CCS) [10,40,48]. However, for most simulation purposes, resolving the full three-dimensional flow field may be too cumbersome, due to the complex interaction between the phases. In particular, this encompasses calculating the temporal evolution of the interface between the phases, and the transfer of mass, heat and momentum across it. Averaging methods (see e.g. Ishii and Hibiki [34] or Drew and Passman [18]) may therefore be applied to avoid direct computation of the interface. The resulting coarse-grained models may often be expressed as hyperbolic relaxation systems with source terms accounting for the interactions between the phases, driving them asymptotically towards equilibrium at a finite rate. In a quasi-linear form, one-dimensional versions of such systems may be written as
∂tU+A(U)∂xU= 1
Q(U), (1)
1 PoreLab, The Njord Centre, University of Oslo, Norway. e-mail:[email protected] 2 Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark.
3 Dept. of Petroleum Engineering, University of Stavanger, Stavanger, Norway. e-mail:[email protected]
c
EDP Sciences, SMAI 2019
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
whereinU(x, t)∈G⊆RN is the (smooth) vector of unknowns andA(U) is a matrix which we shall call the Jacobian of the system, in analogy to conservative systems.1 Further,is a characteristic time associated with
the relaxation process described byQ(U). For an extensive review of the existing literature on such systems, see e.g. Natalini [50], or, for a more concise summary, consider the first few sections of Solem et al. [60] and the references therein.
Two limits of the relaxation system (1) will be considered in this paper:
• Thenon-stiff limit, corresponding to the limit→ ∞. In this case, we may write (1) as
∂tU+A(U)∂xU= 0. (2)
We will refer to (2) as thehomogeneous system.
• The formalequilibrium limit, which is characterized byQ(U)≡0. This defines an equilibrium manifold [12] throughM={U∈G:Q(U) = 0}. We now assume that the reduced vector of variablesu(x, t)∈ Rn, wheren≤N, uniquely defines an equilibrium valueU=E(u)∈ M. We may then express (1) as
∂tu+B(u)∂xu= 0, U=E(u), (3)
whereB(u) =P(u)A(E(u))∂uE(u)is the Jacobian of the reduced system. Herein, the operatorP(u) : RN →Rn satisfiesP(u)∂uE(u) =In, i.e. the identity matrix. We will refer to (3) as the equilibrium system.
We expect solutions of (1) to approach solutions of (3) as → 0, i.e. in thestiff limit, where the relaxation towards equilibrium is assumed to be instantaneous.
1.1.
The subcharacteristic condition
An essential concept which arises in the study of relaxation systems and their stability, is the so-called
subcharacteristic condition. It was first introduced by Leray [38], subsequently independently found by Whitham [68], and later developed by Liu [41] for conservative2×2 systems. For a generalN ×N relaxation system, such as (1), the condition may be formulated as follows.
Definition 1.1 (Subcharacteristic condition). Let the N eigenvalues of the matrix A of the homogeneous system (2) be given byΛi, sorted in ascending order as
Λ1≤. . .≤Λi≤Λi+1≤. . .≤ΛN. (4)
Similarly, let λj be theneigenvalues of the matrix Bof the equilibrium system (3). Herein, the homogeneous
system (2) is applied to a local equilibrium stateU=E(u), such that Λi = Λi(E(u)), andλj =λj(u). Now,
the equilibrium system (3) is said to satisfy thesubcharacteristic condition with respect to the homogeneous system (2) when (i) allλj are real, and (ii) if theλj are sorted in ascending order as
λ1≤. . .≤λj≤λj+1≤. . .≤λn, (5)
then the eigenvalues of the equilibrium system areinterlaced with the eigenvalues of the homogeneous system, in the sense thatλj∈[Λj,Λj+N−n].
For generalN×N conservative relaxation systems, Chen et al. [12] showed that if mathematical entropy is dissipated by the relaxation term, then the subcharacteristic condition follows, and the equilibrium system is endowed with a strictly convex entropy pair. Hence the subcharacteristic condition isnecessaryfor this notion of entropy stability of the relaxation process. Note that the analysis of Chen et al. [12] cannot be straightforwardly
1In systems which can be written on the conservative form∂
applied to non-conservative systems, as it hinges on the symmetry of the Hessian of the relaxation system (1). However, more general results can be obtained from linear analysis. Yong [70] proved that forn=N−1, the subcaracteristic condition is necessary for the linear stability of the equilibrium system. For strictly hyperbolic systems, Solem et al. [60] proved that it is also sufficient. Hence, for strictly hyperbolic relaxation systems wheren=N−1,the subcharacteristic condition is equivalent to linear stability.
The subcharacteristic condition has been shown to be an important trait of many physical models [6,7,23], since the eigenvalues then have a direct physical interpretation as the characteristic wave speeds of the system. In the context of relaxation models for two-phase flow, the fastest wave speeds are the speeds of pressure waves, which involve the fluid-mechanical speeds of sound. The subcharacteristic condition then implies in particular that the sound speeds of an equilibrium model can never exceed that of the relaxation model it is derived from.
This is precisely the observation, well known in the fluid mechanics community, that the “frozen” speed of sound is higher than the equilibrium speed of sound [22,28,54].
1.2.
The model hierarchy
In a general averaged two-phase flow model, the mixture will consist of two fluids which evolve independently.
We assume local thermodynamic equilibrium in each phase, i.e. each of the phases may be described by an
equation of state. Specifying two thermodynamic quantities then completely determines all thermodynamic properties of that phase. Herein lies also the assumption that the thermodynamic quantities are unaffected by the local velocity field. Each phase k may then be thought of as having separate pressures pk, temperatures Tk, chemical potentialsµk, and velocitiesvk. Since the two-phase mixture will move towards phase equilibrium
in each of the mentioned variables, we may model these interactions by employing relaxation source terms
corresponding to the followingrelaxation processes:
p− volume transfer. Relaxation towards mechanical equilibrium due to pressure differences between the phases, i.e. expansion or compression.
T − heat transfer. Relaxation towards thermal equilibrium, due to temperature differences between the phases.
µ− mass transfer. Relaxation towards chemical equilibrium due to differences between the phases in chem-ical potential.2
v − momentum transfer. Relaxation towards velocity equilibrium, due to velocity differences between the phases, i.e. interface friction.
The starting point of the forthcoming analysis will be the classical Baer–Nunziato (BN) model [4], which is a general formulation of atwo-fluid model, in the sense that the phases are associated with separate velocity fields. The BN model is endowed with appropriate relaxation terms corresponding to each of these processes presented above. By considering the homogeneous and equilibrium limits of each relaxation process, i.e. assuming all combinations of zero or more of them to be instantaneous, we obtain a hierarchy of models, each with partial equilibrium in one or more of the aforementioned variables.
This hierarchy can be represented as a four-dimensional hypercube, as illustrated in Figure1. Herein, each model is symbolized by a circle, and corresponds to a “corner” of the hypercube. Parallel edges, in turn, correspond to the same instantaneous relaxation assumption, in the direction of the arrow. The basic model, denoted by “0” and shown in red as the leftmost circle of Figure1, is thus reducible to all models in the hierarchy. Many of the models in the hierarchy have already been derived, explicitly expressed and thoroughly analyzed, and in this respect, the current paper builds heavily on previous work [2,8,15,25,35,37,58,59,71].
The models shown in yellow circles in Figure1constitute thev-branch of the hierarchy, i.e. thehomogeneous flow models, wherein the phase velocities are equal. Such models are a subclass of the so-called drift-flux models, where the phasic velocities are related by an algebraic expression. Herein, the v-model was derived by [59], the vp-model is due to [35] (see also Refs. [2,49]), and the vpT-model was studied e.g. by [23]. The
vpT µ-model is known as thehomogeneous equilibrium model and has been studied by several authors, see e.g.
0
v
p
T
µ
vp
vT
pT
vµ
pµ
T µ
vpT
vpµ
pT µ vT µ
vpT µ
7 eqn. 6 eqn. 5 eqn. 4 eqn. 3 eqn. v
p
T
µ
Figure 1. The 4-dimensional hypercube representing the model hierarchy. Parallel edges correspond to the same relaxation processes, and each vertex signifies a unique model in the hierarchy, assuming instantaneous relaxation in zero or more of the variables p(pressure), T
(temperature), µ (chemical potential) and v (velocity). The leftmost, red circle denoted by “0” represents the Baer–Nunziato model [4]. The colored edges represent relaxation processes where a subcharacteristic condition has previously been explicitly established in the literature; models described in [22] and [42] are shown in yellow, whereas models described by [21] are shown in green. Subcharacteristic conditions were obtained in [46] for the model represented by the blue circle.
Refs. [13,20,32,33,44,62,67]. Flåtten and Lund [22] collected results on thev-, vp-, vpT-, andvpT µ-models, derived thevpµ-model, and showed that the subcharacteristic condition was satisfied for all relaxation processes within this branch of the hierarchy. Lund [42] completed the v-hierarchy by deriving the vT-, vµ- andvT µ -models, and established the subcharacteristic condition in the remainder of thev-branch, given only physically fundamental assumptions.
With regards to thetwo-fluid models in the hierarchy, several of these models have been derived, employed in simulation [9,11], and analyzed. Here, thep-model was analyzed e.g. in Refs. [14,62], and thepT-model was studied e.g. in Refs. [21,29].
An important issue withp-relaxed (one-pressure) two-fluid models is that they develop complex eigenvalues when the velocity difference between the phases exceeds a critical value, i.e. they become non-hyperbolic [17,
24,46,62,66]. This obviously entails a violation of the subcharacteristic condition, and may lead to the lack of stable mathematical and numerical solutions. Nevertheless, these models are extensively used for practical applications; and in numerical simulations they are often mitigated by specifying a regularizing interfacial pressure (see [11,51,63]). Further, estimates of fluid-mechanical sound speeds is of practical importance for the construction of efficient numerical schemes [29,55]. For relations between two-fluid models, we therefore find, as in Ref. [21], the need to state a weaker formulation of the subcharacteristic condition.
Definition 1.2 (Weak subcharacteristic condition). When the subcharacteristic condition of Definition 1.1
holds with the additional equilibrium condition of equal phasic velocities, the weak subcharacteristic condition
is said to be satisfied.
Thep- and pT-models were analyzed by Martínez Ferrer et al. [21], who showed that the subcharacteristic condition, in a weak or strong sense, is satisfied with respect to existing neighbouring models. Similarly, Morin and Flåtten [46] studied thepT µ-model, and showed that subcharacteristic conditions were satisfied in relation
to existing neighbouring models. The highlighted edges in Figure 1 summarize the relations between models
where a subcharacteristic condition has already been shown to be satisfied.
1.3.
Contributions of this paper
The objective of the current paper is to complete the study of the subcharacteristic condition in the full hierarchy of two-phase flow models, proving the remaining subcharacteristic conditions. In this respect, a generalization of the work by [22] and [42] is provided, extending the hierarchy to encompass also two-fluid models, i.e. models with separate momentum equations for the two phases. Herein, the inclusion of the two-fluidT-,µ-,pµ- andT µ-models represent original contributions. A similar hierarchical derivation of two-phase relaxation models was done in the thesis of Labois [37], who focused primarily on the stiffened gas equation-of-state. In our current work, expressions for the sound speeds of the models are provided, valid for general equations of state. Moreover, we show that the remaining 15 subcharactistic conditions are satisfied, i.e. that the subcharacteristic condition is everywhere respected in the hierarchy, either in a strong or in a weak sense. This is done by comparing the new expressions for the sound speeds to many known results from the literature, and by using techniques involving writing the difference of wave velocities as sums of squares (cf. [22,42]). We present each of the models for which we prove at least one subcharacteristic condition.
1.4.
Outline
The organization of the current paper is as follows. In Section2we present the basic model with all possible source terms, derive evolution equations for the primitive variables, and state a parameter set which suffices to satisfy the laws of thermodynamics. In Sections 3 to 8, we present in turn thev-, p-, T-, µ-, pµ- andT µ -models. For each model we give explicit analytic expressions for the sound speeds, and prove the remaining subcharacterisic conditions with respect to related models. In Section9we show plots of the sound speeds in the different models, and briefly discuss physical and mathematical properties of models in the hierarchy. Finally, in Section10, we draw conclusions and suggest possible future work.
2.
Basic model
In this section, we present the basic BN model [4]. In this model, which is hyperbolic, the two phases have separate pressures, temperatures, chemical potentials and velocities. We state the model in a form reminiscent of that proposed by Saurel and Abgrall [56], but with all four possible relaxation source terms accounting for the interaction between the phases. From this, we determine the evolution equations of the primitive variables. Based on the evolution equations, we derive a parameter set which suffices for the model to satisfy fundamental physical laws.
2.1.
Governing equations
In the following, we present the governing equations in the basic model, supplemented with physically appro-priate relaxation terms. We letαk denote volume fraction,vk velocity,ρk density,pk pressure,Tk temperature, µk chemical potential,ek internal energy per mass, for each phasek∈ {g, `}, wheregdenotes gas and`denotes
liquid.
2.1.1. Volume advection
We assume that apart from advection, the interface between the phases can only move due to pressure differences. This is commonly formulated as
whereinvi is an interface velocity and I is the pressure relaxation parameter. Hence, the volume fraction is
advected with the velocityvi. There are several discussions available on how to choose this interface velocity,
see e.g. [15,57]. In Section2.3.2we propose to model it using a temperature-dependent average of the phasic velocites, derived from the second law of thermodynamics.
The local volume transfer must occur so that the phase with the lowest pressure is compressed, and the phase with the highest pressure is expanded. This is enforced throughI ≥0. Moreover, the volume fractions must satisfyαg+α`= 1, whereαk ∈(0,1), and hence only one evolution equation for the volume fractions is needed.
2.1.2. Mass balance
The evolution of the mass of each phase is contained in the balance equations
∂tαgρg+∂xαgρgvg=K(µ`−µg), (7) ∂tα`ρ`+∂xα`ρ`v`=K(µg−µ`), (8)
wherein K is the mass relaxation parameter, and the source terms on the right hand sides of (7) and (8) account for mass transfer between the phases [26,27]. The mass transfer occurs from the phase with the highest
chemical potential towards the phase with the lowest, which is ensured through the assumption K ≥0. We
observe thatconservation of total mass is contained by summing (7) and (8):
∂tρ+∂x(αgρgvg+α`ρ`v`) = 0, (9)
wherein we have defined the mixture densityρ=αgρg+α`ρ`.
2.1.3. Momentum balance
Similar balance laws apply for the momentum of each phase:
∂tαgρgvg+∂x(αgρgvg2+αgpg)−pi∂xαg=viK(µ`−µg) +M(v`−vg), (10) ∂tα`ρ`v`+∂x(α`ρ`v2`+α`p`)−pi∂xα`=viK(µg−µ`) +M(vg−v`). (11)
Herein,pi is an interface pressure andM is the momentum relaxation parameter. In Section2.3.2, we propose
a thermodynamically consistent expression for pi, given as a temperature-dependent average of the phasic
pressures.
Note that from the averaging procedure resulting in these models, the interface velocity vi in (10) and (11)
need not be the same as that in (6) (see e.g. Ref. [45]). However, we have chosen these to be equal to keep the notation to a minimum, as this will not influence the main conclusions of this paper. The source terms associated with vi on the right hand sides of (10) and (11) represent the momentum of the condensating or vaporizing
fluid, which is transferred to the other phase. The source terms associated withM represent interfacial friction, and are assumed to cause momentum transfer from the phase with highest velocity towards the one with lowest velocity, which is ensured by requiringM ≥0. We observe thatconservation of total momentum is ensured by summing (10) and (11):
∂t(αgρgvg+α`ρ`v`) +∂x αgρgv2g+α`ρ`v`2+αgpg+α`p`= 0. (12)
2.1.4. Energy balance
The balance laws for the energy of each phase may be stated as
∂tEg+∂x(Egvg+αgvgpg)−pivi∂xαg
∂tE`+∂x(E`v`+α`v`p`)−pivi∂xα`
=−piI(p`−pg) + µi+12vi2
K
(µg−µ`) +viM(vg−v`) +H(Tg−T`). (14)
Herein,µiis an interface chemical potential,H is the temperature relaxation parameter, and we have introduced
the total phasic energy per volumeEk =Ekint+E
kin
k , where the phasic internal and kinetic energies are given
by, respectively,
Eint
k =αkρkek, (15)
Ekin
k = 12αkρkv 2
k. (16)
On the right hand side of (13) and (14), the terms associated withI represent energy transfer due to expansion– compression work, the terms associated withK represent the energy which the condensating or vaporizing fluid brings into the other phase, the terms associated withM represent energy transfer due to frictious momentum transfer, and the terms associated with H represent pure heat flow. The latter should flow from the hotter to the colder phase, which is ensured through the assumption H ≥0. Moreover, we see thattotal energy is conserved by summing (13) and (14),
∂tE+∂x(Egvg+E`v`+αgvgpg+α`v`p`) = 0, (17)
where we have introduced the mixed total energy per volume E =Eg+E`. Note that the same observation
on the interfacial velocity as pointed out in Section2.1.3applies to (13) and (14). The interface velocity is for simplicity of notation chosen to be the samevi in (13) and (14) as in (6) and (10) and (11), but the choice does
not have consequences for our main conclusions.
2.1.5. Phase independent form
With all possible relaxation terms, the BN model [4], as presented in (6) to (8), (10), (11), (13) and (14), can be stated compactly as
∂tαk+vi∂xαk=Ik, (18) ∂tαkρk+∂xαkρkvk=Kk, (19) ∂tαkρkvk+∂x(αkρkvk2+αkpk)−pi∂xαk=viKk+Mk, (20)
∂tEk+∂x(Ekvk+αkvkpk)−pivi∂xαk=−piIk+ µi+12vi2
Kk+viMk+Hk, (21)
for each phasek∈ {g, `}. Herein, the shorthand forms of the relaxation source terms,Ik,Kk,HkandMk, have
been defined such that Ig = −I` =I(pg−p`), Kg =−K` = K(µ`−µg), Hg =−H` =H(T`−Tg), and Mg=−M`=M(v`−vg).
2.2.
Evolution of primitive variables
In order to systematically derive other models in the hierarchy, and to derive a physically valid parameter set for the basic model, we now seek the evolution equations forprimitive variables, such as phasic velocityvk,
densityρk, pressurepk, temperatureTk, entropysk and chemical potentialµk. To simplify the notation in the
forthcoming, we introduce the phasicmaterial derivative, defined by
Dk(·)≡∂t(·) +vk∂x(·), (22)
In the forthcoming calculations, the following relation will prove useful. For an arbitrary quantityf, we have from (19) and (22) that
αkρkDkf =∂tαkρkf+∂xαkρkvkf −f Kk. (23)
2.2.1. Volume fraction
For clarity we state the evolution equation for the volume fraction. Using (18), we have that
Dkαk=Ik+ (vk−vi)∂xαk. (24)
2.2.2. Velocity
We now seek the evolution equation for the phasic velocity. Using f =vk in (23), and (20), we obtain Dkvk = (αkρk)−1((pi−pk)∂xαk−αk∂xpk+ (vi−vk)Kk+Mk). (25)
2.2.3. Density
The density evolution equation is found by combining (19) and (24),
Dkρk =− ρk αk
(vk−vi)∂xαk−ρk∂xvk− ρk αk
Ik+ 1 αk
Kk. (26)
2.2.4. Kinetic energy
In order to obtain the evolution equation for the specific internal energy, we start by finding the evolution equations for the kinetic energy. Using f =v2
k/2 in (23), and (16) and (25), we obtain
∂tEkkin+∂xEkkinvk+αkvk∂xpk+vk(pk−pi)∂xαk= vivk−12v 2
k
Kk+vkMk. (27)
2.2.5. Internal energy
We obtain the evolution equation for the internal energy by subtracting (27) from (21), expanding and collecting terms:
∂tEkint+∂xEkintvk+αkpk∂xvk+pi(vk−vi)∂xαk =−piIk+gkKk+ (vi−vk)Mk+Hk, (28)
where we have introduced a shorthand interface energygk =µi+12(vi−vk)
2
. Now, by using (15) and (28) and
f =ek in (23), we obtain Dkek= α1
kρk −pi(Ik+ (vk−vi)∂xαk)−αkpk∂xvk+ (gk−ek)Kk+ (vi−vk)Mk+Hk
. (29)
2.2.6. Entropy
The fundamental thermodynamic differential reads
dek=Tkdsk+pkρ−k2dρk, (30)
wheresk is the specific entropy of phasek. By writing (30) in terms of material derivatives, and inserting (26)
and (29), we obtain the evolution equation for the phasic entropy as
Dksk = (αkρkTk)−1
(pk−pi) (Ik+ (vk−vi)∂xαk) + (gk−hk)Kk+ (vi−vk)Mk+Hk
. (31)
Herein, the phasic specific enthalpy is defined as hk = ek+pk/ρk. By using f =sk in (23) along with the
identityµk =hk−Tksk, (31) may be written in the balance form
∂tSk+∂xSkvk=Tk−1
h
(pk−pi) (Ik+ (vk−vi)∂xαk) + (gk−µk)Kk+ (vi−vk)Mk+Hk
i
where we have defined the phasic entropy per volumeSk=αkρksk.
2.2.7. Pressure
The pressure differential in terms of the density and entropy differentials may be written as
dpk=c2kdρk+ ΓkρkTkdsk, (33)
where we have introduced the phasicthermodynamic speed of sound and the first Grüneisen coefficient, respec-tively defined by
c2
k= (∂pk/∂ρk)sk and Γk =ρ
−1
k (∂pk/∂ek)ρk. (34) By writing (33) in terms of the phasic material derivative, and inserting (26) and (29), we arrive at
Dkpk =
Γk(pk−pi)−ρkc2k
αk (Ik+ (vk−vi)∂xαk)−ρkc
2
k∂xvk+
Γk(gk−hk)+c2k
αk Kk+
Γk
αk(vi−vk)Mk+
Γk
αkHk. (35)
2.2.8. Temperature
We now seek the equation governing the phasic temperature evolution. The temperature differential may in terms of the pressure and entropy differentials be written as
dTk= ΓkTkρ−k1ck−2dpk+TkCp,k−1dsk, (36)
where the specific isobaric heat capacity is defined by Cp,k = Tk(∂sk/∂Tk)pk. Now, writing (36) in terms of phasic material derivatives, and inserting (31) and (35), we obtain
DkTk=
1 +Γ2kCp,kTk
c2
k
αkρkCp,k
(pk−pi)− ΓkTk
αk
(Ik+ (vk−vi)∂xαk)−ΓkTk∂xvk
+
ΓkTk αkρk
+1 + Γ2
kCp,kTk
c2
k
αkρkCp,k
(gk−hk)
Kk+
1 + Γ2kCp,kTk
c2
k
αkρkCp,k
[(vi−vk)Mk+Hk]. (37)
2.2.9. Chemical potential
The natural differential of the phasic chemical potential reads
dµk=ρ−k1dpk−skdTk. (38)
Therefore, writing (38) in terms of phasic material derivatives, and inserting (35) and (37), we obtain
Dkµk= α1
k h
Γk−Csk
p,k−
Γ2kTksk
c2
k (p
k−pi) ρk −c
2
k+ ΓkTksk
i
(Ik+ (vk−vi)∂xαk)
− c2
k−ΓkTksk∂xvk+α1
kρk h
c2
k−ΓkTksk+
Γk−Csk
p,k −
Γ2
kTksk
c2
k
(gk−hk)
i
Kk
+ 1
αkρk
Γk−Csk
p,k −
Γ2
kTksk
c2
k
[(vi−vk)Mk+Hk]. (39)
2.3.
Laws of thermodynamics
2.3.1. Total entropy evolution
The total entropy per volume is given by S = Sg+S`. The evolution equation for the total entropy is
therefore found by summing (32) overk∈ {g, `}:
∂tS+∂x(Sgvg+S`v`) =Sp+Sµ+Sv+ST =S, (40)
where we have defined theentropy source terms
Sp=
p
g−pi
Tg − p`−pi
T`
Ig+
h(p
g−pi)(vg−vi)
Tg −
(p`−pi)(v`−vi) T`
i
∂xαg, (41)
Sµ=
µi−µg+12(vi−vg)2
T−1
g − µi−µ`+12(vi−v`)2
T−1
`
Kg, (42)
Sv=
h
(vi−vg)Tg−1−(vi−v`)T`−1
i
Mg, (43)
ST = (Tg−1−T −1
` )Hg. (44)
2.3.2. The second law of thermodynamics
We define theglobal entropy as
Ω(t) =
Z
CS(x, t) dx,
(45)
where C ⊆R is some closed region. Note that this is thethermodynamic entropy, and not the mathematical
entropy discussed by Chen et al. [12]. These are in general not interchangable, although thermodynamic entropy can in some cases play the role as a mathematical entropy. This relationship is highly relevant for the stability analysis of our two-phase flow relaxation models, and we refer to [16] for some discussions. For our present purpose, we limit our interest to the physical validityof our models, i.e. they should satisfy the second law of thermodynamics, which for our models can be formulated as follows.
Definition 2.1. Thesecond law of thermodynamics states that the global entropy is non-decreasing, i.e.,
dΩ
dt ≥0 ∀t, (46)
in our context.
Proposition 2.2. Sufficient conditions for the relaxation model given by (6)to (8),(10),(11),(13) and (14)
to satisfy the second law of thermodynamics (Definition 2.1) are
I,K,M,H ≥0, (47)
min{µg, µ`} ≤µi ≤max{µg, µ`}, (48) pi=
√T
`pg+√Tgp`
√T
g+√T`
, (49)
vi = √T
`vg+√Tgv`
√T
g+√T`
, (50)
given only the physically fundamental assumption Tk ≥0fork∈ {g, `}.
Proof. By temporal differentiation of (45), in combination with (40) and (46), we obtain
Z
CdxS(x, t)≥0,
where we have assumed that the entropy flux of (40), Sgvg+S`v`, vanishes at the boundary ofC. For (51)
to be satisfied, clearlyS ≥0 is a sufficient criterion, for which statement to hold the non-negativity ofall the partial source termsSp,Sµ,ST andSvis in turn sufficient. We now show this for each of the terms under the
conditions of (47) to (50).
Firstly, the conditions (49) and (50) inserted into (41) yields
Sp=I(pg−p`)
2
(TgT`)−1/2≥0. (52)
Now, (48) is equivalent toµi =βµµg+ (1−βµ)µ`, withβµ∈[0,1]. Hence, combination of (42) and (50) yields
Sµ=K(µ`−µg)2
h
(1−βµ)Tg−1+βµT`−1
i
≥0. (53)
Next, (50) inserted into (43) yields
Sv=M(v`−vg)2(TgT`)−1/2≥0. (54)
Finally, (44) becomes
ST =H(T`−Tg)2(TgT`)−1≥0, (55)
and hence all the source terms are non-negative.
Remark 2.3. The interface conditions (49) and (50) are sufficient, not necessary, and the square-root-of-temperature weighted average between the phasic values differs from choices in the literature, e.g. the initial choices by [4]. The reason for this particular weighting is that we enforced the interface velocities in (6), (10), (11), (13) and (14) to be equal. Allowing these to differ would enable other linear combinations of the phasic quantities, which could possibly be more suitable for numerical simulations [57]. These differences, however, do not have implications for the main conclusions of this paper.
2.4.
Wave velocities
We now consider the homogeneous limit of the BN model, where the source terms I,K,M,H → 0.
The resulting model has previously been extensively studied by several authors, see e.g. [56,71]. The model has two fluid-mechanical sound speeds; one for each of the phases. The seven wave velocities are given by
λ0={vi, vg, v`, vg−cg, vg+cg, v`−c`, v`+c`}[56].
In typical applications, the flow is subsonic, i.e. |vg−v`| cg, c` may be a valid approximation. Evaluated
in the velocity equilibrium limit, takingv≡vg=v`, the eigenvalues are, sorted in ascending order,
λ(0)0 ={v−c0,+, v−c0,−, v, v, v, v+c0,−, v+c0,+} (56)
where we have defined c0,+ = max{cg, c`} and c0,− = min{cg, c`} as the higher and lower sound speeds,
respectively.
3.
The
v
-model
We now study the model that arises upon imposing instantaneous equilibrium in velocity, i.e. letting the
velocity relaxation parameterM → ∞, which we expect corresponds to
Simultaneously, we require the termMg=M(v`−vg)to remain finite. By noting that for a general function f, the phasic material derivatives are equal for the two phases, i.e. Dkf =∂tf+v∂xf ≡Df, then the system
that results from evaluating (25) for the two phasesk∈ {g, `}can be solved to yield
Mg= (Ygp`+Y`pg−pi)∂xαg+αgY`∂xpg−α`Yg∂xp`, (58)
where we have introduced the phasic mass fractions Yk =αkρk/ρ. The model that now results from inserting
(57) and (58) into the basic model of Section 2, was analyzed by Flåtten and Lund [22,42], as it constitutes the basic model of thev-branch of the hierarchy. The model is hyperbolic and has previously been studied by many authors [35,52,59].
3.1.
Wave velocities
The wave velocities of the velocity equilibrium model, in the homogeneous limit whereI,K,H →0, are given by [22]
λv={v−cv, v, v, v, v, v+cv}. (59)
Herein, the sound speed of this model is defined by
c2v=Ygcg2+Y`c2`. (60)
Proposition 3.1. The v-model satisfies the subcharacteristic condition with respect to the basic model, given only the physically fundamental conditionsρk, c2k>0, for k∈ {g, `}.
Proof. We observe that Yg+Y` = 1, and due to the given positivity conditions, we have that Yk ∈ (0,1).
Therefore, (60) implies thatmin{cg, c`} ≤cv≤max{cg, c`}. It then follows trivially that the wave velocities of
thev-model areinterlaced in the wave velocities (56) of the basic model evaluated in the velocity equilibrium
state (57). Hence, the associated subcharacteristic condition of Definition1.1 is satisfied.
4.
The
p
-model
In this section, we consider the mechanical equilibrium model, which arises when we assume instantaneous
mechanical equilibrium in the basic model of Section 2. We let the pressure relaxation parameter I → ∞,
which we expect to correspond to pg =p` ≡ p. Simultaneously, the product Ig = I(pg−p`) should remain
finite. The mechanical equilibrium model is found by using (35) evaluated for each of the two phases. From this, we may find an expression forIg without temporal derivatives, and insert it into the basic model of Section2.
The resulting model has been extensively studied previously [21,56]. Like other one-pressure two-fluid models, the model is not hyperbolic.
4.1.
Wave velocities
We consider now the homogeneous limit, whereK,M,H →0. The eigenvalues to the lowest order in the
small parameterε=vg−v`, i.e. evaluated in the equilibrium state defined by (57), are given by [21]
λ(0)p ={v−cp, v, v, v, v, v+cp}, (61)
where the sound speed in thep-model is given by
c2
p=
α
g ρg +
α`
ρ`
αg ρgc2g +
α`
ρ`c2` −1
Proposition 4.1. The p-model satisfies the weak subcharacteristic condition of Definition 1.2 with respect to the basic model of Section2, subject only to the physically fundamental conditionsρk, c2k >0, for k∈ {g, `}, in the equilibrium state defined by (57).
Proof. We see from (62) thatc2
pis a convex combination
c2
p=ϕgc2g+ϕ`c2`, where ϕk=
αk
ρkc2k
αg ρgc2g +
α`
ρ`c2` −1
, (63)
sinceϕg+ϕ`= 1, andϕk∈(0,1), due to the given conditions. This implies that
min{cg, c`} ≤cp≤max{cg, c`}, (64)
and hence the weak subcharacteristic condition is fullfilled with respect to the basic model, whose local
eigen-values evaluated in the same state are given by (56).
5.
The
T
-model
In this section, we investigate the thermal-equilibrium model (T-model), which emerges from assuming
instantaneous thermal equilibrium in the basic model of Section 2. To this end, we let H → ∞ herein, which we expect corresponds to
Tg=T`≡T, (65)
in such a way thatHg=H(T`−Tg)remains finite. In the following we present the governing equations.
5.1.
Governing equations
The fullT-model may be stated as the basic model of Section2, in which (13) and (14) are replaced by (17) and the thermal equilibrium condition (65).
In order to establish the impact of instantaneous thermal relaxation on the wave velocities, we need to express the model in a quasi-linear form, and thus obtain the velocities as the eigenvalues of the associated Jacobian. This is most easily done by exploiting the primitive variables, which is what we now turn to do.
Firstly, we have that the phasic pressure differential in terms of density and temperature may be written as
dpk =c2kζ −1
k dρk+ ΓkρkCp,kζ −1
k dT. (66)
where we have introduced the ratio of specific heats ζk = 1 + Γ2kCp,kT /c2k, and used (65). With (66), (25)
becomes
Dkvk= ∆mipk
k ∂xαk−
c2
k
ρkζk∂xρk−
ΓkCp,k
ζk ∂xT+
∆ivk
mk Kk+
1
mkMk, (67)
where we have defined the phasic mass per volumemk =αkρk, the phasic interface pressure jump∆ipk =pi−pk,
and the phasic interface velocity difference∆ivk =vi−vk. Furthermore, (37) becomes
DkT =−
h
ζk∆ipk
˜
Cp,k +
ΓkT
αk i
(Ik−∆ivk∂xαk)−ΓkT ∂xvk +hΓkT
mk +
ζk
˜
Cp,k(gk−hk) i
Kk+C˜ζp,kk ∆ivkMk+
ζk
˜
Cp,kHk, (68)
where we have introduced the extensive heat capacity at constant pressure C˜p,k =mkCp,k. We now define the
(68) byθk, and summing over the phases yields
∂tT+ (θgvg+θ`v`)∂xT =−
hθ
gΓgT
αg + θ`Γ`T
α` iv
g−v`
2 ∂xαg−θgΓgT ∂xvg−θ`Γ`T ∂xv`
+
pg−p`
˜
Cp,g ζg +
˜
Cp,`
ζ`
−θgαΓgT g
+θ`Γ`T α`
Ig+
h`−hg ˜
Cp,g ζg +
˜
Cp,`
ζ`
+θgΓgT mg −
θ`Γ`T m`
Kg
+ ˜v`−vg Cp,g
ζg +
˜
Cp,`
ζ`
Mg, (69)
where have used the interface parameter definitions of (49) and (50) evaluated in thermal equilibrium (65) to simplify.
5.2.
Wave velocities
We now seek the wave velocities, i.e. eigenvalues, in the homogeneous limit, where the relaxation source terms
I,K,M →0. From (24), it is then clear thatαg is a characteristic variable of the system, since the volume
fraction is advected with the velocityvi in the absence of relaxation source terms. By using (26), (67) and (69),
the remaining, reduced system may now be expressed in the quasi-linear form∂tu˜T+ ˜AT(˜uT)∂xu˜T = 0, where ˜
uT = [ρg, ρ`, vg, v`, T], and the associated Jacobian is given by
˜ AT =
vg 0 ρg 0 0 0 v` 0 ρ` 0 c2g
ρgζg 0 vg 0
ΓgCp,g ζg
0 c2`
ρ`ζ` 0 v`
Γ`Cp,`
ζ`
0 0 θgΓgT θ`Γ`T θgvg+θ`v`
, (70)
from which we can find the remaining five eigenvalues. The characteristic polynomial of the latter is a fifth-degree polynomial, for which in general no closed-form solution can be obtained. We now note that we may writeA˜T = ˜A
(0)
T +εA˜
(1)
T , whereε=vg−v`. The matrices are given by
˜ A(0)T =
¯
v 0 ρg 0 0
0 v¯ 0 ρ` 0 c2g
ρgζg 0 ¯v 0
ΓgCp,g ζg
0 c2`
ρ`ζ` 0 v¯
Γ`Cp,`
ζ`
0 0 θgΓgT θ`Γ`T ¯v
, (71)
andA˜(1)T = diag (θ`,−θg, θ`,−θg,0),where we have taken¯v=θgvg+θ`v`. Hence, we approximate the
eigenval-ues by means of a perturbation expansion in the small parameterε. To the lowest order inε,vg=v`= ¯v=v,
and the eigenvalues of theT-model are given by
λ(0)T ={v−cT ,+, v−cT ,−, v, v, v+cT ,−, v+cT ,+} (72)
where the two distinct sound speeds of the model are given by
c2
T ,±= c2
g+c2`
T
1 ˜
Cp,g +
1 ˜
Cp,`
+ Γ2gc2`
mgc2g +
Γ2
`c2g m`c2` ±
r hc2
g−c2`
T
1 ˜
Cp,g +
1 ˜
Cp,`
− Γ2gc2`
mgc2g +
Γ2
`c2g m`c2`
i2
+ 4Γ2gΓ2`
mgm`
2h Γ
2 g mgc2g +
Γ2`
m`c2` +
1
T
1 ˜
Cp,g +
1 ˜
Cp,`
Proposition 5.1. TheT-model satisfies the weak subcharacteristic condition with respect to the basic model of Section2, subject only to the physically fundamental conditionsρk, Cp,k, T >0, fork∈ {g, `}, in the equilibrium state defined by (57).
Proof. We first show that the sound speeds are real. We note that on the given conditions, clearly c2
T ,± ∈R,
and moreover,c2
T ,+≥0. The product of the sound speeds may be written as
c−2
T ,+c −2
T ,−=c −2 0,+c
−2
0,−+ZT0, where Z
0
T =
T Γ
2 g mgc2g +
Γ2
`
m`c2`
c2 0,+c
2 0,−
1 ˜
Cp,g +
1 ˜
Cp,`
. (74)
Based on the given conditions, it is clear thatZ0
T ≥0 and therefore 0≤c2
T ,+c 2
T ,−≤c20,+c 2
0,−, (75)
and hence alsoc2
T ,− ≥0, and thuscT ,± are real, and by definition, positive. Now, using the definitions of c0,±
and (73), it follows that
(c2 0,+−c
2
T ,+)(c 2 0,+−c
2
T ,−)(c20,−−c2T ,+)(c 2
0,−−c2T ,−) =−Q0T, (76)
where
Q0
T = c
2 g−c
2
`
2 Γ2gΓ2`
mgm` h Γ2
g mgc2g +
Γ2
`
m`c2`
+ 1
T
1 ˜
Cp,g +
1 ˜
Cp,` i−2
. (77)
The given conditions ensure thatQ0
T ≥0. The only ordering of sound speeds compatible with (75) and (76) is 0≤cT ,−≤c0,−≤cT ,+≤c0,+, and hence the subcharacteristic condition of Definition 1.1is satisfied. Proposition 5.2. The vT-model of Lund [42] satisfies the subcharacteristic condition with respect to the T -model, given the physically fundamental assumptions ρk, Cp,k, T >0,fork∈ {g, `}.
Proof. The sound speed of thevT-model is given by [42]
c2
vT = 1 ρ
mgc2gm`c2`
Γ
g mgc2g +
Γ`
m`c2` 2 + 1 T 1 ˜
Cp,g +
1 ˜
Cp,` mgc
2 g+m`c2`
Γ2 g mgc2g +
Γ2
`
m`c2` +
1
T
1 ˜
Cp,g +
1 ˜
Cp,`
. (78)
Now, using (73), we can write the product of the differences as
c2
T ,+−c 2
vT
c2
T ,−−c2vT
=−QT
vT, (79)
where
QT
vT =YgY`
1 T 1 ˜
Cp,g +
1 ˜
Cp,` c
2 g−c2`
− Γ 2 gc 2 `
mgc2g +
Γ2`c2g m`c2` +
1
mg −
1
m`
ΓgΓ`
Γ2 g mgc2g +
Γ2
`
m`c2` +
1
T
1 ˜
Cp,g +
1 ˜ Cp,` 2 . (80)
With the given conditions, clearly QT
vT ≥0. Hence exactly one of the factors on the left hand side of (79) is
negative, and combined with Proposition5.1we realize thatcT ,− ≤cvT ≤cT ,+, and hence the subcharacteristic
Proposition 5.3. The pT-model satisfies the weak subcharacteristic condition with respect to the T-model, given the physically fundamental assumptions ρk, Cp,k, T >0 fork ∈ {g, `} in the equilibrium state defined by
(57).
Proof. The sound speed of thepT-model is given by [21]
c2pT =
α
g ρg +
α`
ρ`
1 ˜
Cp,g +
1 ˜
Cp,`
α
g ρgc2g +
α`
ρ`c2`
1 ˜
Cp,g +
1 ˜
Cp,`
+T Γg ρgc2g −
Γ`
ρ`c2`
2 (81)
We may now write
c2
T ,+−c 2
pT
c2
T ,−−c2pT
=−QT
pT, (82)
where
QT pT =
αgα` ρgc2gρ`c2`T
1 ˜ Cp,g
+ 1 ˜ Cp,`
! "
Γ2 g mgc2g
+ Γ 2
` m`c2`
+ 1 T
1 ˜ Cp,g
+ 1 ˜ Cp,`
!#−1
×
1 ˜
Cp,g +
1 ˜
Cp,` c
2 g−c2`
−T Γg ρgc2g −
Γ`
ρ`c2`
Γgc2`
αg +
Γ`c2g α`
α
g ρgc2g +
α`
ρ`c2`
1 ˜
Cp,g +
1 ˜
Cp,`
+T Γg ρgc2g −
Γ`
ρ`c2` 2
2
. (83)
Clearly QT
pT ≥0, on the given conditions. Hence exactly one factor on the left hand side of (82) is negative,
yieldingcT ,−≤cpT ≤cT ,+, and the weak subcharacteristic condition is satisfied.
6.
The
µ
-model
We now proceed to investigate the chemical-equilibrium model (theµ-model), which arises when we assume
instantaneous chemical equilibrium, i.e. let the chemical relaxation parameter K → ∞, which we expect
corresponds to
µg=µ`≡µ. (84)
Simultaneously, we require the productKg =K(µ`−µg)to remain finite, and in the forthcoming we seek to
express this without any temporal derivatives.
Remark 6.1. In our frameworkK must be set to zero when transition to one-phase flow occurs. Otherwise, strictly enforcing (84) will lead to unphysical results [3,5,43] in the form of negative volumes. Consequently, we strongly emphasize that all our results concerning nonzero phase transfer are meaningful only for the genuine two-phase region where the flow variables reside on the saturation curve.
With Remark6.1in mind, we can write the chemical potential evolution equation (39) as
Dkµ=−ψk∆ipk+ξkα−k1
(Ik−∆ivk∂xαk)−ξk∂xvk+χkKk+ψk∆ivkMk+ψkHk. (85)
where we have used (84), and defined the shorthands
ξk=c2k−ΓkTksk, ψk =mΓkξk
kc2k −
sk
˜
Cp,k, χk =
ξ2k mkc2k+
Tks2k
˜
Cp,k +
1 2(∆ivk)
2
By using (85) evaluated for each of the phases, and subtracting these expressions from each other, we obtain
Kg=κ−µ1(ξg∂xvg−ξ`∂xv`−(ψg+ψ`)Hg)−Aµ∂xαg+ (vg−v`)κ−µ1∂xµ+KpµIg−KvµMg (87)
where we have defined the shorthands
κµ= Tgs2g
˜
Cp,g + T`s2`
˜
Cp,` +
ξ2 g mgc2g +
ξ2
`
m`c2`
+1
2 ψg(∆ivg) 2+ψ
`(∆iv`)2
, (88)
Aµ=κ−1
µ
h
ψg∆ipg+αξg
g
∆ivg+
ψ`∆ip`+αξ`
`
∆iv`
i
, (89)
Kpµ=κ −1
µ
h
ψg∆ipg+ψ`∆ip`+αξg
g +
ξ`
α` i
, Kvµ =κ −1
µ (ψg∆ivg+ψ`∆iv`). (90)
6.1.
Governing equations
By using the expression (87) to insert for Kg in the basic model of Section 2, the µ-model can now be
summarized with the following set of equations:
• Volume advection: ∂tαg+vi∂xαg=Ig,
• Conservation of mass: ∂tρ+∂x(αgρgvg+α`ρ`v`) = 0,
• Momentum balance:
∂tαgρgvg+∂x(αgρgvg2+αgpg)−(pi−viAµ)∂xαg−viξgκ−µ1∂xvg+viξ`κ−µ1∂xv`
−vi(vg−v`)κ−µ1∂xµ=viKpµIg+ (1−viKvµ)Mg−vi(ψg+ψ`)κ−µ1Hg, (91)
• Energy balance:
∂tEg+∂x(Egvg+αgvgpg)−pivi− µ+12vi2Aµ∂xαg − µ+1
2vi 2
κ−1
µ [ξg∂xvg−ξ`∂xv`+ (vg−v`)∂xµ] =
µ+1 2vi
2
Kpµ−piIg+vi− µ+12vi2Kvµ
Mg+1− µ+12vi2(ψg+ψ`)κ−µ1
Hg, (92)
Momentum and energy equations for the liquid phase are found by phase symmetry; interchanging indices g
and`.
6.2.
Evolution of primitive variables
In order to write the system in a quasi-linear form, and thereby find the wave speeds of theµ-model, we use the evolution equations for the primitive variables. We therefore now seek the evolution of some of the primitive variables under the assumption of instantaneous chemical equilibrium.
We first define the weighting factor φk =χ−k1/(χ−
1 g +χ−
1
` ). Multiplying (85) byφk and summing over the
phases, we get for the chemical potential
∂tµ+ (φgvg+φ`v`)∂xµ+Gµαg∂xαg+φgξg∂xvg+φ`ξ`∂xv`
=h−φg(ψg∆ipg+ξgαg−1) +φ`(ψ`∆ip`+ξ`α−`1)
i
Ig
+ (φgψg∆ivg−φ`ψ`∆iv`)Mg+ (φgψg−φ`ψ`)Hg, (93)
where we have defined the shorthand coefficient
Gµαg =−φg ψg∆ipg+ξgα
−1 g
∆ivg+φ`
ψ`∆ip`+ξ`α−`1
For the phasic velocity vg, we find from (25) the evolution equation
∂tvg+
h
vg−ξmg∆ivg
gκµ
i
∂xvg+ξm`∆ivg
gκµ ∂xv`+
∆ivgAµ−∆ipg mg ∂xαg+
1
ρg∂xpg−
∆ivg(vg−v`)
mgκµ ∂xµ
= ∆ivg mg K
µ pIg+m1
g (1−∆ivgK
µ v)Mg−
∆ivg mg
ψg+ψ`
κµ Hg, (95)
andv` is found by phase symmetry.
The phasic pressure evolution is found from (35). For the gas phase, it reads
∂tpg+vg∂xpg+Pgµ,αg∂xαg+P µ
g,vg∂xvg+P µ
g,v`∂xv`+P
µ
g,µ∂xµ =α−1
g
− Γg∆ipg+ρgc2g
+ ξg+12Γg(∆ivg)2
Kpµ
Ig
+α−1 g
Γg∆ivg− ξg+12Γg(∆ivg)2
Kvµ
Mg
+α−1 g
Γg− ξg+12Γg(∆ivg)2
(ψg+ψ`)κ−µ1
Hg. (96)
wherein we have defined the coefficients
Pµ
g,αg =α
−1 g
ξg+12Γg(∆ivg)2Aµ− Γg∆ipg+ρgc2g
∆ivg, (97) Pgµ,vg =ρgc
2
g− ξg+12Γg(∆ivg)2
ξgα−g1κ− 1
µ , (98)
Pµ
g,v` = ξg+
1
2Γg(∆ivg) 2
ξ`α−g1κ −1
µ , (99)
Pgµ,µ=− ξg+12Γg(∆ivg)2
(vg−v`)α−g1κ− 1
µ . (100)
The corresponding expressions related to the evolution ofp` are found by phase symmetry.
6.3.
Wave velocities
We now wish to derive the wave velocities of the µ-model in the homogeneous limit, where I,H,M →
0. In this limit, the volume fraction αg is a characteristic variable with the associated eigenvalue vi. The
remaining, reduced model, i.e. (93), (95) and (96) for both phases, may then be expressed in the quasi-linear form∂tu˜µ+ ˜Aµ(˜uµ)∂xu˜µ= 0, where the reduced vector of unknowns isu˜µ = [µ, vg, v`, pg, p`], and the reduced
Jacobian reads
˜ Aµ =
φgvg+φ`v` φgξg φ`ξ` 0 0
−∆ivg(vg−v`)
mgκµ vg−
ξg∆ivg mgκµ
ξ`∆ivg mgκµ ρ
−1
g 0
∆iv`(vg−v`)
m`κµ
ξg∆iv`
m`κµ v`−
ξ`∆iv`
m`κµ 0 ρ
−1
` Pµ
g,µ Pgµ,vg P
µ
g,v` vg 0
P`,µµ P`,vµ g P`,vµ
` 0 v`
. (101)
Again the eigenvalues λ are given the roots of a fifth degree polynomial, for which in general no closed-form solution exists. We therefore expand in the small parameter ε =vg−v`, i.e. A˜µ = ˜A
(0)
µ +εA˜
(1)
µ +. . ., and λ=λ(0)+ελ(1)+. . .. Herein, the lowest-order system matrix reads, takingv¯=φ
gvg+φ`v`,
˜ A(0)µ =
¯
v φgξg φ`ξ` 0 0 0 v¯ 0 ρ−1
g 0
0 0 v¯ 0 ρ−`1 0 ρgc2g−ξ2g/(αgκ
(0)
µ ) ξgξ`/(αgκ (0)
µ ) ¯v 0 0 ξgξ`/(α`κ
(0)
µ ) ρ`c2`−ξ2`/(α`κ
(0)
µ ) 0 ¯v
where we have used the lowest-order term ofκµ, as defined in (88):
κ(0)
µ = Tgs2g
˜
Cp,g + T`s2`
˜
Cp,` +
ξ2g mgc2g +
ξ`2 m`c2`.
(103)
To the lowest order in ε, vg =v` = ¯v =v, and thus the eigenvalue problem consists in finding the roots of det( ˜A(0)µ −λ(0)I) = 0. Hence, the full vector of eigenvalues is given by
λ(0)µ ={v−cµ,+, v−cµ,−, v, v, v+cµ,−, v+cµ,+} (104)
where the two sound speeds in theµ-model are given by
c2
µ,± =
T
gs2g
˜
Cp,g + T`s2`
˜
Cp,`
(c2
g+c2`) + ξ2
`c2g m`c2` +
ξ2
gc2`
mgc2g ±
r hT
gs2g
˜
Cp,g + T`s2`
˜
Cp,`
(c2
g−c2`) + ξ2
`c2g m`c2` −
ξ2
gc2`
mgc2g
i2
+ 4 ξ2gξ2`
mgm`
2hTgs2g
˜
Cp,g + T`s2`
˜
Cp,` +
ξg2 mgc2g +
ξ2` m`c2`
i (105)
Proposition 6.2. Theµ-model satisfies the weak subcharacteristic condition with respect to the basic model of Section 2, given only the physically fundamental conditions ρk, Cp,k, Tk >0 for k∈ {g, `}, in the equilibrium state defined by (57).
Proof. We first note thatc2
µ,±∈Ron the given conditions, and thatc2µ,+ ≥0. The product of the sound speeds
may be written as
c−µ,2+c −2
µ,−=c− 2 0,+c
−2
0,−+Zµ0, where Z
0
µ=c −2 g c
−2
` ξ2
g mgc2g +
ξ2
`
m`c2`
Tgs2g
˜
Cp,g + T`s2`
˜
Cp,`
. (106)
Given the conditions we have thatZ0
µ≥0, and hence 0≤c2
µ,+c2µ,− ≤c02,+c20,−. (107)
Therefore alsoc2
0,− is positive, and thus we have thatc0,± are real and, by choice, positive.
Now, the product of the differences of the sound speeds may be written as
(c2 0,+−c
2
µ,+)(c 2 0,+−c
2
µ,−)(c20,−−c2µ,+)(c 2
0,−−c2µ,−) =−Q0µ, (108)
where
Q0
µ= c
2 g−c
2
`
2 ξg2ξ2`
mgm` hT
gs2g
˜
Cp,g + T`s2`
˜
Cp,` +
ξ2 g mgc2g +
ξ2
`
m`c2` i−2
. (109)
Clearly, with the given conditions, Q0µ ≥ 0, and hence the only ordering of sound speeds compatible with
(107) and (108) is 0 ≤cµ,− ≤ c0,− ≤cµ,+ ≤c0,+, which means that the weak subcharacteristic condition is
satisfied.
Proposition 6.3. The vµ-model satisfies the subcharacteristic condition with respect to the µ-model, subject only to the physically fundamental conditionsρk, Cp,k, Tk >0, for k∈ {g, `}.
Proof. The sound speed in thevµ-model is given by [42]
c2
vµ= 1 ρ
mgc2gm`c2`
ξ
g mgc2g +
ξ`
m`c2` 2
+Tgs2g
˜
Cp,g + T`s2`
˜
Cp,` mgc
2 g+m`c2`
Tgs2g
˜
Cp,g + T`s2`
˜
Cp,` +
ξ2 g mgc2g +
ξ2
`
m`c2`
We now consider the product of the differences in the sound speeds of the two models, which may be written as
(c2
µ,+−c2vµ)(c2µ,−−c2vµ) =−Qµvµ, (111)
where
Qµ
vµ=YgY`
Tgs2 g mgCp,g +
T`s2`
m`Cp,`
(c2
g−c2`)− ξg2c2` mgc2g +
ξ`c2g m`c2` +
1
mg −
1
m`
ξgξ`
Tgs2g
˜
Cp,g + T`s2`
˜
Cp,` +
ξ2 g mgc2g +
ξ2
`
m`c2`
2
. (112)
ClearlyQµ
vµ≥0. Hence, exactly one of the factors on the left hand side of (111) must be negative, which gives
cµ,− ≤cvµ≤cµ,+, i.e. the subcharacteristic condition is satisfied.
7.
The
pµ
-model
We now consider the model which arises when we impose instantaneous mechanical-chemical equilibrium, i.e. we let the relaxation parametersI,K → ∞, which we expect corresponds to
pg=p`≡p and µg=µ`≡µ. (113)
Simultaneously,Ig=I(pg−p`)andKg=K(µ`−µg)should remain finite. We now seek explicit expressions
for these terms in order to find the governing equations of the model.
In the following analysis we use the parameter set stated in Section2and therefore let the interfacial pressure jump∆ip=pi−p= 0. From (35) and (85) we have
Dkp=− ρkc2k
αk
˜
Ik+∂xαkvk
+ξk+
1
2Γk(∆ivk)2
αk Kk+
Γk
αk∆ivkMk+
Γk
αkHk, (114)
Dkµ=−αξk
k
˜
Ik+∂xαkvk
+h ξk2
mkc2k +
Tks2k
˜
Cp,k +
1 2(∆ivk)
2ψ
k
i
Kk+ψk∆ivkMk+ψkHk. (115)
where we have definedI˜k=Ik−vi∂xαk =∂tαk.
Eqs. (114) and (115) evaluated for each phase now constitute a4×4system which is straightforward to solve for the four unknowns∂p/∂t,∂µ/∂t,I˜g, andKg, in terms of spatial derivatives and the remaining source terms.
The final expressions for the latter two are
˜
Ig=−Pppµ(vg−v`)∂xp−Gppµ(vg−v`)∂xµ−Φg∂xαgvg+ Φ`∂xα`v`+IvpµMg+ITpµHg, (116)
Kg=−Pµpµ(vg−v`)∂xp−Gµpµ(vg−v`)∂xµ−Vµ,pµg∂xv¯+KvpµMg+K
pµ
T Hg, (117)
where the coefficients are given in Appendix A.
7.1.
Governing equations
Inserting the expressions (116) and (117) into the basic model of Section2, we are now in a position to state the full model. The mechanical–chemical equilibrium model may thus be formulated as follows.
• Conservation of mass: ∂tρ+∂x(mgvg+m`v`) = 0,
• Momentum balance:
∂tmgvg+∂xmgvg2+ αg+viPµpµ(vg−v`)
∂xp+viGµpµ(vg−v`)∂xµ+viVµ,pµg∂x¯v
• Energy balance:
∂tEg+∂xEgvg+αgvg+ µ+12vi2Pµpµ−pP pµ p
(vg−v`)∂xp +
µ+1 2vi
2
Gµpµ−pG pµ p
(vg−v`)∂xµ+ µ+12vi2Vµ,pµg+pΦ`∂x¯v =
vi+ µ+12vi2Kvpµ−pI pµ v
Mg+1 + µ+12vi2KTpµ−pI pµ T
Hg. (119)
The momentum and energy equations for the liquid phase are found by phase symmetry. Note that, like other
p-relaxed models, thepµ-model is expected to be non-hyperbolic for nonzero difference in the velocity unless a regularising interfacial pressurepi is defined.
7.2.
Wave velocities
We now wish to write the system in a quasilinear form, in order to find the wave speeds of the system, in the homogeneous limit where we let the relaxation termsM,H →0. To this end, we will express the model in the vector of unknownsupµ = [p, µ,v, v¯ g, v`]. We therefore seek the evolution equations for the elements ofupµ.
For the volume evolution, we find, using (24) and (116), that
∂tαg+Pppµ(vg−v`)∂xp+Gppµ(vg−v`)∂xµ+ Φg∂xαgvg−Φ`∂xα`v`= 0, (120)
For the volume-averaged velocityv¯we find, using (25), (116), (117) and (120), that
∂tv¯+ (αgρg−1+α`ρ−`1+P pµ
¯
v ε2)∂xp+Gpµv¯ ε2∂xµ+αgε∂xvg
−α`ε∂xv`+ Φgvg+ Φ`v`−Vv,¯pµgε
∂x¯v= 0, (121)
where we have defined the shorthand coefficients P¯vpµ, G pµ
¯
v , V pµ
¯
v,g (for which expressions are given in Appendix A), used ε = vg−v`, and inserted βg = 1−β` =
p
T`/(
p
Tg+
p
T`). Now, for the pressure and chemical
potentials, we get from (114) and (115) that
∂tp+
Ψp
gvg+ Ψp`v`
∂xp+Gpµp ε∂xµ+Vppµ∂xv¯= 0, (122)
∂tµ+Pµpµε∂xp+
Ψµ
gvg+ Ψµ`v`
∂xµ+Vµpµ∂x¯v= 0. (123)
Again, the coefficients are given in AppendixA.
The homogeneous system in a quasilinear form thus reads ∂tupµ+Apµ(upµ)∂xupµ = 0, where the system
Jacobian is given by
Apµ =
Ψp
gvg+ Ψp`v` Gpµp ε Vppµ 0 0
Ppµ
µ ε Ψ
µ
gvg+ Ψµ`v` Vµpµ 0 0
αg ρg +
α`
ρ` +P
pµ
¯
v ε2 G pµ
¯
v ε2 Φgvg+ Φ`v`−Vv,¯pµgε αgε −α`ε
1
ρg − β`Pµpµ
mg ε
2
−β`G
pµ µ
mg ε
2
−β`V
pµ µ,g
mg ε vg 0
1
ρ` −
βgPµpµ
m` ε
2
−βgGµpµ
m` ε
2 βgVµ,pµg
m` ε 0 v`
. (124)
Obtaining the assocated eigenvalues exactly by analytic means is again unfeasible, as the problem consists in finding the roots of a fifth-degree polynomial. We therefore expand in ε: Apµ =A
(0)
pµ +εA
(1)
pµ +ε2A
(2)
pµ +. . .,
v=vg=v`, we get the matrix
A(0)pµ =
v 0 Vppµ,(0) 0 0 0 v Vµpµ,(0) 0 0 αg
ρg + α`
ρ` 0 v 0 0
1
ρg 0 0 v 0
1
ρ` 0 0 0 v
, (125)
where the superscript “(0)” on the coefficients signifies the zeroth-order expansion in ε, such that
Vpµ,(0)
p =
Tgs2g
˜
Cp,g + T`s2`
˜
Cp,` α
g ρgc2g +
α`
ρ`c2`
Tgs2g
˜
Cp,g + T`s2`
˜
Cp,`
+ ξg ρgc2g −
ξ`
ρ`c2`
2. (126)
The eigenvalues in thepµ-model are, to the lowest order inε,
λ(0)pµ ={v−cpµ, v, v, v, v+cpµ}, (127)
where we have identified the sound speedcpµ of the model, given by
c2
pµ=
α
g ρg +
α`
ρ`
Tgs2g
˜
Cp,g + T`s2`
˜
Cp,`
α
g ρgc2g +
α`
ρ`c2`
Tgs2g
˜
Cp,g + T`s2`
˜
Cp,`
+ ξg ρgc2g −
ξ`
ρ`c2`
2. (128)
Proposition 7.1. Thepµ-model satisfies the weak subcharacteristic condition with respect to thep-model, given only the physically fundamental conditions ρk, Cp,k, Tk >0, for k ∈ {g, `}, in the equilibrium state defined by
(57).
Proof. From (62) and (128), we observe that we may write
c−2
pµ =c −2
p +Z p
pµ, where Z
p pµ =
ξ
g ρgc2g −
ξ`
ρ`c2` 2
α
g ρg +
α`
ρ`
Tgs2g
˜
Cp,g + T`s2`
˜
Cp,`
. (129)
Due to the given physical conditions,Zpµp ≥0, and hence0≤cpµ ≤cp, i.e. the weak subcharacteristic condition
is satisfied.
Proposition 7.2. Thepµ-model satisfies the weak subcharacteristic condition with respect to theµ-model, under the physically fundamental conditions ρk, Cp,k, Tk>0, for k∈ {g, `}, in the equilibrium state defined by (57). Proof. Using the expressions (105) and (128) for the sound speeds in the two models, we may write
(c2
µ,+−c 2
pµ)(c
2
µ,−−c2pµ) =−Q µ
pµ, (130)
where
Qµ pµ=
αgα`
ρgc2gρ`c2` Tgs2
g
˜
Cp,g + T`s2`
˜
Cp,`
hTgs2 g
˜
Cp,g + T`s2`
˜
Cp,` c
2 g−c
2
`
− ξg ρgc2g −
ξ`
ρ`c2`
ξgc2`
αg + ξ`c2g
α` i2
hT
gs2g
˜
Cp,g + T`s2`
˜
Cp,` +
ξ2 g mgc2g +
ξ2
`
m`c2` i α
g ρgc2g +
α`
ρ`c2`
Tgs2g
˜
Cp,g + T`s2`
˜
Cp,`
+ ξg ρgc2g −
ξ`
ρ`c2` 22
. (131)
Clearly, on the given conditions, Qµ
pµ ≥ 0. Therefore, exactly one factor on the left hand side of (130) is
Proposition 7.3. The vpµ-model satisfies the subcharacteristic condition with respect to the pµ-model, given the physically fundamental conditions ρk, Cp,k, Tk>0.
Proof. The sound speed in thevpµ-model is given by [22,42]
c2
vpµ= 1 ρ
Tgs2g
˜
Cp,g + T`s2`
˜
Cp,` α
g ρgc2g +
α`
ρ`c2`
Tgs2g
˜
Cp,g + T`s2`
˜
Cp,`
+ ξg ρgc2g −
ξ`
ρ`c2`
2. (132)
Now, we may write
c−vpµ2 =c−
2
pµ +Z pµ
vpµ, where Z
pµ vpµ=
αgα`
ρgρ` (ρ`−ρg)
2
c−pµ2, (133)
which is clearly positive, due to the given conditions. Thus,0≤cvpµ≤cpµ, i.e. the subcharacteristic condition
is satisfied.
Remark 7.4. By direct comparison of (128) and (132), we find the ratio
cpµ cvpµ
=
s
ρ
α
g ρg
+α` ρ`
. (134)
This is exactly the same ratio as has been shown to hold for other models associated withv-relaxation in the
p-branch of the hierarchy [21,46]. We can thus extend the relation
cp cvp
= cpT cvpT
= cpT µ cvpT µ
= cpµ cvpµ
, (135)
by the newly obtained ratio (134) between the sound speeds of thevpµ- andpµ-models.
Proposition 7.5. The pT µ-model satisfies the weak subcharacteristic condition with respect to thepµ-model, given the physically fundamental conditions ρk, Cp,k, T >0, in the equilibrium state defined by (57).
Proof. In the equilibrium state defined by the pT µ-model, we have Tg =T` ≡ T. The sound velocity in the pT µ-model is given in [46], and may be rewritten as
c2pT µ=
αg ρg +
α`
ρ`
αg ρgc2g +
α`
ρ`c2` + ˜Cp,gT h
1 ∆h
1
ρ` −
1
ρg
+ Γg ρgc2g
i2
+ ˜Cp,`T
h
1 ∆h
1
ρg −
1
ρ`
− Γ`
ρ`c2`
i2, (136)
where we have introduced the enthalpy difference∆h=hg−h`.
We may reorganize the last equality in (135) to yield
cpµ cpT µ
= cvpµ cvpT µ
. (137)
Flåtten and Lund [22] showed that the subcharacteristic condition is satisfied between the models on the right hand side, i.e. that 0 ≤ cvpT µ ≤ cvpµ. The same must hold for the models on the left hand side of (137),
i.e.0≤cpT µ≤cpµ, and hence the weak subcharacteristic condition is satisfied. In particular, we may write the
sound speed as
c−2
pT µ=c −2
pµ +Z pµ
where
ZpT µpµ = ˜Cp,gC˜p,`T
h
1 ∆h
1
ρ` −
1
ρg sg
˜
Cp,g + s`
˜
Cp,`
+ Γg ρgc2g
s`
˜
Cp,` +
Γ`
ρ`c2`
sg
˜
Cp,g
i2
α
g ρg +
α`
ρ`
s2 g
˜
Cp,g + s2
`
˜
Cp,`
. (139)
Clearly,ZpT µpµ ≥0based on the given conditions.
8.
The
T µ
-model
We now investigate the model which arises when we assume instantaneous thermal-chemical equilibrium, i.e. let the relaxation parametersK,H → ∞, which expectedly corresponds to
Tg=T`≡T and µg=µ`≡µ. (140)
The productsHg=H(T`−Tg)andKg=K(µ`−µg)remain finite, and may be expressed in terms of spatial
derivatives and remaining source terms. In the forthcoming, we seek explicit expressions for these terms to insert into the basic model of Section2.
The equilibrium conditions are contained in (68) and (85). These may be combined to yield
Kg=−AµT µ∂xαg−GµT µε∂xµ−TµT µε∂xT −Vµ,T µg∂xαgvg+Vµ,`T µ∂xα`v`+KpT µI˜g+KvT µεMg (141)
where the coefficients are given in Appendix B.
8.1.
Governing equations
We are now in a position to state the T µ-model in its entirety, by inserting (141) into the basic model of Section2. The model can be expressed by the following equation set:
• Volume advection: ∂tαg+vi∂xαg=Ig,
• Conservation of mass: ∂tρ+∂x(αgρgvg+α`ρ`v`) = 0,
• Conservation of momentum:
∂tαgρgvg+∂x(αgρgvg2+αgpg) +vi
GµT µ(vg−v`)∂xµ+TµT µ(vg−v`)∂xT+Vµ,T µg∂xαgvg−V
T µ µ,`∂xα`v`
+vi2
Vµ,T µg+V
T µ µ,`
−pi
∂xαg=viKpT µIg+ 1 +viKvT µ(vg−v`)
Mg, (142)
• Conservation of energy: ∂tE+∂x(Egvg+E`v`+αgvgpg+α`v`p`) = 0.
8.2.
Wave velocities
We now seek the wave velocities of the model in the homogeneous limit, where I,M → 0. As usual, we
are interested in the zeroth-order expansion in ε=vg−v`.3 We may therefore directly evaluate the evolution
equations in this limit, and takevg=v`=v if they are outside the differential operator.
After some tedious, but fairly straightforward algebra, we find that to the lowest order inε, the wave velocities of theT µ-model are given by
λ(0)T µ={v−cT µ,+, v−cT µ,−, v, v+cT µ,−, v+cT µ,+}. (143) 3Strictly speaking, exact eigenvalues may be found analytically in this model, since noting thatα
g is a characteristic variable
