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A configurational numerical viscosity for thermodynamically compatible finite volume schemes of compressible multiphase flows

Davide Ferrari, Ilya Peshkov, Michael Dumbser

arXiv:2610.23247v1math.NAcs.NA

TL;DR

Scalar viscosity can produce negative entropy when the energy is not convex in the evolved variables, while acoustic-scale dissipation can smear material interfaces. The paper constructs a phase-variable configurational viscosity; its schemes satisfy the stated conservation, entropy, positivity, contact-equilibrium, and formulation-equivalence properties.

  • Problem

    Scalar viscosity on evolved variables need not produce non-negative entropy when the energy is non-convex, and reducing interface dissipation without decoupling phase extensive variables remains a central challenge.

    Source: §1 · pp. 1–2

  • Method

    The method splits phase jumps into configurational and intrinsic parts, dissipates them at material and acoustic speeds, and transfers dissipation side-wise through each cell’s chain rule.

    Source: §1 · pp. 3–4, §3.3 · p. 11

  • Results

    The configurational viscosity restores thermodynamic compatibility where Rusanov viscosity produces negative entropy and reduces interface spreading compared with scalar Rusanov viscosity.

    Source: §8 · pp. 39–40

  • Takeaways & Limitations

    The semi-discrete schemes conserve phase masses, mixture momentum, and total energy, ensure non-negative face- and phase-wise entropy production and positivity under explicit time-step conditions, preserve contact equilibria, and are equivalent to a density-weighted formulation.

    Source: §1 · p. 4, §4 · p. 14, §5.5 · p. 27

    The transferred scheme is not generally conservative in φa, although phase masses, mixture momentum, and total energy remain conserved and extra terms are at least quadratic for small jumps.

    Source: §6.4 · p. 33

Abstract

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We consider the fluid subsystem of a variationally derived hyperbolic thermodynamically compatible (HTC) model of compressible multiphase flows. Its total energy is in general not convex in the natural state variables, so a scalar numerical viscosity acting on them need not produce non-negative entropy. Reformulating the equations in terms of the volume fractions separates changes in the amount of each phase from changes in its intrinsic state. This guides the construction of a configurational numerical viscosity in phase variables, in which the energy is convex, while the scheme retains the structure of the HTC model. At the semi-discrete level, the new schemes satisfy: I) local conservation of the phase masses, the mixture momentum and the total energy; II) a cell entropy inequality whose production is non-negative face by face and phase by phase, with no convexity assumption in the evolved variables and no clipping; III) positivity of the partial densities and volume fractions under explicit time-step conditions; IV) exact preservation of contact equilibria; V) equivalence with a scheme for the density-weighted volume fractions. The key ingredients are: i) an exact decomposition of the jump of each phase across a face into a configurational part, dissipated with the material speed, and an intrinsic part, dissipated with the acoustic speed; ii) an exchange form of this dissipation between neighbouring cells, with non-negative weights; iii) a side-wise transfer of the dissipation to the evolved variables through the chain rule of each cell, which turns the energy defects into Bregman divergences of the single-fluid energies; iv) a thermodynamic correction of the reversible part built in the phase variables. Properties I)-V) are proved and verified numerically, including a shock tube in which the Rusanov viscosity produces negative entropy and a shock-water column interaction.

1. Introduction

A scalar viscosity can smear material interfaces and produce negative entropy when the energy is not convex in the evolved variables. The paper develops a phase-variable configurational viscosity and a thermodynamic correction to address these issues within the HTC framework.

1. Introduction

  • Scalar viscosity can smear interfaces at the acoustic scale even though they move with the material velocity, and can produce negative entropy when the energy is non-convex in evolved variables.In water, acoustic speed can exceed material speed by an order of magnitude or more; clipping negative production breaks total-energy conservation.Source: §1 · pp. 2–3
  • The central design problem is reducing volume-fraction dissipation below the acoustic scale without decoupling it from phase extensive variables, while preserving energy conservation and non-negative entropy production.Volume fractions are transported with the material velocity, but uncoupled changes to phase mass, momentum, and entropy can strongly alter intrinsic states when a phase is present only in traces.Source: §1 · p. 3
  • The method splits each phase jump into configurational and intrinsic parts, dissipated respectively at material and acoustic speeds, then transfers the exchange side-wise through each cell’s chain rule.The resulting cell energy defects are sums of Bregman divergences of single-fluid energies.Source: §1 · p. 4
  • The proposed dissipation has non-negative entropy production face by face and phase by phase without evolved-variable convexity, preserves phase-mass and volume-fraction positivity, and preserves contact equilibria.Across contact equilibria it reduces to upwinding driven by the material velocity.Source: §1 · p. 4
  • The phase-variable direction for the Abgrall correction gives a non-negative Bregman-divergence denominator; the original evolved-variable direction may vanish across slipping interfaces and collapse the time step.The collapse occurred in the shock–water column computation.Source: §1 · p. 4
  • Side-wise transfer also yields a discrete product rule and semi-discrete equivalence with the density-weighted volume-fraction formulation, whose flux and averaged face velocity follow from the construction.These are consequences of the same chain-rule principle used to transfer dissipation.Source: §1 · p. 4

2. The fluid multiphase HTC model and its volume fraction formulation

The section recalls the fluid HTC mixture model and its volume-fraction formulation, then explains why the numerical method evolves volume fractions. It establishes continuous equivalence with density-weighted volume fractions and motivates the later discrete product rule.

2.1. Variables, kinematics and energy

  • The model distinguishes intrinsic density ρa from partial density ϱa=αaρa and density-weighted volume fraction φa=αρ; partial density tracks phase mass, while αa tracks geometry.Across an air–water interface, water partial density changes by roughly the density ratio, whereas φa follows the volume fraction.Source: §2.1 · p. 5
  • The mixture energy is phase-wise structured as mass times single-fluid energy, while HTC variables mix phases and obscure that structure.The kinetic energy separates centre-of-mass motion from relative motion.Source: §2.1 · pp. 5–6

2.2. Thermodynamic balances and relaxation

  • The fluid subsystem evolves mixture momentum, phase partial densities, relative velocities, phase entropies, mixture density, and density-weighted volume fractions, with relaxation sources for interfacial friction and pressure.Phase masses move with their respective phase velocities, while volume fractions and phase entropies are transported with the mixture velocity.Source: §2.2 · p. 6
  • Interfacial friction removes relative motion and pressure relaxation equalizes phase pressures; their energy removal is balanced by phase entropy production, yielding total-energy conservation and non-negative source production.The model’s reversible part conserves energy, while its irreversible part converts dissipated mechanical energy into phase entropy.Source: §2.2 · p. 7
  • The model’s limiting regimes include frozen relaxation, stiff relaxation, and the pure-phase limit; the latter reduces to that phase’s compressible Euler equations.In the stiff-relaxation limit, relative velocities vanish, pressures equalize, and long acoustic waves propagate at Wood’s speed.Source: §2.2 · p. 7

2.3. Why we solve for the volume fractions

  • The volume-fraction formulation is preferred because its admissible set is the independent simplex αa≥0, Σαa=1, and its phase energies expose a perspective structure.The corresponding φa constraints couple composition to mixture density; convexity of the total energy still requires phase momenta.Source: §2.3 · p. 8
  • Volume fractions encode phase geometry and are materially transported: uniform intrinsic states with common velocity and pressure reduce to advection of the interface.This transported equilibrium motivates a discrete requirement on the numerical dissipation.Source: §2.3 · p. 8
  • For smooth fields with ρ>0 and φa=ραa, the volume-fraction and density-weighted formulations are equivalent, with matching energy residuals and unchanged energy conservation and entropy production.The identity relating their residuals is a product rule; the paper’s dissipation construction supplies its discrete counterpart.Source: §2.3 · pp. 8–9

3. Energy geometry and configurational structure

The method uses different representations for model derivation, equation solving, and transparent energy geometry. Phase variables make the energy convex and guide dissipation and thermodynamic correction, which are then transferred to the evolved variables.

3.1. Intrinsic and extensive phase variables

  • Each phase is represented by an intrinsic state Ya, extensive variables Xa = αaYa, and phase variables Za = (αa, Xa), separating its intrinsic state from its amount.The intrinsic state describes the fluid within a phase; the extensive variables also encode how much of that phase is present.Source: §3.1 · p. 9

3.2. The perspective energy

  • Under thermodynamically stable phase equations of state, the perspective energy is convex in Za and positively homogeneous of degree one.Its linearity along rays means changing phase amount at fixed intrinsic state is a zero-curvature direction.Source: §3.2 · pp. 10–11

3.3. The configurational direction

  • A configurational change varies phase amount while holding its intrinsic state fixed, so αa and Xa must change together to avoid altering the phase's intrinsic state.Independent changes can produce a large intrinsic-state change when the phase fraction is small.Source: §3.3 · p. 11

3.4. One energy, three representations

  • Although the internal energy is convex in phase variables, the total energy in evolved variables q has an indefinite Hessian whenever two phases have nonzero relative velocity.The result holds for arbitrarily small nonzero relative velocity; in q, the configurational direction couples to mixture momentum.Source: §3.4 · p. 12

3.5. Implication for numerical dissipation

  • Because scalar viscosity in q can have negative entropy production, the method constructs viscous exchange in Z and transfers it side-wise to q through each cell's chain rule.The phase-variable energy pairing is non-negative by convexity; the resulting energy defects are Bregman divergences of the single-fluid energies.Source: §3.5 · p. 14

3. Energy geometry and configurational structure

  • The method builds fluxes and non-conservative products in q, while constructing dissipation and thermodynamic correction in convex phase variables Z before transferring them to q.The same transfer reaches the original variables qφ, yielding identical semi-discrete solutions.Source: §3 · p. 9, §3.5 · p. 14

4. Semi-discrete HTC framework

The semi-discrete HTC framework separates reversible energy-compatible fluxes from dissipative entropy production, correcting reversible fluctuations in phase variables and assigning energy defects non-negatively to cells. Its multiphase dissipation must also preserve positivity, phase structure, two dissipation scales, and contact equilibria, which clipping negative production may compromise.

4. Semi-discrete HTC framework

  • The semi-discrete framework imposes discrete conditions on reversible fluxes and non-conservative products, relaxation sources, and numerical dissipation, assigning energy restoration to the reversible correction and heat production to dissipation.The framework changes the direction of the standard thermodynamic correction because its denominator need not be non-negative in the evolved variables.Source: §4 · p. 14

4.1. Cell and face notation

  • Each face contributes a reversible fluctuation, a dissipative contribution, and phase-entropy production; dissipative terms may differ between cells on non-conservative rows.On conserved rows, the dissipative contribution is a flux.Source: §4.1 · p. 15

4.2. Conservative fluxes and non-conservative contributions

  • When reversible fluctuations satisfy the face-wise energy condition, their implied single-valued energy flux yields a local semi-discrete energy balance and conserves total energy, which is not evolved.The central flux and midpoint path rule generally violate this condition, motivating the thermodynamic correction.Source: §4.2 · pp. 15–16, §4.3 · p. 16

4.3. Thermodynamic correction of the reversible part

  • The phase-variable correction enforces the face-wise energy identity; its denominator is a sum of non-negative phase-curvature integrals, vanishing only along flat directions with identical intrinsic phase states.The correction remains reversible and produces no entropy; with slip, identical intrinsic states can still leave a nonzero residual, so the common degeneracy remains.Source: §4.3 · pp. 17–18

4.4. Relaxation sources

  • Pointwise relaxation sources preserve the cellwise identity between entropy production and energy removed by friction and pressure relaxation, including when sources are evaluated implicitly.The identity holds at whatever state the sources are evaluated.Source: §4.4 · p. 18

4.5. Energy compatibility of the dissipative part

  • Under the stated boundary and compatibility conditions, the scheme conserves total energy, while non-negative energy defects and productions give each cell an entropy balance with non-negative production.The defects can be partitioned as Bregman divergences using an exact average of the forces; the configurational viscosity applies this construction in phase variables, despite energy non-convexity in the evolved variables.Source: §4.5 · pp. 18, 20
  • Clipping negative dissipative production preserves its sign but leaves the corresponding energy defect uncompensated, breaking compatibility unless the dissipation or energy flux is also modified.Source: §4.5 · p. 20
  • A positive-semidefinite dissipation G = K ∆p cannot dissipate volume-fraction jumps at contact or produce entropy-neutral phase exchange when ∆p = 0.At contact, uniform intrinsic states, common velocity, and common pressure make all thermodynamic forces continuous; central discretization alone does not preserve volume-fraction positivity.Source: §4.5 · p. 20
  • R1–R2 require positive partial densities and volume fractions under an explicit time-step condition, same-phase mass exchange between neighbours, and preservation of intrinsic states during configurational changes.The dissipation must respect the configurational direction.Source: §4.5 · pp. 20–21
  • R3–R4 require volume-fraction dissipation to be weaker than acoustic dissipation without violating R1–R2, while contact equilibria remain intact and their interface is only transported.Source: §4.5 · p. 21
  • The proposed dissipation satisfies the energy requirement without convexity in q and is designed to meet the additional multiphase requirements R1–R4.Section 5 constructs this dissipation with non-negative energy defects.Source: §4.5 · p. 21

5. A thermodynamically compatible configurational viscosity

The section builds dissipation phase by phase, separating changes in phase amount from changes in intrinsic state and transferring the exchange through each cell’s chain rule. A common energy flux yields non-negative cell defects and phase entropy production under the stated convexity conditions.

5.1. An exact decomposition of the phase jump

  • The finite phase jump splits into an amount change at a volume-weighted mixed intrinsic state and an intrinsic-state change weighted by the harmonic mean of volume fractions.Source: §5.1 · p. 21

5.2. Two dissipative scales

  • Configurational changes use material speed while intrinsic changes retain acoustic speed, reducing interface diffusion when intrinsic states match without breaking the product relation.Source: §5.2 · p. 22

5.3. Exchange form with non-negative weights

  • The dissipation exchanges each phase’s own mass, momentum, entropy, and volume with non-negative weights, preserving partial-density and volume-fraction positivity and saturation under the stated condition.Source: §5.3 · p. 23

5.4. Side-wise transfer to the model variables

  • A single phase-variable exchange passes through each cell’s own chain rule; conserved linear rows remain equal-and-opposite, while relative-velocity rows need not.Source: §5.4 · pp. 24–25

5.5. Energy flux and entropy production

  • With positive phase volumes and densities and convex single-fluid energies, both cell defects are non-negative, yielding phase-wise entropy production without clipping.Source: §5.5 · pp. 27–28

6. Properties of the scheme

The semi-discrete scheme combines energy conservation and non-negative entropy production with positivity, contact preservation, and equivalence between volume-fraction and density-weighted formulations. These properties have specific time-step and admissibility conditions, and the transferred formulation is not generally conservative in its density-weighted volume-fraction rows.

6.1. Energy conservation and entropy production

  • The semi-discrete scheme satisfies a local energy balance and a cell entropy balance with non-negative production.The entropy production is non-negative for each phase and hence for the mixture.Source: §6.1 · p. 28

6.2. Positivity of phase masses and volume fractions

  • Under the stated forward Euler time-step restrictions, phase partial densities remain non-negative and volume fractions obey a local maximum principle, preserving [0, 1].The same properties extend to SSP Runge–Kutta methods when each forward Euler substep satisfies the restrictions.Source: §6.2 · p. 29

6.3. Preservation of contact equilibria

  • With constant intrinsic density and specific entropy in each phase and common velocity and pressure, the scheme preserves those states and produces no entropy.Stationary contacts are exact; advected contacts use material-velocity upwind diffusion, so the volume-fraction profile may broaden.Source: §6.3 · pp. 30–31

6.4. Discrete product rule and equivalence of the two formulations

  • The volume-fraction and density-weighted formulations have corresponding semi-discrete solutions with identical other state variables while solutions remain admissible and mixture density stays positive.The transfer also preserves energy balance, entropy production, phase masses, volume fractions, and contact equilibria.Source: §6.4 · p. 33
  • The transferred scheme is not generally conservative in the density-weighted volume-fraction rows, although phase masses, mixture momentum, and total energy remain conserved.The additional terms are at least quadratic in the jumps for small jumps.Source: §6.4 · p. 33
  • Equivalence is semi-discrete: independently applying the same time integrator in each representation need not preserve the product rule.Fully discrete equivalence and transfer of time-step-dependent positivity statements require a compatible time update.Source: §6.4 · p. 33

7. Numerical results

The tests compare configurational and Rusanov viscosities across smooth flow, transported contacts, shocks, and a shock–water interaction. They show reduced interface diffusion with configurational viscosity and expose negative Rusanov entropy production in the shock tube.

7.1. Convergence: a stationary two-phase vortex

  • The smooth-vortex test gives second-order convergence without viscosity and orders approaching one with either viscosity; configurational viscosity reduces volume-fraction error by about a factor of three on every grid.Errors in the other variables remain comparable, and the test observes no negative Rusanov production despite non-convexity at slipping states.Source: §7.1 · pp. 34–35, Table 2 · p. 34, Figure 2 · p. 35

7.2. Transported contact

  • Both schemes preserve the transported contact with small deviations, but configurational viscosity reduces interface diffusion by reducing to material-speed upwinding; Rusanov diffuses on the acoustic timescale.Mixture pressure and velocity remain uniform to about 10^-13; phase pressure difference and relative velocity remain of order 10^-9 and 10^-8, respectively.Source: Figure 3 · p. 35, §7.2 · p. 36

7.3. Multiphase shock tube

  • Both shock-tube viscosities reproduce the reference wave structure and keep partial densities and volume fractions positive, while configurational viscosity gives a narrower volume-fraction transition and smaller L1 differences in density, velocity, and pressure.Both reproduce the entropy increase across the shock with similar entropy-density profiles; numerical profiles also show small positive η1 in the rarefaction and near the contact, where the reference has η1 = 0.Source: §7.3 · pp. 36–37, Figure 4 · p. 36
  • −0.87: Rusanov’s pairing becomes negative where a trace phase slips; clipping then causes positive energy drift, while configurational viscosity needs no clipping and has a relative total-energy change of about 1.2 × 10^-8.The energy compatibility of configurational viscosity is semi-discrete and does not imply exact conservation after time integration.Source: §7.3 · p. 37, Figure 5 · p. 37

7.4. Shock–water column interaction

  • 3.18 mm versus 2.75 mm: the equivalent radius of the water region remains at its initial value with configurational viscosity but decreases with Rusanov.At t = 8 µs, the relative difference between state energy and independently transported energy is 3.5 × 10^-8 with configurational viscosity and 1.4 × 10^-7 with Rusanov.Source: §7.4 · p. 37, Figure 6 · p. 38
  • After 2 µs, the original correction direction increases the largest exchange rate by more than two orders of magnitude and reduces the time step to about 10^-11 s; the phase-variable direction reaches 8 µs in 855 steps.The original-direction run stopped at 3.8 µs after 982 steps.Source: §7.4 · p. 38, Figure 7 · p. 38

8. Conclusions

The paper constructs configurational viscosity in phase variables for a multiphase HTC model whose energy is not convex in the evolved variables. Its analysis and tests establish thermodynamic compatibility without clipping, while the applicability to a complete unified HTC mixture model remains open.

8. Conclusions

  • The phase-variable construction separates configurational and intrinsic changes, dissipates them at material and acoustic speeds, and transfers dissipation through each cell’s chain rule.Non-negative exchange weights and a common viscous energy flux yield non-negative Bregman defects and semi-discrete energy and entropy relations without clipping.Source: §8 · p. 39
  • The construction preserves positive partial densities, admissible volume fractions, and contact equilibria under stated explicit time-step conditions, and is semi-discretely equivalent to a density-weighted volume-fraction formulation.The numerical tests report reduced interface spreading and show Rusanov clipping can introduce an energy defect; phase-variable correction also avoids the severe time-step reduction of the original direction in the water-column test.Source: §8 · pp. 39–40
  • Applicability to the complete unified HTC model of mixtures remains to be investigated.The authors also identify possible model formulations of relative motion addressing energy convexity as a topic for further study.Source: §8 · p. 40

Appendix A. Convexity in the different representations

The appendix explains how nonlinear changes of variables can alter energy convexity through a curvature term weighted by thermodynamic forces. It applies this result to the model’s representations, identifying convexity conditions and cases where Hessians become indefinite.

Appendix A.1. How a change of variables acts on convexity

  • A change of variables transforms the Hessian into a congruence term plus a curvature term weighted by thermodynamic forces; the latter vanishes for affine maps but can change convexity for nonlinear maps.For the model, the nonlinear changes are ratios, including αa = φa/ρ and va = Ma/ϱa.Source: Appendix A · p. 40
  • The condition that the relevant thermodynamic force vanish is sufficient, not necessary, for convexity to transfer between representations; away from it, curvature may destroy or restore convexity.For the changes qφ ↔ α, the force is the pressure difference; for Z ↔ q, it is the relative momentum, and both vanish at mechanical equilibrium.Source: Appendix A · p. 41

Appendix A.2. A single phase and its perspective

  • The phase energy is convex in Z but never strictly convex because extensivity creates flat directions; strict convexity would require an additional strictly convex interfacial contribution.The single-phase energy uses the perspective of the specific internal energy, which preserves convexity under the stated thermodynamic assumption.Source: Appendix A · p. 41

Appendix A.3. Volume fractions versus density-weighted volume fractions

  • The internal energy is convex in q, whereas its Hessian in qφ is indefinite when (p1 − p2)(ρ1 − ρ2) < 0.At pressure equilibrium, the Hessian in qφ is positive semi-definite.Source: Appendix A · pp. 41–42
  • The map qφ → q is the identity except for α1 = φ1/ρ, its only nonlinear component; at p1 = p2, only the congruence remains, and (i) gives semi-definiteness.Source: Appendix A · p. 42
  • For (p1 − p2)(ρ1 − ρ2) > 0, the curvature term can restore convexity along a direction flat in q, but this alone does not establish convexity overall.The appendix gives an example with an indefinite total-energy Hessian in q and a positive definite one in qφ; at pressure equilibrium and nonzero slip, both are indefinite.Source: Appendix A · p. 42

Appendix A.4. Relative velocities

The appendix compares the energy Hessians in q and Z and examines whether a configurational flat direction survives the change of variables. Its conclusion depends on relative slip.

Appendix A.4. Relative velocities

  • At zero relative velocity, the Hessians in q and Z are congruent, and the configurational direction lies in the kernel of the positive semi-definite Hessian.The configurational variation changes phase amounts at fixed intrinsic states and is flat in Z.Source: Appendix A.4 · p. 43
  • With nonzero slip, the Hessian in q is indefinite and the configurational direction no longer belongs to its kernel.The nonlinear rows of Z → q are those for the relative velocities; their curvature contribution changes the Hessian relation.Source: Appendix A.4 · p. 43
  • When all relative velocities vanish, the argument for positive semi-definiteness applies across the N-phase extension as well.For nonzero slip, the appendix identifies a configurational direction and a mixture-momentum direction that expose the indefinite case.Source: Appendix A.4 · p. 43

Appendix A.5. The pairing of two states

The pairing of two states in the evolved variables can differ from the pairing in phase variables by a remainder tied to changes in phase mass and velocity under slip. This remainder can have either sign, so smaller jumps alone do not guarantee a non-negative pairing.

Appendix A.5. The pairing of two states

  • The remainder T_w depends on products of phase-mass and phase-velocity jumps weighted by endpoint drifts, and can change the sign of the pairing.It is quadratic in small jumps, like the pairing itself, and vanishes when all drifts vanish.Source: Appendix A.5 · p. 44
  • Zero relative velocity at both states is sufficient for T_w = 0, regardless of whether the phase pressures are in equilibrium..Source: Appendix A.5 · pp. 44–45
  • For two phases, negative pairings occur arbitrarily close to any admissible state with nonzero relative velocity.The construction takes q_R = q_L + hξ for sufficiently small h and yields a negative pairing.Source: Appendix A.5 · p. 45
  • Reducing the jump size alone does not ensure a non-negative pairing in the evolved variables..Source: Appendix A.5 · p. 45

Appendix B. Characteristic structure and asymptotic limits

This appendix examines two complementary properties of the two-phase fluid subsystem: its frozen characteristic structure at equal phase velocities and its mechanically relaxed acoustic response. It also describes the dispersion spectrum connecting these limits.

Appendix B. Characteristic structure and asymptotic limits

  • The frozen characteristic structure is considered at equal phase velocities, while the relaxed response is obtained with stiff interfacial friction and pressure relaxation..Source: Appendix B · p. 45
  • The appendix presents the dispersion spectrum between the frozen and mechanically relaxed acoustic limits..Source: Appendix B · p. 45
  • The results are stated for the original formulation and apply equally to the volume-fraction formulation used by the numerical method.This equivalence follows from Proposition 2.1.Source: Appendix B · p. 45

Appendix B.1. Frozen characteristic structure for two phases

For two phases without relaxation, the appendix characterizes the frozen system, including its equal-velocity spectrum and pure-phase limits. It distinguishes the dominant-phase Euler subsystem from the full spectrum, which retains waves associated with a vanishing phase.

Appendix B.1. Frozen characteristic structure for two phases

  • At equal phase velocities, the system has seven linearly degenerate eigenvalues equal to V1 and four mixed thermo-acoustic eigenvalues.The four transverse components join the seven waves moving with the fluid.Source: Appendix B.1 · pp. 45–46
  • When both phase entropies are zero, the four thermo-acoustic speeds reduce to V1 ± C1 and V1 ± C2 at every composition..Source: Appendix B.1 · p. 46
  • At α1 = 1, the dominant phase decouples as an Euler system with sound speed C1, while the full spectrum still contains a pair for the vanishing phase.The full spectrum retains V1 ± C1 and V1 ± eC2; the corresponding vanishing-phase pair is also present as α1 → 0.Source: Appendix B.1 · p. 46
  • With stiff friction and pressure relaxation, the response is that of a single mechanically equilibrated mixture with Wood’s sound speed.The speeds also depend on the phase entropies through the products s_a e_a.Source: Appendix B.1 · p. 46

Appendix B.2. Relaxed acoustic limit and dispersion

The dispersion analysis connects frozen short-wave propagation with the mechanically relaxed, Wood-speed acoustic response at long wavelengths. Between these limits, relaxation and mixture composition determine whether acoustic branches propagate, become overdamped, and how strongly trace-phase waves are damped.

Appendix B.2. Relaxed acoustic limit and dispersion

  • The short-wave propagative branches approach the frozen thermo-acoustic speeds, and relaxation damps these waves at bounded rates without changing their speeds.The frozen speeds are relevant to a CFL restriction based on the unreduced hyperbolic subsystem.Source: Appendix B.2 · p. 47
  • The long-wave acoustic perturbation lies tangent to the mechanical-equilibrium manifold, with δw1 = 0 and δp1 = δp2, and has Wood’s sound speed.This is the phase-wise adiabatic, mechanically equilibrated binary-mixture response; equal phase entropies are not required.Source: Appendix B.2 · p. 48
  • At α1 = 0.5, long waves propagate at Wood’s speed (about 24 m s−1) with γ ∝ k2 damping, while an intermediate band is overdamped and short-wave branches approach C1 and C2 with finite saturated damping.The gas branch re-emerges first; the liquid branch re-emerges at k ≈ 10^4 m−1.Source: Appendix B.2 · p. 48, Figure B.9 · p. 49
  • Near pure-phase limits, the continuous-phase branch propagates from Wood’s to its frozen speed, while the dispersed-phase branch remains overdamped to large wavenumbers.The γ ∝ k2 regime ends near k ≈ 10−4 m−1 in bubbly liquid and extends to k ≈ 10^2–10^3 m−1 in mist.Source: Appendix B.2 · p. 49, Figure B.10 · p. 49
  • Across the composition range, the long-wave propagative root coincides with Wood’s speed, which tends to C2 and C1 in the pure-phase limits.Relaxation times set where the long-wave regime is reached, not its limiting speed.Source: Figure B.11 · p. 50, Appendix B.2 · p. 50
  • As α1 approaches 0 or 1, the high-frequency damping rate of the wave pair belonging to the vanishing phase grows by several orders of magnitude.That pair remains in the frozen spectrum even as relaxation damps it increasingly strongly.Source: Appendix B.2 · p. 50, Figure B.11 · p. 50

Appendix C. Constitutive relations and robust evaluation of the Bregman defects

Appendix C gives constitutive relations and closed-form Bregman divergences for the ideal- and stiffened-gas equations of state. It shows the divergence is a sum of separately non-negative terms and describes robust evaluation for small jumps.

Appendix C. Constitutive relations and robust evaluation of the Bregman defects

  • For the stiffened gas, the stiffness pressure enters as a constant in ε(ρ, σ) and does not contribute to the divergence.The ideal-gas case has B = 0.Source: Appendix C · p. 50
  • The closed-form divergence is a sum of three separately non-negative terms, so its sign requires no cancellation.The functions ϕ(z) and ψ(t) are non-negative and vanish quadratically near their minima.Source: Appendix C · pp. 50–51
  • For small jumps, the divergences are evaluated by series to retain full relative accuracy; integral quadrature is avoided near phase traces.Near traces, the integrand grows like 1/αa(s), and quadrature would leave the energy identities subject to quadrature error.Source: Appendix C · p. 51

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