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May  2021, 26(5): 2371-2409. doi: 10.3934/dcdsb.2020183

A nonisothermal thermodynamical model of liquid-vapor interaction with metastability

Hala Ghazi 1,, , François James 2, and  Hélène Mathis 1,

1. 

Université de Nantes & CNRS UMR 6629, Laboratoire de Mathematiques Jean Leray, 2, rue de la Houssinière, BP 92208, F-44322 Nantes Cedex 3, France
2. 

Université d'Orléans, Université de Tours & CNRS UMR 7013, Institut Denis Poisson, Rue de Chartres, BP 6759, F-45067 Orléans Cedex 2, France

* Corresponding author: hala.ghazi@univ-nantes.fr

Received  October 2019 Revised  March 2020 Published  June 2020

The paper concerns the construction of a compressible liquid-vapor relaxation model which is able to capture the metastable states of the non isothermal van der Waals model as well as saturation states. Starting from the Gibbs formalism, we propose a dynamical system which complies with the second law of thermodynamics. Numerical simulations illustrate the expected behaviour of metastable states: an initial metastable condition submitted to a certain perturbation may stay in the metastable state or reaches a saturation state. The dynamical system is then coupled to the dynamics of the compressible fluid using an Euler set of equations supplemented by convection equations on the fractions of volume, mass and energy of one of the phases.

Keywords: Thermodynamics of phase transition, metastable states, van der Waals EoS, dynamical systems, homogeneous relaxation model, numerical simulations.
Mathematics Subject Classification: 80A10, 37N10, 76T10, 35L40.
Citation: Hala Ghazi, François James, Hélène Mathis. A nonisothermal thermodynamical model of liquid-vapor interaction with metastability. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2371-2409. doi: 10.3934/dcdsb.2020183
References:
[1]

M. R. Baer and J. W. Nunziato, A two phase mixture theory for the deflagration to detonation (ddt) transition in reactive granular materials, Int. J. Multiphase Flow, 12 (1986), 861-889.  doi: 10.1016/0301-9322(86)90033-9.  Google Scholar

[2]

D. W. Ball, Physical Chemistry, Cengage Learning, 2002. Google Scholar

[3]

T. Barberon and P. Helluy, Finite volume simulation of cavitating flows, Computers and Fluids, 34 (2005), 832-858.  doi: 10.1016/j.compfluid.2004.06.004.  Google Scholar

[4]

J. Bartak, A study of the rapid depressurization of hot water and the dynamics of vapour bubble generation in superheated water, Int. J. Multiph. Flow, 16 (1990), 789-98.  doi: 10.1016/0301-9322(90)90004-3.  Google Scholar

[5]

H. B. Callen, Thermodynamics and an Introduction to Thermostatistics, 2$^{nd}$ edition, Wiley and Sons, 1985. doi: 10.1119/1.19071.  Google Scholar

[6]

F. Caro, Modélisation et simulation numérique des transitions de phase liquide vapeur, PhD thesis, Ecole Polytechnique X, 2004. Google Scholar

[7]

M. De Lorenzo, Modelling and numerical simulation of metastable two-phase flows, PhD thesis, Université Paris-Saclay, 2018. Google Scholar

[8]

M. De Lorenzo, P. Lafon, M. Di Matteo, M. Pelanti, J.-M. Seynhaeve and Y. Bartosiewicz, Homogeneous two-phase flow models and accurate steam-water table look-up method for fast transient simulations, Int. J. Multiph. Flow, 95 (2017), 199-219.  doi: 10.1016/j.ijmultiphaseflow.2017.06.001.  Google Scholar

[9]

M. De Lorenzo, P. Lafon and M. Pelanti, A hyperbolic phase-transition model with non-instantaneous EoS-independent relaxation procedures, J. Comput. Phys., 379 (2019), 279-308.  doi: 10.1016/j.jcp.2018.12.002.  Google Scholar

[10]

G. Faccanoni, S. Kokh and G. Allaire, Modelling and simulation of liquid-vapor phase transition in compressible flows based on thermodynamical equilibrium, ESAIM Math. Model. Numer. Anal., 46 (2012), 1029-1054.  doi: 10.1051/m2an/2011069.  Google Scholar

[11]

S. Fechter, C.-D. Munz, C. Rohde and C. Zeiler, A sharp interface method for compressible liquid-vapor flow with phase transition and surface tension, J. Comput. Phys., 336 (2017), 347-374.  doi: 10.1016/j.jcp.2017.02.001.  Google Scholar

[12]

T. Gallouët, J.-M. Hérard and N. Seguin, Some recent finite volume schemes to compute Euler equations using real gas EOS, Internat. J. Numer. Methods Fluids, 39 (2002), 1073–1138. doi: 10.1002/fld.346.  Google Scholar

[13]

S. Gavrilyuk, The structure of pressure relaxation terms: The one-velocity case, Technical Report, EDF, (2014), H-I83-2014-0276-EN. Google Scholar

[14]

H. Ghazi, Modélisation d'écoulements compressibles avec transition de phase et prise en compte des états métastables, PhD thesis, 2018. Google Scholar

[15]

H. Ghazi, F. James and H. Mathis, Vapour-liquid phase transition and metastability, ESAIM: Proceedings and Surveys, 66 (2019), 22-41.  doi: 10.1051/proc/201966002.  Google Scholar

[16]

J. W. Gibbs, The Collected Works of J. Willard Gibbs, vol I: Thermodynamics, Yale University Press, 1948. doi: 10.2307/3609900.  Google Scholar

[17]

E. Godlewski and P. A. Raviart, Numerical approximation of hyperbolic systems of conservation laws, Applied Mathematical Sciences, 118 Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-0713-9.  Google Scholar

[18]

P. Helluy, O. Hurisse and E. Le Coupanec, Verification of a two-phase flow code based on an homogeneous model, Int. J. Finite Vol., 24 (2015).  Google Scholar

[19]

P. Helluy and H. Mathis, Pressure laws and fast Legendre transform, Math. Models Methods Appl. Sci., 21 (2011), 745-775.  doi: 10.1142/S0218202511005209.  Google Scholar

[20]

J.-B. Hiriart-Urruty and C. Lemaréchal, Fundamentals of convex analysis, in Grundlehren Text Editions, Springer-Verlag, Berlin, 2001. doi: 10.1007/978-3-642-56468-0.  Google Scholar

[21]

O. Hurisse, Application of an homogeneous model to simulate the heating of two-phase flows, Int. J. Finite Vol., 11 (2014), 37 pp.  Google Scholar

[22]

O. Hurisse, Numerical simulations of steady and unsteady two-phase flows using a homogeneous model, Comput. & Fluids, 152 (2017), 88–103, 2017. doi: 10.1016/j.compfluid.2017.04.007.  Google Scholar

[23]

O. Hurisse and L. Quibel, A homogeneous model for compressible three-phase flows involving heat and mass transfer, ESAIM: Proceedings and Surveys, (2019), 84 – 108. doi: 10.1051/proc/201966005.  Google Scholar

[24]

F. James and H. Mathis, A relaxation model for liquid-vapor phase change with metastability, Commun. Math. Sci., 14 (2016), 2179-2214.  doi: 10.4310/CMS.2016.v14.n8.a4.  Google Scholar

[25]

A. K. Kapila, R. Menikoff, J. B. Bdzil, S. F. Son and D. S. Stewart, Two-phase modelling of DDT in granular materials: Reduced equations, Phys. Fluids, 13 (2001), 3002-3024.   Google Scholar

[26]

L. D. Landau and E. M. Lifshitz, Statistical Physics: V. 5. Course of Theoretical Physics, Pergamon Press, 1969.  Google Scholar

[27]

B. J. Lee, E. F. Toro, C. E. Castro and N. Nikiforakis, Adaptive Osher-type scheme for the Euler equations with highly nonlinear equations of state, J. Comput. Phys., 246 (2013), 165-183.  doi: 10.1016/j.jcp.2013.03.046.  Google Scholar

[28]

R. G. Mortimer, Physical Chemistry, Academic Press Elsevier, 2008. Google Scholar

[29]

D. Y. Peng and D. B. Robinson, A new two-constant equation of state, Industrial and Engineering Chemistry: Fundamentals, 15 (1976), 59-64.  doi: 10.1021/i160057a011.  Google Scholar

[30]

R. T. Rockafellar, Convex Analysis, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1997.  Google Scholar

[31]

R. Saurel, F. Petitpas and R. Abgrall, Modelling phase transition in metastable liquids: Application to cavitating and flashing flows, J. Fluid Mech., 607 (2008), 313-350.  doi: 10.1017/S0022112008002061.  Google Scholar

[32]

N. Shamsundarl and J. H. Lienhard, Equations of state and spinodal lines–A review, Nuclear Engineering and Design, 141 (1993), 269-287.  doi: 10.1016/0029-5493(93)90106-J.  Google Scholar

[33]

G. Soave, Equilibrium constants from a modified Redlich-Kwong equation of state, Chemical Engineering Science, 27 (1972), 1197-1203.   Google Scholar

[34]

E. F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction, 3$^{rd}$ edition, Springer-Verlag, Berlin, 2009. doi: 10.1007/b79761.  Google Scholar

[35]

A. Zein, M. Hantke and G. Warnecke, Modeling phase transition for compressible two-phase flows applied to metastable liquids, J. Comp. Phys., 229 (2010), 1964-2998.  doi: 10.1016/j.jcp.2009.12.026.  Google Scholar

show all references

References:
[1]

M. R. Baer and J. W. Nunziato, A two phase mixture theory for the deflagration to detonation (ddt) transition in reactive granular materials, Int. J. Multiphase Flow, 12 (1986), 861-889.  doi: 10.1016/0301-9322(86)90033-9.  Google Scholar

[2]

D. W. Ball, Physical Chemistry, Cengage Learning, 2002. Google Scholar

[3]

T. Barberon and P. Helluy, Finite volume simulation of cavitating flows, Computers and Fluids, 34 (2005), 832-858.  doi: 10.1016/j.compfluid.2004.06.004.  Google Scholar

[4]

J. Bartak, A study of the rapid depressurization of hot water and the dynamics of vapour bubble generation in superheated water, Int. J. Multiph. Flow, 16 (1990), 789-98.  doi: 10.1016/0301-9322(90)90004-3.  Google Scholar

[5]

H. B. Callen, Thermodynamics and an Introduction to Thermostatistics, 2$^{nd}$ edition, Wiley and Sons, 1985. doi: 10.1119/1.19071.  Google Scholar

[6]

F. Caro, Modélisation et simulation numérique des transitions de phase liquide vapeur, PhD thesis, Ecole Polytechnique X, 2004. Google Scholar

[7]

M. De Lorenzo, Modelling and numerical simulation of metastable two-phase flows, PhD thesis, Université Paris-Saclay, 2018. Google Scholar

[8]

M. De Lorenzo, P. Lafon, M. Di Matteo, M. Pelanti, J.-M. Seynhaeve and Y. Bartosiewicz, Homogeneous two-phase flow models and accurate steam-water table look-up method for fast transient simulations, Int. J. Multiph. Flow, 95 (2017), 199-219.  doi: 10.1016/j.ijmultiphaseflow.2017.06.001.  Google Scholar

[9]

M. De Lorenzo, P. Lafon and M. Pelanti, A hyperbolic phase-transition model with non-instantaneous EoS-independent relaxation procedures, J. Comput. Phys., 379 (2019), 279-308.  doi: 10.1016/j.jcp.2018.12.002.  Google Scholar

[10]

G. Faccanoni, S. Kokh and G. Allaire, Modelling and simulation of liquid-vapor phase transition in compressible flows based on thermodynamical equilibrium, ESAIM Math. Model. Numer. Anal., 46 (2012), 1029-1054.  doi: 10.1051/m2an/2011069.  Google Scholar

[11]

S. Fechter, C.-D. Munz, C. Rohde and C. Zeiler, A sharp interface method for compressible liquid-vapor flow with phase transition and surface tension, J. Comput. Phys., 336 (2017), 347-374.  doi: 10.1016/j.jcp.2017.02.001.  Google Scholar

[12]

T. Gallouët, J.-M. Hérard and N. Seguin, Some recent finite volume schemes to compute Euler equations using real gas EOS, Internat. J. Numer. Methods Fluids, 39 (2002), 1073–1138. doi: 10.1002/fld.346.  Google Scholar

[13]

S. Gavrilyuk, The structure of pressure relaxation terms: The one-velocity case, Technical Report, EDF, (2014), H-I83-2014-0276-EN. Google Scholar

[14]

H. Ghazi, Modélisation d'écoulements compressibles avec transition de phase et prise en compte des états métastables, PhD thesis, 2018. Google Scholar

[15]

H. Ghazi, F. James and H. Mathis, Vapour-liquid phase transition and metastability, ESAIM: Proceedings and Surveys, 66 (2019), 22-41.  doi: 10.1051/proc/201966002.  Google Scholar

[16]

J. W. Gibbs, The Collected Works of J. Willard Gibbs, vol I: Thermodynamics, Yale University Press, 1948. doi: 10.2307/3609900.  Google Scholar

[17]

E. Godlewski and P. A. Raviart, Numerical approximation of hyperbolic systems of conservation laws, Applied Mathematical Sciences, 118 Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-0713-9.  Google Scholar

[18]

P. Helluy, O. Hurisse and E. Le Coupanec, Verification of a two-phase flow code based on an homogeneous model, Int. J. Finite Vol., 24 (2015).  Google Scholar

[19]

P. Helluy and H. Mathis, Pressure laws and fast Legendre transform, Math. Models Methods Appl. Sci., 21 (2011), 745-775.  doi: 10.1142/S0218202511005209.  Google Scholar

[20]

J.-B. Hiriart-Urruty and C. Lemaréchal, Fundamentals of convex analysis, in Grundlehren Text Editions, Springer-Verlag, Berlin, 2001. doi: 10.1007/978-3-642-56468-0.  Google Scholar

[21]

O. Hurisse, Application of an homogeneous model to simulate the heating of two-phase flows, Int. J. Finite Vol., 11 (2014), 37 pp.  Google Scholar

[22]

O. Hurisse, Numerical simulations of steady and unsteady two-phase flows using a homogeneous model, Comput. & Fluids, 152 (2017), 88–103, 2017. doi: 10.1016/j.compfluid.2017.04.007.  Google Scholar

[23]

O. Hurisse and L. Quibel, A homogeneous model for compressible three-phase flows involving heat and mass transfer, ESAIM: Proceedings and Surveys, (2019), 84 – 108. doi: 10.1051/proc/201966005.  Google Scholar

[24]

F. James and H. Mathis, A relaxation model for liquid-vapor phase change with metastability, Commun. Math. Sci., 14 (2016), 2179-2214.  doi: 10.4310/CMS.2016.v14.n8.a4.  Google Scholar

[25]

A. K. Kapila, R. Menikoff, J. B. Bdzil, S. F. Son and D. S. Stewart, Two-phase modelling of DDT in granular materials: Reduced equations, Phys. Fluids, 13 (2001), 3002-3024.   Google Scholar

[26]

L. D. Landau and E. M. Lifshitz, Statistical Physics: V. 5. Course of Theoretical Physics, Pergamon Press, 1969.  Google Scholar

[27]

B. J. Lee, E. F. Toro, C. E. Castro and N. Nikiforakis, Adaptive Osher-type scheme for the Euler equations with highly nonlinear equations of state, J. Comput. Phys., 246 (2013), 165-183.  doi: 10.1016/j.jcp.2013.03.046.  Google Scholar

[28]

R. G. Mortimer, Physical Chemistry, Academic Press Elsevier, 2008. Google Scholar

[29]

D. Y. Peng and D. B. Robinson, A new two-constant equation of state, Industrial and Engineering Chemistry: Fundamentals, 15 (1976), 59-64.  doi: 10.1021/i160057a011.  Google Scholar

[30]

R. T. Rockafellar, Convex Analysis, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1997.  Google Scholar

[31]

R. Saurel, F. Petitpas and R. Abgrall, Modelling phase transition in metastable liquids: Application to cavitating and flashing flows, J. Fluid Mech., 607 (2008), 313-350.  doi: 10.1017/S0022112008002061.  Google Scholar

[32]

N. Shamsundarl and J. H. Lienhard, Equations of state and spinodal lines–A review, Nuclear Engineering and Design, 141 (1993), 269-287.  doi: 10.1016/0029-5493(93)90106-J.  Google Scholar

[33]

G. Soave, Equilibrium constants from a modified Redlich-Kwong equation of state, Chemical Engineering Science, 27 (1972), 1197-1203.   Google Scholar

[34]

E. F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction, 3$^{rd}$ edition, Springer-Verlag, Berlin, 2009. doi: 10.1007/b79761.  Google Scholar

[35]

A. Zein, M. Hantke and G. Warnecke, Modeling phase transition for compressible two-phase flows applied to metastable liquids, J. Comp. Phys., 229 (2010), 1964-2998.  doi: 10.1016/j.jcp.2009.12.026.  Google Scholar

Figure 1.  Isothermal curves of the van der Waals EoS in the $ (\tau, p) $ plane. Isothermal curves $ p(\tau, T) $ are plotted in black. The isothermal curve at critical temperature $ T = T_c $ is plotted in green. Below the critical isothermal curve, the pressure is not monotone with respect to the specific volume and increases in the spinodal zone of non admissible states. This zone is delimited by the blue curve representing the set of minima $ (\tau_-, p(\tau_-, T)) $ and maxima $ (\tau_+, p(\tau_+, T)) $ of the pressure for each temperature $ T<T_c $. The Maxwell equal area rule construction allows to replace the non physically admissible increasing branch of an isothermal curve by computing two volumes $ \tau_1^* $ and $ \tau_2^* $ at each temperature $ T<T_c $, such that $ p(\tau_1^*, T) = p(\tau_2^*, T) $. The set of these volumes is represented in red in the graph and corresponds to the saturation dome. The states belonging to decreasing branches of isothermal curves, below the saturation dome (in red) and above the spinodal zone (in blue), are called metastable states
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Figure 2.  Isothermal curves of the van der Waals EoS in the $ (\tau, e) $-plane. The black lines correspond to isothermal curves $ e(\tau, T) $. The isothermal curve at the critical temperature $ T = T_c $ is plotted in green. States belonging to the zone above the critical isothermal curve are supercritical states. The spinodal zone is delimited by the blue curve, which is the graph of the function $ g $ defined in (24). The saturation dome is represented by the set of red points. Stable states belong to the areas below the critical isothermal curve (in green) and above the saturation dome (in red). The metastable areas correspond to zones above the spinodal zone (in blue) and below the saturation dome (in red)
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Figure 9.  Metastable zone: vector field of the dynamical system (53) (light blue arrows). The red line corresponds to the line $ \alpha = \varphi = \xi $. For any initial condition $ \mathbf{r}(0) $, the trajectories converge either towards a point belonging to the line $ \alpha = \varphi = \xi $, corresponding to the state $ (\tau, e) $ (yellow trajectories), or to the point $ \mathbf{r}^* = (\alpha^*, \varphi^*, \xi^*) $, which concurs with a state belonging to the saturation dome (green trajectories)
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Figure 3.  Spinodal zone: vector field of the dynamical system (53) (light blue arrows). The red line corresponds to the line $ \alpha = \varphi = \xi $. Depending on the initial condition $ \mathbf{r}(0) $, the trajectories converge either towards the equilibrium $ \mathbf{r}^* = (\alpha^*, \varphi^*, \xi^*) $ (green lines) or towards $ \mathbf{r}^\# = (1-\alpha^*, 1-\varphi^*, 1-\xi^*) $ (yellow lines). In both case, the asymptotic regime corresponds to the state $ (\tau_i^*, e_i^*) $, $ i = 1, 2 $, defined by (48), belonging to the saturation dome
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Figure 4.  Spinodal zone, from top to bottom. Trajectories of the dynamical system (53) in the $ (\tau, e) $ plane. Starting from an initial state $ (\tau_1(\mathbf{r}), e_1(\mathbf{r}))(0) $ in the stable liquid region (on the magenta isothermal curve), the trajectory $ (\tau_1(\mathbf{r}), e_1(\mathbf{r}))(t) $ is represented with a dashed magenta line and converges towards the saturation dome. The trajectory $ (\tau_2(\mathbf{r}), e_2(\mathbf{r}))(t) $ is represented in orange. Middle and bottom figures: zoom of trajectories $ (\tau_1(\mathbf{r}), e_1(\mathbf{r}))(t) $ and $ (\tau_2(\mathbf{r}), e_2(\mathbf{r}))(t) $ respectively
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Figure 5.  Spinodal zone, from top to bottom. Trajectories of the dynamical system (53) in the $ (\tau, p) $ plane. Starting from an initial state $ (\tau_1(\mathbf{r}), p(\tau_1(\mathbf{r}), e_1(\mathbf{r})))(0) $ in the liquid region (on the isothermal curve in magenta), the trajectory $ (\tau_1(\mathbf{r}), p(\tau_1(\mathbf{r}), e_1(\mathbf{r})))(t) $ is represented with a dashed magenta line and converges towards the saturation dome. The trajectory $ (\tau_2(\mathbf{r}), p(\tau_2(\mathbf{r}), e_2(\mathbf{r})))(t) $ is represented in orange. Middle and bottom figures: zoom of trajectories $ (\tau_1(\mathbf{r}), p(\tau_1(\mathbf{r}), e_1(\mathbf{r})))(t) $ and $ (\tau_2(\mathbf{r}), p(\tau_2(\mathbf{r}), e_2(\mathbf{r})))(t) $ respectively
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Figure 6.  Stable phase zone: vector field of the dynamical system (53) (light blue arrows). The red line corresponds to the line $ \alpha = \varphi = \xi $. For any initial condition $ \mathbf{r}(0) $, the trajectories converge towards a point belonging to the line $ \alpha = \varphi = \xi $, corresponding to the state $ (\tau, e) $
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Figure 7.  Stable phase zone. Trajectories of the dynamical system (53) in the $ (\tau, e) $ plane. Starting from an initial state $ (\tau_1(\mathbf{r}), e_1(\mathbf{r}))(0) $ in the stable liquid region (on the isothermal curve in magenta), the trajectory $ (\tau_1(\mathbf{r}), e_1(\mathbf{r}))(t) $ is represented with a dashed magenta line and converges towards the state $ (\tau, e) $. The trajectory $ (\tau_2(\mathbf{r}(t)), e_2(\mathbf{r}))(t) $ is represented in orange. Middle and bottom figures: zoom of trajectories $ (\tau_1(\mathbf{r}), e_1(\mathbf{r}))(t) $ and $ (\tau_2(\mathbf{r}), e_2(\mathbf{r}))(t) $ respectively
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Figure 8.  Stable phase zone, from top to bottom. Trajectories of the dynamical system (53) in the $ (\tau, p) $ plane. Starting from an initial state $ (\tau_1(\mathbf{r}), p(\tau_1(\mathbf{r}), e_1(\mathbf{r})))(0) $ in the stable liquid region (on the isothermal curve in magenta), the trajectory $ (\tau_1(\mathbf{r}), p(\tau_1(\mathbf{r}), e_1(\mathbf{r})))(t) $ is represented with a dashed magenta line and converges towards the point $ (\tau, e) $. The trajectory $ (\tau_2(\mathbf{r}), p(\tau_2(\mathbf{r}), e_2(\mathbf{r})))(t) $ is represented in orange. Middle and bottom figures: zoom of trajectories $ (\tau_1(\mathbf{r}), p(\tau_1(\mathbf{r}), e_1(\mathbf{r})))(t) $ and $ (\tau_2(\mathbf{r}), p(\tau_2(\mathbf{r}), e_2(\mathbf{r})))(t) $ respectively
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Figure 10.  Metastable state and perturbation within the phase. Top figure: trajectories of the dynamical system (53) in the $ (\tau, e) $ plane. Starting from an initial state $ (\tau_1(\mathbf{r}), e_1(\mathbf{r}))(0) $ in the metastable vapor region (on the magenta isothermal curve), the trajectory $ (\tau_1(\mathbf{r}), e_1(\mathbf{r}))(t) $ is represented with a dashed magenta line and converges towards the state $ (\tau, e) $. The trajectory $ (\tau_2(\mathbf{r}), e_2(\mathbf{r}))(t) $ is represented in orange and starts with an initial condition in the spinodal zone. Bottom figure: zoom of trajectories $ (\tau_i(\mathbf{r}), e_i(\mathbf{r}))(t) $
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Figure 11.  Metastable state and perturbation within the phase. Top figure: trajectories of the dynamical system (53) in the $ (\tau, p) $ plane. Starting from an initial state $ (\tau_1(\mathbf{r}), p(\tau_1(\mathbf{r}), e_1(\mathbf{r})))(0) $ in the metastable vapor region (on the isothermal curve in magenta), the trajectory $ (\tau_1(\mathbf{r}), p(\tau_1(\mathbf{r}, e_1(\mathbf{r}))))(t) $ is represented with a dashed magenta line and converges towards the point $ (\tau, e) $. The counterpart for the state $ (\tau_2(\mathbf{r}), p(\tau_2(\mathbf{r}), e_2(\mathbf{r})))(t) $ is represented in orange, and starts from an initial datum in the spinodal zone. Bottom figure: zoom of trajectories $ (\tau_1(\mathbf{r}), p(\tau_1(\mathbf{r}), e_1(\mathbf{r})))(t) $ and $ (\tau_2(\mathbf{r}), p(\tau_2(\mathbf{r}), e_2(\mathbf{r})))(t) $ respectively
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Figure 12.  Metastable state and perturbation outside the phase. From top to bottom: trajectories of the dynamical system (53) in the $ (\tau, e) $ plane. Starting from an initial state $ (\tau_1(\mathbf{r}), e_1(\mathbf{r}))(0) $ in the metastable vapor region (on the magenta isothermal curve), the trajectory $ (\tau_1(\mathbf{r}), e_1(\mathbf{r}))(t) $ is represented with a dashed magenta line and converges towards the state $ (\tau, e) $. The trajectory $ (\tau_2(\mathbf{r}), e_2(\mathbf{r}))(t) $ is represented in orange and starts with an initial condition in the spinodal zone. Middle and bottom figures: zoom of trajectories $ (\tau_i(\mathbf{r}), e_i(\mathbf{r}))(t) $
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Figure 13.  Metastable state and perturbation outside the phase. From top to bottom: trajectories of the dynamical system (53) in the $ (\tau, p) $ plane. Starting from an initial state $ (\tau_1(\mathbf{r}), p(\tau_1(\mathbf{r}), e_1(\mathbf{r})))(0) $ in the stable liquid region (on the magenta isothermal curve), the trajectory $ (\tau_1(\mathbf{r}), p(\tau_1(\mathbf{r}), e_1(\mathbf{r})))(t) $ is represented with a dashed magenta line and converges towards the point $ (\tau, e) $. The trajectory $ (\tau_2(\mathbf{r}), p(\tau_2(\mathbf{r}), e_2(\mathbf{r})))(t) $ is represented in orange. Middle and bottom figures: zoom of trajectories $ (\tau_1(\mathbf{r}), p(\tau_1(\mathbf{r}), e_1(\mathbf{r})))(t) $ and $ (\tau_2(\mathbf{r}), p(\tau_2(\mathbf{r}), e_2(\mathbf{r})))(t) $ respectively
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Figure 14.  Initial data for the stable diphasic test in plane $ (\tau, p) $
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Figure 15.  Stable diphasic test}. From top left to bottom right: density profile, velocity, pressure, internal energy and temperature profile with respect to the space variable
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Figure 16.  Initial $ (\tau_i, e_i) $ and intermediate $ (\tau_*, e_*) $ data for a weak acoustic perturbation of a metastable vapor state in the plane $ (\tau, p) $
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Figure 17.  Weak acoustic perturbation of a metastable vapor state. From top left to bottom right: density profile, velocity, pressure, internal energy, temperature and fractions profile with respect to the space variable.
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Figure 18.  Initial $ (\tau_i, e_i) $ and intermediate $ (\tau_*, e_*) $ data for a strong acoustic perturbation of a metastable vapor state in plane $ (\tau, p) $
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Figure 19.  Strong acoustic perturbation of a metastable vapor state with a compression with velocity $ 0.9 $. From top left to bottom right: density profile, velocity, pressure, internal energy, temperature and fractions profile with respect to the space variable.
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