A nonisothermal thermodynamical model of liquid-vapor interaction with metastability

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

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)

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)

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

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

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

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) $

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

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

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) $

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

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) $

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

Figure 14.  Initial data for the stable diphasic test in plane $ (\tau, p) $

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

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) $

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.

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) $

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.