Kinetic model for the phase transition of the van der Waals fluid

This is a continuation of previous works [S. Takata and T. Noguchi, J. Stat. Phys. 172, 880 (2018); S. Takata, T. Matsumoto, A. Hirahara, and M. Hattori, Phys. Rev. E 98, 052123 (2018)]. The simple model proposed in the previous works is extended to be free from the isothermal assumption. The new model conserves the total mass, momentum, and energy in the periodic domain. A monotone functional is found, assuring the H theorem for the new model. Different approaches are taken to tell apart the stable, the metastable, and the unstable uniform equilibrium state. Numerical simulations are also conducted for spatially one-dimensional cases to demonstrate various features occurring in the time evolution process. A prediction method for the profile at the stationary state is discussed as well.

Figure 1

The density, temperature, and volume fraction in the stationary state in the limit $\kappa \to 0$. (a) ${\rho }_{\mathrm{A}}$, (b) ${\rho }_{\mathrm{B}}$, (c) $T$, and (d) ${\chi }_{\mathrm{A}}$. The dashed line indicates the curve (26), while the solid line the curve (35). The symbols ($◯$ or $\times$) with the label C1, L1, R1, etc. indicate the cases where the numerical simulation has been carried out.

Figure 2

Time evolution of the density and temperature: Case C4 with $K=1.0\times {10}^{-4}$, $k=0.1$, and ${\epsilon }_{\rho }=0.1$. (a) Density and (b) temperature.

Figure 3

Time evolution of the density: Case L2 with $K=1.0\times {10}^{-5}$ and $k=0.1$ from different initial disturbances. (a) ${\epsilon }_{\rho }=0.01$ and (b) ${\epsilon }_{\rho }=0.1$.

Figure 4

Time evolution of the density for large and small disturbances. (a) Case L1 with ${\epsilon }_{\rho }=0.9$ and (b) case L3 with ${\epsilon }_{\rho }=0.01$. The other parameters are commonly set as $K=1.0\times {10}^{-5}$ and $k=0.1$.

Figure 5

Time evolution of the density: Cases L2 and L3. (a) Case L2 with $K=1.0\times {10}^{-4}$, $k=0.1$, and ${\epsilon }_{\rho }=0.9$. (b) Case L3 with $K=1.0\times {10}^{-3}$, $k=0.1$, and ${\epsilon }_{\rho }=0.9$.

Figure 6

Comparisons of density profile between the simulation and the prediction result. (a) Case C4, (b) case R2, and (c) case L4. Here $K$ is commonly put as $K=1.0\times {10}^{-4}$. The solid line indicates the prediction result, while the bullets the simulation result. The prediction result is shifted to adjust the position of the interface with the simulation result.

Figure 7

Comparisons of the density profile in the stationary state between the previous and the present model. (a) Case C2 and (b) Case C4. In both cases, $K=1.0\times {10}^{-4}$.

Figure 8

The equiarea rule applied to $\tilde{g}$ for the previous and the present model. (a) Case C2 and (b) case C4. The horizontal coordinate of the open circle and that of the closed circle indicate ${\rho }_{\mathrm{A}}/{\rho }_0$ and ${\rho }_{\mathrm{B}}/{\rho }_0$, respectively.