Characterization of rarefaction waves in van der Waals fluids

We calculate the isentropic evolution of an instantaneously heated foil, assuming a van der Waals equation of state with the Maxwell construction. The analysis by Yuen and Barnard [Phys. Rev. E 92, 033019 (2015)] is extended for the particular case of three degrees of freedom. We assume heating to temperatures in the vicinity of the critical point. The self-similar profiles of the rarefaction waves describing the evolution of the foil display plateaus in density and temperature due to a phase transition from the single-phase to the two-phase regime. The hydrodynamic equations are expressed in a dimensionless form and the solutions form a set of universal curves, depending on a single parameter: the dimensionless initial entropy. We characterize the rarefaction waves by calculating how the plateau length, density, pressure, temperature, velocity, internal energy, and sound speed vary with dimensionless initial entropy.

Figure 1

($\tilde{\rho },\tilde{T}$) diagram of isentropic expansions. The gray dashed curve represents the Maxwell-constructed binodal. The full lines are isentropes, all starting from $\tilde{\rho }=2.7$ and initial temperature ${\tilde{T}}_0=1.8, 3.5, 6.8$, 13, and 26, representing entropies ${\tilde{s}}_0=-2$ (dark blue), $-1$ (light blue), 0 (black), 1 (light brown), and 2 (dark brown).

Figure 2

The isentropes from Fig. 1 are represented and follow the same color notation: as a function of $\tilde{\xi }$, (a) fluid density profile $\tilde{\rho }$, (b) fluid pressure profile $\tilde{p}$, (c) fluid temperature profile $\tilde{T}$, (d) fluid energy density profile $\tilde{\epsilon }$, (e) fluid velocity profile $\tilde{v}$, and (f) fluid sound speed profile ${\tilde{c}}_s$ (note the discontinuity of the sound speed at the binodal). In each plot, an initial density ${\tilde{\rho }}_0=2.7$ is assumed for practical reasons, but each curve can in principle be extended to the left, reaching an asymptotic value of 3. The dots represent the numerical calculations using the 1D planar Lagrangian hydrodynamic code dish. A expanded representation of this figure can be found in Fig. 3.

Figure 3

Expanded representation of Fig. 2. The dots are removed for clarity.

Figure 4

Sound speed ${\tilde{c}}_{s,b}$ at the vicinity of the binodal as a function of entropy ${\tilde{s}}_0$: ${\tilde{c}}_{s,b}^{+}$ in the single-phase regime, ${\tilde{c}}_{s,b}^{-}$ in the two-phase regime. The discontinuity of ${\tilde{c}}_{s,b}$ yields the length in $\tilde{\xi }$ of the plateau, denoted $\mathrm{\Delta }{\tilde{\xi }}_b$ [see Fig. 5].

Figure 5

For each value of ${\tilde{s}}_0$ can be associated at the binodal a unique value of (a) fluid density ${\tilde{\rho }}_b$, (b) fluid pressure ${\tilde{p}}_b$, (c) fluid temperature ${\tilde{T}}_b$, (d) fluid energy density ${\tilde{\epsilon }}_b$, (e) fluid velocity ${\tilde{v}}_b$, and (f) plateau in $\xi$, denoted as $\mathrm{\Delta }{\tilde{\xi }}_b$.

Figure 6

In the (${\tilde{s}}_0, {\tilde{\rho }}_0$) diagram, we traced the curves of constant ratios $R_{\rho }=0.89,0.6,0.2,0.02,0.002$, and $R_{\xi }=0.45,0.1,0.01,0.001$. A given $R_{\rho }$ curve intersects with a given $R_{\xi }$ curve only once. The blue area is the space where the fluid is already in the two-phase regime at the initial point. The curves $R_{\rho }=0.89$ and $R_{\xi }=0.45$, found in Appendix pp2, intersect at ${\tilde{s}}_0=-3.01$ and ${\tilde{\rho }}_0=2.70$, represented by the black dot point.

Figure 7

A single density profile generated by the dish code to test the algorithm of Appendix pp1. The input parameters are revealed in the end of Appendix pp2 to determine how accurate the algorithm is.