تأثیر مقیاس زمان سرمایش و غیر ایده‌آل بودن گاز در امواج ضربه ای

According to the suddenly compression of the matters in some regions of the compressible fluids, the density and temperature suddenly increases, and shockwaves can be produced. The cooling of post-shock region and non-idealness of the equation of state, $p=(k_B/mu m_p)rho T (1+brho) equivmathcal{K}rho T (1+eta R)$, where $mu m_p$ is the relative density of the post-shock gas and $Requiv rho_2 / rho_1$ is the non-idealness parameter, may affect on the shocked gases. In this article, we study the effects of both cooling timescale and non-idealness of the shocked gases, on the relative density of the post-shock region. For simplicity, the shock is assumed planar and steady in which the deceleration is negligible and there is no any instabilities through the cooling layer. Conservation of mass, momentum, and energy across the shock front are given by the Rankine-Hugoniot conditions. The most important factor through the shock is the energy lost per unit mass during the shock process, $Q=frac{n_2 Lambda}{mu_2 m_p} t_{dur}$, where $Lambda (erg cm^{-3} s^{-1}$ is the cooling function at the post-shock region with density $n_2} and mean particle mass $mu_2 m_p$, and $t_{dur}$ is the duration time of the post-shock process. Accurate determination of the cooling timescale requires specifying the elemental abundance of the post-shock region, but a simple estimate can be obtained using $t_{cool}approx k_B T_2/(n_2Lambda)$. Eliminating the $n_2 Lambda$, we approximately have $Q/c^2approx lambda T$, where $c equiv sqrt{K_1 T_1}$ is the pre-shock sound speed, $lambda  equiv t_{dur}/t_{cool}$ and $T equiv K_2 T_2/K_1 T_1$. We would be interested to consider the collision of two gas sheets with velocities $v_0$ in the rest frame of the laboratory. Defining the Mach number as $M_0 equiv v_0/c$, we obtain a third degree polynomial equation for $R$, with coefficients as functions of the three parameters $eta$, $lambda$, and $M_0$. We numerically solved this three degree polynomial equation to obtain $R$.The results for adiabatic case ($lambda=0$), with ideal ($eta=0$) and non-ideal ($eta neq 0$) mono-atomic ($gamma_1 = gamma_2 = 5/3$) gas are shown in the Fig.1. In the ideal case, the strong supersonic shockwave ($M_0 rightarrow infty$) leads to $R approx 4$. Considering of non-ideal parameter ($eta neq 0$) increases the pressure of the post-shock region so that the shock fronts move faster. In this way, for each $M_0$, the relative density of the post-shock non-ideal gas decreases in respect to the ideal case. The cooling shockwaves with low cooling timescale ($lambda=1$) and fast cooling timescale ($lambda=10$) are shown in the Fig.2. The results show that the relative density of post-shock gas, $R$, increases with increasing the Mach number, $M_0$, and asymptotically reaches to a value which depends on the two other parameters $eta$ and $lambda$. With increasing of the energy lost per unit mass during the shock process, $Q$ (i.e., increasing of $lambda$), the post-shock gas has more chance for condensation and increasing of its relative density, while including the non-ideal effects (i.e., increasing of $eta$) reduces this chance.