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One dimensional hydrodynamic model of nuclear collisions

Recently a hydrodynamical description of low energy nuclear collisions between heavy ions is very popular. Nuclear matter is a symmetric homogeneous mixture of two kind of nuclear constituents: protons (positively charged) and neutrons (not charged). In Lagrangian form the Navier-Stokes system is given by: Let $ u(x,t)=1/n(x,t)$ the specific volume, $v(x,t) $ the velocity, $e(x,t)$ the internal energy, $\theta(x,t)$ the temperature, $Q(x,t)$ the termal flux and $\sigma(x,t)$ the stress then the compressible Navier-stokes system reads as follows:
\begin{eqnarray*}u_t&=&v_x\\v_t&=&\sigma_x\\(e+1/2v^2)_t&=&(\sigma v)_x-Q_x\end{eqnarray*}
for $t\ge0$ and $x\in [0,M]$ where $M$ is the conserved mass on a slab. Further, it is necessary to assume the so-called Clausius-Duhem inequality
\begin{eqnarray*}S_t+\left(\frac{Q}{\theta}\right)_x\ge 0\end{eqnarray*}
where $S$ is the specific entropy. The stress is given by
\begin{eqnarray*}\sigma(u,v, \theta)=-p(u,\theta)+\nu\frac{v_x}{u}\end{eqnarray*}
The first term is the effective pressure and the second is the phenomenological viscous stress (cf. Hasse, R.W. (1978) Approaches to nuclear friction, Reports on Progress in Physics 41, 1027 - 1101. ). The functions $p$ and $e$ are completely defined by
\begin{eqnarray*}p(u,\theta)=\frac{3}{8}t_0u^{-2}+\frac18t_3u^{-3}+\frac{{\hbar}^2}{5m}\left(\frac{3\pi^2}{2}\right)u^{-5/3}+\frac{k\theta}{u}.\end{eqnarray*}
Here $t_0<0$ and $t_3>0$ are parameters of the so-called Skyrme interaction. $k$ is the Boltzmann constant for our monoparticular gas and $\hbar$ is the Planck' s constant. The internal energy is given by
\begin{eqnarray*}e(u,\theta)=\frac38t_0u^{-1}+\frac{1}{16}t_3u^{-2}+\frac{3\hba......eft(\frac{3\pi^2}{2}\right)^{2/3}u^{-2/3}+ e_m + \frac32 k\theta\end{eqnarray*}
$e_m$ is a suitable chosen positive constant such that $e(u,\theta\ge 0$ for $0<u<\infty$ and $0\le\theta< \infty$
 

A late reference: B. Ducomet Asymptotic behaviour for a non-monotone fluid in one dimension: the positive temperature case (2001) MMAS, 24, ???-???.


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collected and worked out by Wolfgang Sprößig, TU-Bergakademie Freiberg