constraint of a well-posed boundary value problem requires that the initial (IC) and boundary conditions (BC) of the PDEs coincide [@Ames1965]. The PDE \[eq:origsys\_c,eq:origsys\_i\] requires one IC ($f\left(x, t=0\right)=f_\mathrm{i}$) and two BCs ($f\left(x_{0\left(\infty\right)}\right)=f_{0\left(\infty\right)}$) for each $f\in\left\{u,\rho\right\}$ yielding six auxiliary conditions. The limits of the similarity variable \[eq:simvar\] $$\begin{aligned}
    \xi\left(r,\tau\right)\rightarrow \left\{
    \begin{array}{ll}
      0,  & \mbox{if\ }\quad r\rightarrow 0~\mbox{or\ } \tau\rightarrow\infty\\[2pt]
      \infty, & \mbox{if\ }\quad r\rightarrow\infty~\mbox{or\ } \tau\rightarrow0\label{eq:iccases}
    \end{array}\right.
  \end{aligned}$$ combine the spatial conditions at infinity with the conditions for $\tau=0$, or more specifically, the IC with the outer BC $$\begin{aligned}
   u_\mathrm{i}\left(r\right)=\lim_{\tau\to0}u\left(r,\tau\right)&=\lim_{\xi\to\infty}u\left(\xi\right)=u_\infty\\
   \rho_\mathrm{i}\left(r\right)=\lim_{\tau\to0}\rho\left(r,\tau\right)&=r^{-\kappa}\lim_{\xi\to\infty}S\left(\xi\right)=r^{-\kappa} S_\infty.\label{eq:IC_out}
  \end{aligned}$$ A constant initial velocity field for the dynamical system \[eq:origsys\_c,eq:origsys\_i\] only exists in the limit $\lim_{\tau\to0}u\left(r,\tau\right)\rightarrow u_\mathrm{i}$. This limit may be explained by e.g. \[fig:figure7\] showing the time evolution of the velocity in the $u\left(r, \tau\right)$-plane. The shorter the time, the smaller the osculating circle to the function describing the upstream region until it is ending up in a $\delta$-distribution-like initial condition. The initial density distribution equation \[eq:densdist\] is given by a power law depending on radius and the constant $S_{\infty}$. We may argue, that the physically relevant profiles are due to only non-negative values of $\kappa$, because we want the majority of the fluid located at 