Asymptotic self-similar solutions with a characteristic time-scale
For a wide variety of initial and boundary conditions, adiabatic one dimensional flows of an ideal gas approach self-similar behavior when the characteristic length scale over which the flow takes place, $R$, diverges or tends to zero. It is commonly assumed that self-similarity is approached since in the $R\to\infty(0)$ limit the flow becomes independent of any characteristic length or time scales. In this case the flow fields $f(r,t)$ must be of the form $f(r,t)=t^{α_f}F(r/R)$ with $R\propto(\pm t)^α$. We show that requiring the asymptotic flow to be independent only of characteristic length scales imply a more general form of self-similar solutions, $f(r,t)=R^{δ_f}F(r/R)$ with $\dot{R}\propto R^δ$, which includes the exponential ($δ=1$) solutions, $R\propto e^{t/τ}$. We demonstrate that the latter, less restrictive, requirement is the physically relevant one by showing that the asymptotic behavior of accelerating blast-waves, driven by the release of energy at the center of a cold gas sphere of initial density $ρ\propto r^{-ω}$, changes its character at large $ω$: The flow is described by $0\leδ<1$, $R\propto t^{1/(1-δ)}$, solutions for $ω<ω_c$, by $δ>1$ solutions with $R\propto (-t)^{1/(δ-1)}$ diverging at finite time ($t=0$) for $ω>ω_c$, and by exponential solutions for $ω=ω_c$ ($ω_c$ depends on the adiabatic index of the gas, $ω_c\sim8$ for $4/3<γ<5/3$). The properties of the new solutions obtained here for $ω\geω_c$ are analyzed, and self-similar solutions describing the $t>0$ behavior for $ω>ω_c$ are also derived.