Self-similar flows in spherical geometry

If we are looking at the implosion of a sphere starting with a strong shock, the study of self-similar flows is a classical problem. We will assume that: - The sphere contains a perfect gas with a polytropic coefficient $\gamma $=5/3. - The shock follows the equation: r$_{c}$=A(-t)$^{\alpha}$ with t$_{0}<$t$<$0. There are two known solutions to that problem: - The G.Guderley solution corresponding to $\alpha =\alpha _{ref}$ = 0.68838. In this solution, the outer implosion velocity is almost constant and the compression rate at focalization is $\rho $/$\rho _{0}$=9.6. - The Y. Saillard solution corresponding to the same value of $\alpha $ (see SCCM-2005 Proceedings p1515). In this solution, the outer velocity is increasing and the compression rate is tending to infinity. We will exhibit a new family of solutions: there is one solution for each value of $\alpha $ from 0 to $\alpha _{ref}$. As in the Y. Saillard solution, outer velocity and compressing rate are tending to infinity. These new solutions (with two parameters, initial outer velocity and shock shape coefficient $\alpha)$ can provide us with benchmarks and perhaps also with ICF target design tool.