Asymptotic Scaling Laws for Imploding Thin Fluid Shells

Scaling laws governing implosions of thin shells in converging flows are established by analyzing the implosion trajectories in the ( $A,M$) parametric plane, where $A$ is the in-flight aspect ratio, and $M$ is the implosion Mach number. Three asymptotic branches, corresponding to three implosion phases, are identified for each trajectory in the limit of $A,M\gg 1$. It is shown that there exists a critical value ${\gamma }_{\mathrm{cr}}=1+2/\nu$ ( $\nu =1,2\mathrm{}$ for, respectively, cylindrical and spherical flows) of the adiabatic index $\gamma$, which separates two qualitatively different patterns of the density buildup in the last phase of implosion. The scaling of the stagnation density ${\rho }_s$ and pressure $P_s$ with the peak value $M_0$ of the Mach number is obtained.