Solution to the Nemchinov-Dyson Problem in $2$-D Axial Geometry

The purpose of this work is to examine the solutions to the $2$-D inviscid compressible flow (Euler) equations in axial geometry subject to an ideal gas equation of state (EOS) constrained by the Nemchinov-Dyson assumption on the included velocity field. Assuming a separable solution for the flow velocities $u_r$ and $u_z$ which is defined as a linear spatial component and an arbitrary time function $R_r(t)$ and $R_z(t)$, respectively, we find we find several solution sets for density ($\rho$) , pressure ($P$), and specific internal energy (SIE) ($I$) that are constrained by two ordinary differential equations and arbitrary spatial dependence. These spatial functions are defined as $\Pi\left(\xi,\eta\right)$, $\beta\left(\xi,\eta\right)$, $\Upsilon\left(\xi,\eta\right)$ for $\rho$, $P$, and $I$, respectively, for similarity variables $\xi=r/R_r(t)$ and $\eta=z/R_z(t)$. Using various physically-relevant initial conditions, we find 11 unique numerical solutions to the functional form of $R_r(t)$ and $R_z(t)$. Using different initial density profiles, with assumptions connecting back to uniform thermodynamic properties, we derive specific unique spatial functions for $\rho$, $P$, and $I$. Finally, we show the overall solutions to $\rho$, $P$, and $I$.