Splash in an inhomogeneous gas in one dimension: Exact analysis and molecular dynamics simulations

We investigate the splash phenomenon resulting from the energy input at the interface between a vacuum and an inhomogeneous gas with density profile $\rho(r) = \rho_0 r^{-\beta}$. The energy input causes the formation of ballistic spatters that propagate into the vacuum, leading to a decay of the total energy in the inhomogeneous medium following a power law, $E(t) \sim t^{-\delta_s}$. We determine exactly the exponents $\delta_s$ by solving the Euler equation using a self-similar solution of second kind for different values of $\beta$. These exponents are further validated through event-driven molecular dynamics simulations. The determination of these exponents also allows us to numerically determine the spatio-temporal dependence of the density, velocity and temperature.