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 the 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.