Wave structure in the contact line region during high speed droplet impact on a surface: Solution of the Riemann problem for the stiffened gas equation of state

Abstract

Upon impact of a liquid droplet on a rigid substrate at sufficiently high velocity, a wave front adjacent to the impacted surface is created, separating the compressed area from the undisturbed bulk of the liquid. At a certain point in the evolution of the phenomenon, the free surface at the vicinity of the contact line yields and an intense lateral jetting phenomenon occurs. The often employed assumption that the wave front is composed of a single shock wave across its entire surface leads to the emergence of an anomaly, prior to the jetting eruption. This anomaly can be removed by the proposition of the existence of a multiple wave structure at the contact line region. In this article, the wave structure at the contact line region is explored analytically. The exact one-dimensional Riemann problem solution for the stiffened gas equation of state is presented, under the assumption of an isentropic flow in the smooth flow region (i.e., in the region with no contact discontinuities). For the case of impact velocity V=500 m/s and a water droplet with radius R=0.1 mm, we show that in addition to the single shock structure, solutions consisting of an expansion fan and shock wave are physically admissible. In this scenario, the pressure created at the contact line decreases compared to the pressure of the single shock structure. It is demonstrated that the liquid particle velocity in the intermediate region between the waves is lower than the velocity in the bulk of compressed liquid. An algorithm for estimating the intermediate region state, when the left and right states are known, is also presented.