Early-time jet formation in liquid-liquid impact problems: theory and simulations

We perform a thorough qualitative and quantitative comparison of theoretical predictions and direct numerical simulations for the two-dimensional, vertical impact of two droplets of the same fluid. In particular, we show that the theoretical predictions for the location and velocity of the jet root are excellent in the early stages of the impact, while the predicted jet velocity and thickness profiles are also in good agreement with the computations before the jet begins to bend. By neglecting the role of the surrounding gas both before and after impact, we are able to use Wagner theory to describe the early-time structure of the impact. We derive the model for general droplet velocities and radii, which encompasses a wide range of impact scenarios from the symmetric impact of identical drops to liquid drops impacting a deep pool. The leading-order solution is sufficient to predict the curve along which the root of the high-speed jet travels. After moving into a frame fixed in this curve, we are able to derive the zero-gravity shallow-water equations governing the leading-order thickness and velocity of the jet. Our numerical simulations are performed in the open-source software Gerris, which allows for the level of local grid refinement necessary for a problem with such a wide variety of length scales. The numerical simulations incorporate more of the physics of the problem, in particular the surrounding gas, the fluid viscosities, gravity and surface tension. We compare the computed and predicted solutions for a range of droplet radii and velocities, finding excellent agreement in the early stage. In light of these successful comparisons, we discuss the tangible benefits of using Wagner theory to confidently track properties such as the jet-root location, jet thickness and jet velocity in future studies of splash jet/ejecta evolution.