Introducing pre-impact air-cushioning effects into the Wagner model of impact theory

TL;DR

This paper extends the Wagner impact model by incorporating pre-impact air-water interactions, demonstrating that air cushioning reduces impact forces and modifies initial impact conditions, with effects diminishing over time.

Contribution

It introduces a novel modification to the Wagner model accounting for pre-impact air effects, providing analytical expressions for impact dynamics with air cushioning.

Findings

Pre-impact air forces cushion the impactor, reducing contact size and force.

The model derives asymptotic solutions showing air effects diminish over time.

Classical Wagner impact behavior is recovered at large times.

Abstract

In this analysis, we consider the effects of non-quiescent initial conditions driven by pre-impact air-water interactions on the classical Wagner model of impact theory. We consider the problem of a rigid, solid impactor moving vertically towards a liquid pool. Prior to impact, viscous forces in the air act to deform the liquid free surface, inducing a flow in the pool. These interactions are then incorporated as initial conditions in the post-impact analysis. We derive expressions for the size of the effective contact set, the leading-order pressure and force on the impactor, and the speed and thickness of the jet at its base. In all cases, we show that the effect of the pre-impact behaviour is to cushion the impactor, reducing the size of the effective contact set and, hence, the force on the impactor. Small- and large-time asymptotic solutions are derived for general power-law impactors, and we show that the effects of the air die away as the impact progresses, so that we approach the classical Wagner solution.