Analytic solution of liquid-drop impact problems

The pressure in an impacting liquid drop against both a rigid and an elastic target is calculated for the period when the contact region is expanding faster than the wave speed in the liquid. For very low speed impact a geometrical-acoustics model is shown to give a good representation of the solution, until the edge speed approaches the wave speed. A self-similar solution, that takes account of nonlinear effects, is used in the neighbourhood of the contact edge. Comparisons are made with linear theory and numerical calculations. It is shown that linear theory is totally inadequate in predicting the escape of the shock system from the contact edge and that numerical calculations have used too large a time step to calculate the time of escape correctly. The delay in escape time from the previous theoretical predictions of Heymann (1969) is attributed to the elasticity of the target, an effect that is taken into account in the present work.