The mathematics of ship slamming
Motivated by the motion of a ship in a heavy sea, a mathematical model for the vertical impact of a two-dimensional solid body onto a half-space of quiescent, inviscid, incompressible fluid is formulated. No solutions to the full problem are known, but in the case when the impacting body has small deadrise angle (meaning that the angle between the tangent to the profile and the horizontal is everywhere small) a uniformly valid solution is obtained by using the method of matched asymptotic expansions. The pressure on the body is calculated and is in fair agreement with experimental results. The model is generalised for more complicated impacts and the justifications for the model are discussed. The method is extended to three-dimensional bodies with small deadrise angle and solutions are obtained in some special cases. A variations! formulation of the leading order outer problem is derived, which gives information about the solution and leads to an fixed domain scheme for calculating solutions numerically. A partial linear stability analysis of the outer problem is given which indicates that entry problems are stable but exit problems are unstable to small perturbations. A mathematical model for the effect of a cushioning air layer between the body and the fluid is presented and analysed both numerically and in appropriate asymptotic limits. Finally, the limitations of the models are discussed and directions for future work indicated.