Liquid impact, kinetic energy loss and compressibility: Lagrangian, Eulerian and acoustic viewpoints

From Lagrange's equations of incompressible fluid motion a model is derived for the collision between a liquid mass and a solid surface. The classical idea of pressure impulse, P, is re-expressed as a quantity following the fluid-particle motion. It is shown that within this formulation P=0 is the exact free-surface boundary condition and the domain of definition of P is unambiguously time-independent. Some of the total kinetic energy of the fluid is lost during impact and this is associated with the usual choice of boundary condition for inelastic impact. With elastic impact, in which the fluid rebounds from the solid target, there is no kinetic energy loss. Some simple potentials are used to express P for incompressible fluid impacts, which have non-singular velocity fields: (i) in an acute wedge; (ii) in a cylindrical container; and (iii) in an idealised sea-wave impact. In the last the impact of a triangular fluid domain, T, illustrates kinetic energy loss from an impacting sea wave. Impact is also investigated for the collision of T with a movable solid block. The subsequent displacement of the block, with friction, is also calculated. Lastly a solution is obtained within T composed of a compressible fluid impacting a rigid wall. Standing compression-waves store within T some of the kinetic energy lost from the incident wave water.