Nonlinear problem of flat-plate entry into an incompressible liquid

The self-similar flow and free-surface shape induced by a flat plate entering an inviscid and incompressible liquid are investigated for arbitrary initial conditions. An analytical solution, which is based on two governing expressions, namely the complex velocity and the derivative of the complex potential, is obtained. These expressions are derived in an auxiliary parameter plane using integral formulae proposed for the determination of an analytical function from its modulus and argument given on the boundary of the parameter region. We derive a system of an integral and an integro-differential equation in terms of the velocity modulus and the velocity angle at the free surface, which are determined by the dynamic and kinematic boundary conditions. A numerical procedure for solving these equations is carefully validated by comparisons with results available in the literature. The results are presented in terms of the free surface shape, the angles at the tip of the splash jet, the contact angles at the intersection with the plate surface, pressure distribution and force coefficients. New features caused by the flow unsteadiness are found and discussed.