Hydrodynamic arrest of a flat body moving towards a parallel surface at arbitrary Reynolds number

Abstract

The motion of a flat body towards a parallel plane surface in incompressible fluid is considered both in the presence and absence of an applied force for a non-vanishing initial velocity. In the inviscid limit, a first integral of the equations is obtained and analytic solutions are presented for the cases of finite body inertia with zero applied force and finite applied force with negligible body inertia. In the former case when the ratio of body inertia to fluid inertia is large, a singular behaviour is observed in the arrest of the body before impact wherein the time-dependent pressure and radial velocity of the fluid exhibit a sharp peak and there is a large transfer of kinetic energy from the body to the thin fluid layer. For a real fluid, a general procedure is described to obtain solutions at arbitrary Reynolds number for naturally occurring initial velocity conditions. Solutions to the full Navier-Stokes equations are obtained for an arbitrary Reynolds number based on gap height which are valid provided the flow remains laminar and the gap height is small. In general, the equations of motion of the body and fluid are both dynamically and kinematically coupled. The dynamic coupling, however, is removed when the body inertia is neglected. In particular, the cases of hydrodynamic arrest with zero applied force, and draining of the fluid under a constant applied force are considered. The natural initial conditions lead to a new exact similarity solution of the Navier-Stokes equations which is valid for an instantaneous time-dependent Re based on gap height of greater than approximately 100, wherein the top and bottom boundary layers remain distinct. The longer time portions of the motion and the final arrest are described by a numerical calculation for intermediate Reynolds number and a low-Reynolds-number analysis.