Unsteady flow between rectangular plates and between circular plates

The unsteady flow of a viscous incompressible fluid between two parallel, infinitely long rectangular plates and between two parallel circular plates is analyzed. In this analysis, the lower plate is stationary, and the upper plate moves toward the lower plate. Use is made of the full Navier-Stokes equations in order to obtain the pressure distribution as a function of the film thickness and the velocity of the upper plate. Allowance is made for the inertia, and for a given load on the upper plate, the sinkage relation between the thickness and the time is determined for various values of the Reynolds. For various values of the two dimensionless parameters involved (one characterizing the load, the other gravity), the departure from the classical inertialess solution is exhibited.