Falling Plate over Flat Surface

Consider the 2-D unsteady motion of a plate of width 2L falling on a flat surface a very small distance h apart where h/L ^ 1. The gap is occupied by air. The air viscosity is neglected and the flow is assumed incompressible, see Fig. 5.12.

Apply conservation of mass theorem to the control volume and show that the flow field corresponds to a stagnation point flow with a time dependent multiplication constant

u = K(t) x, w = – K(t) z

Find the relationship between K(t) and h(t).

Find the pressure in the gap by integrating the x – and z-momentum equations, and find the pressure on the lower side of the plate by enforcing the condition that the pressure should be continuous and equal to patm at the edge x = L, z = h(t).

Let M’ be the mass per unit span of the plate and assume that M’ ^ pL3/h(0), where p is the air density. Show that the plate motion is described by

h(t) _ 1

Подпись: where Подпись: C Подпись: 3 M' g 2 PL3

h(0) = cosh VCt

Fig. 5.12 Falling plate