Fluid reaction on a vibrating disc in a viscous medium

This article studies the fluid reaction on a vibrating disc immersed in a viscous fluid. The fluid is considered incompressible and Newtonian. The disc which is of negligible thickness vibrates harmonically in the direction perpendicular to its surface with an amplitude much smaller than the radius of the disc, in such a way that the non-linear terms can be neglected. The flow is axisymmetric and the velocity tends to zero away from the disc. Different approaches to this problem are presented. The first method consists in solving numerically an integral equation obtained from the Navier–Stokes equation. The second method calculates in an analytic fashion the asymptotic series for the pressure differential across the plate for large values of the dimensionless parameter β, equal to the frequency times the radius squared divided by the kinematic viscosity. The limit when β tends to zero is also studied. The analytical expressions give more reliable results when approaching the limits β large and β small than the numerical solution.