Capillary oscillations at the exit of nozzle

Abstract

We compute the natural frequencies for the oscillation of the free boundary of a perfect incompressible fluid in the presence of capillary forces and partially in contact with a solid container. First, we study the case when the fluid occupies the domain {(x, y) : y <= h(x, t)} with h(x, t) = 0 for |x|> a and h(x, t) the free boundary for |x| <= a. We deduce an integrodifferential evolutionary equation for the linearized free boundary and impose two different boundary conditions: the condition that the contact line between free boundary and the solid is pinned and the condition that the contact line can move vertically with a contact angle pi/2. For both cases, we compute the natural oscillation frequencies for the free surface and compare the results with the frequencies of oscillation in the absence of solid walls. Secondly, we study the effect of having two parallel solid walls at |x| = b + a on the natural frequencies of oscillation of the free boundary.