The effect of pressure fluctuations on the shapes of thinning liquid curtains

Abstract

We consider the time-dependent response of a gravitationally-thinning inviscid liquid sheet (a coating curtain) leaving a vertical slot to sinusoidal ambient pressure disturbances. The theoretical investigation employs the hyperbolic partial differential equation developed by \cite{weinp1}. The response of the curtain is characterized by the slot Weber number, $W_{e_0} = ρq V/2σ$, where $V$ is the speed of the curtain at the slot, $q$ is the volumetric flow rate per unit width, $σ$ is the surface tension, and $ρ$ is the fluid density. Flow disturbances travel along characteristics with speeds relative to the curtain of $\pm \sqrt{uV/W_{e_0}}$, where $u = \sqrt{V^2 + 2gx}$ is the curtain speed at a distance $x$ downstream from the slot. When the flow is subcritical ($W_{e_0} < 1$), upstream traveling disturbances near the slot affect the curtain centerline, and the slope of the curtain centerline at the slot oscillates with an amplitude that is a function of $W_{e_0}$. In contrast, all disturbances travel downstream in supercritical curtains ($W_{e_0} > 1$) and the slope of the curtain at the slot is vertical. Here, we specifically examine the curtain response under supercritical and subcritical flow conditions near $W_{e_0} = 1$ to deduce whether there is a substantial change in the overall shape and magnitude of the curtain responses. Despite the local differences in the curtain solution near the slot, we find that subcritical and supercritical curtains have similar responses for all imposed sinusoidal frequencies.