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

Published online by Cambridge University Press:  15 January 2021

Bridget M. Torsey Open the ORCID record for Bridget M. Torsey [Opens in a new window] ,
Steven J. Weinstein Open the ORCID record for Steven J. Weinstein [Opens in a new window] ,
David S. Ross Open the ORCID record for David S. Ross [Opens in a new window]  and
Nathaniel S. Barlow Open the ORCID record for Nathaniel S. Barlow [Opens in a new window]
Bridget M. Torsey*
Affiliation:
School of Mathematical Sciences, Rochester Institute of Technology, Rochester, NY14623, USA
Steven J. Weinstein
Affiliation:
School of Mathematical Sciences, Rochester Institute of Technology, Rochester, NY14623, USA Department of Chemical Engineering, Rochester Institute of Technology, Rochester, NY14623, USA
David S. Ross
Affiliation:
School of Mathematical Sciences, Rochester Institute of Technology, Rochester, NY14623, USA
Nathaniel S. Barlow
Affiliation:
School of Mathematical Sciences, Rochester Institute of Technology, Rochester, NY14623, USA
*
†Email address for correspondence: [email protected]
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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 Weinstein et al. (Phys. Fluids, vol. 9, issue 12, 1997, pp. 3625–3636). The response of the curtain is characterized by the slot Weber number, $W_{e_0} = \rho q V/2\sigma$, where $V$ is the speed of the curtain at the slot, $q$ is the volumetric flow rate per unit width, $\sigma$ is the surface tension and $\rho$ 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. Here g is the acceleration of gravity. When the flow is subcritical ($W_{e_0} < 1$), upstream travelling disturbances near the slot affect the curtain centreline, and the slope of the curtain centreline 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.

JFM classification

Interfacial Flows (free surface): Thin films Materials Processing Flows: Coating Mathematical Foundations: Computational methods
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 910 , 10 March 2021 , A38
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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