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On the effect of electrostatic surface forces on dielectric falling films

Published online by Cambridge University Press:  13 November 2020

Wilko Rohlfs Open the ORCID record for Wilko Rohlfs [Opens in a new window] ,
Liam M. F. Cammiade Open the ORCID record for Liam M. F. Cammiade [Opens in a new window] ,
Manuel Rietz Open the ORCID record for Manuel Rietz [Opens in a new window]  and
Benoit Scheid Open the ORCID record for Benoit Scheid [Opens in a new window]
Wilko Rohlfs*
Affiliation:
Institute of Heat and Mass Transfer, RWTH Aachen University, Augustinerbach 6, 52056Aachen, Germany
Liam M. F. Cammiade
Affiliation:
Institute of Heat and Mass Transfer, RWTH Aachen University, Augustinerbach 6, 52056Aachen, Germany
Manuel Rietz
Affiliation:
Institute of Heat and Mass Transfer, RWTH Aachen University, Augustinerbach 6, 52056Aachen, Germany
Benoit Scheid
Affiliation:
TIPs, Université Libre de Bruxelles C.P. 165/67, Avenue F.D. Roosevelt 50, 1050Bruxelles, Belgium
*
†Email address for correspondence: [email protected]
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Abstract

The destabilization of a dielectric film flow due to an electrostatic surface force is investigated. A weighted residuals integral boundary-layer (WIBL) model is derived and validated against full numerical simulations. The equations of the WIBL model indicate that the electrostatic surface force contributes to the evolution equations in a similar mathematical way as the volumetric gravitational force. Contrary to gravity, an additional electrostatic contribution ($\chi _2$) arises, whose impact increases nonlinearly with decreasing capacitor plate distance. This nonlinear contribution causes a fold of the branch of solutions of the dynamical system and, thus, the co-existence of a low amplitude solution that is stable against infinitesimal disturbances and an unstable high amplitude solution. In time-dependent simulations, the fold coincides with the limit in the parameter space beyond which a finite-time blow-up occurs with an unsaturated growth of the main wave hump leading to wave pinch-off and drop formation. Thus, a phase diagram can be constructed by tracking this fold. The shape of the main wave prior to blow-up depends on the electrostatic parameter $\chi _2$. If this parameter is zero, the force is equivalent to a hanging film flow configuration and dripping occurs with a drop-shaped structure. With an increasing contribution of the parameter $\chi _2$, Taylor-cone waves occur prior to finite-time blow-up, leading to jetting. Finally, the transition from stable to unstable waves is investigated in terms of the two dimensionless electric parameters, the Reynolds, and viscous dissipation numbers. Imposing the most amplified wavelength, a transition border between stable solutions and jetting is identified.

JFM classification

Interfacial Flows (free surface): Thin films
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 906 , 10 January 2021 , A18
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

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