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Moffatt eddies in electrohydrodynamics flows: numerical simulations and analyses

Published online by Cambridge University Press:  06 December 2022

Xuerao He Open the ORCID record for Xuerao He [Opens in a new window] ,
Zhihao Sun  and
Mengqi Zhang Open the ORCID record for Mengqi Zhang [Opens in a new window]
Xuerao He
Affiliation:
Department of Mechanical Engineering, National University of Singapore, 9 Engineering Drive 1, 117575 Singapore
Zhihao Sun
Affiliation:
Department of Mechanical Engineering, National University of Singapore, 9 Engineering Drive 1, 117575 Singapore School of Energy Science and Engineering, Harbin Institute of Technology, Harbin 150001, PR China
Mengqi Zhang*
Affiliation:
Department of Mechanical Engineering, National University of Singapore, 9 Engineering Drive 1, 117575 Singapore
*
†Email address for correspondence: [email protected]
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Abstract

We study numerically a sequence of eddies in two-dimensional electrohydrodynamics (EHD) flows of a dielectric liquid, driven by an electric potential difference between a hyperbolic blade electrode and a flat plate electrode (or the blade–plate configuration). The electrically driven flow impinges on the plate to generate vortices, which resemble Moffatt eddies (Moffatt, J. Fluid Mech., vol. 18, 1964, pp. 1–18). Such a phenomenon in EHD was first reported in the experimental work of Perri et al. (J. Fluid Mech., vol. 900, 2020, A12). We conduct direct numerical simulations of the EHD flow with three Moffatt-type eddies in a large computational domain at moderate electric Rayleigh numbers ($T$, quantifying the strength of the electric field). The ratios of size and intensity of the adjacent eddies are examined, and they can be compared favourably to the theoretical prediction of Moffatt; interestingly, the quantitative comparison is remarkably accurate for the two eddies in the far field. Our investigation also shows that a larger $T$ strengthens the vortex intensity, and a stronger charge diffusion effect enlarges the vortex size. A sufficiently large $T$ can further result in an oscillating flow, consistent with the experimental observation. In addition, a global stability analysis of the steady blade–plate EHD flow is conducted. The global mode is characterised in detail at different values of $T$. When $T$ is large, the confinement effect of the geometry in the centre region may lead to an increased oscillation frequency. This work contributes to the quantitative characterisation of the Moffatt-type eddies in EHD flows.

JFM classification

Vortex Flows: Vortex dynamics Complex Fluids: Dielectrics
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 953 , 25 December 2022 , A14
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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References

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He et al. supplementary movie 1

Time-periodic oscillation of charge density at the electric Taylor number T=30,000. The other parameters are C=5, M=50, Fe=5,000, R=0.05.

Download He et al. supplementary movie 1(Video)
Video 5 MB

He et al. supplementary movie 2

Time-periodic oscillation of charge density at the electric Taylor number T=40,000. The other parameters are C=5, M=50, Fe=5,000, R=0.05.

Download He et al. supplementary movie 2(Video)
Video 780.3 KB