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Two-layer electrified pressure-driven flow in topographically structured channels

Published online by Cambridge University Press:  02 February 2017

Elizaveta Dubrovina ,
Richard V. Craster  and
Demetrios T. Papageorgiou
Elizaveta Dubrovina
Affiliation:
Department of Mathematics, Imperial College London, London SW7 2AZ, UK
Richard V. Craster
Affiliation:
Department of Mathematics, Imperial College London, London SW7 2AZ, UK
Demetrios T. Papageorgiou*
Affiliation:
Department of Mathematics, Imperial College London, London SW7 2AZ, UK
*
†Email address for correspondence: [email protected]
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Abstract

The flow of two stratified viscous immiscible perfect dielectric fluids in a channel with topographically structured walls is investigated. The flow is driven by a streamwise pressure gradient and an electric field across the channel gap. This problem is explored in detail by deriving and studying a nonlinear evolution equation for the interface valid for large-amplitude long waves in the Stokes flow regime. For flat walls, the electrified flow is long-wave unstable with a critical cutoff wavenumber that increases linearly with the magnitude of the applied voltage. In the nonlinear regime, it is found that the presence of pressure-driven flow prevents electrostatically induced interface touchdown that has been observed previously – time-modulated nonlinear travelling waves emerge instead. When topography is present, linearly stable uniform flows become non-uniform spatially periodic steady states; a small-amplitude asymptotic theory is carried out and compared with computations. In the linearly unstable regime, intricate nonlinear structures emerge that depend, among other things, on the magnitude of the wall corrugations. For a low-amplitude sinusoidal boundary, time-modulated travelling waves are observed that are similar to those found for flat walls but are influenced by the geometry of the wall and slide over it without touching. The flow over a high-amplitude sinusoidal pattern is also examined in detail and it is found that for sufficiently large voltages the interface evolves to large-amplitude waves that span the channel and are subharmonic relative to the wall. A type of ‘walking’ motion emerges that causes the lower fluid to wash through the troughs and create strong vortices over the peaks of the lower boundary. Non-uniform steady states induced by the topography are calculated numerically for moderate and large values of the flow rate, and their stability is analysed using Floquet theory. The effect of large flow rates is also considered asymptotically to find solutions that compare very well with numerical computations.

JFM classification

Interfacial Flows (free surface): Interfacial Flows (free surface) Low-Reynolds-number flows: Low-Reynolds-number flows Interfacial Flows (free surface): Thin films
Type
Papers
Information
Journal of Fluid Mechanics , Volume 814 , 10 March 2017 , pp. 222 - 248
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Copyright
© 2017 Cambridge University Press 

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Dubrovina et al. supplementary movie

Flow past a low amplitude sinusoidal wall showing with travelling waves spanning the channel and following the topography.

Download Dubrovina et al. supplementary movie(Video)
Video 385.7 KB

Dubrovina et al. supplementary movie

Flow past a large amplitude sinusoidal wall at a voltage that is just below the bifurcation of the second unstable mode - a "walking" solution is found at large time.

Download Dubrovina et al. supplementary movie(Video)
Video 785.9 KB

Dubrovina et al. supplementary movie

Flow past a sinusoidal wall of amplitude 0.4 at a voltage $V_b=5.6$ that is just above the bifurcation of the second unstable mode - a travelling wave forms following the topography with time periodic oscillations that reach the upper wall.

Download Dubrovina et al. supplementary movie(Video)
Video 1 MB

Dubrovina et al. supplementary movie

Flow past a sinusoidal wall of amplitude 0.4 at a voltage $V_b=8.5$, before the bifurcation of the third unstable mode - the wave gets caught by the topography and oscillates in time before loosing stability to a "walking" solution spanning the whole channel.

Download Dubrovina et al. supplementary movie(Video)
Video 668.2 KB