### Abstract

The flow of a gravity-driven thin liquid film over a substrate containing topography, in the presence of a normal electric field, is investigated. The liquid is assumed to be a perfect conductor and the air above it a perfect dielectric. Of particular interest is the interplay between inertia, for finite values of the Reynolds number, Re, and electric field strength, expressed in terms of the Weber number, We, on the resultant free-surface disturbance away from planarity. The hydrodynamics of the film are modelled via a depth-averaged form of the Navier-Stokes equations which is coupled to a Fourier series separable solution of Laplace’s equation for the electric potential: detailed steady-state solutions of the coupled equation set are obtained numerically.

The case of two-dimensional flow over different forms of discrete and periodically varying spanwise topography is explored. In the case of the familiar free-surface capillary peaks and depressions that arise for steep topography, and become more pronounced with increasing Re, greater electric field strength affects them differently. In particular, it is found that for topography heights commensurate with the long-wave approximation: (i) the capillary ridge associated with a step-down topography at first increases before decreasing, both monotonically, with increasing We; (ii) the free-surface hump which arises at a step-up topography continues to increase monotonically with increasing We, the increase achieved being smaller the larger the value of Re.

A series of results for the more practically relevant problem of three-dimensional film flow over substrate containing a localised square trench topography is provided. These exhibit behaviour and features consistent with those observed for two-dimensional flow, in that as We is increased the primary free-surface capillary ridges and depressions are at first enhanced, with a corresponding narrowing, before becoming suppressed. In addition, it is found that, while the well-known horse-shoe shaped disturbance characteristic of such flows continues to persist with increasing Re in the absence of an electric field, when the latter is present and We increased in value the associated comet tail disappears as does the related downstream surge. The phenomenon is explained with reference to the competition between the corresponding capillary pressure and Maxwell stress distributions

The case of two-dimensional flow over different forms of discrete and periodically varying spanwise topography is explored. In the case of the familiar free-surface capillary peaks and depressions that arise for steep topography, and become more pronounced with increasing Re, greater electric field strength affects them differently. In particular, it is found that for topography heights commensurate with the long-wave approximation: (i) the capillary ridge associated with a step-down topography at first increases before decreasing, both monotonically, with increasing We; (ii) the free-surface hump which arises at a step-up topography continues to increase monotonically with increasing We, the increase achieved being smaller the larger the value of Re.

A series of results for the more practically relevant problem of three-dimensional film flow over substrate containing a localised square trench topography is provided. These exhibit behaviour and features consistent with those observed for two-dimensional flow, in that as We is increased the primary free-surface capillary ridges and depressions are at first enhanced, with a corresponding narrowing, before becoming suppressed. In addition, it is found that, while the well-known horse-shoe shaped disturbance characteristic of such flows continues to persist with increasing Re in the absence of an electric field, when the latter is present and We increased in value the associated comet tail disappears as does the related downstream surge. The phenomenon is explained with reference to the competition between the corresponding capillary pressure and Maxwell stress distributions

Original language | English |
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Pages (from-to) | 48-69 |

Journal | International Journal of Multiphase Flow |

Volume | 44 |

Issue number | n/a |

Early online date | 30 Mar 2012 |

DOIs | |

Publication status | Published - Sep 2012 |

### Keywords

- Thin liquid films
- Free surface flow
- Inertia
- Electrohydrodynamics
- Numerical solutions
- Topography

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## Cite this

Veremieiev, S., Thompson, H. M., Scholle, M., Lee, Y. C., & Gaskell, P. H. (2012). Electrified thin film flow at finite Reynolds number on planar substrates featuring topography.

*International Journal of Multiphase Flow*,*44*(n/a), 48-69. https://doi.org/10.1016/j.ijmultiphaseflow.2012.03.010