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IMA Journal of Applied Mathematics Advance Access originally published online on October 18, 2007
IMA Journal of Applied Mathematics 2007 72(6):832-853; doi:10.1093/imamat/hxm040
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© The Author 2007. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Numerical and analytical studies of non-linear gravity–capillary waves in fluid layers under normal electric fields

Demetrios T. Papageorgiou{dagger}

Department of Mathematical Sciences and Center for Applied Mathematics and Statistics, New Jersey Institute of Technology, University Heights, Newark, NJ 07102, USA

J.-M. Vanden-Broeck{ddagger}

School of Mathematics, University of East Anglia, Norwich NR4 7TJ, UK

{dagger} Email: depapa{at}oak.njit.edu

{ddagger} Corresponding author. Present address: Department of Mathematics, University College London, Gower Street, London WC1E 6BT, UK. Email: j.vanden-broeck{at}uea.ac.uk

Received on June 13, 2007; Accepted on August 29, 2007

Non-linear gravity–capillary waves travelling at constant speed are considered in the presence of a normal electric field. The fluid, which is assumed to be inviscid, irrotational and a perfect dielectric, is bounded below by a solid plate electrode held at constant voltage, and the region above the free surface is a hydrodynamically passive perfect dielectric, e.g. air. A second parallel flat plate electrode is placed laterally far away and drives a uniform normal electric field there. Electrohydrodynamic coupling occurs at the free surface through the Maxwell stresses which act to modify the normal stress balance and consequently the Bernoulli equation boundary condition there. Three harmonic problems in deforming domains need to be solved, one for the hydrodynamics and one each for the electrostatics above and below the free surface, respectively. We derive and implement an accurate boundary integral method to compute travelling waves of arbitrary wavelength and amplitude. In addition, we consider a long-wave non-linear model of the full problem and compare solutions with the direct simulations. In both cases, we establish the existence of multiple families of solutions extending the classical theory of gravity–capillary waves. An asymptotic theory is also developed to construct periodic waves with ripples. These are used in comparisons with the numerical calculations and the agreement is very good.

Keywords: gravity-capillary waves; electric fields; boundary integral; equation methods; Wilton ripples.


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