The influence of electric fields and surface tension on Kelvin–Helmholtz instability in twodimensional jets
 Scott Grandison,
 Demetrios T. Papageorgiou,
 JeanMarc VandenBroeck
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We consider nonlinear aspects of the flow of an inviscid twodimensional jet into a second immiscible fluid of different density and unbounded extent. Velocity jumps are supported at the interface, and the flow is susceptible to the Kelvin–Helmholtz instability. We investigate theoretically the effects of horizontal electric fields and surface tension on the nonlinear evolution of the jet. This is accomplished by developing a longwave matched asymptotic analysis that incorporates the influence of the outer regions on the dynamics of the jet. The result is a coupled system of longwave nonlinear, nonlocal evolution equations governing the interfacial amplitude and corresponding horizontal velocity, for symmetric interfacial deformations. The theory allows for amplitudes that scale with the undisturbed jet thickness and is therefore capable of predicting singular events such as jet pinching. In the absence of surface tension, a sufficiently strong electric field completely stabilizes (linearly) the Kelvin–Helmholtz instability at all wavelengths by the introduction of a dispersive regularization of a nonlocal origin. The dispersion relation has the same functional form as the destabilizing Kelvin–Helmholtz terms, but is of a different sign. If the electric field is weak or absent, then surface tension is included to regularize Kelvin–Helmholtz instability and to provide a wellposed nonlinear problem. We address the nonlinear problems numerically using spectral methods and establish two distinct dynamical behaviors. In cases where the linear theory predicts dispersive regularization (whether surface tension is present or not), then relatively large initial conditions induce a nonlinear flow that is oscillatory in time (in fact quasiperiodic) with a basic oscillation predicted well by linear theory and a second nonlinearly induced lower frequency that is responsible for quasiperiodic modulations of the spatiotemporal dynamics. If the parameters are chosen so that the linear theory predicts a band of long unstable waves (surface tension now ensures that short waves are dispersively regularized), then the flow generically evolves to a finitetime rupture singularity. This has been established numerically for rather general initial conditions.
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 Title
 The influence of electric fields and surface tension on Kelvin–Helmholtz instability in twodimensional jets
 Journal

Zeitschrift für angewandte Mathematik und Physik
Volume 63, Issue 1 , pp 125144
 Cover Date
 20120201
 DOI
 10.1007/s0003301101766
 Print ISSN
 00442275
 Online ISSN
 14209039
 Publisher
 SP Birkhäuser Verlag Basel
 Additional Links
 Topics
 Keywords

 76B07
 76B15
 76B45
 Industry Sectors
 Authors

 Scott Grandison ^{(1)}
 Demetrios T. Papageorgiou ^{(2)}
 JeanMarc VandenBroeck ^{(3)}
 Author Affiliations

 1. Department of Computer Science, University of East Anglia, Norwich, UK
 2. Department of Mathematics, Imperial College London, London, SW7 2AZ, UK
 3. Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, UK