TY - JOUR

T1 - The influence of electric fields and surface tension on Kelvin–Helmholtz instability in two-dimensional jets

AU - Grandison, Scott

AU - Papageorgiou, Demetrios T.

AU - Vanden-Broeck, Jean-Marc

PY - 2012

Y1 - 2012

N2 - We consider nonlinear aspects of the flow of an inviscid two-dimensional 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 long-wave matched asymptotic analysis that incorporates
the influence of the outer regions on the dynamics of the jet. The result is a coupled system of long-wave 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 well-posed 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 quasi-periodic) with a basic oscillation predicted well by linear theory and a second nonlinearly induced
lower frequency that is responsible for quasi-periodic modulations of the spatio-temporal 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 finite-time rupture singularity. This has been established
numerically for rather general initial conditions.

AB - We consider nonlinear aspects of the flow of an inviscid two-dimensional 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 long-wave matched asymptotic analysis that incorporates
the influence of the outer regions on the dynamics of the jet. The result is a coupled system of long-wave 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 well-posed 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 quasi-periodic) with a basic oscillation predicted well by linear theory and a second nonlinearly induced
lower frequency that is responsible for quasi-periodic modulations of the spatio-temporal 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 finite-time rupture singularity. This has been established
numerically for rather general initial conditions.

U2 - 10.1007/s00033-011-0176-6

DO - 10.1007/s00033-011-0176-6

M3 - Article

VL - 63

SP - 125

EP - 144

JO - Zeitschrift für angewandte Mathematik und Physik

JF - Zeitschrift für angewandte Mathematik und Physik

SN - 0044-2275

IS - 1

ER -