Interfacial capillary waves in the presence of electric fields

Grandison, Scott; Papageorgiou, Demetrios T.; Vanden‐Broeck, J.‐M.

doi:10.1016/j.euromechflu.2006.06.005

Cited by 24 publications

(41 citation statements)

References 20 publications

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“…The boundary conditions at the free surfaces y = S(x, t) and y =S(x, t) are the kinematic condition, continuity of the normal stresses, the continuity of the normal component of the displacement field (εE) and the continuity of the tangential components of the electric field (see [12,23,24] for details of these conditions). In what follows we consider symmetric interfacial deflections so thatS(x, t) = −S(x, t), and it is sufficient to consider y > 0 with the symmetry conditions…”

Section: Governing Equationsmentioning

confidence: 99%

“…In [12] traveling wave solutions of arbitrary amplitude and wavelength were computed using boundary integral equation methods. In the present study, we analyze the time-dependent problem using long-wave asymptotics that result in a coupled system of integro-differential partial differential equations; the initial value problem is addressed numerically to identify parameter values where the flow may rupture or not.…”

Section: Governing Equationsmentioning

confidence: 99%

“…In general, such a problem must be addressed numerically, and by analogy with nonlinear Kelvin-Helmholtz instability, we expect singularities to form after finite time in the absence of surface tension (see for example [7,8,16,19]). As already found in [12], a sufficiently strong electric field provides a dispersive regularization of the Kelvin-Helmholtz instability even in the absence of surface tension, and one of our main aims is to understand and quantify the nonlinear dynamics of such systems. In particular, we not only compare traveling wave solutions of reduced long-wave model equations with those of the full numerical computations of [12] but also consider the initial value problem in the linearly unstable regime (surface tension included).…”

Section: Nondimensionalizationmentioning

confidence: 99%

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

Grandison

Papageorgiou

Vanden‐Broeck

2011

Z. Angew. Math. Phys.

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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

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Section: Governing Equationsmentioning

confidence: 99%

Section: Governing Equationsmentioning

confidence: 99%

Section: Nondimensionalizationmentioning

confidence: 99%

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

Grandison

Papageorgiou

Vanden‐Broeck

2011

Z. Angew. Math. Phys.

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“…The upper fluid, which can be a gas or a liquid, with constant density ρ (1) and velocity (V (1) , 0, 0) is taken to be nonconducting, and the lower fluid, which is invariably a liquid, with density ρ (2) (> ρ (1) ) and velocity (V (2) , 0, 0) is conductive of electricity. The xoy plane is taken to be coincide with the unperturbed middle level separating the two fluids, and the positive z-axis in the upward direction normal to the unperturbed fluid surfaces.…”

Section: Basic Equations and Equilibrium Statementioning

confidence: 99%

“…The xoy plane is taken to be coincide with the unperturbed middle level separating the two fluids, and the positive z-axis in the upward direction normal to the unperturbed fluid surfaces. The upper dielectric fluid is bounded above by an electrode with potential φ (1) and below by the interface, has depth h 1 , and the lower conducting fluid is bounded below by an electrode with potential [35,36] φ (2) …”

Section: Basic Equations and Equilibrium Statementioning

confidence: 99%

Instability of two streaming conducting and dielectric bounded fluids in porous medium under time-varying electric field

El-Sayed

2008

Arch Appl Mech

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In this paper, we consider the instability of the interface between two superposed streaming conducting and dielectric fluids of finite depths through porous medium in a vertical electric field varying periodically with time. A damped Mathieu equation with complex coefficients is obtained. The method of multiple scales is used to obtain an approximate solution of this equation, and then to analyze the stability criteria of the system. We distinguish between the non-resonance case, and the resonance case, respectively. It is found, in the first case, that both the porosity of porous medium, and the kinematic viscosities have stabilizing effects, and the medium permeability has a destabilizing effect on the system. While in the second case, it is found that each of the frequency of the electric field, and the fluid velocities, as well as the medium permeability, has a stabilizing effect, and decreases the value of the resonance point, while each of the porosity of the porous medium, and the kinematic viscosities has a destabilizing effect, and increases the value of the resonance point. In the absence of both streaming velocities and porous medium, we obtain the canonical form of the Mathieu equation. It is found that the fluid depth and the surface tension have a destabilizing effect on the system. This instability sets in for any value of the fluid depth, and by increasing the depth, the instability holds for higher values of the electric potential; while the surface tension has no effect on the instability region for small wavenumber values. Finally, the case of a steady electric field in the presence of a porous medium is also investigated, and the stability conditions show that each of the fluid depths and the porosity of the porous medium ε has a destabilizing effect, while the fluid velocities have stabilizing effect. The stability conditions for two limiting cases of interest, the case of purely fluids), and the case of absence of streaming, are also obtained and discussed in detail.

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Axisymmetric waves in electrohydrodynamic flows

et al. 2007

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The formation of nonlinear axisymmetric waves on inviscid irrotational liquid jets in the presence of radial electric fields is considered. Gravity is neglected but surface tension is considered. Electrohydrodynamic waves of arbitrary amplitude and wavelength are computed using finite-difference methods. Particular attention is paid to nonlinear traveling waves. In the first class of problems, an electric field generated by placing the liquid jet inside a hollowcylindrical electrode held at constant voltage, its axis coinciding with that of the jet, is studied. The jet is assumed to be a perfect conductor whose free surface is stressed by the electric field acting in the hydrodynamically passive annulus. In the second class of problems, the annular gas is a perfect conductor that transmits a constant voltage onto the liquid/gas surface. The liquid axisymmetrically wets a constant-radius cylindrical rod electrode placed coaxially with respect to the hollow outer electrode, and held at a different constant voltage. The fluid dynamics and electrostatics need to be addressed simultaneously in the inner region. Axisymmetric interfacial waves influenced by surface tension and electrical stresses are computed in both cases. The computations are capable of following highly nonlinear solutions and predict, for certain parameter values, the onset of interface pinching accompanied with the formation of toroidal bubbles. For given wave amplitudes, the results suggest that, for the former case, the electric field delays bubble formation and reduces wave steepness, while for the latter case the electric field promotes bubble formation, all other parameters being equal

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Interfacial capillary waves in the presence of electric fields

Cited by 24 publications

References 20 publications

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

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

Instability of two streaming conducting and dielectric bounded fluids in porous medium under time-varying electric field

Axisymmetric waves in electrohydrodynamic flows

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