Nonlinear waves at the interface between two dielectric fluids in a horizontal electric field for unmatched density and permittivity ratios

Two immiscible, inviscid, non-conducting (dielectric) fluids separated by an interface and subjected to a strong horizontal electric field support electrohydrodynamic interfacial waves. Zubarev and Kochurin (2013) proved that in the weakly nonlinear, small-slope regime with electrostatic forces dominating gravity and capillarity, the interface admits exact three-dimensional traveling waves of arbitrary profile that propagate without distortion along or against the field direction — but only under the material matching condition that the density ratio equals the permittivity ratio, $\rho_1/\rho_2=\varepsilon_1/\varepsilon_2$ — and derived an explicit quadratic-order law for the interaction of counter-propagating waves. The open problem is to determine the interfacial wave dynamics when the matching condition is lifted while the strong-field, weakly nonlinear setting is retained: for unmatched material parameters $\rho_1/\rho_2\ne\varepsilon_1/\varepsilon_2$, the source's own general evolution equation (its Eq. (8)) cannot be solved analytically, and the fate of coherent one-way three-dimensional waves and the interaction law for counter-propagating waves are unknown. Later work has settled the free-surface limits, long-wave gravity-capillary regimes, and shear-flow (Kelvin–Helmholtz-suppressed) variants, but the general quiescent two-fluid interface in the dispersionless strong-field regime without the matching condition remains unsolved.

Background

Electrohydrodynamics (EHD) studies the coupling of electric fields and fluid motion. A dielectric fluid is an electrical insulator that polarizes in an applied field; its response is characterized by the permittivity $\varepsilon$. When two immiscible dielectric fluids meet at an interface, any deformation of the interface distorts the field, and the electrostatic Maxwell stresses push back on the interface. A field applied parallel to the undisturbed interface (a horizontal field, the interface being the plane $z=0$) is stabilizing: it raises the threshold of the Kelvin–Helmholtz instability (the instability of a sheared interface) and of gravity–capillary disturbances. Classical EHD studies of tangential fields go back to Melcher and coworkers, who analyzed interfacial waves and the destabilizing role of finite conductivity; the ideal limit of perfectly insulating, inviscid fluids is the clean setting in which exact nonlinear results are possible.

In this ideal setting each fluid is modeled as inviscid, incompressible and irrotational, so the flow in fluid $j$ ($j=1,2$, occupying the half-space below or above the interface) is described by a velocity potential $\Phi_j$ satisfying Laplace's equation, and the electric field in each fluid derives from an electrostatic potential $\varphi_j$ also satisfying Laplace's equation. The electrostatic matching conditions at the interface are continuity of the potential, $\varphi_1=\varphi_2$, and of the normal electric displacement, $\varepsilon_1\partial_n\varphi_1=\varepsilon_2\partial_n\varphi_2$, appropriate to perfect dielectrics with no free charge. Following Zakharov's Hamiltonian formulation of deep-water gravity waves, the interfacial dynamics can be written in the canonical variables $\eta(x,y,t)$ (the interface displacement) and $\psi$ (the velocity potential at the interface), with the total energy — kinetic plus electrostatic, plus gravitational and capillary terms when retained — serving as the Hamiltonian.

For a single dielectric free surface in a strong tangential field, Zubarev (2004, 2009) showed that finite-amplitude surface waves can propagate without distortion, in the limit of large relative permittivity, along or against the field. Zubarev and Kochurin (2013) extended this to a genuine two-fluid interface. They worked in the strong-field regime, defined by $\varepsilon_0 E^2\gg\sqrt{g\sigma|\rho_1-\rho_2|}$ (with $\varepsilon_0$ the electric constant, $E$ the field magnitude, $g$ gravity, $\sigma$ surface tension and $\rho_{1,2}$ the densities), in which electrostatic forces dominate capillary and gravitational forces, which are dropped. Expanding the Hamiltonian to cubic order in the small-slope parameter $\alpha$ (interface slopes $|\nabla_\perp\eta|\sim\alpha\ll1$, nonlinearities retained through $O(\alpha^2)$), they derived, for general density and permittivity Atwood numbers $A=(\rho_1-\rho_2)/(\rho_1+\rho_2)$ and $A_E=(\varepsilon_1-\varepsilon_2)/(\varepsilon_1+\varepsilon_2)$, a (2+1)-dimensional weakly nonlinear evolution equation with quadratic nonlinearities (their Eq. (8)), in which linear waves propagate at the speed $v_0=A_E\sqrt{(1+A)/(1+A_E)}$. The matching condition is imposed only afterwards: the paper states that in the general case the nonlinear equation (8) cannot be solved analytically, and that substantial progress in solving it is possible only when the quadratic nonlinear terms compensate, which happens precisely when $A=A_E$, i.e. $\rho_1/\rho_2=\varepsilon_1/\varepsilon_2$ (their Eq. (9)), and only then. Under this matching condition the reduced equation admits exact three-dimensional traveling waves $\eta(x,y,t)=f(x-At,y)$ and $\eta(x,y,t)=g(x+At,y)$ with arbitrary profiles $f,g$ propagating at the speed $A$ (under the matching condition the common value $A=A_E$ is itself the wave speed), plus an explicit $O(\alpha^2)$ formula for the interaction of counter-propagating disturbances involving the Hilbert-transform-type operator $\hat{k}$ defined by $\hat{k}e^{i\mathbf{k}\cdot\mathbf{r}}=|\mathbf{k}|e^{i\mathbf{k}\cdot\mathbf{r}}$. In the limits $\rho_2/\rho_1\to0$, $\varepsilon_2/\varepsilon_1\to0$ (so $A=A_E=1$) these results reduce to the earlier free-surface exact solutions. The paper's conclusion further records that the same results apply, with the electric field $E$ replaced by the magnetic field $H$ and the permittivities $\varepsilon_{1,2}$ replaced by the magnetic permeabilities $\mu_{1,2}$, to waves along the interface of ferromagnetic liquids in a horizontal magnetic field, where the matching condition (their Eq. (14)) takes the form $\rho_1/\rho_2=\mu_1/\mu_2$.

What the 2013 theory does not cover defines the problem below: interfaces whose density and permittivity ratios do not match, for which the paper's own general evolution equation (Eq. (8)) is left analytically unresolved. Field strengths at which gravity and capillarity are not negligible, nonlinearities beyond the quadratic/cubic truncation at a two-fluid interface, unmatched permeability ratios in the magnetic analog, and any viscous or conducting (non-ideal) physics lie beyond the problem posed by the source.

Problem Statement

Determine the nonlinear dynamics of waves at the interface between two immiscible ideal (inviscid, incompressible, non-conducting) dielectric fluids under a strong horizontal electric field in the dispersionless regime $\varepsilon_0 E^2\gg\sqrt{g\sigma|\rho_1-\rho_2|}$ when the matching condition $\rho_1/\rho_2=\varepsilon_1/\varepsilon_2$ (equivalently $A=A_E$, with $A=(\rho_1-\rho_2)/(\rho_1+\rho_2)$ and $A_E=(\varepsilon_1-\varepsilon_2)/(\varepsilon_1+\varepsilon_2)$) is not satisfied. The weakly nonlinear (2+1)-dimensional evolution equation for the interface displacement $\eta(x,y,t)$, quadratic in the small-slope parameter $\alpha$ and valid for general $A$, $A_E$, is already given by Eq. (8) of Zubarev and Kochurin (2013); the open target is the dynamics of that equation for $A\ne A_E$, which the source itself leaves analytically unresolved:

1. The fate of coherent one-way three-dimensional interfacial waves: whether distortionless propagation of arbitrary profiles survives in modified form, or the uncompensated quadratic nonlinear terms lead to cumulative deformation, collapse, or wave breaking.

2. The interaction law for counter-propagating waves.

The characterization is to be achieved by justified analysis of the source's Eq. (8) and/or validated fully nonlinear computations, and must be consistent with the matched-case exact traveling-wave solutions (Eq. (11)) and the $O(\alpha^2)$ counter-propagating interaction formula (Eq. (12)) of the 2013 theory in the limit $A=A_E$.

Viscous, conducting, or otherwise non-ideal fluids lie outside the scope of the source and of this problem; treatments of them would require abandoning the potential-flow Hamiltonian framework. Extensions to finite field strength (where gravity and capillarity make the interfacial waves dispersive), to nonlinearities beyond the quadratic/cubic truncation, and to general (unmatched) permeability ratios in the magnetic analog are explicitly proposed generalizations that lie beyond the open problem posed by the source and are not part of the target above; the matched ferromagnetic case $\rho_1/\rho_2=\mu_1/\mu_2$ already follows from the 2013 results by the substitution $E\to H$, $\varepsilon_{1,2}\to\mu_{1,2}$ recorded in the source's conclusion (Eq. (14)).

References

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