Role of conductivity in the electrohydrodynamic patterning of air-liquid interfaces

The effect of electrical conductivity on the wavelength of an electrohydrodynamic instability of a leaky dielectric-perfect dielectric (LD-PD) fluid interface is investigated. For instabilities induced by dc fields, two models, namely the PD-PD model, which is independent of the conductivity, and the LD-PD model, which shows very weak dependence on the conductivity of the LD fluid, have been previously suggested. In the past, experiments have been compared with either of these two models. In the present work, experiments, analytical theory, and simulations are used to elucidate the dependence of the wavelength obtained under dc fields on the ratio of the instability time (τs=1/smax) and the charge relaxation time (τc=εε0/σ, where ε0 is the permittivity of vacuum, ε is the dielectric constant, and σ is the electrical conductivity). Sensitive dependence of the wavelength on the nondimensional conductivity S2=σ2μ2h0(2)/(ε0(2)φ0(2)δ2) (where σ2 is the electrical conductivity, μ2 is the viscosity, h0 is the thickness of the thin liquid film, φ0 is the rms value of the applied field, and δ is a small parameter) is observed and the PD-PD and the LD-PD cases are observed only as limiting behaviors at very low and very high values of S2, respectively. Under an alternating field, the frequency of the applied voltage can be altered to realize several regimes of relative magnitudes of the three time scales inherent to the system, namely τc, τs, and the time period of the applied field, τf. The wavelength in the various regimes that result from a systematic variation of these three time scales is studied. It is observed that the linear Floquet theory is invalid in most of these regimes and nonlinear analysis is used to complement it. Systematic dependence of the wavelength of the instability on the frequency of the applied field is presented and it is demonstrated that nonlinear simulations are necessary to explain the experimental results.