In this work, we analyze the influence of a vertical electric field on the stability of the free surface of a perfectly conducting fluid in contact with a perfect dielectric (Taylor & Mcewan, 1965). The fluids are confined between two rigid plates, where the top plate is maintained at a constant voltage (D) and the bottom plate is grounded. The applied voltage difference counteracts gravity. This work has its application to patterning of thin polymer films (Chou & Zhuang, 1999), where the conducting fluid self assembles into arrays of pillars. The pillar patterns observed are a consequence of an instability which arises from a competition between the applied constant voltage on the one hand and gravity and surface tension on the other hand (Wu et al., 2005; Papageorgiou, 2019; Ward et al., 2019). These competing effects lead to a minimum in a plot of D versus the wavenumber of the disturbance at the onset of the instability. Neutral stability analysis indicates that the applied voltage is balanced by gravity for low wavenumbers and is balanced by surface tension for high wavenumbers. The competition with surface tension at high wavenumbers is reminiscent of the Rayleigh-Taylor (R-T) problem where, there surface tension is balanced by destabilizing gravitational forces. The R-T instability is always subcritical in nature. However, the presence of a minimum, in the D versus wavenumber plot is indicative of a supercritical instability, much like the Bénard problem. This suggests that there is a supercritical to subcritical transition at some wavenumber in the problem of this study. This transition shows that there can be an instability that saturates to steady wave patterns for low wavenumbers and for high wavenumbers an instability that could ultimately lead to rupture. To find this transition from supercritical to subcritical states, we consider a weakly nonlinear analysis about the neutral state where the applied voltage difference is advanced slightly beyond the critical point. Plots of this transition are presented as a function of the Bond number.
Acknowledgment: Support from NASA NNX17AL27G, NASA 80NSSC18K1173 and FSGC-08 TO No NNX15_023 are gratefully acknowledged.
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