Nature of branching in electrohydrodynamic instability

An electric field imposed on a bilayer of fluids that are stably stratified in the presence of gravity leads to an instability manifested by interfacial deflections. For the case of a perfect conductor underlying a perfect dielectric, an analytical expression obtained from weak nonlinear analysis shows that sinusoidal deflections can only lead to subcritical breakup. While this expression indicates that there is a transition wave number below which supercritical saturation ought to occur, it can be shown that such wave numbers cannot be geometrically accessed, thus precluding any supercritical saturation to steady waves.

Figure 1

Schematic of electrostatic instability. The bottom fluid is a perfect conductor and the top fluid is a perfect dielectric. The bottom plate maintains a constant voltage $D$ and the top plate is grounded.

Figure 2

${\mathcal{D}}_0\mathrm{Ca}$ vs $k^2$ obtained from Eq. (10). The case of air on top of water is assumed. Panel (a) is drawn for $\mathcal{H}=1$ and $\mathcal{B}=0$, 1, and 5. Panel (b) is drawn for $\mathcal{H}=1$ and $\mathcal{B}=5$, taking $k=\frac{n\pi }{w}$, with $n$ being the horizontal mode index.

Figure 3

Graph of ${\mathcal{D}}_0\mathrm{Ca}$ vs $k_{\mathrm{transit}}^2$ on plots of the neutral stability curves for various horizontal modes $n$ for the case of $\mathcal{B}=5$ on the left and $\mathcal{B}=1$ on the right. Allowable wave numbers are to the right of the intersection in (a) and are subcritical. In (b) the $k_{\mathrm{transit}}^2$ line lies above the neutral stability curve for low $\mathcal{B}$ implying that all wave numbers are subcritical.

Figure 4

Figure 3 redrawn for the case of deep layers and $\mathcal{B}=5$.