Branching dynamics in electrohydrodynamic instabilities of viscoelastic soft gels

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. The layer of perfect conductor is simulated using a linear viscoelastic model and the perfect dielectric is considered to be a layer of air. Under neutral conditions, the key dimensionless groups are the dimensionless electric potential, Bond number and the Weissenberg number. The branching behavior upon instability to sinusoidal disturbances is determined by weak nonlinear analysis with the dimensionless potential advanced from its critical value at neutral stability. An analytical expression obtained from weak nonlinear analysis leads to the unintuitive result that sinusoidal deflections can either lead to supercritical saturated waves or lead to subcritical breakup depending on the elasticity of the perfect conductor. The analytical expression also indicates that there is a transition wave number below which supercritical saturation ought to occur, it can be shown that such wave numbers can be geometrically accessed, thus permitting any supercritical saturation to steady waves. In contrast, our results demonstrate that when the perfect conductor is modeled as an Oldroyd-B fluid, the branching remains subcritical in nature, ultimately leading to interface rupture-mirroring the behavior observed in the Newtonian fluid case (as demonstrated by B. Dinesh and R. Narayanan, <i>Phys. Rev. Fluids</i>, 2021, <b>6</b>, 054001).