Self-Similar Taylor Cone Formation in Conducting Viscous Films: Computational Study of the Influence of Reynolds Number

Previous studies by Zubarev (2001) and Suvorov and Zubarev (2004) have shown that above a critical field strength, an \textit{ideal} (inviscid) conducting fluid film will deform into a singular profile characterized by a conic cusp. The governing equations for the electrohydrodynamic response beneath the cusp admit self-similar solutions leading to so-called blow-up behavior in the Maxwell pressure, capillary pressure and kinetic energy density. The runaway behavior in these variables reflects divergence in time characterized by an exponent of -2/3. Here we extend the physical system to include viscous effects and conduct a computational study of the cusp region as a function of increasing electrical Reynolds number $Re_E$. We employ a finite element, moving mesh algorithm to examine the behavior of the film shape, Maxwell pressure and capillary pressure upon approach to the blow-up event. Our study indicates that self-similarity establishes at relatively low $Re_E$ despite the presence of vorticity, which is localized to the cusp surface region. With increasing $Re_E$, the period of self-similiarity extends further in time as the exponent changes from about -4/5 to the ideal value of -2/3, with slightly different values distinguishing the Maxwell and capillary stresses.