Influence of Capillary Number and Substrate Curvature on Taylor Cone Formation in Thin Conductive Viscous Films

We previously demonstrated by direct numerical simulation that Taylor cone formation in liquid metals confined between parallel substrates held at constant voltage difference proceeds by a self-similar process irrespective of the Reynolds number Re. The power law exponents characterizing the Maxwell and capillary pressure at the conical tip vary smoothly with Re at fixed capillary number Ca. The observed behavior smoothly bridges the inviscid prediction by Zubarev (2001) to the Stokes flow prediction by Fontelos, Kindelan and Vantzos (2008). In this work, we focus on the thin film limit in axisymmetric geometry and explore two additional aspects, namely the influence of Ca on surface excitation and growth and the influence of substrate curvature on single and multimode protrusions. Generally speaking, we find that increasing Ca at fixed Re for flow on curved surfaces generates undulations whose wavelength closely approximate the fastest growing mode associated with the fundamental planar linear instability. More interestingly, certain types of substrate curvature trigger sinusoidal traveling waves with the potential for both on- and off-axis emission.