The effect of viscosity on the self-similar growth of conic cusps on the surface of a conducting liquid in an electric field: Limiting cone angle

Abstract

The dynamics of the formation of conic cusps on an initially smooth surface of a perfectly conducting liquid (liquid metal) in an external electric field is analytically studied. When the singularity is formed, the apex curvature radius of the accelerating protrusion, local electric field strength, and fluid velocity become infinite in a finite time. It has been demonstrated that two scales with different types of fluid behavior can be distinguished in this process. At the nanoscale (the curvature radius of the conic apex is tens of nanometers or less; the electric field strength at the apex is about 108 V/cm and higher), viscous effects play a decisive role, and a cone with the limiting opening angle of 33.1° is formed. On the macroscopic scale (the local field strength is less than 108 V/cm for liquid metals), the ideal fluid approximation is applicable, and a cone of the opening angle 98.6° (Taylor's angle) develops. In both cases, self-similar fluid flow regimes are realized, for which the spatial scale decreases with time following the power law (t0 – t)2/3, where t0 is the blowup time. In this process, the Weber number remains practically unchanged and, according to our estimates, approximately equal to 102; at the same time, the Reynolds number decreases as (t0 – t)1/3.