Electrified cone formation in perfectly conducting viscous liquids: Self-similar growth irrespective of Reynolds number

Abstract

Above a critical field strength, the free surface of an electrified, perfectly conducting viscous liquid, such as a liquid metal, is known to develop an accelerating protrusion resembling a cusp with a conic tip. Field self-enhancement from tip sharpening is reported to generate divergent power law growth in finite time of the forces acting in that region. Previous studies have established that tip sharpening proceeds via a self-similar process in two distinct limits -the Stokes regime at Re = 0 and the inviscid regime Re → ∞. Using finite element simulations to track the acceleration of an electrified protrusion in a perfectly conducting Newtonian liquid in vacuum held at constant capillary number, we demonstrate that the conic tip always undergoes self-similar growth irrespective of Reynolds number. The computed blow up exponents at the tip for the terms in the Navier-Stokes equation and interface normal stress condition reveal the different forces at play as Re increases. Rescaling of the tip shape by the capillary stress exponent yields excellent collapse onto a universal conic tip shape with interior half-angle dependent on the magnitude of the Maxwell stress. The rapid acceleration of the liquid interface also generates a thin surface boundary layer with very high local strain rate. Additional details of the modeled flow, applicable to cone growth in systems such as liquid metal ion sources, help dispel prevailing misconceptions that dynamic cones resemble conventional Taylor cones or that viscous stresses at finite Re can be neglected.Credit line: This article has been submitted to the Physics of Fluids -a link to the final version will be provided once published.

Thermocapillary Patterning of Highly Uniform Microarrays by Resonant Wavelength Excitation

“…For cases involving continual growth without formation of an intermediate saturated periodic state, theoretical predictions from Eqs. ( 6) and (20) for the maximum exponential growth rates β(K/K max ) and β(K/K ext ) based on a noise-free initial condition are directly compared to For studies focused on the linear stability of Bloch periodic states, which tended to form at intermediate times, theoretical predictions for the maximum growth rates (Q, K/K ext ) are computed from Eq. (47).…”

“…There is substantial discrepancy between the simulation results and Eq. (20), which increases significantly as Q approaches K ext . The results suggest that the presence of white noise, likely enhanced by resonant excitations near K ext , triggers amplification of modes that would otherwise decay in time due to capillary leveling.…”

“…More recent theoretical work examining late time dynamics in the nonlinear regime has also demonstrated how fluid protrusions driven by significant thermocapillary stresses will sharpen in time due to a selfsimilar runaway process [19] that leads to cusplike shapes. Runaway dynamics leading to self-similar shapes is also known to occur in perfectly electrically conducting liquids subjected to strong electric field gradients [20,21].…”

Unmasking of Novel Conic Modes in Electrically Stressed Perfectly Conducting Liquids

“…3 of Ref. [43]. Those simulations [44], however, were conducted using moving mesh, finite element simulations of the full Navier-Stokes equation, while the approach adopted in this current study relies on solution of the inviscid interface Bernoulli equation.…”

Loading a High-Viscous Droplet via the Cone-Shaped Liquid Bridge Induced by an Electrostatic Force

“…From the figure, it is evident that the contours of the liquid velocity are nearly a series of concentric arcs. Accordingly, the flow inside the cone is almost spherically symmetric. , …”