Finite-time singularity formation in drop electrohydrodynamics

In the Taylor-Melcher leaky-dielectric model, a suspended drop with an imposed external electric field may attain a singular steady state, in which the surface-charge density blows up (anti-symmetrically) near the equator as q ~ x^(-1/3), where x is the distance to the equator [Peng, Brandão, Yariv and Schnitzer, Phys. Rev. Fluids 9, 083701 (2024)]. This is associated with an anti-parallel surface-charge polarization driving a surface flow v ~ x^(1/3) which advects opposite charges towards the equator where they "annihilate" at a rate qv, which is non-zero despite both q and v being odd functions of x.

We use finite-difference numerical simulations of a symmetric two-dimensional circular drop to study how an initially smooth state evolves towards the singular steady state, and find that a singularity develops in finite time, but with q ~ x^(-0.28). The singularity formation can be described with a local similarity solution, in which the universal exponent emerges from the condition that the solution approaches x=0 linearly rather than with a non-integer power. By introducing an extremely small amount of artificial surface-charge diffusion, we simulate the evolution beyond the finite-time singularity, which is described by another similarity solution, and the approach to the ultimate profile with q ~ x^(-1/3).

*This work was funded by the Leverhulme Trust Research Project Grant RPG-2021-161.