### Equatorial blowup and polar caps in drop electrohydrodynamics

Fluids and Materials Seminar

1st February 2024, 2:00 pm – 3:00 pm

Fry Building, Fry 2.04

A classical problem in electrohydrodynamics is that of a liquid drop suspended in an immiscible background liquid and exposed to an external electric field. In a seminal paper (Proc. R. Soc. A, 291 1425 159-166, 1966), G. I. Taylor provided a complete theory in the weak-field limit wherein deformation and surface-charge advection are both negligible. In particular, Taylor’s theory resolved a discrepancy between experiments and earlier theories, explaining why some drops become flattened rather than elongated, and introduced the key physics underlying Taylor and Melcher’s celebrated “leaky-dielectric” model, which is widely used in electrohydrodynamics modelling.

Drop electrohydrodynamics beyond weak fields is an extremely complex problem. In this talk, I will focus on two fundamental effects of surface-charge advection that we have discovered in the course of a detailed numerical and asymptotic analysis of the symmetric steady state of a non-deformable drop. The first is singularity formation. I will show by local analysis that the leaky-dielectric equations universally support an interfacial singularity structure wherein the surface-charge density diverges; and employ a numerical singularity-capturing scheme as well as asymptotic analysis to illuminate how this blowup singularity emerges and evolves in the drop problem. I will also present a self-similar analysis of the unsteady (finite-time blowup) version of the local singularity structure, and its subtle manifestation in the drop problem. The second effect is the formation of surface-charge “stagnant caps,” which I will demonstrate by asymptotic analysis in an appropriate strong-field limit of the drop problem. I will also discuss connections between these findings and open questions arising from recent strong-field experiments and simulations.

Joint work with Gunnar G. Peng, Rodolfo Brandão and Ehud Yariv

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