Maxwell v Marangoni: a study of stagnant surfactant cap formation in electrohydrodynamics

The competition between Maxwell stresses and Marangoni stresses on the surface of a leaky-dielectric viscous droplet is examined. The droplet, contaminated with insoluble surfactant, is embedded in another viscous leaky dielectric fluid through which a uniform electric field is imposed. It is shown that a two-dimensional model of the nonlinear multiphysics problem for the surfactant as it evolves under the influence of interfacial Maxwell stresses while giving rise to concomitant Marangoni stresses has interesting mathematical structure. In particular, it is shown that the dynamics can be encoded by an analytic function satisfying a complex nonlinear partial differential equation of Burgers type which, moreover, can be linearized by a generalization of the classical Cole-Hopf transformation. The theoretical formulation is used to gain insight into the steady states, including stagnant surfactant caps when surface diffusion is absent, and the unsteady dynamical evolution to those states.