  The Taylor-Melcher (TM) model is the standard model for describing the
dynamics of poorly conducting leaky dielectric fluids under an electric field.
The TM model treats the fluids as Ohmic conductors, without modeling the
underlying ion dynamics. On the other hand, electrodiffusion models, which have
been successful in describing electrokinetic phenomena, incorporate ionic
concentration dynamics. Mathematical reconciliation of the electrodiffusion
picture and the TM model has been a major issue for electrohydrodynamic theory.
Here, we derive the TM model from an electrodiffusion model in which we
explicitly model the electrochemistry of ion dissociation. We introduce salt
dissociation reaction terms in the bulk electrodiffusion equations and take the
limit in which the salt dissociation is weak; the assumption of weak
dissociation corresponds to the fact that the TM model describes poor
conductors. Together with the assumption that the Debye length is small, we
derive the TM model with or without the surface charge convection term
depending upon the scaling of relevant dimensionless parameters. An important
quantity that emerges is the Galvani potential (GP), the jump in voltage across
the liquid-liquid interface between the two leaky dielectric media; the GP
arises as a natural consequence of the interfacial boundary conditions for the
ionic concentrations, and is absent under certain parametric conditions. When
the GP is absent, we recover the TM model. Our analysis also reveals the
structure of the Debye layer at the liquid-liquid interface, which suggests how
interfacial singularities may arise under strong imposed electric fields. In
the presence of a non-zero GP, our model predicts that the liquid droplet will
drift under an imposed electric field, the velocity of which is computed
explicitly to leading order.
