From Electrodiffusion Theory to the Electrohydrodynamics of Leaky Dielectrics through the Weak Electrolyte Limit

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 ionic electrodiffusion. 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 the electrochemistry of ion dissociation is modeled explicitly. We introduce salt dissociation reaction terms in the bulk electrodiffusion equations and take the limit in which the salt dissociation is weak (i.e., poorly conducting media). Together with the assumption of small Debye length, we derive the TM model with the surface charge convection term. 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. When the GP is absent, we recover the TM model. 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.