Free surface problem in electrohydrodynamics

[hunt_mat]

So do you agree that the authors meant to write $\nabla$ and that their eq.(2) is therefore not nonsense?

I don't know. It's certainly something that I'm not going to be concerned about when I deal with variable charge density, although I am curious where the term $\rho_{v}\mathbf{u}$ comes from. That would couple the variable charge density very nicely.