Electrohydrodynamic pump and equations (1 Viewer)

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Hi all. I am reading a paper about electrohydrodynamic pump and I encounter the following equation. Can anyone explain me the physicl meaning of these two equations? I find no clue in understanding them.

The first equation is from the manipulation of Maxwell's equation in an electroquasistatic system (BTW, what i an electroquasistatic system?)

The author says that from the drive term on the right hand side of the first equation, it can be concluded that a requirement for the existence of free charge is the presence of a spatial gradient in conductivity or permittivity. Free charge injected into a region without such a gradient would relax in a time characterized by the charge relaxation time epsion/sigma.

Can anyone provide me some of the background knowldege and explain me why the author says so?
 

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