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How about integrating Faraday's law ##\nabla\times\vec{E}=-\frac{\partial\vec{B}}{\partial t}## w.r.t. time ##t## to get ##\vec{B}=-\int\nabla\times\vec{E}\,dt\,##?
I think that ##\nabla^2## is just a typo; it should instead be the vector-operator ##\nabla## according to Fick's laws of diffusion. The Fick diffusion electric current is distinct from the ohmic drift electric current. See: https://en.wikipedia.org/wiki/Diffusion_current.
OK, I can see from your attached paper that you are quite knowledgeable regarding EHD, so providing a detailed answer to your question is clearly above my pay grade!
I will only comment that, just as the shape and evolution of a free surface must be computed from the fluid dynamics, I would...
Here's an upcoming presentation from the grad school I attended about applying AI to fundamental physics, namely, analyzing high-energy scattering data:
(https://universitydevelopment.cmail20.com/t/d-e-gamhk-hliyjihltk-u/)
You say in the first line of your abstract:
Research on free-surface flows in electrohydrodynamics has predominantly focused on the limiting cases of perfectly conducting and perfectly insulating fluids.
But magnetohydrodynamics (MHD), which is the shortened name of what Alfvén originally called...
The OP posits a fluid medium with a constant bulk charge density and so it carries a nonzero total charge. Assuming that the fluid does not extend to infinity in any direction, it can therefore be completely surrounded by a closed surface. The integral of the normal electric field over this...
Is this material available anywhere else? Regarding your link, my security software states:
"Your connection to this web page is not safe due to an unmatching security certificate."
A tragedy indeed.
And I must say I did a double-take at the birth- and death-dates. I hadn't before encountered Buddhist Era (BE) dates, which are apparently 543 years ahead of the Gregorian calendar.
No, the two expressions ##\nabla\cdot\vec{E}## and ##\vec{E}\cdot\nabla## are very different. The first is simply the scalar quantity ##\nabla\cdot\vec{E}=\frac{\partial E_{x}}{\partial x}+\frac{\partial E_{y}}{\partial y}+\frac{\partial E_{z}}{\partial z}##, whereas the second is the...