You are not assuming that the electric field is stationary. Indeed, from Maxwell-Faraday’s law, you recover a time dependence. You are just neglecting the displacement current. This amounts to setting the speed of light to infinity. More realistically, it’s rather about saying that the typical velocity of your fluid is small compared to the speed of light.

Hope this helps.

You are not assuming that the electric field is stationary. Indeed, from Maxwell-Faraday’s law, you recover a time dependence. You are just neglecting the displacement current (as indicated by the wikipedia article). Physically, you are cutting part of the feedback loop between the EM fields, so you are neglecting radiation.

Pedestrianly, this amounts to setting the speed of light to infinity. More realistically, it’s rather about saying that the typical velocity of your fluid is small compared to the speed of light. Note that this is not about the actual velocity field of the fluid. Any combination of the parameters which gives a velocity works just as well. For example at large distance with fixed frequency this approximation would necessarily break down.

Hope this helps.