Using data from this link, I found the viscosity of water is ~0.001 Pa s and the viscosity of castor oil is ~0.7 Pa s. I know that castor oil is thicker than water, so higher value means less runny. Which is good since viscosity is, roughly, resistance to flow.

From this link, I found that the viscosity of the Earth's lower mantle is ~10^22 Pa s.

So to answer your question as to whether the pressure and temperature make it runny: No. I mean, well, it's runny compared to solid Iron.

Crap. Then I found this link, which is a scholarly 2 page paper and not for the feint of heart in the physics zone, yadda yadda.
"Our overall conclusion is that the shear viscosity of liquid iron in the Earth's core is ~0.015 Pa s, with an uncertainty factor of 3."

So yeah... The outer core is about as runny as ethelene glycol (IDK, it's from the link), and about 10 times thicker than mercury.

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If we're talking currents in the outer core, then the inner core and the lower mantle are the surrounding material. Yes, it's liquid moving against those solid surfaces (which there is no reason to assume are rotating in sync with each other if the layer in between isn't).

But why aren't they in sync? What source of energy is keeping that stuff flowing after 3.5 billion years? Why hasn't the drag against the mantle bled off all currents which weren't in sync?

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If not charge, but gravity, then the effects are just too weak to account for the lack of energy dissipation. Tidal effects from the moon are significant to ocean currents, but can't account for the numbers involved. (Sorry. A bit hand-wavey, there.)

Tidal effects from the sun are difficult to detect. Tidal effects from Jupiter are likely too small to detect. While Jupiter makes up 3/4 of the mass of the solar system which is NOT the sun. The sun still makes up ~99.8% of the mass of the solar system, Jupiter included. Also, Jupiter is further away than the sun.