# Would a fluid with zero viscosity reach an equilibrium

My hunch is yes, but I can't think how to prove it. An argument against this is you have a box with a divider in the middle, one side is filled up higher than the other. The divider is removed, you will get waves in the fluid, when the viscosity is lower the waves will continue for longer, so when the viscosity is zero nothing will stop the waves so they will never stop, ignoring friction with the sides of the box. Is this right?

• No viscosity $\Rightarrow$ energy conservation. So it wouldn't reach equilibrium. Check the "Euler equations" at wikipedia. – nabla Dec 10 '15 at 0:18
• If there was irregularity in the container, wouldn't turbulence be generated? And wouldn't that turbulence (due to entropy) become more and more subdivided and distributed over time, right? What would the end point be? – Daniel Griscom Dec 10 '15 at 1:27

## 1 Answer

Viscosity is the only dissipating mechanism in a fluid absent outside forces (gravity, magnetic/electric fields if the fluid responds to those, etc.). So if there are no outside forces, it will continue moving forever.

Of course, this is only academic because a truly zero-viscosity fluid doesn't exist (well -- superfluid helium is a thing, but it would require constant energy removal for it to stay in that state, otherwise perpetual motion would be possible).

But, this all hinges on how you define "equilibrium". For example, would a stationary vortex be in equilibrium? It would be time-invariant, so I would call that an equilibrium but maybe you don't. It's hard to say without being more precise. However, it is possible to find many time-invariant solutions to the Euler equations (ie. the equations that govern a fluid when viscosity is neglected) so it boils down to definitions.

• Why would the superfluid helium need energy removal; is it that the surrounding environment would inevitably be warmer? – Daniel Griscom Dec 10 '15 at 1:37
• In my example by equilibrium I ment the fluid in the box becoming level, I should have been clearer. What about a gas with uneven concentrations, would the pressure act as a limiting force, causing the concentration to become even? – DanOc004 Dec 10 '15 at 19:41
• @DanOc004 I am not sure what you mean by "pressure act as a limiting force." However, to your question about gas mixing -- that has absolutely nothing to do with viscosity. Viscosity comes from velocity non-equilibrium and species diffusion comes from species non-equilibrium. So it all comes back to how you want to define your terms -- when you say "zero viscosity" are you also removing the species diffusion from the equations? Because some people do that when they say inviscid. Others do not. – tpg2114 Dec 11 '15 at 0:18
• You are accumulating close votes for being unclear and I think it's because you aren't being very precise in the case you are considering, the assumptions you are making, and what the motivation behind the question is (it seems like you may be asking an X-Y problem -- you ask about X, but you really want to know about Y; you just don't know that yet). – tpg2114 Dec 11 '15 at 0:19
• Sorry my question is badly written, should I delete it or let it get closed. – DanOc004 Dec 15 '15 at 20:38