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I was reading about Feynman's sprinkler problem, and came across a paper that discussed irreversibility in ideal fluids. It quoted the fact that

When a real fluid is expelled quickly from a tube, it forms a jet separated from the surrounding fluid by a thin, turbulent layer. On the other hand, when the same fluid is sucked into the tube, it comes in from all directions, forming a sink-like flow.

And then came to the conclusion that

The asymmetry between outflow and inflow therefore does not depend on viscous dissipation, but rather on the experimenter’s limited control of initial and boundary conditions. This illustrates ... how irreversibility may arise in systems whose microscopic dynamics are fully reversible.

I am not sure what irreversibility means here. Is it just that the system isn't invariant under time reversal? But I think water being expelled and water being sucked in are not "the same process under time reversal" to begin with, since in the former there is angular momentum, while in the latter there is none.

Also, when viscosity is not considered (as in the paper's statement), ideal fluids are described by the Euler equation: $$ \left( \frac{\partial}{\partial t} + \boldsymbol{u}\cdot\nabla \right) \boldsymbol{u} = -\frac{\nabla P}{\rho} + \boldsymbol{g}$$ Which is definitely time-invariant to me, as the transformation $t\mapsto -t$ doesn't change the equation. Then how can there be irreversible processes arising from this assumption?

Qmechanic
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Jono94
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  • Reversible equations are arising from our habit of modeling nature with reversible equations. In reality absolutely every observation you will ever make is irreversible to some degree (even if for some systems that irreversibility is extremely small). So why are we making this approximation? Because it works so well for many physical systems. The Euler equation simply neglects viscosity and heat transfer in real liquids to make the modeling of an "idea' fluid considerably easier. – FlatterMann May 08 '23 at 17:25
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    @FlatterMann Please use comments to ask for clarification or point to useful resources; answers should be posted as answers. – Chemomechanics May 08 '23 at 19:11
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    @Chemomechanics if you pay some attention on the comments sections, you will easily see that, while FlatterMann and John Doty are very competent experimental physicists, they keep posting such "reversibility is an approximation", "maths is not physics", style of comments basically everywhere they can. – naturallyInconsistent May 09 '23 at 05:52
  • @naturallyInconsistent We are simply trying to make nature and its physical description easier to understand for the beginner. There is a strong emphasis on theory these days, which follows from the fact that modern experiments are extremely expensive and that only very few people will ever get to play with the required hardware. This can lead to a very skewed idea of what is actually happening in reality. The OP asked a good question that has a good answer. He is welcome to look at the solution theory of the irreversible equations... oh, wait... there is no known solution theory for them! – FlatterMann May 09 '23 at 15:34

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The paper is only making the simplistic argument that, you have a system that obeys microscopic reversiblity, and a pair of operating conditions that ought to be time reversal versions of each other (one sprouting water and one sucking where the water is being sprouted in the other case).

Yet, the former case the sprinkler rotates, whereas the latter case the sprinkler does not. His argument is then that this mismatch is an appearance of irreversibility, and that he claims this is caused by lack of control of initial and boundary conditions.

Note that in both cases, angular momentum is conserved (and zero). The rotating sprinkler gets the opposite angular momentum of the ejected water.

I am not very convinced that the paper is being insightful, but I also do not have a good refutation. Quite meh.

naturallyInconsistent
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