The equations of motion for a very simple ideal fluid (specifically a calorically perfect, monatomic, ideal gas) are \begin{align*}\dot{\rho}+\nabla \cdot (\rho u)=0 \;&\text{(mass conservation)} \\ \dot{(\rho u)}+\nabla \cdot (\rho u u) + \nabla p=0 \;&\text{(momentum conservation)} \\ \dot{(\rho e)} +\nabla \cdot (\rho ue+\rho p)=0 \;&\text{(energy conservation)} \\ e=\frac{1}{2}u^2+\frac{3}{2}p \;&\text{(equation of state)}\end{align*} where $\rho$ is the density, $u$ the velocity, $p$ the pressure, and $e$ the total energy (including the internal energy).

I've noticed that these equations are time reversible, i.e. if we have a solution on the time interval $[0,T]$, then by simply sending $u \to -u$, $t \to -t$ we get a solution on $[-T,0]$. From the point of view of thermodynamics, specifically the fact that total entropy is (weakly) increasing, this only makes sense if entropy is constant.

EDIT: In response to some of the comments I've deleted the example of gas expansion, since as pointed out, this wasn't strange. However I'd like to mention that a very simple model of particle collisions in a gas gives rise to the above equations:

Assume particles are interacting through collisions only (i.e. not through 'long range' forces), and there are sufficient collisions occuring that to a good degree of accuracy, the distribution of velocities of particles at any point is isotropic (after subtracting the mean velocity). For example if the velocity distribution of all particles at a given point is always a spherical gaussian this would be the case. Under just this assumption, the above equations follow.

I'm not denying that they are time reversible, and so must have constant entropy. It's just that I have no intuitive explanation for this, other than simply computing the equations. The statistical model mentioned is not time reversible, in fact it relies heavily on frequent collisions, and so time reversibility is a rather suprising fact.

Is there some other explanation, perhaps more intuitive than simply computing the equations, that explains time reversibility?

thermodynamic. There is no reference to heat or temperature, nor to any sort of statistical distribution. Just describing a macroscopic matter flow doesn't yet imply the generation of entropy. $\endgroup$3more comments