This raises a question I hadn't thought of before. Real-world fluid flow is ultimately well-modeled by the equations of many-body Newtonian mechanics, right (atoms bumping around)? Are those equations vulnerable to blow-ups?
If you model atoms as dimensionless points (1), then any kind of force law with the distance between atoms in the denominator can lead to a singularity when that distance is zero. In practice, you write the simulator to disallow this, but it's still there in the equations, you're just ignoring it.
If you model your atoms as finite-sized but incompressible billiard balls, then when they hit each other it's a discontinuity, since they instantly change direction when they collide. These collisions conserve total momentum and energy, but they're unphysical because real physical quantities can't jump from one value to another (in classical physics).
Even if you model your atoms as little rubber balls, the model can still be singular. Linear elasticity (the most common choice) allows you to compress a finite-sized object down to zero size with finite energy, which yields infinite energy density. Again, you'd have to disallow that in the simulator, which is very practical, but not theoretically satisfying.
1) https://en.wikipedia.org/wiki/Molecular_dynamics is the typical method of atomistic simulation.
For example, if you're simulating charged particles moving around, and you use a force equation F = k q1 q2 / d^2 (1), then when d approaches 0 (i.e. when the distance between particles approaches zero), then the force F goes to infinity.
For atoms, it works the same way. If you use a force law like Lennard-Jones (2), it also has the interatomic distance in the denominator, so the equation has a singularity baked right in.
You could always adopt a more complex force equation that doesn't have a singularity in it. But in practice, it's easier to use a simple but singular equation, and then selectively ignore its bad behavior.
Here's a cool expository article about blow-ups in classical mechanics and elsewhere: https://arxiv.org/abs/1609.01421
1) "Struggles with the Continuum" by John Baez
https://en.wikipedia.org/wiki/Sonoluminescence
I would think that nothing in reality is infinite, but allegedly sound waves collapsing bubbles in a fluid can cause a very small amount of plasma to become hotter than the sun and emit light. Some controversial research claims it might even be possible to create atomic fusion this way.