[Terence] Tao speculated that the Navier–Stokes equations might be able to simulate a Turing complete system, and that as a consequence it might be possible to (negatively) resolve the existence and smoothness problem [...] However, such results remain (as of 2022) conjectural.
Witchcraft. And if it's true, it would totally explain why is fluid dynamics so damn difficult...This suggests an ambitious (but not obviously impossible) program (in both senses of the word) to achieve the same effect for the true Navier-Stokes equations [...]
Define an ideal (incompressible, inviscid) fluid [...] Somewhat analogously to how [...] a Turing machine can be constructed from cellular automata such as Conway’s “Game of Life” [...] one could hope to design logic gates entirely out of ideal fluid [...] If these gates were sufficiently “Turing complete”, and also “noise-tolerant”, one could then hope to combine enough of these gates together to “program” a von Neumann machine consisting of ideal fluid
http://cba.mit.edu/docs/theses/08.09.Prakash.pdf is a _practical_ attempt at gates, memory, and synchronization primitives (for signals that enter gates) using real-world water with nitrogen bubbles. I expect that you'd want to do standard error correction on top of that (incl. error correction by doing the same computation multiple times and comparing results).
What was the thing that suggested this (referred to in the first quoted sentence)?
@robryk@qoto.org Terence Tao proved a "finite time blowup" result for a modified version of the Navier-Stokes equations, he then speculated that it might be possible to apply the same result to the real Navier-Stokes equations and negatively resolve the open problem of existence and smoothness, using the unconventional approach of constructing a Turing machine using an ideal fluid to demonstrate the same properties as his proven results. https://gilkalai.wordpress.com/2014/02/07/navier-stokes-fluid-computers/