  In 1991, Moore [20] raised a question about whether hydrodynamics is capable
of performing computations. Similarly, in 2016, Tao [25] asked whether a
mechanical system, including a fluid flow, can simulate a universal Turing
machine. In this expository article, we review the construction in [8] of a
"Fluid computer" in dimension 3 that combines techniques in symbolic dynamics
with the connection between steady Euler flows and contact geometry unveiled by
Etnyre and Ghrist. In addition, we argue that the metric that renders the
vector field Beltrami cannot be critical in the Chern-Hamilton sense [9]. We
also sketch the completely different construction for the Euclidean metric in
$\mathbb R^3$ as given in [7]. These results reveal the existence of
undecidable fluid particle paths. We conclude the article with a list of open
problems.
