Warren D. Smith: On the uncomputability of hydrodynamics 17 pages. http://www.neci.nj.nec.com/homepages/wds/navstokes.ps not patentable. ----------Abstract:------------------------------------ We construct a rigid and bounded 3D container $C$ with bounded surface area and whose boundary, although complicated, is smooth everywhere except at a single point. It is partly filled with a fluid of constant density and viscosity, having bounded kinetic energy, and indeed, uniformly bounded flow velocities. The container's shape and the initial location-set and velocity field for the fluid (both of which are as smooth as possible) all have finite-length mathematical descriptions. We demonstrate that, e.g., predicting which of the two alternatives $\ge 2 \mathrm{cm}^3$ of fluid will flow into basin $A$ during the next minute'' and $\le 1 \mathrm{cm}^3$ will flow into basin $A$, ever'' will happen (one of these may be guaranteed to happen) is at least as hard as solving Turing's general halting problem,'' i.e. UNDECIDABLE. But if there is a physical system corresponding to $C$, it would solve the problem in 1 minute. (This demonstrates the falsity of Church's thesis'' under these laws of physics.) My demonstration'' is not a mathematical proof since it depends on certain unproved -- but empirically very well confirmed -- assumptions. (It also shamelessly exploits certain mathematical, but unphysical, features of the equations of hydrodynamics, namely: assumption of a perfect continuum all the way down to zero length scale, perfect wall rigidity, and exact constancy of viscosity and density despite any temperature and pressure changes.) Nevertheless I produce a genuine theorem whose statement (I think) signifies the failure of hydrodynamics. --------------------------------------------------------- -------------------Keywords:---------------------------- Fluidics, hydrodynamics, undecidability, Church's thesis, Turing's halting problem, algorithmization of physics, non-existence of hydrodynamic limit, failure of hydrodynamics, water hammer, slug flow, Toom rule, fault tolerance. -----------------------------------------------------------