# Navier–Stokes equations The Navier–Stokes equations are Newton's second law written for a fluid: mass times acceleration equals the sum of pressure, viscous, and body [[Force|forces]], applied to every parcel at once. Paired with the [[Continuity_equation]] they describe essentially all everyday [[Fluid_dynamics]] — and they have resisted general solution since the 1840s. ## Known law, unknown behaviour The difficulty is the nonlinear advection term, which lets a flow feed on itself; take the curl and it becomes [[Vortex_stretching]] in the [[Vorticity_equation]]. Whether smooth three-dimensional solutions always exist is an unclaimed Clay Millennium problem, and the practical consequence is that [[Turbulence]] must be modelled rather than derived — every code in [[List_of_computational_fluid_dynamics_software|the CFD catalogue]] is an approximation strategy. That gap between knowing the law and knowing the behaviour is the standing lesson of [[Nonlinear_system|nonlinear]] [[Dynamical_system|dynamics]], and the reason [[Chaos_theory]] found its first example in a convection model. **Reads with:** *All Things Flow: Fluid Mechanics for the Natural Sciences (2019)* — [OTL record](https://open.umn.edu/opentextbooks/textbooks/all-things-flow-fluid-mechanics-for-the-natural-sciences) · [download](https://blogs.oregonstate.edu/salty/all-things-flow-fluid-mechanics-for-the-natural-sciences/). Seeded from the Thury Hydrodynamics Apex Spine book shelf. **On the spine:** [[Continuity_equation]] · [[Vorticity_equation]] · [[Turbulence]] · [[WT!Thury_Hydrodynamics_Compendium]]. <!-- COMPENDIUMLINK:BEGIN g19 — generated from _registry/plans/THURY_COMPENDIUM_SECTIONS.md; do not hand-edit inside --> *Linked from the [[WT!Thury_Hydrodynamics_Compendium|Thury Hydrodynamics Compendium]], section 1, Fluid dynamics.* <!-- COMPENDIUMLINK:END --> ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Navier–Stokes_equations) : [Wikitube](https://en.wikitube.io/wiki/Navier–Stokes_equations) ## Previous hub tags Hubs: `Life_Physics`, `Systems`. Portals: [[PORTAL_Thury_Hydrodynamics_Apex_Spine]], [[PORTAL_Physics]], [[PORTAL_Dynamical_system]]. --- *Apex Spine wave · 2026-09-10 · stub · microsim layer deferred.*