# Continuity equation The continuity equation is conservation of mass written for a flowing fluid: what enters a region either leaves it or accumulates inside. For an incompressible flow it reduces to the statement that the [[Velocity]] field has zero divergence, which in a pipe becomes the familiar rule that narrowing the bore speeds the flow in exact proportion. ## The constraint that shapes everything Because it is a constraint rather than a force law, continuity works silently behind every other result. [[Bernoulli's_principle]] needs it to convert a speed change into a pressure change; the [[Stream_function]] exists precisely because it satisfies continuity automatically; and the pressure field in an incompressible [[Simulation]] is really just the enforcement of this equation at every step. It also generalises: the same accounting applied to charge gives conservation of current, and applied to [[Population_dynamics|populations]] or [[Stock_and_flow|stocks]] gives the balance equations of [[System_dynamics]]. Continuity is the reason a [[Vortex]] tube cannot simply end. **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:** [[Navier–Stokes_equations]] · [[Bernoulli's_principle]] · [[Stream_function]] · [[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 22, Energy along a streamline.* <!-- COMPENDIUMLINK:END --> <!-- THURYSIM:BEGIN g21 — Thury Compendium microsim (framework build, specs/variants/Continuity_equation.json); do not hand-edit inside --> **Microsim — three.js (Wikitube framework):** *Continuity equation* <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/thury/Continuity_equation.html" data-title="Continuity equation"></div> *Built from `MICROSIM_GUIDE/specs/variants/Continuity_equation.json`; part of the [[WT!Thury_Hydrodynamics_Compendium|Thury Hydrodynamics Compendium]] set.* <!-- THURYSIM:END --> ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Continuity_equation) : [Wikitube](https://en.wikitube.io/wiki/Continuity_equation) ## Previous hub tags Hubs: `Life_Physics`, `Systems`. Portals: [[PORTAL_Thury_Hydrodynamics_Apex_Spine]], [[PORTAL_Physics]]. --- *Apex Spine wave · 2026-09-10 · stub · microsim layer deferred.*