Editing Continuity equation (section)

== Fluid dynamics ==

{{see also|Mass flux|Mass flow rate|Vorticity equation}}

In [[fluid dynamics]], the continuity equation states that the rate at which mass enters a system is equal to the rate at which mass leaves the system plus the accumulation of mass within the system.<ref name=Pedlosky>{{Cite book | publisher = [[Springer Science+Business Media|Springer]] | isbn = 978-0-387-96387-7 | last = Pedlosky | first = Joseph | title = Geophysical fluid dynamics | year = 1987 | pages = [https://archive.org/details/geophysicalfluid00jose/page/10 10–13] | url = https://archive.org/details/geophysicalfluid00jose/page/10 }}</ref><ref>Clancy, L.J.(1975), ''Aerodynamics'', Section 3.3, Pitman Publishing Limited, London</ref>
The differential form of the continuity equation is:<ref name=Pedlosky/>
<math display="block"> \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{u}) = 0</math>
where
* {{math|''ρ''}} is fluid [[density]],
* {{math|''t''}} is time,
* {{math|'''u'''}} is the [[flow velocity]] [[vector field]].

The time derivative can be understood as the accumulation (or loss) of mass in the system, while the [[divergence]] term represents the difference in flow in versus flow out. In this context, this equation is also one of the [[Euler equations (fluid dynamics)]]. The [[Navier–Stokes equations]] form a vector continuity equation describing the conservation of [[linear momentum]].

If the fluid is [[Incompressible flow|incompressible]] (volumetric strain rate is zero), the mass continuity equation simplifies to a volume continuity equation:<ref name="Fielding">{{cite web |last1=Fielding |first1=Suzanne |title=The Basics of Fluid Dynamics |url=https://community.dur.ac.uk/suzanne.fielding/teaching/BLT/sec1.pdf |website=Durham University |access-date=22 December 2019}}</ref>
<math display="block">\nabla \cdot \mathbf{u} = 0,</math>
which means that the [[divergence]] of the velocity field is zero everywhere. Physically, this is equivalent to saying that the local volume dilation rate is zero, hence a flow of water through a converging pipe will adjust solely by increasing its velocity as water is largely incompressible.