# Continuity equation

(Difference between revisions)
 Revision as of 17:11, 24 May 2007 (view source)Jola (Talk | contribs)m (Continuity Equation moved to Continuity equation: Corrected case on title)← Older edit Latest revision as of 16:30, 25 May 2007 (view source)Jasond (Talk | contribs) Line 1: Line 1: - In [[fluid dynamics]], a '''continuity equation''' is an equation of [[conservation of matter]].  Its differential form is + In [[fluid dynamics]], the '''continuity equation''' is an expression of conservation of mass.  In (vector) differential form, it is written as - :${\partial \rho \over \partial t} + \nabla \cdot (\rho \mathbf{u}) = 0$ + :${\partial \rho \over \partial t} + \nabla \cdot (\rho \vec{u}) = 0.$ - where $\rho$ is density, t is time, and '''u''' is fluid velocity. + where $\rho$ is density, $t$ is time, and $\vec{u}$ is fluid velocity.  In cartesian tensor notation, it is written as + + :${\partial \rho \over \partial t} + {\partial \over \partial x_j}(\rho u_j) = 0.$ + + For incompressible flow, the density drops out, and the resulting equation is + + :${\partial u_j\over \partial x_j} = 0$ + + in tensor form or + + :$\nabla \cdot \vec{u} = 0$ + + in vector form.  The left-hand side is the divergence of velocity, and it is sometimes said that an incompressible flow is divergence free. -->

## Latest revision as of 16:30, 25 May 2007

In fluid dynamics, the continuity equation is an expression of conservation of mass. In (vector) differential form, it is written as

${\partial \rho \over \partial t} + \nabla \cdot (\rho \vec{u}) = 0.$

where $\rho$ is density, $t$ is time, and $\vec{u}$ is fluid velocity. In cartesian tensor notation, it is written as

${\partial \rho \over \partial t} + {\partial \over \partial x_j}(\rho u_j) = 0.$

For incompressible flow, the density drops out, and the resulting equation is

${\partial u_j\over \partial x_j} = 0$

in tensor form or

$\nabla \cdot \vec{u} = 0$

in vector form. The left-hand side is the divergence of velocity, and it is sometimes said that an incompressible flow is divergence free.