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Continuity equation

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In [[fluid dynamics]], a '''continuity equation''' is an equation of [[conservation of matter]]Its differential form is
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In [[fluid dynamics]], the '''continuity equation''' is an expression of conservation of massIn (vector) differential form, it is written as
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:<math> {\partial \rho \over \partial t} + \nabla \cdot (\rho \mathbf{u}) = 0</math>
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:<math> {\partial \rho \over \partial t} + \nabla \cdot (\rho \vec{u}) = 0.</math>
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where <math> \rho </math> is density, t is time, and '''u''' is fluid velocity.
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where <math> \rho </math> is density, <math>t</math> is time, and <math>\vec{u}</math> is fluid velocity.  In cartesian tensor notation, it is written as
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:<math> {\partial \rho \over \partial t} + {\partial \over \partial x_j}(\rho u_j) = 0.</math>
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For incompressible flow, the density drops out, and the resulting equation is
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:<math> {\partial u_j\over \partial x_j} = 0</math>
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in tensor form or
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:<math> \nabla \cdot \vec{u} = 0</math>
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in vector form.  The left-hand side is the divergence of velocity, and it is sometimes said that an incompressible flow is divergence free.
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The '''law of conservation of mass/matter''' (The [[Lomonosov]]-[[Lavoisier]] law) states that the [[mass]] of a [[closed system]] of substances will remain constant, regardless of the processes acting inside the system. An equivalent statement is that [[matter]] changes form, but cannot be created or destroyed. This implies that for any chemical process in a closed system, the mass of the reactants must equal the mass of the products'''.  
The '''law of conservation of mass/matter''' (The [[Lomonosov]]-[[Lavoisier]] law) states that the [[mass]] of a [[closed system]] of substances will remain constant, regardless of the processes acting inside the system. An equivalent statement is that [[matter]] changes form, but cannot be created or destroyed. This implies that for any chemical process in a closed system, the mass of the reactants must equal the mass of the products'''.  
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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.


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