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Continuity equation

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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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