# Generic scalar transport equation

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For example, the conservation equation of a chemical species $c_i$ is
$\frac{\partial{\rho c_i}}{\partial t} + \nabla \cdot (\rho \vec u c_i + \vec J) = R_i$ where $\vec u$ denotes the velocity field, $\vec J$ denotes the diffusion flux the of the chemical species, and $R_i$ denotes the rate of generation of $R_i$ caused by the chemical reaction.
$\frac{\partial{\rho u}}{\partial t} + \nabla \cdot (\rho \vec u u ) =\nabla \cdot (\mu \nabla u ) - \frac {\partial p}{\partial x} + B_x + V_x$
where $B_x$ is the body force in the x-direction and $V_x$ includes the viscous terms that are not expressed by $\nabla \cdot (\mu \nabla u )$