65 Advective Flux Divergence
This is a derivation of the form of the advective flux divergence that is most useful. Recall that the full advective flux is 
. Let’s start by writing out all the terms in its divergence
![\[ \nabla \cdot \left( \underline{\textbf{u}}C \right) = \frac{\partial (uC)}{\partial x}+\frac{\partial (vC)}{\partial y}+\frac{\partial (wC)}{\partial z} \]](https://uw.pressbooks.pub/app/uploads/quicklatex/quicklatex.com-783ed1b5c35c6ce2bfbbd8ec518168cf_l3.png "Rendered by QuickLaTeX.com")
Now let’s use the chain rule for each one of those 3 terms on the right:
![\[ \nabla \cdot \left( \underline{\textbf{u}}C \right) = C\frac{\partial u}{\partial x}+u\frac{\partial C}{\partial x}+C\frac{\partial v}{\partial y}+v\frac{\partial C}{\partial y}+C\frac{\partial w}{\partial z}+w\frac{\partial C}{\partial z} \]](https://uw.pressbooks.pub/app/uploads/quicklatex/quicklatex.com-ced3ddbb77753ccc55b6734300756dd8_l3.png "Rendered by QuickLaTeX.com")
And rearrange them:
![\[ \nabla \cdot \left( \underline{\textbf{u}}C \right) = u\frac{\partial C}{\partial x}+v\frac{\partial C}{\partial y}+w\frac{\partial C}{\partial z}+C\frac{\partial u}{\partial x}+C\frac{\partial v}{\partial y}+C\frac{\partial w}{\partial z} \]](https://uw.pressbooks.pub/app/uploads/quicklatex/quicklatex.com-539f930a335024a1822e1fdc67140e0e_l3.png "Rendered by QuickLaTeX.com")
The recognize that the last three terms are just concentration times the divergence of the velocity, and the latter is zero, so that:
![\[ \nabla \cdot \left( \underline{\textbf{u}}C \right) = u\frac{\partial C}{\partial x}+v\frac{\partial C}{\partial y}+w\frac{\partial C}{\partial z}= \underline{\textbf{u}} \cdot \nabla C \]](https://uw.pressbooks.pub/app/uploads/quicklatex/quicklatex.com-ea2bbfa923d9818b62ad851325ea28a2_l3.png "Rendered by QuickLaTeX.com")
