65 Advective Flux Divergence

This is a derivation of the form of the advective flux divergence that is most useful. ┬áRecall that the full advective flux is \underline{\textbf{u}}C=(uC,vC,wC). ┬áLet’s start by writing out all the terms in its divergence

    \[ \nabla \cdot \left( \underline{\textbf{u}}C \right) = \frac{\partial (uC)}{\partial x}+\frac{\partial (vC)}{\partial y}+\frac{\partial (wC)}{\partial z} \]

Now let’s use the chain rule for each one of those 3 terms on the right:

    \[  \nabla \cdot \left( \underline{\textbf{u}}C \right) = C\frac{\partial u}{\partial x}+u\frac{\partial C}{\partial x}+C\frac{\partial v}{\partial y}+v\frac{\partial C}{\partial y}+C\frac{\partial w}{\partial z}+w\frac{\partial C}{\partial z} \]

And rearrange them:

    \[  \nabla \cdot \left( \underline{\textbf{u}}C \right) = u\frac{\partial C}{\partial x}+v\frac{\partial C}{\partial y}+w\frac{\partial C}{\partial z}+C\frac{\partial u}{\partial x}+C\frac{\partial v}{\partial y}+C\frac{\partial w}{\partial z} \]

The recognize that the last three terms are just concentration times the divergence of the velocity, and the latter is zero, so that:

    \[  \nabla \cdot \left( \underline{\textbf{u}}C \right) = u\frac{\partial C}{\partial x}+v\frac{\partial C}{\partial y}+w\frac{\partial C}{\partial z}= \underline{\textbf{u}} \cdot \nabla C \]