Math Help - Total derivative manipulation

1. Total derivative manipulation

Hey guys, I think I'm making this harder than it needs to be but anyway,

I have two transport equations, one in k (the turbulent kinetic energy) and one in $\phi$ (a dimensionaly independant variable). So two equations in the form
$\frac{Dk}{Dt} = ...$
$\frac{D \phi}{Dt} = ...$

I am also given a relation to $\epsilon$ (the rate of dissipation of turbulent kinetic energy) which is
$\phi = k^m \epsilon^n$
where m and n are constants.

I need to get the equation in $\phi$ into one in $\epsilon$, e.g.
$\frac{D \epsilon}{Dt} = ...$

I assumed to just substitute for $\phi$ into the transport equation and then expand it out using the product rule but in getting a term as $\frac{D \epsilon}{Dt}$ I end up with terms like
$\frac{D \epsilon^{n-1}}{Dt}$ and $\frac{D k^m}{Dt}$.

Anyone got any ideas how I'd get rid of these unwanted total derivatives?

Thanks,

James

2. If I've understood your problem then, by taking logs to base e and then differentiating, you should get:

$\frac{1}{\phi} \, \frac{D \phi}{D t} = \frac{m}{k} \, \frac{D k}{D t} +\frac{n}{\epsilon} \, \frac{D \epsilon}{D t}$ .

Is that what you got? If not, perhaps this will help.