A conservative equation for the general property $\phi$ can be written as:
$$ \frac{D \rho \phi}{Dt}=\nabla\cdot(\Gamma_\phi \nabla\phi)+S_\phi $$
where $\frac{D}{Dt}$ is the material derivative, $\Gamma_\phi$ is the diffusion coefficient and $S_\phi$ is the source term of variable $\phi$. $\nabla$ is the del operator, used to calculate the gradient of a scalar field and the divergence of a vectorial field. The conservative equation as defined above is usually written in cartesian coordinates, so it is useful to use the $\nabla$ operator to write in a compact form the conservative equation for the property $\phi$.
My question revolves around the material derivate. For cartesian coordinates, it can be written as:
$$ \frac{D \rho \phi}{Dt}=\frac{\partial \rho \phi}{\partial t}+\vec{u}\cdot\nabla{\rho \phi} $$
So the general conservative equation for the property $\phi$ is written in a more convenient way for finite volume analysis as
$$ \frac{\partial \phi}{\partial t}+\nabla(\rho \vec{u} \phi) = \nabla\cdot(\Gamma_\phi \nabla\phi)+S_\phi $$
Is this general conservative equation only valid for cartesian coordinates? The material derivative in cylindrical coordinates is not written as $\frac{\partial \rho \phi}{\partial t}+\vec{u}\cdot\nabla{\rho \phi}$, as is shown here.
