The convective analog to Fourier's equation of conduction? The differential form of the thermal conduction law is given by 
$J=-\kappa\nabla T$
where $\kappa$ is the thermal conduction coefficient, or in the one-dimensional case,
$J=-\kappa \frac{\partial T}{\partial z}$
I was wondering whether there is a convective analog of this equation that describes the heat flux due to convection only. I've searched around and Newton's law of cooling comes up, but there isn't much written about whether there exists a differential form for it. I am interested in finding out whether there is some relation between $J$ and $\partial_z T$ for strictly convective transport.
 A: To my knowledge, there is no "analogue", i.e. differential equation,  in convection which is a far more complicated process. Convection is the process of fluid motion resulting from density differences, in this case due to heating and is typically very efficient at removing heat from an object, particularly if the fluid is moving (think about wind chill). However, close to the object, a boundary layer develops in which heat is transferred through the fluid via conduction, and further away heat is then transported away through bulk advection.
Newton's law of cooling is written as,
$$J_{conv} = h_c (T - T_0),$$
where $T$ and $T_0$ are the surface and free-stream temperatures, respectively and $h_c$ is the convection heat-transfer coefficient. This very straightforward equation does a pretty good job of predicting heat transfer as it hides all the complexities in the $h_c$ factor, though this can be notoriously difficult to compute accurately.
EDIT:
To answer your question backwards, it struck me that for 1D linear conduction, i.e. constant thermal gradient, Fourier's law simplifies to:
$$ J_{cond} = \dfrac{k}{L}\left(T_0 - T_1\right), $$
which is the complete analogue of the convection equation above.
