is there some kind of governing equation for analyzing nuclear fission? Something like Navier stokes equations, which probably connects density of active substance, to heat across a cross section of the fuel rod, to the effect of moderators, or something like that. Or is just all based on experimental data.
 A: Neutron transport equations are a place to start. Once you know how neutrons diffuse and interact in a material, you can predict where the neutron flux is highest, which tells you where heat gets deposited and where new neutrons are born (fission).
A basic differential form looks something like:
$$-\nabla \cdot D\nabla{\phi(\vec{r})} + \Sigma_a \phi(\vec{r}) = S + \left(\nu\Sigma_f\right)\phi(\vec{r}) $$
Neutron flux $\phi$ depends on just position here, and $\Sigma$ is a probability-per-flux of some interaction. The left represents losses, and the right represents sources - from left to right we have:


*

*How many neutrons leave the system, based on diffusion coefficient $D(\vec{r})$

*How many neutrons are absorbed, based on the surrounding material

*Source term describing any arbitrary source of neutrons

*Production term, based on fission producing $\nu$ neutrons per fission


Starting here, you can build the model by adding energy dependency and discretizing the domain (and a lot of other neat tricks for simplifying calculations, etc). 
Other resources:


*

*Neutron Transport wiki

*MIT lectures (starting around chapter 8) for a more aggressive analysis

*"Nuclear Reactor Analysis" by Duderstadt and Hamilton 

A: Yes, there are governing equations.
The neutron density is governed by the neutron Boltzmann transport equation, which is described in the this Wikipedia Page.  The full equation accounts for the neutron density as a function of time, energy, space, and direction. There are several simplifications you can make, depending on the problem you are solving.   The inputs to the equation are the surrounding material density and probabilities of interaction (cross sections).
One you know the neutron density, you can calculate the number of fission and absorption reactions that occur.  Each reaction releases a certain amount of energy, so this will give you the total power released as a function of space.
You then usually need to solve a heat transfer problem to determine the temperatures of all the materials given the amount of power generated and the amount of power removed from the system.
To make things more complicated, the temperatures feed back to the cross sections, so the whole system is usually solved iteratively.
There are several nuclear engineering textbooks that cover these topics.
