I am solving the equations describing the reaction in a tubular reactor where terms reading something like this show up: $$\frac{d}{dz} \dot{n}_i = \frac{d}{dz} u c_i. $$ Here $u$ denotes velocity and $c$ concentration. Now by the chain rule I can write it as $$u \frac{d}{dz} c_i + c_i \frac{d}{dz} u$$ and all is fine. However, the term $\frac{d}{dz} u $ is nonzero and extremely hard to close since velocity depends on a lot of parmateters (i.e. by the ideal gas law) which further down the line explodes into a mess of derivative terms that are again hard to close by themselves.
As an engineer I was told the following "technique", which I know is questionable at least, but I can not find any reference to any name or underlying assumption as to what is going on. So here's the trick:
When solving this equation in a ODE solver, we first omit the term containing $\frac{d}{dz} u $. Thereby we are not obeying differentiation rules, obviously. To not be too wrong though, we calculate $u$ at every position $z$ from known values (which is significantly easier then carying around the gradients). So basically we neglect the gradient but somehow explicitly do consider $u$ changing over $z$ This approach is often implemented but never documented. I only realized how wrong it is when working on a problem like this myself. I was never told why or how we do it and most of my colleagues experience this too.
Does this have a name? Is it valid under certain conditions? If so, when? Your help is very much appreciated.