# Dimensional analysis with space-distributed variables

It frequently happens that both the "governed" variable and some of the "governing" variables are entities which are "distributed" ones, meaning that their values can physically change with the position. Some examples are (in fluid dynamics): pressure, temperature, density, velocity etc

However, in nearly all examples that I could find on the books, it is assumed that these variables only have one specific "average" value for the problem under investigation. For example: it is shown how to derive the relation which links the average velocity of the fluid and its average density (plus some other variables), which is valid for the specific geometry and conditions under investigation.

I wonder: is it possible to use Dimensional Analysis to derive a relation which takes into account the variation with position of all the "distributed" variables involved? (Avoiding a brutal averaging operation).

A possible solution could be considering the position "x" as an additional governing variable. However, this fixes the problem only if the "governed" variable is the only one which actually depends on the position. In case some of the "governing" variables depend on the position too (as I stated in the example abobe), I still don't know how to approach the problem.

Can anybody give me any insight?