Can I scale (non-dimensionalize) a variable with another variable? For example I have a velocity function, u(x,t), which gives velocity of a flow field and has spatial and temporal variations (function of time and position). Suppose that I scaled with a velocity Sl which is a constant.
What if my Sl velocity has a spatial variation such as Sl(x). Still can I perform a scaling as u(x,t)/Sl(x)? Or rules are strict and scaling can be done only with a constant?
Thank you!
Of course you can formally make any scale transformation you wish, even "local" scale transformations. However, the equations of motion will acquire "fictitious forces" stemming from the non-constant transformation you have applied.
Let's take as an example a 3-body problem, in which we are looking for the motion of a light third body, and have the other two bodies travelling in a mutual circular orbit. Then scaling to use $\vec{s}_3 = \vec{x}_3/|r_{12}(t)|$ makes good sense and gets rid of a non-vital parameter; $r_{12}(t)|$ remains constant so no mysterious apparent forces arise.
But if bodies $1$ and $2$ are in a mutual elliptical orbit and you re-scale, using $$ \vec{s}_3(t) \equiv\vec{x}_3(t)/|r_{12}(t)|$$ you will find that the equations of motion for the third body have become much more complicated. It would be much simpler to work in a scale where the semi-major axis is taken to be one; that is constant, and makes your work easier.
