As we usually on dimensionalize the governing equation and the boundary conditions to get the small parameter? What if we already have the small parameter without non-dimensionalising?
I think it is wise to non-dimensionalize the governing equations and BCs. This way you can be consistent in evaluating the order of magnitude of the various BCs and terms in the differential equation, even if your small parameter does not have units.
Non-dimensionalizing alone may be misleading. If one of your variables is of magnitude 1000 and another one of magnitude 0.001 (before or after non-dimensionalizing does not matter) and perturbing both by an amount of 0.0001 then the effect is essentially that of non-perturbing the first variable, the result does nothing say about this variable. (Here, I assumed that perturbations are by an absolute amount, not a relative one.) So the point is to rescale the variables such a way that all variables have the same magnitude (and the size of perturbation is to be chosen according to the common magnitude). The process of rescaling may also non-dimensionalize the variables.
- Badges
- Science topic
- Similar topics
- Mathematics
- Applied Mathematics
- Differential Equations