A rough idea of the dependence of various constants in analysis can be obtained by considering the dimensions of the terms involved. For example, taking the dimensions of both sides of the Poincar Friedrichs inequality gives

To simplify, let U denote units of velocity. We then have

so we expect CPF to scale like L = diam(). Pick five inequalities in analysis which you feel are key to the numerical analysis of the NSE. Repeat the above heuristic analysis to get insight into the scaling of their constants.