Deborah Number for harmonic excitation I think I do not understand well the concept of Deborah number.
It is presented in the sources available to me as the ratio between the relaxation time of a fluid and a characteristic time scale of the flow. In other sources the denominator of the ratio is occupied by the "observation time scale".
Firstly, I struggle in front of the definition of a "relaxation time" for a "real world" fluid, characterised by a continuous spectrum of relaxation times.
But the worst is to come, so let us assume a fluid with a clearly defined relaxation time is at hand.
If an harmonic excitation with, say, frequency $\omega$, is applied to the viscoelastic and the flow observed for a duration of time $t$, how to define Deborah number? The relaxation time is fixed as a material property, but what to use in the denominator? The time scale linked to the frequency applied, or the observational time scale?
 A: Firstly, some general answer on nondimensional groups such as the Deborah number, but also the Reynolds number e.g.: they do not generally characterise the flow as a whole, but a feature that you choose in the flow. If the flow is not an academic problem, you will have several such features, which have different lengths, velocities...
In the case of the Deborah number, it compares a relaxation time (property of the fluid) to an "external" time. You choose which external time: it can be the period of your forcing, if you want to focus on the steady harmonic response, or the time of establishment of it, if that's the process for which you want to see whether the Deborah might be small and therefore a simplification can be made. 
If you have several relaxation times, you also choose which you use: in general it will be the longest, because you want to see whether your fluid has time to relax ; but in some cases it may also be relevant to see the Deborah number of the other relaxation times.
