Deborah Number

This is the one number on the site with a human being inside it.

Everywhere else, a dimensionless group is built from properties of the material and the flow. The Deborah number is built from a property of the material — its relaxation time $\lambda$, how long it takes to forget a deformation — divided by how long you decided to watch. Change nothing about the material, change only your patience, and the answer changes from "solid" to "liquid".

That is not a defect in the number. It is the physics, and it is the most important idea in rheology. Solid and liquid are not categories that matter belongs to. They are descriptions of how matter behaves on a particular timescale, and the timescale is supplied by the observer.

The examples are famous because they are so hard to argue with. Pitch — road tar — shatters like glass under a hammer, because the hammer's observation time is milliseconds and $\lambda$ for pitch is of order years, putting De somewhere around $10^{10}$. The same pitch drips through a funnel. The University of Queensland set up a funnel of it in 1927 and it has produced nine drops in almost a century; watch for a decade and De falls below one and it is unmistakably a liquid. Silly putty bounces off the floor and puddles on the shelf overnight. Glass in medieval cathedral windows is often said to have flowed — this one is actually false, the windows are thicker at the bottom because of how they were made, and the relaxation time of window glass at room temperature is longer than the age of the universe — but the reason the story is plausible is that everyone understands the principle even when they get the number wrong. A glacier is a pile of ice crystals, and it flows.

Markus Reiner named the group in 1964, in a short note in Physics Today, and the name is a small joke with a serious point. The Song of Deborah in Judges 5:5 says "the mountains flowed before the Lord". Reiner's reading was that the prophetess was not being poetic but precise: mountains do flow, on a long enough view, and God's observation time is infinite, which puts the Deborah number of the mountains at zero. Everything is a liquid to a patient enough observer. It is one of the very few pieces of scripture that has become a defined quantity in an engineering handbook.

Where $\lambda$ comes from deserves care, because a real material does not have one. A polymer melt has a whole spectrum of relaxation times, from the fast wiggling of a few monomer units to the slow reptation of a whole chain out of its tube. Quoting a single $\lambda$ means you have selected a characteristic value from that spectrum, and the usual selection is either the longest relaxation time or the reciprocal of the frequency at which the storage modulus $G'$ and the loss modulus $G''$ cross in a small-amplitude oscillatory shear sweep. That crossover is De = 1 by construction, which is why the measurement and the definition fit together so neatly. Say which one you used.

Do not confuse this with the Weissenberg number. They look alike and are constantly conflated. Weissenberg is $\mathrm{Wi} = \lambda \dot\gamma$, the relaxation time times a deformation rate; Deborah is $\lambda / t_{obs}$, the relaxation time over an observation time. In steady simple shear the observation time is naturally $1/\dot\gamma$ and the two coincide. In an extensional flow, a start-up transient, or a flow through a contraction, they do not, and the distinction matters: Weissenberg tells you how strongly the material is being stretched, Deborah tells you whether it has had time to respond. A great deal of argument in the rheology literature turns out, on inspection, to be about which of the two somebody meant.

The practical payoff is in processing. In an extruder die you want De above one, so the melt still has its elasticity when it exits and the extrudate holds its shape; that same elasticity is what causes die swell and, pushed too far, melt fracture. In the mould you want De below one for long enough that the frozen-in orientation relaxes out before the part solidifies — and injection moulding never quite manages it, which is why mouldings have residual stress, why they warp, and why the shrinkage is different along the flow direction and across it. The whole business of the trade is living on both sides of De = 1 at once.