Tuesday, August 24, 2010

The Deborah and Weissenberg Numbers

Engineers have always loved dimensionless numbers [*], groups of variables where the units cancel leaving them free from the chosen system of measurement. (i.e., the Reynold number for a system has the same value when measured in SI or English units.) Rheology is generally lacking in dimensionless numbers except for two - the Deborah Number and the Weissenberg number.

I must confess that I never really understood there to be much difference between the two, but a new article in the Rheology Bulletin (pdf file, open access, article starts on p. 14) makes clear the difference. The author is John Dealy of McGill University. John is someone who's ability to write clearly on rheology I have always greatly admired and envied. The article, not one of his best, has a number of fine details that I will not discuss here, such as how to work with a single relaxation time when it is known that the system has a distribution of such times. I would highly recommend reading more than just my summary.

As for the numbers, the Deborah number is ratio of the relaxation time of the polymer divided by the observation time. Fine points that I've never caught in the past are that the observation time is not the reciprocal of the deformation rate, and that the Deborah number is identically zero for steady state flows. The deformation rate however, does appear directly in the Weissenberg number, which is defined as the product of the relaxation time and the deformation rate.

It's apparent that I've been using the Deborah number too often when I should have been using the Weissenber number. I'm going to have to work this over in my head a few times as the distinctions clearly exist and I want to get it correct. Fortunately, I've not had to use the numbers too often, and the mistakes I have made have been made by others as well, but I certainly intend to do better in the future.

[*] One of my ChemE professors told us that he loved to taunt chemists by talking to them about his favorite dimensionless number - the 2nd Damköhler number. He was able to double tweak them with this because it also implied that there was a 1st Damköhler number that they also didn't know about.

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