Deborah number

The Deborah number is a dimensionless number, often used in rheology to characterize the fluidity of materials under specific flow conditions. It was originally proposed by Markus Reiner, a professor at Technion in Israel, inspired by a verse in the Bible, stating "The mountains flowed before the Lord" in a song by prophetess Deborah (Judges 5:5). It is based on the premise that given enough time even the hardest material, like mountains, will flow. Thus the flow characteristics is not an inherent property of the material alone, but a relative property that depends on two fundamentally different characteristic times.

Formally, the Deborah number is defined as the ratio of the relaxation time characterizing the time it takes for a material to adjust to applied stresses or deformations, and the characteristic time scale of an experiment (or a computer simulation) probing the response of the material. It incorporates both the elasticity and viscosity of the material. At lower Deborah numbers, the material behaves in a more fluidlike manner, with an associated Newtonian viscous flow. At higher Deborah numbers, the material behavior changes to a non-Newtonian regime, increasingly dominated by elasticity, demonstrating solidlike behavior.

The equation is thus:

$\mathrm{De} = \frac{t_\mathrm{c}}{t_\mathrm{p}}$

where t_c refers to the stress relaxation time (sometimes called the Maxwell relaxation time), and t_p refers to the time scale of observation.

References

• J.S. Vrentas, C.M. Jarzebski, J.L. Dudda (1975) "A Deborah number for diffusion in polymer-solvent systems", AIChE Journal 21(5):894–901, weblink to Wiley Online Library.