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 on the elasticity and viscosity of the material. The smaller the Deborah number, the material behaves more fluid like with an associated Newtonian viscous flow. At higher Deborah numbers, the material behavior changes to non-Newtonian regime, increasingly dominated by elasticity, reaching solid like behavior with very high Deborah numbers.

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.

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