# 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.[1][2]

The equation is thus:

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

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

## References

1. ^ Reiner, M. (1964), "The Deborah Number", Physics Today 17 (1): 62, doi:10.1063/1.3051374
2. ^ The Deborah Number
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