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| The '''Deborah number''' ('''De''') is a [[dimensionless number]], often used in [[rheology]] to characterize the fluidity of materials under specific flow conditions. It is based on the premise that given enough time even a solid-like material will flow. The flow characteristics are not inherent properties of the material alone, but a relative property which depends on two fundamentally different characteristic times.
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| 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<!-- characterizing the intrinsic fluidity of a material: THIS WILL BE A CIRCULAR EXPLANATION-->, 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 enters the non-Newtonian regime, increasingly dominated by elasticity and demonstrating solidlike behavior.<ref>{{citation|first=M. |last=Reiner |year=1964|journal=Physics Today|volume =17|issue= 1| page= 62 |title=The Deborah Number|doi=10.1063/1.3051374}}</ref><ref>[http://rrc.engr.wisc.edu/deborah.html The Deborah Number]</ref>
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| The equation is thus:
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| :<math> \mathrm{De} = \frac{t_\mathrm{c}}{t_\mathrm{p}}</math>
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| where ''t''<sub>c</sub> refers to the [[Stress relaxation|stress relaxation time]] (sometimes called the Maxwell relaxation time), and ''t''<sub>p</sub> refers to the time scale of observation.
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| <!--Note that the Deborah number is relevant for materials that flow on long time scales (like a [[Maxwell material|Maxwell fluid]]) but ''not'' for the reverse kind of materials (like the Voigt or [[Kelvin material|Kelvin model]]) that are viscous on short time scales but solid on the long term.-->
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| '''History'''
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| It was originally proposed by [[Markus Reiner]], a professor at [[Technion – Israel Institute of Technology|Technion]] in [[Israel]], inspired by a verse in the [[Bible]], stating "The mountains flowed before the Lord" in a song by prophetess [[Deborah]] ([[Book of Judges|Judges]] 5:5).
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| '''Time-Temperature Superposition'''
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| The Deborah Number is particularly useful in conceptualizing the [[Time–temperature superposition]] principle. Time-Temperature superposition has to do with altering experimental time scales using reference temperatures to extrapolate on the temperature-dependent mechanical properties of [[polymers]]. A material at low temperature with a long experimental or [[relaxation time]] behaves like the same material at high temperature and short experimental or relaxation time if the Deborah number remains the same. This can be particularly useful when working with materials which relax on a long time scale under a certain temperature. The practical application of this idea arises in the [[Williams–Landel–Ferry equation]]. Time-temperature superposition avoids the inefficiency of measuring a polymer’s behavior over long periods of time at a specified temperature by utilizing the Deborah Number.<ref>Rudin, Alfred, and Phillip Choi. The Elements of Polymer Science and Engineering. 3rd. Oxford: Academic Press, 2013. Print. Page 221.</ref>
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| ==References==
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| {{reflist}}
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| * J.S. Vrentas, C.M. Jarzebski, J.L. Dudda (1975) [http://onlinelibrary.wiley.com/doi/10.1002/aic.690210510/abstract "A Deborah number for diffusion in polymer-solvent systems"], [[AIChE]] Journal 21(5):894–901, weblink to Wiley Online Library.
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| {{NonDimFluMech}}
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| [[Category:Dimensionless numbers of fluid mechanics]]
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| [[Category:Fluid dynamics]]
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