A Non-Newtonian Fluid is one for which stress is not linearly related to strain-rate. All non-Newtonian fluids are *elasticoviscous*, that is they combine elastic and viscous properties. When the time-scale of a flow t_{f} is much less than the relaxation time tr of an elasticoviscous material, elastic effects dominate. When, on the other hand, t_{f} is much greater than t_{r}, elastic effects relax sufficiently for viscous effects to dominate. The ratio t_{f}/t_{r} is a dimensionless number of particular significance in the study of flow of non-Newtonian fluids: depending on the circumstances, this number is called the Deborah number or the Weissenberg Number. The Deborah number, De, is named after the prophetess Deborah: "... the mountains flowed before the Lord ...," implying that all materials flow (are viscous) on a sufficiently long time-scale. Thus glass, normally considered to be elastic, flows on a time-scale of order centuries as the windows of medieval cathedrals testify. Similarly water, normally considered to be viscous, behaves elastically on a time-scale of order nanoseconds. The precise definition of De depends on the circumstances, though t_{f} is always taken to be a characteristic residence time. Thus, for flow at mean velocity U through a pipe of length L, t_{f} = L/U and so De = t_{r}U/L. If the Deborah number is small, elastic effects can be neglected and the non-Newtonian fluid treated as a purely viscous material, albeit with a non-constant viscosity.

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Heat & Mass Transfer, and Fluids Engineering

# DEBORAH NUMBER

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Article added: 2 February 2011
Article last modified: 9 February 2011
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