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DEBORAH NUMBER

DOI: 10.1615/AtoZ.d.deborah_number

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 tf is much less than the relaxation time tr of an elasticoviscous material, elastic effects dominate. When, on the other hand, tf is much greater than tr, elastic effects relax sufficiently for viscous effects to dominate. The ratio tf/tr 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 tf is always taken to be a characteristic residence time. Thus, for flow at mean velocity U through a pipe of length L, tf = L/U and so De = trU/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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