Weissenberg Number
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The Weissenberg number (Wi) is a
dimensionless number Dimensionless quantities, or quantities of dimension one, are quantities implicitly defined in a manner that prevents their aggregation into unit of measurement, units of measurement. ISBN 978-92-822-2272-0. Typically expressed as ratios that a ...
used in the study of
viscoelastic In materials science and continuum mechanics, viscoelasticity is the property of materials that exhibit both Viscosity, viscous and Elasticity (physics), elastic characteristics when undergoing deformation (engineering), deformation. Viscous mate ...
flows. It is named after
Karl Weissenberg Karl Weissenberg (11 June 1893, Vienna – 6 April 1976, The Hague) was an Austrian physicist, notable for his contributions to rheology and crystallography.stress relaxation In materials science, stress relaxation is the observed decrease in stress in response to strain generated in the structure. This is primarily due to keeping the structure in a strained condition for some finite interval of time hence causing som ...
time of the fluid and a specific process time. For instance, in simple steady shear, the Weissenberg number, often abbreviated as Wi or We, is defined as the
shear rate In physics, mechanics and other areas of science, shear rate is the rate at which a progressive shear strain is applied to some material, causing shearing to the material. Shear rate is a measure of how the velocity changes with distance. Simple ...
\dot times the relaxation time \lambda. Using the
Maxwell model A Maxwell model is the most simple model viscoelastic material showing properties of a typical liquid. It shows viscous flow on the long timescale, but additional elastic resistance to fast deformations. It is named for James Clerk Maxwell who p ...
and the
Oldroyd-B model The Oldroyd-B model is a constitutive model used to describe the flow of viscoelastic fluids. This model can be regarded as an extension of the upper-convected Maxwell model and is equivalent to a fluid filled with elastic bead and spring dumbbells. ...
, the elastic forces can be written as the first Normal force (N1). :\text = \dfrac = \frac = \frac= 2 \dot \lambda.\, Since this number is obtained from scaling the evolution of the stress, it contains choices for the shear or elongation rate, and the length-scale. Therefore the exact definition of all non dimensional numbers should be given as well as the number itself. While Wi is similar to the
Deborah number The Deborah number (De) is a dimensionless number, often used in rheology to characterize the fluidity of materials under specific flow conditions. It quantifies the observation that given enough time even a solid-like material might flow, or a f ...
and is often confused with it in technical literature, they have different physical interpretations. The Weissenberg number indicates the degree of anisotropy or orientation generated by the deformation, and is appropriate to describe flows with a constant stretch history, such as simple shear. In contrast, the Deborah number should be used to describe flows with a non-constant stretch history, and physically represents the rate at which elastic energy is stored or released.


References

{{Dimensionless numbers in fluid mechanics Dimensionless numbers of fluid mechanics Fluid dynamics Non-Newtonian fluids Rheology