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The Weissenberg number (Wi) is a
dimensionless number A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1) ...
used in the study of
viscoelastic In materials science and continuum mechanics, viscoelasticity is the property of materials that exhibit both viscous and elastic characteristics when undergoing deformation. Viscous materials, like water, resist shear flow and strain linear ...
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, shear rate is the rate at which a progressive shearing deformation is applied to some material. Simple shear The shear rate for a fluid flowing between two parallel plates, one moving at a constant speed and the other one stationary ...
\dot times the relaxation time \lambda. Using the Maxwell model 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 dumbbell ...
, 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 fl ...
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