Weissenberg number explained

• \gamma times the relaxation time λ . Using the Maxwell model and the Oldroyd-B model, the elastic forces can be written as the first Normal force (N₁).Wi = \dfrac{ elasticforces }{ viscousforces } = (\tau _{xx} - \tau _{yy})/(\tau _{xy}) = (• 2λ \mu \gamma ^{2})/(• \mu \gamma) = 2 • \gamma λ . ^[1]

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 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.

Notes and References

1. Poole. R J. 2012. The Deborah and Weissenberg numbers. Rheology Bulletin. 53 . 2. 32–39.