Weissenberg number

Remarks

Example: for the upper-convected Maxwell model in steady simple shear with shear rate $\dot\gamma$, the shear stress is $\tau_{12} = \eta_0 \dot\gamma$ and the first normal stress difference is $N_1 = 2 \lambda \eta_0 \dot\gamma^2$. So $N_1 / \tau_{12} = 2 \lambda \dot\gamma = 2 \, \mathrm{Wi}$: the ratio of elastic to viscous stress is measured by $\mathrm{Wi}$.

Non-example: the quantity $\lambda \dot\gamma^2$ is not the Weissenberg number; it has the unit of an inverse time, so it is not dimensionless.

The Weissenberg number is different from the Deborah number, which compares the relaxation time with the observation time of the process rather than with the inverse rate of deformation. For a Newtonian fluid one has $\lambda = 0$ and so $\mathrm{Wi} = 0$.