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 tr/tf 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 Weissenberg Number or the Deborah Number. The Weissenberg number Ws is named after Karl Weissenberg, an early worker in the field of non-Newtonian fluids. The definition of Ws depends on that of tf which is given by:
where tr() denotes trace and e denotes strain-rate given in terms of velocity u by:
For a simple shear flow with shear-rate γ:
while for a simple extensional flow with extension rate ε:
Thus for flow at mean velocity U through a pipe of diameter D and length L, and so . By contrast the Deborah number . Thus Ws is larger than De by the ratio L/D.