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 t_{f} is much less than the relaxation time t_{r} of an elasticoviscous material, elastic effects dominate. When on the other hand t_{f} is much greater than t_{r}, elastic effects relax sufficiently for viscous effects to dominate. The ratio t_{r}/t_{f} 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 t_{f} 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.

Heat & Mass Transfer, and Fluids Engineering