# POLYFLOW - How to recover the Trouton ratio of 3? How to model strain hardening with a generalised Newtonian (non-Newtonian inelastic) fluid model?

 From the analytical point of view, when one uses tensorial form for the constitutive equation of a Newtonian fluid, one actually obtains the uniaxial elongational viscosity as predicted by Trouton (i.e. thrice the zero shear viscosity). A quick calculation in POLYFLOW for a simple uniaxial elongational flow may show that. So, this is an intrinsic property of the model. However, this Trouton ratio of 3 holds only for a simple uniaxial elongational flow of a Newtonian fluid; it can be modelled as an axisymmetric flow; the result differs in a planar flow.It is possible to define a generalised Newtonian flow where the Trouton ratio differs from 3 and increases with the strain rate, i.e. with an increasing viscosity. For this, a UDF is defined on the viscosity, as visc = visc0 * (1 + 0.1 III^2). An even function is used, for making sure that the viscosity remains positive. Other laws are suggested in an example.Finally, it is worth mentionning that the strain hardening viscosity defined on the basis of the third invariant produces an effect only for 2D axisymmetric and 3D flows. Indeed, the third invariant is the determinant of the rate-of-deformation tensor, and it vanishes when the flow has no 3D character. Consequently, for 2D planar flows, no specific effect should be expected. This can be understood when one considers that a Newtonian fluid is unable to discriminate between the shear and planar extension.