In this example, we would investigate the axisymmetric contraction of a viscoelastic liquid at a high Weissenberg number. Non-dimesionnal values are used here.

*Weissenberg == U*Lambda / l*

U: caracteristic speed of the flow

Lambda: caracteristic time scale of the fluid

l: caracteristic space scale of the flow

We will use an other adimensionnal number to define the viscoelastic
liquid :

*Deborah number = lambda / tflow
*Thus for this example,

We select a Phan Tien - Tanner viscoelastic constitutive model. The material parameters are :

*eta1 + eta2 = 1
eta2/(eta1+eta2)=1/9
lambda = 5
ksi = 0.2
eps = 0.015*

Four boundary sets are considered :

- an inflow and an outflow both with the same rate Q=pi
- a rigid wall
- an axis of symetry

In viscolelastic, calculations, the stress field is also unknown and must be computed together with the velocity and pressur fields. The hyperboloic character of the viscolelastic constitutive equation requires essantial boundary conditions for the stress at the inlet.

For the values of *Q* and *lambda* given previously, the Weissenberg
number is large in the downstream section and the problem is highly non-linear.

There are no terms of inertia and gravity ( i.e. Re=0).