Description of the problem
Contraction flow of a viscoelastic liquid

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, De = 2.lamda

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

Four boundary sets are considered :

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).