Non newtonian viscoelastic fluid

1 - A classification of fluids

2 -Descrition of viscoelastic fluids

There are both viscous and elastic : elasticity generates a memory effect whereas viscosity reduces it in time. This phenomenom is called evanescent memory. Thus we can define a characteristic time and a memory function for these fluids.

• charateristic time : among differents expressions, there is the relax time defined by :

• memory function : the behaviour law of a linear viscoelastic fluid can be written

where M(t-t') is a decreasing memory function.

Examples ( R.I. Tanner)

 Fluid type Temperature (K) Relax time (s) Water 293 10-12 High-density polyethylene 453 0.07 Glass 300 >105

3 - Behaviour laws of viscoelastic fluids

As there is a huge diversity of fluids, plenty of laws exists and so makes modelling difficult. Here we gives three examples of differential law ( an other form is the integral written used to express the memory function).

• Maxwell's model

This comes from an approach of fluid such a continous material.
Newton laws for viscous material

Hooke's law for elastic material

Maxwell's law - synthesis of the previous expressions

that became

• Note : derivation of tensor

a = -1 : under-convected derivate
a = 0 : Jaumann's derivate
a = 1 : over-convected derivate

• Oldroyd's law

Macromolecules can be modelled by "elastic dumbbell". We use :

The over-convected Oldroyd's law is :

• Phan tien- Tanner's law

It comes from modelling of high-density polymeres with "networks models".