It is known that the first normal stress difference is mainly responsible for enhanced extrudate swell in extrusion flow. This is typically a viscoelastic property. With respect to this, the simplified viscoelastic model is an extension of existing Newtonian fluid models, where a normal stress difference has been incorporated into the force balance. That is, in simple shear flow along the first axis and with a shear rate , the total extra-stress tensor is given by:

(6–81)

In this tensor, is the shear stress component, which involves the shear rate dependent viscosity . Several laws are available for describing the shear viscosity (see Generalized Newtonian Flow for more details), for instance, the constant law (Equation 6–7), the Bird-Carreau law (Equation 6–8), the Power law (Equation 6–9), the Cross law (Equation 6–14), the modified Cross law (Equation 6–15), and the Carreau-Yasuda law (Equation 6–17).

The first normal stress is given by . This quantity involves the viscoelastic variable , a quantity that can be referred to as the first normal viscosity, and a weighting coefficient .

The viscoelastic variable obeys a transport equation that involves a characteristic or relaxation time and is given by:

(6–82)

The equation is such that you recover the solution in simple shear flow. The first normal viscosity found in Equation 6–81 is described by means of functions similar to those available for the shear viscosity , where is presently replaced by . In order to facilitate the set up of a flow simulation involving the simplified viscoelastic model, identical dependences for and are considered by default. However, it is important to note that different functions can be selected for the shear and first normal viscosities.

Three algebraic models are available for the relaxation time function:

Finally, for non-isothermal flows, temperature dependence laws can be selected for the shear and first normal viscosities (see Temperature Dependence of Viscosity for more details). For instance, there is the Arrhenius law Equation 6–21), the approximate Arrhenius law (Equation 6–22), and the WLF law (Equation 6–24).

When defining a non-isothermal case, a single function is used to describe the temperature dependence of the material functions , , , and optionally of .