where τ is the extra stress tensor, which is defined as a sum of a viscous and a viscoelastic or elastic contribution as
where μs is the solvent viscosity,
S is the strain-rate tensor, and
Te is the elastic (or viscoelastic) stress tensor. To adequately describe a flow of fluid with a complex rheological behavior, the symmetric stress tensor
Te is represented as a sum of the individual modes:
where the relaxation function frm and the viscosity factor
fpm are model-specific functions of stress,
λem is a relaxation time,
μem is a polymer viscosity, and the upper convective derivative operator is defined as
where αem is the dimensionless mobility factor.
The theory about boundary conditions is found in the section Theory for the Single-Phase Flow Interfaces. Note that for the viscoelastic models, the extra stress tensor is defined as a sum of a viscous and an elastic contribution:
τ = K + Te. Therefore, an additional term should be added to the expression for the normal extra stress:
Kn = Kn + Ten.