General Viscoelastic Flow Theory
The Viscoelastic Flow Interface is used to simulate incompressible and isothermal flow of viscoelastic fluids. It solves the continuity equation, the momentum balance equation, and a constitutive equation that defines the extra elastic stress contribution. The continuity and momentum balance can be expressed as
(3-242)
(3-243)
where τ is the extra stress tensor, which is defined as a sum of a viscous and a viscoelastic or elastic contribution as
(3-244)
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:
(3-245)
To close the equation system, the constitutive relation for each mode is required.
Several commonly used constitutive models can be written as a hyperbolic partial differential transport equation of the form
(3-246)
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
(3-247)
The first two terms on the right-hand side represent the material derivative, and the other two terms represent the deformation. For more information, see Ref. 1.
Oldroyd-B Model
For the Oldroyd B model, the relaxation function and the viscosity factor are given by
(3-248)
The Oldroyd-B model can be derived from the kinetic theory representing the polymer molecules as suspensions of the Hookean spring in a Newtonian solvent. While demonstrating some basic features of viscoelasticity, the model can only predict a constant shear viscosity and gives unrealistic results for purely extensional flows due to the lack of a mechanism that limits the extensibility.
FENE-P Model
The finitely extensible nonlinear elastic model (FENE) is based on the kinetic theory that describes the polymer chains as a non-linear elastic springs and account for finite extension of the polymers monoculars. The FENE model with Peterlin closure (FENE-P) shows a finite extensibility and a shear-thinning behavior. The expressions for the relaxation function and the viscosity factor are given by
(3-249)
where Lem is the extensibility.
Giesekus Model
The Giesekus model is often used to model the flow of the semi-diluted and concentrated polymers. It adds the quadratic nonlinearity that is attributed to the effect of the hydrodynamic drag induced by the polymer-polymer interactions. The corresponding relaxation function and the viscosity factor are given by
(3-250)
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.