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
(3-56)
(3-57)
where τ is the extra stress tensor, which is defined as a sum of a viscous and a viscoelastic or elastic contribution as
(3-58)
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:
(3-59)
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
(3-60)
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
(3-61)
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
(3-62)
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
(3-63)
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
(3-64)
where αem is the dimensionless mobility factor.
LPTT and EPTT Model
The Phan–Thien–Tanner viscoelastic model is derived from the kinetic theory of an elastic network representing a polymeric melt. The rates of creation and destruction of junctions between the polymer strands depend on stress in the network. The expressions for the relaxation function and the viscosity factor for the linear Phan–Thien–Tanner model (LPTT) are given by
(3-65)
where εem is the extensibility. The expressions for the relaxation function for the exponential Phan–Thien–Tanner model (EPTT) is given by
(3-66)
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.