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.
The constitutive relation can be formulated using different dependent variables. Several commonly used constitutive models can be written using stress formulation 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 bead-spring dumbbell and accounts for finite extension of the polymer molecules. 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.
FENE-CR Model
Several different closures for FENE types models exists The expressions for FENE-CR (Finitely Extensible Nonlinear Elastic-Chilcott Rallison) model are given by
(3-64)
Rolie–Poly Model
The Rouse Linear Entangled Polymers (Rolie–Poly) model is a model that derived based on the tube theory. The model incorporates the molecular motion mechanisms of reptation in the tubes, the stretching of polymer chains, and convective constraint release (CCR) due to the release of entanglements as a result of the relaxation of the other chains surrounding the reference chain.
(3-65)
where λem is the reptation relaxation time, λRm is the Rouse relaxation time, parameters βm and δm control the convective constrain release.
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
(3-66)
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
(3-67)
where εem is the extensibility. The expressions for the relaxation function for the exponential Phan–Thien–Tanner model (EPTT) is given by
(3-68)
Conformation Formulation
The conformation tensor is a macroscopic representation of the polymer molecules structure, for example a representation of the average orientation of the polymer chains. The conformation tensor is a positive definite symmetric tensor that is equal to the identity matrix in the absence of deformation
(3-69)
where frm is a the relaxation function. The polymer stress is related to the conformation tensor through the strain function
(3-70)
For more information, see Ref. 3. For many popular constitutive models, the strain and relaxation functions are polynomials of A, usually of first or second orders, with coefficients which may depend of the invariants of A. The strain and relaxation functions for the models that are available in the viscoelastic flow interface are given in the Table 3-2.
Table 3-2: Strain and relaxation functions.
Constitutive model
 fs
 fr
Oldroyd B
FENE-P
FENE-CR
LPTT
EPTT
Giesekus
Rolie–Poly
Conformation formulation vs elastic tensor formulation
Note that, for FENE-CR, FENE-P, and Rolie–Poly models conformation and elastic stress formulations are not fully equivalent. Several terms are not included in the elastic stress formulation. For example, the complete equation for FENE-CR model is
The time derivative of fpm is dropped in the elastic formulation in the viscoelastic interface (Equation 3-64).
Boundary Conditions
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.