Viscoelastic Flow Model
Standard CFD models treat fluids as purely viscous, whereas many complex materials exhibit time-dependent viscoelastic constitutive behaviors, where stress response depends on both the instantaneous shear rate and the deformation history. Polymer melts resist rapid deformation like a solid yet flow easily under slow shear. Filled suspensions like cement, food products, or slurries behave rigidly up to a stress threshold, then yield and flow. Accurately simulating these materials requires models that capture both the viscous and elastic contributions to stress, and their interaction over time.
FLOW-3D provides a family of models for non-Newtonian fluids with elastic behavior, covering both viscoelastic materials, where viscous and elastic stresses interact continuously, and elasto-viscoplastic materials, where the material transitions from solid-like to liquid-like behavior when a yield stress is exceeded.
Viscoelastic Models
Viscoelasticity can be well explained with the classic oscillatory shear experiment simulated below with FLOW-3D. A rectangular fluid element is placed between two plates. The top plate moves with a sinusoidal velocity, and the horizontal force is measured at the bottom plate. We can observe that the maximum force for a Newtonian fluid is measured when the top plate velocity is maximum when the shear rate is the highest. For purely elastic material (such as solid), the maximum force is in the most pivoted position of the top plate, i.e. when the displacement (or strain) is the highest. Viscoelastic fluid response is a mixture of those two ideal fluids, when both viscosity and elasticity play a role for the overall rheological response.
Viscoelastic materials exhibit time-dependent mechanical behavior: they respond like an elastic solid under rapid deformation and like a viscous liquid under slow deformation. The classic example is a polymer melt or solution, where long molecular chains become entangled under rapid shear and resist deformation, but slide past each other easily under slow flow. The relative contribution of viscous and elastic stress changes continuously depending on the local strain rate.
Two built-in viscoelastic constitutive models are available in FLOW-3D:
Oldroyd-B Model
The Oldroyd-B model is the simplest viscoelastic model and is appropriate for dilute polymer solutions. It represents the polymer contribution to stress as a Hookean spring in series with a dashpot, producing a fluid that has a constant viscosity but exhibits elastic stress that decays exponentially after flow stops. It is a good starting point for viscoelastic flow problems where material data is limited. The Oldroyd-B model is suited for viscoelastic flows with a low Weissenberg number, that is when the characteristic strain rate is lower than the reciprocal of the relaxation time.
Giesekus Model
Custom Viscoelastic Models
For materials that do not fit either built-in model, FLOW-3D supports custom viscoelastic constitutive equations through a user-modifiable Fortran routine. This allows additional terms to be added to the elastic stress formulation without modifying the core solver.
Elasto-Viscoplastic Materials
Elasto-viscoplastic materials behave as elastic solids up to a yield stress threshold, beyond which they flow as viscous fluids. This behavior describes a wide range of industrially important materials including fresh concrete, food products, muds, and highly filled suspensions such as toothpaste or solder paste. Below the yield stress, the material stores energy elastically and deforms without flowing. Above it, the material yields and flows with plastic deformation.
The elasto-viscoplastic model in FLOW-3D uses an incremental Hookean elastic stress formulation, computing the change in elastic stress from one time step to the next based on the local incremental strain. Because each increment is computed over a small time step where the deformation is small, the model can accurately predict large, highly nonlinear deformations over long time scales. Temperature-dependent elastic shear modulus and yield stress can be specified using tabular data when the Heat Transfer model is active.
Constitutive Law Formulations
For both the Oldroyd-B and Giesekus viscoelastic models, two formulations of the constitutive law are available:
- Stress tensor formulation: solves for the elastic stress tensor. Lower computational cost per time step, appropriate for low Weissenberg number flows.
- Log-conformation tensor formulation: solves for the logarithm of the conformation tensor, which is a representation of elastic strain, rather than the stress tensor directly. This formulation is more robust in high Weissenberg number flows, where the stress tensor formulation can develop numerical instabilities. Recommended when elastic stress is large relative to viscous stress.
The simulation below is a classic case of 4:1 contraction flow that features a sharp corner. While the stress tensor formulation becomes unstable after 3s of the flow due to exponential growth of the elastic stress, log-conformation tensor formulation remain stable and enables robust simulation.
The log-conformation tensor formulation is a significant improvement for demanding polymer flow simulations, such as high-speed injection molding, extrusion through sharp dies, and other flows where elastic stress dominates.
Applications
The Viscoelastic Flow Model in FLOW-3D is used for:
- Polymer melt injection molding: predicting fill patterns, weld line locations, and die swell effects that viscous models cannot capture
- Extrusion die design: modeling extrudate swell and flow instabilities in profile and film extrusion
- Food processing: simulating flow of non-Newtonian food products including doughs, gels, and emulsions
- Cement and concrete placement: predicting flow and self-leveling behavior of fresh concrete and grout
- Pharmaceutical manufacturing: modeling filled suspensions, ointments, and other yield-stress formulations
- Solder paste printing and dispensing in electronics manufacturing
- Biofluid mechanics: simulating blood and other biological fluids with viscoelastic rheology