Newtonian and Non-Newtonian Fluids
A fluid may be characterized according to its response under the action of a shear stress. A Newtonian fluid has a linear relationship between shear stress and shear rate, with the line passing through the origin. The constant of proportionality is referred to as the viscosity of the fluid and may depend on temperature, pressure, and composition. For a non-Newtonian fluid, the curve for shear stress versus shear rate is nonlinear or does not pass through the origin. If the curve is shifted away from the origin, the fluid has a yield stress. If the curve bends toward the shear-rate axis, the fluid is said to be shear thinning (pseudoplastic), but if it instead bends toward the shear-stress axis, it is said to be shear thickening (dilatant). The relationship may also contain time derivatives of the shear rate to model memory effects (thixotropy), and even parallel viscous and elastic responses. We refer to the latter as viscoelastic non-Newtonian fluids and all others as inelastic non-Newtonian fluids. For the inelastic non-Newtonian models, it is possible to define an apparent viscosity from a generalized Newtonian relationship between the deviatoric stress tensor and the strain-rate tensor.
Figure 2: Conceptual behavior of shear stress versus shear rate for Newtonian, yield-stress, shear-thinning, and shear-thickening fluids.
The apparent viscosity will then be a function of the first invariant of the strain-rate tensor. In simple shear flows, this invariant reduces to the shear rate.
The simplest viscoelastic model in the Polymer Flow Module is the Oldroyd-B model which has a constant viscosity and a constant first normal-stress coefficient. This is conceptually visualized as a Hookean spring in series with a dashpot in Figure 3.
Figure 3: A Hookean spring connected to a dashpot.
To ensure material-frame indifference, an upper-convected time derivative is used to describe the transport of the additional stress due to the spring and dashpot. Multiple branches with different relaxation times and viscosities can be added in order to include multimodal responses. In the Oldroyd-B model, the springs are infinitely extensible, which can lead to unphysical behavior. The other viscoelastic models in the module include nonlinear-spring effects and can also model shear-thinning behavior.
When applying a viscoelastic model, it is important to keep in mind that the mathematical formulation may become ill-posed at sharp corners. One remedy is to use a fillet operation to smooth the geometry at such points. As always when working with complex fluid flow simulations, it is advisable to start with a simple model and then switch to a more complicated model if the simple model does not capture the expected flow phenomena. In most industrial applications, inelastic non-Newtonian models produce sufficiently accurate results at a reasonable computational cost.