Non-Newtonian Flow: The Power Law and the Carreau Model
The viscous stress tensor is directly dependent on the shear rate tensor and can be written as:
using the compressible and incompressible formulations. Here denotes the strain-rate tensor defined by:
Its magnitude, the shear rate, is:
where the contraction operator “:” is defined by
For a non-Newtonian fluid, the dynamic viscosity is assumed to be a function of the shear rate:
The Laminar Flow interfaces have the following predefined models to prescribe a non-Newtonian viscosity—the power law and the Carreau model.
Power Law
The power law model is an example of a generalized Newtonian model. It prescribes
(3-19)
where m and n are scalars that can be set to arbitrary values. For n > 1, the power law describes a shear thickening (dilatant) fluid. For n < 1, it describes a shear thinning (pseudoplastic) fluid. A value of n equal to one gives the expression for a Newtonian fluid.
Equation 3-19 predicts an infinite viscosity at zero shear rate for n < 1. This is however never the case physically. Instead, most fluids have a constant viscosity for shear rates smaller than 10-2 s-1 (Ref. 19). Since infinite viscosity also makes models using Equation 3-19 difficult to solve, COMSOL Multiphysics implements the power law model as
(3-20)
where is a lower limit for the evaluation of the shear rate magnitude. The default value for is 10-2 s-1, but can be given an arbitrary value or expression using the corresponding text field.
Carreau Model
The Carreau model defines the viscosity in terms of the following four-parameter expression
(3-21)
where λ is a parameter with the unit of time, μ0 is the zero shear rate viscosity, μinf is the infinite shear-rate viscosity, and n is a dimensionless parameter. This expression is able to describe the viscosity for most stationary polymer flows.