For a non-Newtonian fluid, the dynamic viscosity is assumed to be a function of the shear rate:
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 3-18 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. 20). Since infinite viscosity also makes models using
Equation 3-18 difficult to solve, COMSOL Multiphysics implements the power law model as
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.
Viscoplastic fluid behavior is characterized by existence of the yield stress τy — a limit which must be exceeded before significant deformation can occur. To model the stress-deformation behavior of viscoplastic materials, different constitutive equations have been propose. The Bingham plastic model is written as
where μp is the plastic viscosity.
where mp is a scale that controls the exponential growth of stress.
where m and
n are scalars that can be set to arbitrary values. For a value of
n equal to one, the Bingham-Papanastasiou model is recovered. This equation is valid in both yielded and unyielded regions. The exponent
mp controls the smoothness of the viscosity function.
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.
denotes the strain-rate tensor defined by: