The Laminar Flow interfaces provide various predefined inelastic non-Newtonian constitutive models including Power law, Carreau, Carreau–Yasuda, Cross, Cross–Williamson, Sisko, Bingham–Papanastasiou, Herschel–Bulkley–Papanastasiou, Casson–Papanastasiou, DeKee–Turcotte–Papanastasiou, Robertson–Stiff–Papanastasiou, Ellis, and Houska thixotropy.

where m, n are scalars that can be set to arbitrary values and denotes a reference shear rate for which the default value is 1 s−1. 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 3-13 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. 17). Since infinite viscosity also makes models using Equation 3-13 difficult to solve, COMSOL Multiphysics implements the Power law model as

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.

where λ is a parameter with the unit of time, μ0 is the zero shear rate viscosity, μ∞ 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.

The Carreau–Yasuda model is a generalized version of the Carreau model with the transition parameter a which allows for modifying the stiffness of the transition from constant μ0 to thinning. So, it has five parameters in the expression:

The Cross model is a special case of the Carreau–Yasuda model with a = 1 − n and λ = μ0/τtr where τtr is the critical stress. So, it has the following form:

The Cross–Williamson model is a special case of the Carreau–Yasuda model with μ∞ = ,0 and a = 1. So, its equation has three parameters:

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.

To allow computation in both yielded and unyielded region, the Papanastasiou continuous regularization for the viscosity function is used:

where mp is a scale that controls the exponential growth of stress.

The Herschel–Bulkley –Papanastasiou model combines the effects of the Power law the Papanastasiou continuous regularization:

where m, n are scalars that can be set to arbitrary values, and is the reference shear rate. In case n equals 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.

where μDK and λDK denote the shear rate viscosity and relaxation time respectively. In case λDK equals zero, it recovers the Bingham–Papanastasiou model.

In case n equals one, it also recovers the Bingham–Papanastasiou model.

In the Ellis model, μapp is computed recursively using the following equation

where τ1/2 is the shear stress at which μapp = μ0/2. The exponent ae is the shear thinning index, which is a measure of the degree of nonlinearity.

The Houska thixotropy model takes a similar form as the Herschel–Bulkley–Papanastasiou model, but with a linear dependence of the consistency m and the yield stress τy on the structure field ξ:

where my,0 and τy,0 denote the consistency and yield stress of the fully broken down material, respectively, my,0 + my,t and τy,0 + τy,t indicate the consistency and yield stress of the fully recovered material, respectively.

Here, the structure field ξ obeys the following equation:

where kf and kb denote the rebuild and breakdown coefficients, respectively.

It is also possible to add the thermal effects to the non-Newtonian constitutive models. The following options: None, Arrhenius, Williams–Landel–Ferry (WLF), Exponential, and User defined are expressed in terms of a thermal function αT. The thermal function αT is coupled with the non-Newtonian constitutive models by multiplying by μ0, μp, m, μDK, my,0, and my,0 if they are User defined.

where Q denotes the activation energy, R is the universal gas constant, T and T0 are the temperature and reference temperature, respectively.

where C1WLF and C2WLF are model constants, and TWLF is the reference temperature.

where b denotes a temperature sensitivity, and T0 is the reference temperature.