## 7.3Transport/rheology models

In OpenFOAM, solvers that do not include energy/heat, include a library of models for viscosity . The models typically relate viscosity to strain rate and are speciﬁed by the user in the transportProperties dictionary. The available models are listed in the following sections.

### 7.3.1Newtonian model

The Newtonian model assumes is constant. Viscosity is speciﬁed by a dimensionedScalar nu in transportProperties, e.g.

transportModel Newtonian;

nu             [ 0 2 -1 0 0 0 0 ] 1.5e-05;
Note the units for kinematic viscosity are .

### 7.3.2Bird-Carreau model

The Bird-Carreau model is:

 (7.21)
where the coeﬃcient has a default value of 2. An example speciﬁcation of the model in transportProperties is:

transportModel BirdCarreau;
BirdCarreauCoeffs
{
nu0      [ 0 2 -1 0 0 0 0 ] 1e-03;
nuInf    [ 0 2 -1 0 0 0 0 ] 1e-05;
k        [ 0 0  1 0 0 0 0 ] 1;
n        [ 0 0  0 0 0 0 0 ] 0.5;
}

### 7.3.3Cross Power Law model

The Cross Power Law model is:

 (7.22)
An example speciﬁcation of the model in transportProperties is:

transportModel CrossPowerLaw;
CrossPowerLawCoeffs
{
nu0      [ 0 2 -1 0 0 0 0 ] 1e-03;
nuInf    [ 0 2 -1 0 0 0 0 ] 1e-05;
m        [ 0 0  1 0 0 0 0 ] 1;
n        [ 0 0  0 0 0 0 0 ] 0.5;
}

### 7.3.4Power Law model

The Power Law model provides a function for viscosity, limited by minimum and maximum values, and respectively. The function is:

 (7.23)
An example speciﬁcation of the model in transportProperties is:

transportModel powerLaw;
powerLawCoeffs
{
nuMax    [ 0 2 -1 0 0 0 0 ] 1e-03;
nuMin    [ 0 2 -1 0 0 0 0 ] 1e-05;
k        [ 0 2 -1 0 0 0 0 ] 1e-05;
n        [ 0 0  0 0 0 0 0 ] 1;
}

### 7.3.5Herschel-Bulkley model

The Herschel-Bulkley model combines the eﬀects of Bingham plastic and power-law behavior in a ﬂuid. For low strain rates, the material is modelled as a very viscous ﬂuid with viscosity . Beyond a threshold in strain-rate corresponding to threshold stress , the viscosity is described by a power law. The model is:

 (7.24)
An example speciﬁcation of the model in transportProperties is:

transportModel HerschelBulkley;
HerschelBulkleyCoeffs
{
nu0      [ 0 2 -1 0 0 0 0 ] 1e-03;
tau0     [ 0 2 -2 0 0 0 0 ] 1;
k        [ 0 2 -1 0 0 0 0 ] 1e-05;
n        [ 0 0  0 0 0 0 0 ] 1;
}

### 7.3.6Casson model

The Casson model is a basic model used in blood rheology that speciﬁes minimum and maximum viscosities, and respectively. Beyond a threshold in strain-rate corresponding to threshold stress , the viscosity is described by a “square-root” relationship. The model is:

 (7.25)
An example speciﬁcation of model parameters for blood is:

transportModel Casson;
CassonCoeffs
{
m        [ 0 2 -1 0 0 0 0 ] 3.934986e-6;
tau0     [ 0 2 -2 0 0 0 0 ] 2.9032e-6;
nuMax    [ 0 2 -1 0 0 0 0 ] 13.3333e-6;
nuMin    [ 0 2 -1 0 0 0 0 ] 3.9047e-6;
}

### 7.3.7General strain-rate function

A strainRateFunction model exists that allows a user to specify viscosity as a function of strain rate at run-time. It uses the same Function1 functionality to specify the function of strain-rate, used by time varying properties in boundary conditions described in section 5.2.3.4 . An example speciﬁcation of the model in transportProperties is shown below using the polynomial function:

transportModel  strainRateFunction;
strainRateFunctionCoeffs
{
function polynomial ((0 0.1) (1 1.3));
}
OpenFOAM v9 User Guide - 7.3 Transport/rheology models