7.3 Transport/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.

7.3.1 Newtonian 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 .

7.3.2 Bird-Carreau model

The Bird-Carreau model is:

 (7.16)
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;
}

7.3.3 Cross Power Law model

The Cross Power Law model is:

 (7.17)
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;
}

7.3.4 Power Law model

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

 (7.18)
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;
}

7.3.5 Herschel-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:

 (7.19)
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;
}

7.3.6 Casson 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:

 (7.20)
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;
}

7.3.7 General 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 v7 User Guide - 7.3 Transport/rheology models