7.3 Transport/rheology models

In OpenFOAM, solvers that do not include energy/heat, include a library of models for viscosity ν  \relax \special {t4ht=. The models typically relate viscosity to strain rate γ˙  \relax \special {t4ht= 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 ν  \relax \special {t4ht= 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 L2∕T  \relax \special {t4ht=.

7.3.2 Bird-Carreau model

The Bird-Carreau model is:

                            a (n- 1)∕a
ν = ν∞ +  (ν0 - ν∞ )[1 + (k˙γ) ]
\relax \special {t4ht=
(7.16)
where the coefficient a  \relax \special {t4ht= 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:

           ν0 - ν∞
ν = ν∞ +  ---------n
          1 + (m ˙γ)
\relax \special {t4ht=
(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, νmin   \relax \special {t4ht= and νmax   \relax \special {t4ht= respectively. The function is:

ν = k ˙γn-1  ν    ≤ ν ≤  ν
             min         max
\relax \special {t4ht=
(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 ν0   \relax \special {t4ht=. Beyond a threshold in strain-rate corresponding to threshold stress τ0   \relax \special {t4ht=, the viscosity is described by a power law. The model is:

        (                )
ν = min  ν0,τ0∕˙γ + kγ˙n -1
\relax \special {t4ht=
(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, νmin   \relax \special {t4ht= and νmax   \relax \special {t4ht= respectively. Beyond a threshold in strain-rate corresponding to threshold stress τ0   \relax \special {t4ht=, the viscosity is described by a “square-root” relationship. The model is:

     (∘ -----  √ --)2
ν =     τ0∕γ˙+   m      νmin ≤ ν ≤ νmax
\relax \special {t4ht=
(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 v6 User Guide - 7.3 Transport/rheology models
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