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OpenFOAM v9 User Guide: 7.3 Transport/rheology models

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7.3 Transport/rheology models

In OpenFOAM, solvers that do not include energy/heat, include a library of models for viscosity ν. The models typically relate viscosity to strain rate γ˙ 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 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.

7.3.2 Bird-Carreau model

The Bird-Carreau model is:
a (n-1)∕a ν = ν∞ + (ν0 - ν∞) [1 + (kγ˙) ]
(7.21)
where the coefficient a 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)
(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;
}

7.3.4 Power Law model

The Power Law model provides a function for viscosity, limited by minimum and maximum values, νmin and νmax respectively. The function is:
ν = k ˙γn-1 ν ≤ ν ≤ ν min max
(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;
}

7.3.5 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. Beyond a threshold in strain-rate corresponding to threshold stress τ0, the viscosity is described by a power law. The model is:
( ) ν = min ν0,τ0∕γ˙+ k ˙γn-1
(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;
}

7.3.6 Casson model

The Casson model is a basic model used in blood rheology that specifies minimum and maximum viscosities, νmin and νmax respectively. Beyond a threshold in strain-rate corresponding to threshold stress τ0, the viscosity is described by a “square-root” relationship. The model is:
(∘ –––-- √ --)2 ν = τ0∕γ˙+ m νmin ≤ ν ≤ νmax
(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;
}

7.3.7 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));
    }

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OpenFOAM v9 User Guide: 7.3 Transport/rheology models