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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 + (k˙γ) ]
(7.15)

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)
(7.16)

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
(7.17)

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 + kγ˙n −1
(7.18)

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 (currently OpenFOAM-dev)

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
(7.19)

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;
}

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