OpenFOAM v9 User Guide

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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 $\text{[math]}$ . The models typically relate viscosity to strain rate $\text{[math]}$ 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 $\text{[math]}$ 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 $\text{[math]}$ .

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=

(7.21)

where the coeﬃcient $\text{[math]}$ 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=

(7.22)

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, $\text{[math]}$ and $\text{[math]}$ respectively. The function is:

ν = k˙γn− 1 νmin ≤ ν ≤ νmax \relax \special {t4ht=

(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 $\text{[math]}$ . Beyond a threshold in strain-rate corresponding to threshold stress $\text{[math]}$ , the viscosity is described by a power law. The model is:

ν = min (ν0,τ0∕γ˙+ k˙γn− 1) \relax \special {t4ht=

(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, $\text{[math]}$ and $\text{[math]}$ respectively. Beyond a threshold in strain-rate corresponding to threshold stress $\text{[math]}$ , the viscosity is described by a “square-root” relationship. The model is:

(∘ ----- √--)2 ν = τ0∕˙γ + m νmin ≤ ν ≤ νmax \relax \special {t4ht=

(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));
    }

OpenFOAM v9 User Guide - 7.3 Transport/rheology models