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8.3 Transport/rheology models
The momentumTransport ﬁle includes any model for the viscous stress in a ﬂuid. That includes turbulence models, described in the previous section 8.2 , but also nonNewtonian and viscoelastic models described in this section. These models are described as laminar, located in $FOAM_SRC/MomentumTransportModels/momentumTransportModels/laminar, including:

a family of generalisedNewtonian models for a nonuniform viscosity which is a function of strain rate , described in sections 8.3.1 , 8.3.2, 8.3.3 , 8.3.4 , 8.3.5 and 8.3.6 ;

a set of viscoelastic models, including Maxwell, Giesekus and PTT (PhanThien & Tanner), described in sections 8.3.7 , 8.3.8 and 8.3.9 , respectively;

the lambdaThixotropic model, described in section 8.3.10 .
When turbulence modelling is selected in the momentumTransport ﬁle, the generalisedNewtonian model is used by default to calculate the molecular viscosity. The choice of generalisedNewtonian model, speciﬁed by the viscosityModel keyword, is set to Newtonian by default, which simply uses the viscosity nu speciﬁed in the physicalProperties ﬁle. The following example exposes the default settings used with turbulence modelling.
simulationType RAS
RAS
{
model kEpsilon; // RAS model
turbulence on;
printCoeffs on;
// "laminar" model generalisedNewtonian is used by default
viscosityModel Newtonian; // default
}
When turbulence modelling is not selected, by setting the laminar simulation type, the user can select any of the laminar models through the model keyword entry in the laminar subdictionary, including the viscoelastic models. The laminar models are listed by the following command.
foamToC table laminarincompressibleMomentumTransportModel
foamToC table generalisedNewtonianViscosityModel
simulationType laminar
laminar
{
model generalisedNewtonian;
viscosityModel BirdCarreau;
// ... followed by the BirdCarreau parameters
}
viscosityModel constant;
nu 1.5e05;
8.3.1 BirdCarreau model
The BirdCarreau generalisedNewtonian model is
(8.22) 
viscosityModel BirdCarreau;
nuInf 1e05;
k 1;
n 0.5;
8.3.2 Cross Power Law model
The Cross Power Law generalisedNewtonian model is:
(8.23) 
viscosityModel CrossPowerLaw;
nuInf 1e05;
m 1;
n 0.5;
8.3.3 Power Law model
The Power Law generalisedNewtonian model provides a function for viscosity, limited by minimum and maximum values, and respectively. The function is:
(8.24) 
viscosityModel powerLaw;
nuMax 1e03;
nuMin 1e05;
k 1e05;
n 0.5;
8.3.4 HerschelBulkley model
The HerschelBulkley generalisedNewtonian model combines the eﬀects of Bingham plastic and powerlaw behaviour in a ﬂuid. For low strain rates, the material is modelled as a very viscous ﬂuid with viscosity . Beyond a threshold in strainrate corresponding to threshold stress , the viscosity is described by a power law. The model is:
(8.25) 
viscosityModel HerschelBulkley;
tau0 0.01;
k 0.001;
n 0.5;
8.3.5 Casson model
The Casson generalisedNewtonian model is a basic model used in blood rheology that speciﬁes minimum and maximum viscosities, and respectively. Beyond a threshold in strainrate corresponding to threshold stress , the viscosity is described by a “squareroot” relationship. The model is:
(8.26) 
viscosityModel Casson;
m 3.934986e6;
tau0 2.9032e6;
nuMax 13.3333e6;
nuMin 3.9047e6;
8.3.6 General strainrate function
A strainRateFunction generalisedNewtonian model exists that allows a user to specify viscosity as a function of strain rate at runtime. It uses the same Function1 functionality to specify the function of strainrate, used by time varying properties in boundary conditions described in section 6.4.4 . An example speciﬁcation of the model in momentumTransport is shown below using the polynomial function:
viscosityModel strainRateFunction;
function polynomial ((0 0.1) (1 1.3));
8.3.7 Maxwell model
The Maxwell laminar viscoelastic model solves an equation for the ﬂuid stress tensor :
(8.27) 
simulationType laminar;
laminar
{
model Maxwell;
MaxwellCoeffs
{
nuM 0.002;
lambda 0.03;
}
}
8.3.8 Giesekus model
The Giesekus laminar viscoelastic model is similar to the Maxwell model but includes an additional “mobility” term in the equation for :
(8.28) 
simulationType laminar;
laminar
{
model Giesekus;
GiesekusCoeffs
{
nuM 0.002;
lambda 0.03;
alphaG 0.1;
}
}
8.3.9 PhanThienTanner (PTT) model
The PhanThienTanner (PTT) laminar viscoelastic model is also similar to the Maxwell model but includes an additional “extensibility” term in the equation for , suitable for polymeric liquids:
(8.29) 
simulationType laminar;
laminar
{
model PTT;
PTTCoeffs
{
nuM 0.002;
lambda 0.03;
epsilon 0.25;
}
}
8.3.10 Lambda thixotropic model
The Lambda Thixotropic laminar model calculates the evolution of a structural parameter (lambda) according to:
(8.30) 
(8.31) 
An example speciﬁcation of the model in momentumTransport is:
simulationType laminar;
laminar
{
model lambdaThixotropic;
lambdaThixotropicCoeffs
{
a 1;
b 2;
c 1e3;
d 3;
nu0 0.1;
nuInf 1e4;
}
}