Regularized HerschelBulkley model is a fourparameter model and it is given by the following equation.
This model describes the viscosity using a powerlaw relationship and a yield criterion. The exponent n determines the nature of the power law relationship and the yield criterion is implemented using the regularized model with yield stress t0 and regularization parameter m. Higher the value of m, closer the approximation to the actual yield stress behavior of HerschelBulkley model. For practical purposes, value of m=10000 should be suffice. This regularization parameter controls the exponential growth of stress. The original HerschelBulkley model, which is given by the following equation, suffers from the disadvantage of stress growing very large when the shear rate tends to zero. This will lead to numerical instability and also increase the difficulty of obtaining a converged solution. Hence, this model is regularized using the parameter m.
In addition to the above data, few additional parameters arise due the temperature dependence of material data; these variables are discussed in Section titled Temperature Dependence.
Syntax
Syntax of the data packet Polymer is as follows:
Polymer 
Polymer_name 
{ 

ConstitutiveModel = 
"HerschelBulkley" 

Density = 
ρ 

SpecificHeat = 
Cp(T) 

Conductivity = 
K(T) 

CoeffOfThermalExpansion = 
βΤ 

VolumetricHeatSource = 
Qvol 

Consistency = 
A 

Exponent = 
n 

YieldStress = 
τ0 

StressExponentGrowth = 
m 

TemperatureDependence = 
"None"} 
Parameter 
Description 
Units 
Data Type 
Condition 
Typical Value 
ConstitutiveModel 
Describes the model used 
None 
String 
Required 
"HerschelBulkely" 
Density 
Density of the polymer 
kg/m^3 
Constant 
Required 
995.0 
SpecificHeat 
Specific heat at constant pressure 
J/kg/K 
Constant / F(T) 
Required 
2000.0 
Conductivity 
Thermal conductivity 
W/m/K 
Constant / F(T) 
Required 
0.167 
CoeffOfThermalExpansion 
Indicates the change in volume with change in temperature 
1/K 
Constant 
Required 
1.0e05 
VolumetricHeatSource 
Heat generated/ removed in the volume by methods like electrical heating 
W/m^3 
Constant 
Required 
0.0 
Consistency 
One of the parameters of the Herschel—Bulkley model. When n=1 it is same as viscosity. 
Pa s^n 
Constant 
Required 
1.0e+04 
Exponent 
Power law index, defines the dependency of viscosity on shear rate. 
None 
Constant 
Required 
0.66 
ZeroShearRateLimit 
See Power law model 
1/s 
Constant 
Required 
0.01 
YieldStress 
One of the parameters of the Herschel—Bulkley model. 
Pa 
Constant 
Required 
1.0E+05 
StressExponentGrowth 
One of the parameters of the Herschel—Bulkley model. It is also known as the regularization parameter. It controls the exponential growth of stress 
s 
Constant 
Required 

TemperatureDependence 
None 
String 
Required 
"Exp(Beta(DeltaT))" 

ReferenceTemperature 
Temperature at which data is calculated for the initialization step. 
K 
Constant 
Required only if TD is not "None" 
533 
FreezeTemperature 
This is the no flow temperature. Below this temperature, material ceases to flow. 
K 
Constant 
Required only if TD is not "None" 
350 
ActivationEnergy 
A parameter required by Arrhenius model. 
J/mol 
Constant 
Required only if TD is Exp(Q/RT) 
16628 
UniversalGasConstant 
A parameter from state equation PV = nRT, R is universal Gas constant. 
J/mol/K 
Constant 
Required only if TD is Exp(Q/RT) 
8.314 
TemperatureSensitivity 
A derived parameter which has the same physical meaning as Q/R. 
K 
Constant 
Required only if TD is Exp(Tb/T) 
2000 K 
WLFConstant1 
Constant C1 of WLF model 
None 
Constant 
Required only if TD is WLF 
17.44 
WLFConstant2 
Constant C2 of WLF model. This is like DeltaT, hence the value is same in K and Celsius. 
K 
Constant 
Required only if TD is WLF 
51.6 
GlassTransitionTemperature 
Temperature below with polymer molecules ceases to move (frozen). There are few definitions of this term. 
K 
Constant 
Required only if TD is WLF 
320 
Beta 
Parameter in the relationship Exp(Beta(DeltaT)) 
None 
Constant 
Required only if TD is Exp(Beta(DeltaT)) 
0.005 
F(T)  Function of Temperature. Can be specified as a TABLE1 or TCL function.
TD  TemperatureDependence