A GeneralisedNewtonianConstitutiveEquation class defining a Herschel-Bulkley fluid using Tanner and Milthorpe's (1983) regularisation with a smooth transition using a quadratic. More...

#include <generalised_newtonian_constitutive_models.h>

Inheritance diagram for oomph::HerschelBulkleyTanMilRegWithBlendingConstitutiveEquation< DIM >:

Public Member Functions
	HerschelBulkleyTanMilRegWithBlendingConstitutiveEquation (double yield_stress_pt, double flow_index_pt, double *critical_second_invariant_pt)
	"Cutoff regularised" Herschel Bulkley constitutive equation

double	calculate_cutoff_viscosity ()
	Function that calculates the viscosity at the cutoff invariant Note: this is NOT the viscosity at zero I2.

double	calculate_zero_shear_viscosity ()
	Function that calculates the viscosity at zero I2.

void	report_cut_off_values (double &cut_off_invariant, double &cut_off_viscosity, double &zero_shear_viscosity)
	Report cutoff values.

double	viscosity (const double &second_invariant_of_rate_of_strain_tensor)
	Viscosity ratio as a fct of strain rate invariant.

double	dviscosity_dinvariant (const double &second_invariant_of_rate_of_strain_tensor)
	Deriv of viscosity w.r.t. strain rate invariant.

Public Member Functions inherited from oomph::GeneralisedNewtonianConstitutiveEquation< DIM >
	GeneralisedNewtonianConstitutiveEquation ()
	Empty constructor.

virtual	~GeneralisedNewtonianConstitutiveEquation ()
	Empty virtual destructor.

Private Attributes
double *	Yield_stress_pt
	yield stress tau_y

double *	Flow_index_pt
	power law index n

double *	Critical_second_invariant_pt
	value of the second invariant below which we have constant (Newtonian) viscosity – assumed to be always positive

double	a
	We use a quadratic function to smoothly blend from the Herschel Bulkley model at the cut-off to a constant viscosity as the strain rate /approaches zero; this way we avoid the discontinuity of the gradient at the cut-off, which is present in the classic Tanner Milthorpe regularisation.

double	b

double	c

double	alpha
	Fraction of the cut-off strain rate below which the viscosity is constant 0 <= alpha < 1.

Detailed Description

template<unsigned DIM>
class oomph::HerschelBulkleyTanMilRegWithBlendingConstitutiveEquation< DIM >

A GeneralisedNewtonianConstitutiveEquation class defining a Herschel-Bulkley fluid using Tanner and Milthorpe's (1983) regularisation with a smooth transition using a quadratic.

Definition at line 358 of file generalised_newtonian_constitutive_models.h.

Constructor & Destructor Documentation

◆ HerschelBulkleyTanMilRegWithBlendingConstitutiveEquation()

template<unsigned DIM>

oomph::HerschelBulkleyTanMilRegWithBlendingConstitutiveEquation< DIM >::HerschelBulkleyTanMilRegWithBlendingConstitutiveEquation	(	double *	yield_stress_pt,
		double *	flow_index_pt,
		double *	critical_second_invariant_pt
	)

inline

"Cutoff regularised" Herschel Bulkley constitutive equation

Blend over one order of magnitude

Calculate the coefficients of the quadratic equation

get the cutoff viscosity

get the zero shear viscosity

Definition at line 388 of file generalised_newtonian_constitutive_models.h.

Member Function Documentation

◆ calculate_cutoff_viscosity()

template<unsigned DIM>

double oomph::HerschelBulkleyTanMilRegWithBlendingConstitutiveEquation< DIM >::calculate_cutoff_viscosity ( )

inline

Function that calculates the viscosity at the cutoff invariant Note: this is NOT the viscosity at zero I2.

Definition at line 441 of file generalised_newtonian_constitutive_models.h.

References oomph::HerschelBulkleyTanMilRegWithBlendingConstitutiveEquation< DIM >::Critical_second_invariant_pt, oomph::HerschelBulkleyTanMilRegWithBlendingConstitutiveEquation< DIM >::Flow_index_pt, and oomph::HerschelBulkleyTanMilRegWithBlendingConstitutiveEquation< DIM >::Yield_stress_pt.

Referenced by oomph::HerschelBulkleyTanMilRegWithBlendingConstitutiveEquation< DIM >::calculate_zero_shear_viscosity(), oomph::HerschelBulkleyTanMilRegWithBlendingConstitutiveEquation< DIM >::HerschelBulkleyTanMilRegWithBlendingConstitutiveEquation(), and oomph::HerschelBulkleyTanMilRegWithBlendingConstitutiveEquation< DIM >::report_cut_off_values().

◆ calculate_zero_shear_viscosity()

template<unsigned DIM>

double oomph::HerschelBulkleyTanMilRegWithBlendingConstitutiveEquation< DIM >::calculate_zero_shear_viscosity ( )

inline

Function that calculates the viscosity at zero I2.

Definition at line 454 of file generalised_newtonian_constitutive_models.h.

References oomph::HerschelBulkleyTanMilRegWithBlendingConstitutiveEquation< DIM >::a, oomph::HerschelBulkleyTanMilRegWithBlendingConstitutiveEquation< DIM >::alpha, oomph::HerschelBulkleyTanMilRegWithBlendingConstitutiveEquation< DIM >::b, oomph::HerschelBulkleyTanMilRegWithBlendingConstitutiveEquation< DIM >::c, oomph::HerschelBulkleyTanMilRegWithBlendingConstitutiveEquation< DIM >::calculate_cutoff_viscosity(), and oomph::HerschelBulkleyTanMilRegWithBlendingConstitutiveEquation< DIM >::Critical_second_invariant_pt.

Referenced by oomph::HerschelBulkleyTanMilRegWithBlendingConstitutiveEquation< DIM >::HerschelBulkleyTanMilRegWithBlendingConstitutiveEquation(), oomph::HerschelBulkleyTanMilRegWithBlendingConstitutiveEquation< DIM >::report_cut_off_values(), and oomph::HerschelBulkleyTanMilRegWithBlendingConstitutiveEquation< DIM >::viscosity().

◆ dviscosity_dinvariant()

template<unsigned DIM>

double oomph::HerschelBulkleyTanMilRegWithBlendingConstitutiveEquation< DIM >::dviscosity_dinvariant ( const double & second_invariant_of_rate_of_strain_tensor )

inlinevirtual

Deriv of viscosity w.r.t. strain rate invariant.

Implements oomph::GeneralisedNewtonianConstitutiveEquation< DIM >.

Definition at line 496 of file generalised_newtonian_constitutive_models.h.

References oomph::HerschelBulkleyTanMilRegWithBlendingConstitutiveEquation< DIM >::a, oomph::HerschelBulkleyTanMilRegWithBlendingConstitutiveEquation< DIM >::alpha, oomph::HerschelBulkleyTanMilRegWithBlendingConstitutiveEquation< DIM >::b, oomph::HerschelBulkleyTanMilRegWithBlendingConstitutiveEquation< DIM >::Critical_second_invariant_pt, and oomph::HerschelBulkleyTanMilRegWithBlendingConstitutiveEquation< DIM >::Flow_index_pt.

◆ report_cut_off_values()

template<unsigned DIM>

void oomph::HerschelBulkleyTanMilRegWithBlendingConstitutiveEquation< DIM >::report_cut_off_values	(	double &	cut_off_invariant,
		double &	cut_off_viscosity,
		double &	zero_shear_viscosity
	)

inline

Report cutoff values.

Definition at line 465 of file generalised_newtonian_constitutive_models.h.

References oomph::HerschelBulkleyTanMilRegWithBlendingConstitutiveEquation< DIM >::calculate_cutoff_viscosity(), oomph::HerschelBulkleyTanMilRegWithBlendingConstitutiveEquation< DIM >::calculate_zero_shear_viscosity(), and oomph::HerschelBulkleyTanMilRegWithBlendingConstitutiveEquation< DIM >::Critical_second_invariant_pt.

◆ viscosity()

template<unsigned DIM>

double oomph::HerschelBulkleyTanMilRegWithBlendingConstitutiveEquation< DIM >::viscosity ( const double & second_invariant_of_rate_of_strain_tensor )

inlinevirtual

Viscosity ratio as a fct of strain rate invariant.

Implements oomph::GeneralisedNewtonianConstitutiveEquation< DIM >.

Definition at line 475 of file generalised_newtonian_constitutive_models.h.

Member Data Documentation

◆ a

template<unsigned DIM>

double oomph::HerschelBulkleyTanMilRegWithBlendingConstitutiveEquation< DIM >::a

private

We use a quadratic function to smoothly blend from the Herschel Bulkley model at the cut-off to a constant viscosity as the strain rate /approaches zero; this way we avoid the discontinuity of the gradient at the cut-off, which is present in the classic Tanner Milthorpe regularisation.

Definition at line 377 of file generalised_newtonian_constitutive_models.h.

Referenced by oomph::HerschelBulkleyTanMilRegWithBlendingConstitutiveEquation< DIM >::calculate_zero_shear_viscosity(), oomph::HerschelBulkleyTanMilRegWithBlendingConstitutiveEquation< DIM >::dviscosity_dinvariant(), oomph::HerschelBulkleyTanMilRegWithBlendingConstitutiveEquation< DIM >::HerschelBulkleyTanMilRegWithBlendingConstitutiveEquation(), and oomph::HerschelBulkleyTanMilRegWithBlendingConstitutiveEquation< DIM >::viscosity().