oomph::HerschelBulkleyTanMilRegWithBlendingConstitutiveEquation< DIM > Class Template Reference

#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 More...
double	calculate_cutoff_viscosity ()
double	calculate_zero_shear_viscosity ()
	Function that calculates the viscosity at zero I2. More...
void	report_cut_off_values (double &cut_off_invariant, double &cut_off_viscosity, double &zero_shear_viscosity)
	Report cutoff values. More...
double	viscosity (const double &second_invariant_of_rate_of_strain_tensor)
	Viscosity ratio as a fct of strain rate invariant. More...
double	dviscosity_dinvariant (const double &second_invariant_of_rate_of_strain_tensor)
	Deriv of viscosity w.r.t. strain rate invariant. More...
Public Member Functions inherited from oomph::GeneralisedNewtonianConstitutiveEquation< DIM >
	GeneralisedNewtonianConstitutiveEquation ()
	Empty constructor. More...
virtual	~GeneralisedNewtonianConstitutiveEquation ()
	Empty virtual destructor. More...

Private Attributes
double *	Yield_stress_pt
	yield stress tau_y More...
double *	Flow_index_pt
	power law index n More...
double *	Critical_second_invariant_pt
double	a
double	b
double	c
double	alpha

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

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

      : GeneralisedNewtonianConstitutiveEquation<DIM>(),
        Yield_stress_pt(yield_stress_pt),
        Flow_index_pt(flow_index_pt),
        Critical_second_invariant_pt(critical_second_invariant_pt),
        a(0.0),
        b(0.0),
        c(0.0)
    {
      alpha = 0.1;
 
      if (fabs(*Critical_second_invariant_pt) > 0.0)
      {
        a = (pow(2.0, *Flow_index_pt - 3.0) * (*Flow_index_pt - 1.0) *
               pow(fabs(*Critical_second_invariant_pt),
                   (*Flow_index_pt - 1.0) / 2.0 - 2.0) -
             (*Yield_stress_pt) /
               (8.0 * pow(fabs(*Critical_second_invariant_pt), 5.0 / 2.0))) /
            (1.0 - alpha);
        b = -2.0 * a * alpha * fabs(*Critical_second_invariant_pt);
        c = (*Yield_stress_pt) /
              (2.0 * sqrt(fabs(*Critical_second_invariant_pt))) +
            pow(2.0 * sqrt(fabs(*Critical_second_invariant_pt)),
                *Flow_index_pt - 1.0) -
            a * pow(fabs(*Critical_second_invariant_pt), 2.0) -
            b * fabs(*Critical_second_invariant_pt);
      }
 
      double cut_off_viscosity = calculate_cutoff_viscosity();
 
      double zero_shear_viscosity = calculate_zero_shear_viscosity();
 
      oomph_info << "HerschelBulkleyTanMilRegWithBlendingConstitutiveEquation: "
                 << " zero shear viscosity = " << zero_shear_viscosity
                 << std::endl;
 
      oomph_info << "HerschelBulkleyTanMilRegWithBlendingConstitutiveEquation: "
                 << " cut off viscosity = " << cut_off_viscosity << std::endl;
 
      oomph_info << "                                              "
                 << " cutoff invariant = " << *Critical_second_invariant_pt
                 << std::endl;
    }

References oomph::HerschelBulkleyTanMilRegWithBlendingConstitutiveEquation< DIM >::a, oomph::HerschelBulkleyTanMilRegWithBlendingConstitutiveEquation< DIM >::alpha, oomph::HerschelBulkleyTanMilRegWithBlendingConstitutiveEquation< DIM >::b, oomph::HerschelBulkleyTanMilRegWithBlendingConstitutiveEquation< DIM >::c, oomph::HerschelBulkleyTanMilRegWithBlendingConstitutiveEquation< DIM >::calculate_cutoff_viscosity(), oomph::HerschelBulkleyTanMilRegWithBlendingConstitutiveEquation< DIM >::calculate_zero_shear_viscosity(), oomph::HerschelBulkleyTanMilRegWithBlendingConstitutiveEquation< DIM >::Critical_second_invariant_pt, boost::multiprecision::fabs(), oomph::HerschelBulkleyTanMilRegWithBlendingConstitutiveEquation< DIM >::Flow_index_pt, oomph::oomph_info, Eigen::bfloat16_impl::pow(), and sqrt().

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

    {
      // Calculate the Newtonian cutoff viscosity from the constitutive
      // equation and the cutoff value of the second invariant
      return (*Yield_stress_pt) /
               (2.0 * sqrt(std::max(fabs(*Critical_second_invariant_pt),
                                    DBL_EPSILON))) +
             pow((2.0 * sqrt(std::max(fabs(*Critical_second_invariant_pt),
                                      DBL_EPSILON))),
                 *Flow_index_pt - 1.0);
    }

References oomph::HerschelBulkleyTanMilRegWithBlendingConstitutiveEquation< DIM >::Critical_second_invariant_pt, boost::multiprecision::fabs(), oomph::HerschelBulkleyTanMilRegWithBlendingConstitutiveEquation< DIM >::Flow_index_pt, max, Eigen::bfloat16_impl::pow(), sqrt(), 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.

    {
      // The maximum of either the viscosity at alpha*cut-off or the cut-off
      // viscosity for a cut-off of zero
      return std::max(a * pow(alpha, 2.0) *
                          pow(fabs(*Critical_second_invariant_pt), 2.0) +
                        b * alpha * fabs(*Critical_second_invariant_pt) + c,
                      calculate_cutoff_viscosity());
    }

References oomph::HerschelBulkleyTanMilRegWithBlendingConstitutiveEquation< DIM >::a, oomph::HerschelBulkleyTanMilRegWithBlendingConstitutiveEquation< DIM >::alpha, oomph::HerschelBulkleyTanMilRegWithBlendingConstitutiveEquation< DIM >::b, oomph::HerschelBulkleyTanMilRegWithBlendingConstitutiveEquation< DIM >::c, oomph::HerschelBulkleyTanMilRegWithBlendingConstitutiveEquation< DIM >::calculate_cutoff_viscosity(), oomph::HerschelBulkleyTanMilRegWithBlendingConstitutiveEquation< DIM >::Critical_second_invariant_pt, boost::multiprecision::fabs(), max, and Eigen::bfloat16_impl::pow().

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 >.

    {
      if (fabs(second_invariant_of_rate_of_strain_tensor) <
          alpha * fabs(*Critical_second_invariant_pt))
      {
        return 0.0;
      }
      else if (fabs(second_invariant_of_rate_of_strain_tensor) <
               fabs(*Critical_second_invariant_pt))
      {
        return 2.0 * a * fabs(second_invariant_of_rate_of_strain_tensor) + b;
      }
 
      return pow(2.0, *Flow_index_pt - 2.0) * (*Flow_index_pt - 1.0) *
               pow(fabs(second_invariant_of_rate_of_strain_tensor),
                   (*Flow_index_pt - 1.0) / 2.0 - 1.0) -
             (*Yield_stress_pt) /
               (4.0 * pow(fabs(second_invariant_of_rate_of_strain_tensor),
                          3.0 / 2.0));
    }

References oomph::HerschelBulkleyTanMilRegWithBlendingConstitutiveEquation< DIM >::a, oomph::HerschelBulkleyTanMilRegWithBlendingConstitutiveEquation< DIM >::alpha, oomph::HerschelBulkleyTanMilRegWithBlendingConstitutiveEquation< DIM >::b, oomph::HerschelBulkleyTanMilRegWithBlendingConstitutiveEquation< DIM >::Critical_second_invariant_pt, boost::multiprecision::fabs(), oomph::HerschelBulkleyTanMilRegWithBlendingConstitutiveEquation< DIM >::Flow_index_pt, and Eigen::bfloat16_impl::pow().

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.

    {
      cut_off_invariant = *Critical_second_invariant_pt;
      cut_off_viscosity = calculate_cutoff_viscosity();
      zero_shear_viscosity = calculate_zero_shear_viscosity();
    }

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 >.

    {
      if (fabs(second_invariant_of_rate_of_strain_tensor) <
          alpha * fabs(*Critical_second_invariant_pt))
      {
        return calculate_zero_shear_viscosity();
      }
      else if (fabs(second_invariant_of_rate_of_strain_tensor) <
               fabs(*Critical_second_invariant_pt))
      {
        return a * pow(fabs(second_invariant_of_rate_of_strain_tensor), 2.0) +
               b * fabs(second_invariant_of_rate_of_strain_tensor) + c;
      }
 
      return (*Yield_stress_pt) /
               (2.0 * sqrt(fabs(second_invariant_of_rate_of_strain_tensor))) +
             pow(2.0 * sqrt(fabs(second_invariant_of_rate_of_strain_tensor)),
                 *Flow_index_pt - 1.0);
    }

References oomph::HerschelBulkleyTanMilRegWithBlendingConstitutiveEquation< DIM >::a, oomph::HerschelBulkleyTanMilRegWithBlendingConstitutiveEquation< DIM >::alpha, oomph::HerschelBulkleyTanMilRegWithBlendingConstitutiveEquation< DIM >::b, oomph::HerschelBulkleyTanMilRegWithBlendingConstitutiveEquation< DIM >::c, oomph::HerschelBulkleyTanMilRegWithBlendingConstitutiveEquation< DIM >::calculate_zero_shear_viscosity(), oomph::HerschelBulkleyTanMilRegWithBlendingConstitutiveEquation< DIM >::Critical_second_invariant_pt, boost::multiprecision::fabs(), oomph::HerschelBulkleyTanMilRegWithBlendingConstitutiveEquation< DIM >::Flow_index_pt, Eigen::bfloat16_impl::pow(), sqrt(), and oomph::HerschelBulkleyTanMilRegWithBlendingConstitutiveEquation< DIM >::Yield_stress_pt.

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

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

alpha

template<unsigned DIM>

double oomph::HerschelBulkleyTanMilRegWithBlendingConstitutiveEquation< DIM >::alpha

private

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

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

b

template<unsigned DIM>

double oomph::HerschelBulkleyTanMilRegWithBlendingConstitutiveEquation< DIM >::b

private

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

c

template<unsigned DIM>

double oomph::HerschelBulkleyTanMilRegWithBlendingConstitutiveEquation< DIM >::c

private

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

Critical_second_invariant_pt

template<unsigned DIM>

double* oomph::HerschelBulkleyTanMilRegWithBlendingConstitutiveEquation< DIM >::Critical_second_invariant_pt

private

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

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

Flow_index_pt

template<unsigned DIM>

double* oomph::HerschelBulkleyTanMilRegWithBlendingConstitutiveEquation< DIM >::Flow_index_pt

private

power law index n

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

Yield_stress_pt

template<unsigned DIM>

double* oomph::HerschelBulkleyTanMilRegWithBlendingConstitutiveEquation< DIM >::Yield_stress_pt

private

yield stress tau_y

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

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The documentation for this class was generated from the following file:

• generalised_newtonian_constitutive_models.h