generalised_newtonian_axisym_navier_stokes_elements.h

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// Header file for Navier Stokes elements
 
#ifndef OOMPH_GENERALISED_NEWTONIAN_AXISYMMETRIC_NAVIER_STOKES_ELEMENTS_HEADER
#define OOMPH_GENERALISED_NEWTONIAN_AXISYMMETRIC_NAVIER_STOKES_ELEMENTS_HEADER
 
// Config header
#ifdef HAVE_CONFIG_H
#include <oomph-lib-config.h>
#endif
 
// OOMPH-LIB headers
// #include "generic.h"
 
#include "generic/Qelements.h"
#include "generic/fsi.h"
#include "generic/projection.h"
#include "generic/generalised_newtonian_constitutive_models.h"
 
namespace oomph
{
  //======================================================================
  /// A class for elements that solve the unsteady
  /// axisymmetric Navier--Stokes equations in
  /// cylindrical polar coordinates, \f$ x_0^* = r^*\f$ and \f$ x_1^* = z^* \f$
  /// with \f$ \partial / \partial \theta = 0 \f$. We're solving for the
  /// radial, axial and azimuthal (swirl) velocities,
  /// \f$ u_0^* = u_r^*(r^*,z^*,t^*) = u^*(r^*,z^*,t^*), \ u_1^* = u_z^*(r^*,z^*,t^*) = w^*(r^*,z^*,t^*)\f$
  /// and
  /// \f$ u_2^* = u_\theta^*(r^*,z^*,t^*) = v^*(r^*,z^*,t^*) \f$,
  /// respectively, and the pressure \f$ p(r^*,z^*,t^*) \f$.
  /// This class contains the generic maths -- any concrete
  /// implementation must be derived from this.
  ///
  /// In dimensional form the axisymmetric Navier-Stokes equations are given
  /// by the momentum equations (for the \f$ r^* \f$, \f$ z^* \f$ and \f$ \theta \f$
  /// directions, respectively)
  /// \f[ \rho\left(\frac{\partial u^*}{\partial t^*} + {u^*}\frac{\partial u^*}{\partial r^*} - \frac{{v^*}^2}{r^*} + {w^*}\frac{\partial u^*}{\partial z^*} \right) = B_r^*\left(r^*,z^*,t^*\right)+ \rho G_r^*+ \frac{1}{r^*} \frac{\partial\left({r^*}\sigma_{rr}^*\right)}{\partial r^*} - \frac{\sigma_{\theta\theta}^*}{r^*} + \frac{\partial\sigma_{rz}^*}{\partial z^*}, \f]
  /// \f[ \rho\left(\frac{\partial w^*}{\partial t^*} + {u^*}\frac{\partial w^*}{\partial r^*} + {w^*}\frac{\partial w^*}{\partial z^*} \right) = B_z^*\left(r^*,z^*,t^*\right)+\rho G_z^*+ \frac{1}{r^*}\frac{\partial\left({r^*}\sigma_{zr}^*\right)}{\partial r^*} + \frac{\partial\sigma_{zz}^*}{\partial z^*}, \f]
  /// \f[ \rho\left(\frac{\partial v^*}{\partial t^*} + {u^*}\frac{\partial v^*}{\partial r^*} + \frac{u^* v^*}{r^*} +{w^*}\frac{\partial v^*}{\partial z^*} \right)= B_\theta^*\left(r^*,z^*,t^*\right)+ \rho G_\theta^*+ \frac{1}{r^*}\frac{\partial\left({r^*}\sigma_{\theta r}^*\right)}{\partial r^*} + \frac{\sigma_{r\theta}^*}{r^*} + \frac{\partial\sigma_{\theta z}^*}{\partial z^*}, \f]
  /// and
  /// \f[ \frac{1}{r^*}\frac{\partial\left(r^*u^*\right)}{\partial r^*} + \frac{\partial w^*}{\partial z^*} = Q^*. \f]
  /// The dimensional, symmetric stress tensor is defined as:
  /// \f[ \sigma_{rr}^* = -p^* + 2\mu\frac{\partial u^*}{\partial r^*}, \qquad \sigma_{\theta\theta}^* = -p^* +2\mu\frac{u^*}{r^*}, \f]
  /// \f[ \sigma_{zz}^* = -p^* + 2\mu\frac{\partial w^*}{\partial z^*}, \qquad \sigma_{rz}^* = \mu\left(\frac{\partial u^*}{\partial z^*} + \frac{\partial w^*}{\partial r^*}\right), \f]
  /// \f[ \sigma_{\theta r}^* = \mu r^*\frac{\partial}{\partial r^*} \left(\frac{v^*}{r^*}\right), \qquad \sigma_{\theta z}^* = \mu\frac{\partial v^*}{\partial z^*}. \f]
  /// Here, the (dimensional) velocity components are denoted
  /// by \f$ u^* \f$, \f$ w^* \f$
  /// and \f$ v^* \f$ for the radial, axial and azimuthal velocities,
  /// respectively, and we
  /// have split the body force into two components: A constant
  /// vector \f$ \rho \ G_i^* \f$ which typically represents gravitational
  /// forces; and a variable body force, \f$ B_i^*(r^*,z^*,t^*) \f$.
  /// \f$ Q^*(r^*,z^*,t^*) \f$ is a volumetric source term for the
  /// continuity equation and is typically equal to zero.
  /// \n\n
  /// We non-dimensionalise the equations, using problem-specific reference
  /// quantities for the velocity, \f$ U \f$, length, \f$ L \f$, and time,
  /// \f$ T \f$, and scale the constant body force vector on the
  /// gravitational acceleration, \f$ g \f$, so that
  /// \f[ u^* = U\, u, \qquad w^* = U\, w, \qquad v^* = U\, v, \f]
  /// \f[ r^* = L\, r, \qquad z^* = L\, z, \qquad t^* = T\, t, \f]
  /// \f[ G_i^* = g\, G_i, \qquad B_i^* = \frac{U\mu_{ref}}{L^2}\, B_i, \qquad p^* = \frac{\mu_{ref} U}{L}\, p, \qquad Q^* = \frac{U}{L}\, Q. \f]
  /// where we note that the pressure and the variable body force have
  /// been non-dimensionalised on the viscous scale. \f$ \mu_{ref} \f$
  /// and \f$ \rho_{ref} \f$ (used below) are reference values
  /// for the fluid viscosity and density, respectively. In single-fluid
  /// problems, they are identical to the viscosity \f$ \mu \f$ and
  /// density \f$ \rho \f$ of the (one and only) fluid in the problem.
  /// \n\n
  /// The non-dimensional form of the axisymmetric Navier-Stokes equations
  /// is then given by
  /// \f[ R_{\rho} Re\left(St\frac{\partial u}{\partial t} + {u}\frac{\partial u}{\partial r} - \frac{{v}^2}{r} + {w}\frac{\partial u}{\partial z} \right) = B_r\left(r,z,t\right)+ R_\rho \frac{Re}{Fr} G_r + \frac{1}{r} \frac{\partial\left({r}\sigma_{rr}\right)}{\partial r} - \frac{\sigma_{\theta\theta}}{r} + \frac{\partial\sigma_{rz}}{\partial z}, \f]
  /// \f[ R_{\rho} Re\left(St\frac{\partial w}{\partial t} + {u}\frac{\partial w}{\partial r} + {w}\frac{\partial w}{\partial z} \right) = B_z\left(r,z,t\right)+ R_\rho \frac{Re}{Fr} G_z+ \frac{1}{r}\frac{\partial\left({r}\sigma_{zr}\right)}{\partial r} + \frac{\partial\sigma_{zz}}{\partial z}, \f]
  /// \f[ R_{\rho} Re\left(St\frac{\partial v}{\partial t} + {u}\frac{\partial v}{\partial r} + \frac{u v}{r} +{w}\frac{\partial v}{\partial z} \right)= B_\theta\left(r,z,t\right)+ R_\rho \frac{Re}{Fr} G_\theta+ \frac{1}{r}\frac{\partial\left({r}\sigma_{\theta r}\right)}{\partial r} + \frac{\sigma_{r\theta}}{r} + \frac{\partial\sigma_{\theta z}}{\partial z}, \f]
  /// and
  /// \f[ \frac{1}{r}\frac{\partial\left(ru\right)}{\partial r} + \frac{\partial w}{\partial z} = Q. \f]
  /// Here the non-dimensional, symmetric stress tensor is defined as:
  /// \f[ \sigma_{rr} = -p + 2R_\mu \frac{\partial u}{\partial r}, \qquad \sigma_{\theta\theta} = -p +2R_\mu \frac{u}{r}, \f]
  /// \f[ \sigma_{zz} = -p + 2R_\mu \frac{\partial w}{\partial z}, \qquad \sigma_{rz} = R_\mu \left(\frac{\partial u}{\partial z} + \frac{\partial w}{\partial r}\right), \f]
  /// \f[ \sigma_{\theta r} = R_\mu r \frac{\partial}{\partial r}\left(\frac{v}{r}\right), \qquad \sigma_{\theta z} = R_\mu \frac{\partial v}{\partial z}. \f]
  /// and the dimensionless parameters
  /// \f[ Re = \frac{UL\rho_{ref}}{\mu_{ref}}, \qquad St = \frac{L}{UT}, \qquad Fr = \frac{U^2}{gL}, \f]
  /// are the Reynolds number, Strouhal number and Froude number
  /// respectively. \f$ R_\rho=\rho/\rho_{ref} \f$ and
  /// \f$ R_\mu(T) =\mu(T)/\mu_{ref}\f$ represent the ratios
  /// of the fluid's density and its dynamic viscosity, relative to the
  /// density and viscosity values used to form the non-dimensional
  /// parameters (By default, \f$ R_\rho = R_\mu = 1 \f$; other values
  /// tend to be used in problems involving multiple fluids).
  //======================================================================
  class GeneralisedNewtonianAxisymmetricNavierStokesEquations
    : public virtual FiniteElement,
      public virtual NavierStokesElementWithDiagonalMassMatrices
  {
  private:
    /// Static "magic" number that indicates that the pressure is
    /// not stored at a node
    static int Pressure_not_stored_at_node;
 
    /// Static default value for the physical constants (all initialised to
    /// zero)
    static double Default_Physical_Constant_Value;
 
    /// Static default value for the physical ratios (all are initialised to
    /// one)
    static double Default_Physical_Ratio_Value;
 
    /// Static default value for the gravity vector
    static Vector<double> Default_Gravity_vector;
 
  protected:
    /// Static boolean telling us whether we pre-multiply the pressure and
    /// continuity by the viscosity ratio
    static bool Pre_multiply_by_viscosity_ratio;
 
    // Physical constants
 
    /// Pointer to the viscosity ratio (relative to the
    /// viscosity used in the definition of the Reynolds number)
    double* Viscosity_Ratio_pt;
 
    /// Pointer to the density ratio (relative to the density used in the
    /// definition of the Reynolds number)
    double* Density_Ratio_pt;
 
    // Pointers to global physical constants
 
    /// Pointer to global Reynolds number
    double* Re_pt;
 
    /// Pointer to global Reynolds number x Strouhal number (=Womersley)
    double* ReSt_pt;
 
    /// Pointer to global Reynolds number x inverse Froude number
    /// (= Bond number / Capillary number)
    double* ReInvFr_pt;
 
    /// Pointer to global Reynolds number x inverse Rossby number
    /// (used when in a rotating frame)
    double* ReInvRo_pt;
 
    /// Pointer to global gravity Vector
    Vector<double>* G_pt;
 
    /// Pointer to body force function
    void (*Body_force_fct_pt)(const double& time,
                              const Vector<double>& x,
                              Vector<double>& result);
 
    /// Pointer to volumetric source function
    double (*Source_fct_pt)(const double& time, const Vector<double>& x);
 
    /// Pointer to the generalised Newtonian constitutive equation
    GeneralisedNewtonianConstitutiveEquation<3>* Constitutive_eqn_pt;
 
    // Boolean that indicates whether we use the extrapolated strain rate
    // for calculation of viscosity or not
    bool Use_extrapolated_strainrate_to_compute_second_invariant;
 
    /// Boolean flag to indicate if ALE formulation is disabled when
    /// the time-derivatives are computed. Only set to true if you're sure
    /// that the mesh is stationary
    bool ALE_is_disabled;
 
    /// Access function for the local equation number information for
    /// the pressure.
    /// p_local_eqn[n] = local equation number or < 0 if pinned
    virtual int p_local_eqn(const unsigned& n) const = 0;
 
    /// Compute the shape functions and derivatives
    /// w.r.t. global coords at local coordinate s.
    /// Return Jacobian of mapping between local and global coordinates.
    virtual double dshape_and_dtest_eulerian_axi_nst(const Vector<double>& s,
                                                     Shape& psi,
                                                     DShape& dpsidx,
                                                     Shape& test,
                                                     DShape& dtestdx) const = 0;
 
    /// Compute the shape functions and derivatives
    /// w.r.t. global coords at ipt-th integration point
    /// Return Jacobian of mapping between local and global coordinates.
    virtual double dshape_and_dtest_eulerian_at_knot_axi_nst(
      const unsigned& ipt,
      Shape& psi,
      DShape& dpsidx,
      Shape& test,
      DShape& dtestdx) const = 0;
 
    /// Shape/test functions and derivs w.r.t. to global coords at
    /// integration point ipt; return Jacobian of mapping (J). Also compute
    /// derivatives of dpsidx, dtestdx and J w.r.t. nodal coordinates.
    virtual double dshape_and_dtest_eulerian_at_knot_axi_nst(
      const unsigned& ipt,
      Shape& psi,
      DShape& dpsidx,
      RankFourTensor<double>& d_dpsidx_dX,
      Shape& test,
      DShape& dtestdx,
      RankFourTensor<double>& d_dtestdx_dX,
      DenseMatrix<double>& djacobian_dX) const = 0;
 
    /// Compute the pressure shape functions at local coordinate s
    virtual void pshape_axi_nst(const Vector<double>& s, Shape& psi) const = 0;
 
    /// Compute the pressure shape and test functions
    /// at local coordinate s
    virtual void pshape_axi_nst(const Vector<double>& s,
                                Shape& psi,
                                Shape& test) const = 0;
 
    /// Calculate the body force fct at a given time and Eulerian position
    virtual void get_body_force_axi_nst(const double& time,
                                        const unsigned& ipt,
                                        const Vector<double>& s,
                                        const Vector<double>& x,
                                        Vector<double>& result)
    {
      // If the function pointer is zero return zero
      if (Body_force_fct_pt == 0)
      {
        // Loop over dimensions and set body forces to zero
        for (unsigned i = 0; i < 3; i++)
        {
          result[i] = 0.0;
        }
      }
      // Otherwise call the function
      else
      {
        (*Body_force_fct_pt)(time, x, result);
      }
    }
 
    /// Get gradient of body force term at (Eulerian) position x.
    /// Computed via function pointer (if set) or by finite differencing
    /// (default)
    inline virtual void get_body_force_gradient_axi_nst(
      const double& time,
      const unsigned& ipt,
      const Vector<double>& s,
      const Vector<double>& x,
      DenseMatrix<double>& d_body_force_dx)
    {
      // hierher: Implement function pointer version
      /*    //If no gradient function has been set, FD it */
      /*    if(Body_force_fct_gradient_pt==0) */
      {
        // Reference value
        Vector<double> body_force(3, 0.0);
        get_body_force_axi_nst(time, ipt, s, x, body_force);
 
        // FD it
        const double eps_fd = GeneralisedElement::Default_fd_jacobian_step;
        Vector<double> body_force_pls(3, 0.0);
        Vector<double> x_pls(x);
        for (unsigned i = 0; i < 2; i++)
        {
          x_pls[i] += eps_fd;
          get_body_force_axi_nst(time, ipt, s, x_pls, body_force_pls);
          for (unsigned j = 0; j < 3; j++)
          {
            d_body_force_dx(j, i) =
              (body_force_pls[j] - body_force[j]) / eps_fd;
          }
          x_pls[i] = x[i];
        }
      }
      /*    else */
      /*     { */
      /*      // Get gradient */
      /*      (*Source_fct_gradient_pt)(time,x,gradient); */
      /*     } */
    }
 
    /// Calculate the source fct at given time and Eulerian position
    double get_source_fct(const double& time,
                          const unsigned& ipt,
                          const Vector<double>& x)
    {
      // If the function pointer is zero return zero
      if (Source_fct_pt == 0)
      {
        return 0;
      }
 
      // Otherwise call the function
      else
      {
        return (*Source_fct_pt)(time, x);
      }
    }
 
    /// Get gradient of source term at (Eulerian) position x. Computed
    /// via function pointer (if set) or by finite differencing (default)
    inline virtual void get_source_fct_gradient(const double& time,
                                                const unsigned& ipt,
                                                const Vector<double>& x,
                                                Vector<double>& gradient)
    {
      // hierher: Implement function pointer version
      /*    //If no gradient function has been set, FD it */
      /*    if(Source_fct_gradient_pt==0) */
      {
        // Reference value
        const double source = get_source_fct(time, ipt, x);
 
        // FD it
        const double eps_fd = GeneralisedElement::Default_fd_jacobian_step;
        double source_pls = 0.0;
        Vector<double> x_pls(x);
        for (unsigned i = 0; i < 2; i++)
        {
          x_pls[i] += eps_fd;
          source_pls = get_source_fct(time, ipt, x_pls);
          gradient[i] = (source_pls - source) / eps_fd;
          x_pls[i] = x[i];
        }
      }
      /*    else */
      /*     { */
      /*      // Get gradient */
      /*      (*Source_fct_gradient_pt)(time,x,gradient); */
      /*     } */
    }
 
    /// Calculate the viscosity ratio relative to the
    /// viscosity used in the definition of the Reynolds number
    /// at given time and Eulerian position
    /// This is the equivalent of the Newtonian case and
    /// basically allows the flow of two Generalised Newtonian fluids
    /// with different reference viscosities
    virtual void get_viscosity_ratio_axisym_nst(const unsigned& ipt,
                                                const Vector<double>& s,
                                                const Vector<double>& x,
                                                double& visc_ratio)
    {
      visc_ratio = *Viscosity_Ratio_pt;
    }
 
    /// Compute the residuals for the Navier--Stokes equations;
    /// flag=2 or 1 or 0: compute the Jacobian and/or mass matrix
    /// as well.
    virtual void fill_in_generic_residual_contribution_axi_nst(
      Vector<double>& residuals,
      DenseMatrix<double>& jacobian,
      DenseMatrix<double>& mass_matrix,
      unsigned flag);
 
    /// Compute the derivative of residuals for the
    /// Navier--Stokes equations; with respect to a parameeter
    /// flag=2 or 1 or 0: compute the Jacobian and/or mass matrix as well
    virtual void fill_in_generic_dresidual_contribution_axi_nst(
      double* const& parameter_pt,
      Vector<double>& dres_dparam,
      DenseMatrix<double>& djac_dparam,
      DenseMatrix<double>& dmass_matrix_dparam,
      unsigned flag);
 
    /// Compute the hessian tensor vector products required to
    /// perform continuation of bifurcations analytically
    void fill_in_contribution_to_hessian_vector_products(
      Vector<double> const& Y,
      DenseMatrix<double> const& C,
      DenseMatrix<double>& product);
 
  public:
    /// Constructor: NULL the body force and source function
    GeneralisedNewtonianAxisymmetricNavierStokesEquations()
      : Body_force_fct_pt(0),
        Source_fct_pt(0),
        Constitutive_eqn_pt(new NewtonianConstitutiveEquation<3>),
        Use_extrapolated_strainrate_to_compute_second_invariant(false),
        ALE_is_disabled(false)
    {
      // Set all the Physical parameter pointers to the default value zero
      Re_pt = &Default_Physical_Constant_Value;
      ReSt_pt = &Default_Physical_Constant_Value;
      ReInvFr_pt = &Default_Physical_Constant_Value;
      ReInvRo_pt = &Default_Physical_Constant_Value;
      G_pt = &Default_Gravity_vector;
      // Set the Physical ratios to the default value of 1
      Viscosity_Ratio_pt = &Default_Physical_Ratio_Value;
      Density_Ratio_pt = &Default_Physical_Ratio_Value;
    }
 
    /// Vector to decide whether the stress-divergence form is used or not
    // N.B. This needs to be public so that the intel compiler gets things
    // correct somehow the access function messes things up when going to
    // refineable navier--stokes
    static Vector<double> Gamma;
 
    // Access functions for the physical constants
 
    /// Reynolds number
    const double& re() const
    {
      return *Re_pt;
    }
 
    /// Product of Reynolds and Strouhal number (=Womersley number)
    const double& re_st() const
    {
      return *ReSt_pt;
    }
 
    /// Pointer to Reynolds number
    double*& re_pt()
    {
      return Re_pt;
    }
 
    /// Pointer to product of Reynolds and Strouhal number (=Womersley number)
    double*& re_st_pt()
    {
      return ReSt_pt;
    }
 
    /// Global inverse Froude number
    const double& re_invfr() const
    {
      return *ReInvFr_pt;
    }
 
    /// Pointer to global inverse Froude number
    double*& re_invfr_pt()
    {
      return ReInvFr_pt;
    }
 
    /// Global Reynolds number multiplied by inverse Rossby number
    const double& re_invro() const
    {
      return *ReInvRo_pt;
    }
 
    /// Pointer to global inverse Froude number
    double*& re_invro_pt()
    {
      return ReInvRo_pt;
    }
 
    /// Vector of gravitational components
    const Vector<double>& g() const
    {
      return *G_pt;
    }
 
    /// Pointer to Vector of gravitational components
    Vector<double>*& g_pt()
    {
      return G_pt;
    }
 
    /// Density ratio for element: Element's density relative to the
    ///  viscosity used in the definition of the Reynolds number
    const double& density_ratio() const
    {
      return *Density_Ratio_pt;
    }
 
    /// Pointer to Density ratio
    double*& density_ratio_pt()
    {
      return Density_Ratio_pt;
    }
 
    /// Viscosity ratio for element: Element's viscosity relative to the
    /// viscosity used in the definition of the Reynolds number
    const double& viscosity_ratio() const
    {
      return *Viscosity_Ratio_pt;
    }
 
    /// Pointer to Viscosity Ratio
    double*& viscosity_ratio_pt()
    {
      return Viscosity_Ratio_pt;
    }
 
    /// Access function for the body-force pointer
    void (*&axi_nst_body_force_fct_pt())(const double& time,
                                         const Vector<double>& x,
                                         Vector<double>& f)
    {
      return Body_force_fct_pt;
    }
 
    /// Access function for the source-function pointer
    double (*&source_fct_pt())(const double& time, const Vector<double>& x)
    {
      return Source_fct_pt;
    }
 
    /// Access function for the constitutive equation pointer
    GeneralisedNewtonianConstitutiveEquation<3>*& constitutive_eqn_pt()
    {
      return Constitutive_eqn_pt;
    }
 
    /// Switch to fully implict evaluation of non-Newtonian const eqn
    void use_current_strainrate_to_compute_second_invariant()
    {
      Use_extrapolated_strainrate_to_compute_second_invariant = false;
    }
 
    /// Use extrapolation for non-Newtonian const eqn
    void use_extrapolated_strainrate_to_compute_second_invariant()
    {
      Use_extrapolated_strainrate_to_compute_second_invariant = true;
    }
 
    /// Function to return number of pressure degrees of freedom
    virtual unsigned npres_axi_nst() const = 0;
 
    /// Return the index at which the i-th unknown velocity component
    /// is stored. The default value, i, is appropriate for single-physics
    /// problems.
    /// In derived multi-physics elements, this function should be overloaded
    /// to reflect the chosen storage scheme. Note that these equations require
    /// that the unknowns are always stored at the same indices at each node.
    virtual inline unsigned u_index_axi_nst(const unsigned& i) const
    {
      return i;
    }
 
    /// Return the index at which the i-th unknown velocity component
    /// is stored with a common interface for use in general
    /// FluidInterface and similar elements.
    /// To do: Merge all common storage etc to a common base class for
    /// Navier--Stokes elements in all coordinate systems.
    inline unsigned u_index_nst(const unsigned& i) const
    {
      return this->u_index_axi_nst(i);
    }
 
    /// Return the number of velocity components for use in
    /// general FluidInterface class
    inline unsigned n_u_nst() const
    {
      return 3;
    }
 
    /// i-th component of du/dt at local node n.
    /// Uses suitably interpolated value for hanging nodes.
    double du_dt_axi_nst(const unsigned& n, const unsigned& i) const
    {
      // Get the data's timestepper
      TimeStepper* time_stepper_pt = this->node_pt(n)->time_stepper_pt();
 
      // Initialise dudt
      double dudt = 0.0;
      // Loop over the timesteps, if there is a non Steady timestepper
      if (!time_stepper_pt->is_steady())
      {
        // Get the index at which the velocity is stored
        const unsigned u_nodal_index = u_index_axi_nst(i);
 
        // Number of timsteps (past & present)
        const unsigned n_time = time_stepper_pt->ntstorage();
 
        // Add the contributions to the time derivative
        for (unsigned t = 0; t < n_time; t++)
        {
          dudt +=
            time_stepper_pt->weight(1, t) * nodal_value(t, n, u_nodal_index);
        }
      }
 
      return dudt;
    }
 
    /// Disable ALE, i.e. assert the mesh is not moving -- you do this
    /// at your own risk!
    void disable_ALE()
    {
      ALE_is_disabled = true;
    }
 
    /// (Re-)enable ALE, i.e. take possible mesh motion into account
    /// when evaluating the time-derivative. Note: By default, ALE is
    /// enabled, at the expense of possibly creating unnecessary work
    /// in problems where the mesh is, in fact, stationary.
    void enable_ALE()
    {
      ALE_is_disabled = false;
    }
 
    /// Pressure at local pressure "node" n_p
    /// Uses suitably interpolated value for hanging nodes.
    virtual double p_axi_nst(const unsigned& n_p) const = 0;
 
    /// Which nodal value represents the pressure?
    virtual int p_nodal_index_axi_nst() const
    {
      return Pressure_not_stored_at_node;
    }
 
    /// Integral of pressure over element
    double pressure_integral() const;
 
 
    /// Get max. and min. strain rate invariant and visocosity
    /// over all integration points in element
    void max_and_min_invariant_and_viscosity(double& min_invariant,
                                             double& max_invariant,
                                             double& min_viscosity,
                                             double& max_viscosity) const;
 
    /// Return integral of dissipation over element
    double dissipation() const;
 
    /// Return dissipation at local coordinate s
    double dissipation(const Vector<double>& s) const;
 
    /// Get integral of kinetic energy over element
    double kin_energy() const;
 
    /// Strain-rate tensor: \f$ e_{ij} \f$  where \f$ i,j = r,z,\theta \f$
    /// (in that order)
    void strain_rate(const Vector<double>& s,
                     DenseMatrix<double>& strain_rate) const;
 
    /// Strain-rate tensor: \f$ e_{ij} \f$  where \f$ i,j = r,z,\theta \f$
    /// (in that order) at a specific time history value
    void strain_rate(const unsigned& t,
                     const Vector<double>& s,
                     DenseMatrix<double>& strain_rate) const;
 
    /// Extrapolated strain-rate tensor:
    /// \f$ e_{ij} \f$  where \f$ i,j = r,z,\theta \f$
    /// (in that order) based on the previous time steps evaluated
    /// at local coordinate s
    virtual void extrapolated_strain_rate(
      const Vector<double>& s, DenseMatrix<double>& strain_rate) const;
 
    /// Extrapolated strain-rate tensor:
    /// \f$ e_{ij} \f$  where \f$ i,j = r,z,\theta \f$
    /// (in that order) based on the previous time steps evaluated at
    /// ipt-th integration point
    virtual void extrapolated_strain_rate(
      const unsigned& ipt, DenseMatrix<double>& strain_rate) const
    {
      // Set the Vector to hold local coordinates (two dimensions)
      Vector<double> s(2);
      for (unsigned i = 0; i < 2; i++) s[i] = integral_pt()->knot(ipt, i);
      extrapolated_strain_rate(s, strain_rate);
    }
 
    /// Compute traction (on the viscous scale) at local coordinate s
    /// for outer unit normal N
    void traction(const Vector<double>& s,
                  const Vector<double>& N,
                  Vector<double>& traction) const;
 
    /// Compute the diagonal of the velocity/pressure mass matrices.
    /// If which one=0, both are computed, otherwise only the pressure
    /// (which_one=1) or the velocity mass matrix (which_one=2 -- the
    /// LSC version of the preconditioner only needs that one)
    /// NOTE: pressure versions isn't implemented yet because this
    ///       element has never been tried with Fp preconditoner.
    void get_pressure_and_velocity_mass_matrix_diagonal(
      Vector<double>& press_mass_diag,
      Vector<double>& veloc_mass_diag,
      const unsigned& which_one = 0);
 
    /// Number of scalars/fields output by this element. Reimplements
    /// broken virtual function in base class.
    unsigned nscalar_paraview() const
    {
      return 4;
    }
 
    /// Write values of the i-th scalar field at the plot points. Needs
    /// to be implemented for each new specific element type.
    void scalar_value_paraview(std::ofstream& file_out,
                               const unsigned& i,
                               const unsigned& nplot) const
    {
      // Vector of local coordinates
      Vector<double> s(2);
 
      // Loop over plot points
      unsigned num_plot_points = nplot_points_paraview(nplot);
      for (unsigned iplot = 0; iplot < num_plot_points; iplot++)
      {
        // Get local coordinates of plot point
        get_s_plot(iplot, nplot, s);
 
        // Velocities
        if (i < 3)
        {
          file_out << interpolated_u_axi_nst(s, i) << std::endl;
        }
        // Pressure
        else if (i == 3)
        {
          file_out << interpolated_p_axi_nst(s) << std::endl;
        }
        // Never get here
        else
        {
          std::stringstream error_stream;
          error_stream
            << "Axisymmetric Navier-Stokes Elements only store 4 fields "
            << std::endl;
          throw OomphLibError(error_stream.str(),
                              OOMPH_CURRENT_FUNCTION,
                              OOMPH_EXCEPTION_LOCATION);
        }
      }
    }
 
    /// Name of the i-th scalar field. Default implementation
    /// returns V1 for the first one, V2 for the second etc. Can (should!) be
    /// overloaded with more meaningful names in specific elements.
    std::string scalar_name_paraview(const unsigned& i) const
    {
      // Veloc
      if (i < 3)
      {
        return "Velocity " + StringConversion::to_string(i);
      }
      // Pressure field
      else if (i == 3)
      {
        return "Pressure";
      }
      // Never get here
      else
      {
        std::stringstream error_stream;
        error_stream
          << "Axisymmetric Navier-Stokes Elements only store 4 fields "
          << std::endl;
        throw OomphLibError(
          error_stream.str(), OOMPH_CURRENT_FUNCTION, OOMPH_EXCEPTION_LOCATION);
        return " ";
      }
    }
 
    /// Output solution in data vector at local cordinates s:
    /// r,z,u_r,u_z,u_phi,p
    void point_output_data(const Vector<double>& s, Vector<double>& data)
    {
      // Output the components of the position
      for (unsigned i = 0; i < 2; i++)
      {
        data.push_back(interpolated_x(s, i));
      }
 
      // Output the components of the FE representation of u at s
      for (unsigned i = 0; i < 3; i++)
      {
        data.push_back(interpolated_u_axi_nst(s, i));
      }
 
      // Output FE representation of p at s
      data.push_back(interpolated_p_axi_nst(s));
    }
 
 
    /// Output function: x,y,[z],u,v,[w],p
    /// in tecplot format. Default number of plot points
    void output(std::ostream& outfile)
    {
      unsigned nplot = 5;
      output(outfile, nplot);
    }
 
    /// Output function: x,y,[z],u,v,[w],p
    /// in tecplot format. nplot points in each coordinate direction
    void output(std::ostream& outfile, const unsigned& nplot);
 
 
    /// Output function: x,y,[z],u,v,[w],p
    /// in tecplot format. Default number of plot points
    void output(FILE* file_pt)
    {
      unsigned nplot = 5;
      output(file_pt, nplot);
    }
 
    /// Output function: x,y,[z],u,v,[w],p
    /// in tecplot format. nplot points in each coordinate direction
    void output(FILE* file_pt, const unsigned& nplot);
 
    /// Output function: x,y,[z],u,v,[w] in tecplot format.
    /// nplot points in each coordinate direction at timestep t
    /// (t=0: present; t>0: previous timestep)
    void output_veloc(std::ostream& outfile,
                      const unsigned& nplot,
                      const unsigned& t);
 
    /// Output exact solution specified via function pointer
    /// at a given number of plot points. Function prints as
    /// many components as are returned in solution Vector
    void output_fct(std::ostream& outfile,
                    const unsigned& nplot,
                    FiniteElement::SteadyExactSolutionFctPt exact_soln_pt);
 
    /// Output exact solution specified via function pointer
    /// at a given time and at a given number of plot points.
    /// Function prints as many components as are returned in solution Vector.
    void output_fct(std::ostream& outfile,
                    const unsigned& nplot,
                    const double& time,
                    FiniteElement::UnsteadyExactSolutionFctPt exact_soln_pt);
 
    /// Validate against exact solution at given time
    /// Solution is provided via function pointer.
    /// Plot at a given number of plot points and compute L2 error
    /// and L2 norm of velocity solution over element
    void compute_error(std::ostream& outfile,
                       FiniteElement::UnsteadyExactSolutionFctPt exact_soln_pt,
                       const double& time,
                       double& error,
                       double& norm);
 
    /// Validate against exact solution.
    /// Solution is provided via function pointer.
    /// Plot at a given number of plot points and compute L2 error
    /// and L2 norm of velocity solution over element
    void compute_error(std::ostream& outfile,
                       FiniteElement::SteadyExactSolutionFctPt exact_soln_pt,
                       double& error,
                       double& norm);
 
    /// Compute the element's residual Vector
    void fill_in_contribution_to_residuals(Vector<double>& residuals)
    {
      // Call the generic residuals function with flag set to 0
      // and using a dummy matrix argument
      fill_in_generic_residual_contribution_axi_nst(
        residuals,
        GeneralisedElement::Dummy_matrix,
        GeneralisedElement::Dummy_matrix,
        0);
    }
 
    /// Compute the element's residual Vector and the jacobian matrix
    /// Virtual function can be overloaded by hanging-node version
    void fill_in_contribution_to_jacobian(Vector<double>& residuals,
                                          DenseMatrix<double>& jacobian)
    {
      // Call the generic routine with the flag set to 1
 
      // obacht
      // Specific element routine is commented out and instead the
      // default FD version is used
      fill_in_generic_residual_contribution_axi_nst(
        residuals, jacobian, GeneralisedElement::Dummy_matrix, 1);
      // FiniteElement::fill_in_contribution_to_jacobian(residuals,jacobian);
    }
 
    /// Add the element's contribution to its residuals vector,
    /// jacobian matrix and mass matrix
    void fill_in_contribution_to_jacobian_and_mass_matrix(
      Vector<double>& residuals,
      DenseMatrix<double>& jacobian,
      DenseMatrix<double>& mass_matrix)
    {
      // Call the generic routine with the flag set to 2
      fill_in_generic_residual_contribution_axi_nst(
        residuals, jacobian, mass_matrix, 2);
    }
 
    /// Compute derivatives of elemental residual vector with respect to
    /// nodal coordinates. This function computes these terms analytically and
    /// overwrites the default implementation in the FiniteElement base class.
    /// dresidual_dnodal_coordinates(l,i,j) = d res(l) / dX_{ij}
    virtual void get_dresidual_dnodal_coordinates(
      RankThreeTensor<double>& dresidual_dnodal_coordinates);
 
    /// Compute the element's residual Vector
    void fill_in_contribution_to_dresiduals_dparameter(
      double* const& parameter_pt, Vector<double>& dres_dparam)
    {
      // Call the generic residuals function with flag set to 0
      // and using a dummy matrix argument
      fill_in_generic_dresidual_contribution_axi_nst(
        parameter_pt,
        dres_dparam,
        GeneralisedElement::Dummy_matrix,
        GeneralisedElement::Dummy_matrix,
        0);
    }
 
    /// Compute the element's residual Vector and the jacobian matrix
    /// Virtual function can be overloaded by hanging-node version
    void fill_in_contribution_to_djacobian_dparameter(
      double* const& parameter_pt,
      Vector<double>& dres_dparam,
      DenseMatrix<double>& djac_dparam)
    {
      // Call the generic routine with the flag set to 1
      fill_in_generic_dresidual_contribution_axi_nst(
        parameter_pt,
        dres_dparam,
        djac_dparam,
        GeneralisedElement::Dummy_matrix,
        1);
    }
 
    /// Add the element's contribution to its residuals vector,
    /// jacobian matrix and mass matrix
    void fill_in_contribution_to_djacobian_and_dmass_matrix_dparameter(
      double* const& parameter_pt,
      Vector<double>& dres_dparam,
      DenseMatrix<double>& djac_dparam,
      DenseMatrix<double>& dmass_matrix_dparam)
    {
      // Call the generic routine with the flag set to 2
      fill_in_generic_dresidual_contribution_axi_nst(
        parameter_pt, dres_dparam, djac_dparam, dmass_matrix_dparam, 2);
    }
 
 
    /// Compute vector of FE interpolated velocity u at local coordinate s
    void interpolated_u_axi_nst(const Vector<double>& s,
                                Vector<double>& veloc) const
    {
      // Find number of nodes
      unsigned n_node = nnode();
      // Local shape function
      Shape psi(n_node);
      // Find values of shape function
      shape(s, psi);
 
      for (unsigned i = 0; i < 3; i++)
      {
        // Index at which the nodal value is stored
        unsigned u_nodal_index = u_index_axi_nst(i);
        // Initialise value of u
        veloc[i] = 0.0;
        // Loop over the local nodes and sum
        for (unsigned l = 0; l < n_node; l++)
        {
          veloc[i] += nodal_value(l, u_nodal_index) * psi[l];
        }
      }
    }
 
    /// Return FE interpolated velocity u[i] at local coordinate s
    double interpolated_u_axi_nst(const Vector<double>& s,
                                  const unsigned& i) const
    {
      // Find number of nodes
      unsigned n_node = nnode();
      // Local shape function
      Shape psi(n_node);
      // Find values of shape function
      shape(s, psi);
 
      // Get the index at which the velocity is stored
      unsigned u_nodal_index = u_index_axi_nst(i);
 
      // Initialise value of u
      double interpolated_u = 0.0;
      // Loop over the local nodes and sum
      for (unsigned l = 0; l < n_node; l++)
      {
        interpolated_u += nodal_value(l, u_nodal_index) * psi[l];
      }
 
      return (interpolated_u);
    }
 
 
    /// Return FE interpolated velocity u[i] at local coordinate s
    // at time level t (t=0: present, t>0: history)
    double interpolated_u_axi_nst(const unsigned& t,
                                  const Vector<double>& s,
                                  const unsigned& i) const
    {
      // Find number of nodes
      unsigned n_node = nnode();
      // Local shape function
      Shape psi(n_node);
      // Find values of shape function
      shape(s, psi);
 
      // Get the index at which the velocity is stored
      unsigned u_nodal_index = u_index_axi_nst(i);
 
      // Initialise value of u
      double interpolated_u = 0.0;
      // Loop over the local nodes and sum
      for (unsigned l = 0; l < n_node; l++)
      {
        interpolated_u += nodal_value(t, l, u_nodal_index) * psi[l];
      }
 
      return (interpolated_u);
    }
 
 
    /// Compute the derivatives of the i-th component of
    /// velocity at point s with respect
    /// to all data that can affect its value. In addition, return the global
    /// equation numbers corresponding to the data. The function is virtual
    /// so that it can be overloaded in the refineable version
    virtual void dinterpolated_u_axi_nst_ddata(
      const Vector<double>& s,
      const unsigned& i,
      Vector<double>& du_ddata,
      Vector<unsigned>& global_eqn_number)
    {
      // Find number of nodes
      unsigned n_node = nnode();
      // Local shape function
      Shape psi(n_node);
      // Find values of shape function
      shape(s, psi);
 
      // Find the index at which the velocity component is stored
      const unsigned u_nodal_index = u_index_axi_nst(i);
 
      // Find the number of dofs associated with interpolated u
      unsigned n_u_dof = 0;
      for (unsigned l = 0; l < n_node; l++)
      {
        int global_eqn = this->node_pt(l)->eqn_number(u_nodal_index);
        // If it's positive add to the count
        if (global_eqn >= 0)
        {
          ++n_u_dof;
        }
      }
 
      // Now resize the storage schemes
      du_ddata.resize(n_u_dof, 0.0);
      global_eqn_number.resize(n_u_dof, 0);
 
      // Loop over th nodes again and set the derivatives
      unsigned count = 0;
      // Loop over the local nodes and sum
      for (unsigned l = 0; l < n_node; l++)
      {
        // Get the global equation number
        int global_eqn = this->node_pt(l)->eqn_number(u_nodal_index);
        if (global_eqn >= 0)
        {
          // Set the global equation number
          global_eqn_number[count] = global_eqn;
          // Set the derivative with respect to the unknown
          du_ddata[count] = psi[l];
          // Increase the counter
          ++count;
        }
      }
    }
 
 
    /// Return FE interpolated pressure at local coordinate s
    double interpolated_p_axi_nst(const Vector<double>& s) const
    {
      // Find number of nodes
      unsigned n_pres = npres_axi_nst();
      // Local shape function
      Shape psi(n_pres);
      // Find values of shape function
      pshape_axi_nst(s, psi);
 
      // Initialise value of p
      double interpolated_p = 0.0;
      // Loop over the local nodes and sum
      for (unsigned l = 0; l < n_pres; l++)
      {
        interpolated_p += p_axi_nst(l) * psi[l];
      }
 
      return (interpolated_p);
    }
 
    /// Return FE interpolated derivatives of velocity component u[i]
    /// w.r.t spatial local coordinate direction s[j] at local coordinate s
    double interpolated_duds_axi_nst(const Vector<double>& s,
                                     const unsigned& i,
                                     const unsigned& j) const
    {
      // Determine number of nodes
      const unsigned n_node = nnode();
 
      // Allocate storage for local shape function and its derivatives
      // with respect to space
      Shape psif(n_node);
      DShape dpsifds(n_node, 2);
 
      // Find values of shape function (ignore jacobian)
      (void)this->dshape_local(s, psif, dpsifds);
 
      // Get the index at which the velocity is stored
      const unsigned u_nodal_index = u_index_axi_nst(i);
 
      // Initialise value of duds
      double interpolated_duds = 0.0;
 
      // Loop over the local nodes and sum
      for (unsigned l = 0; l < n_node; l++)
      {
        interpolated_duds += nodal_value(l, u_nodal_index) * dpsifds(l, j);
      }
 
      return (interpolated_duds);
    }
 
    /// Return FE interpolated derivatives of velocity component u[i]
    /// w.r.t spatial global coordinate direction x[j] at local coordinate s
    double interpolated_dudx_axi_nst(const Vector<double>& s,
                                     const unsigned& i,
                                     const unsigned& j) const
    {
      // Determine number of nodes
      const unsigned n_node = nnode();
 
      // Allocate storage for local shape function and its derivatives
      // with respect to space
      Shape psif(n_node);
      DShape dpsifdx(n_node, 2);
 
      // Find values of shape function (ignore jacobian)
      (void)this->dshape_eulerian(s, psif, dpsifdx);
 
      // Get the index at which the velocity is stored
      const unsigned u_nodal_index = u_index_axi_nst(i);
 
      // Initialise value of dudx
      double interpolated_dudx = 0.0;
 
      // Loop over the local nodes and sum
      for (unsigned l = 0; l < n_node; l++)
      {
        interpolated_dudx += nodal_value(l, u_nodal_index) * dpsifdx(l, j);
      }
 
      return (interpolated_dudx);
    }
 
    /// Return FE interpolated derivatives of velocity component u[i]
    /// w.r.t time at local coordinate s
    double interpolated_dudt_axi_nst(const Vector<double>& s,
                                     const unsigned& i) const
    {
      // Determine number of nodes
      const unsigned n_node = nnode();
 
      // Allocate storage for local shape function
      Shape psif(n_node);
 
      // Find values of shape function
      shape(s, psif);
 
      // Initialise value of dudt
      double interpolated_dudt = 0.0;
 
      // Loop over the local nodes and sum
      for (unsigned l = 0; l < n_node; l++)
      {
        interpolated_dudt += du_dt_axi_nst(l, i) * psif[l];
      }
 
      return (interpolated_dudt);
    }
 
    /// Return FE interpolated derivatives w.r.t. nodal coordinates
    /// X_{pq} of the spatial derivatives of the velocity components
    /// du_i/dx_k at local coordinate s
    double interpolated_d_dudx_dX_axi_nst(const Vector<double>& s,
                                          const unsigned& p,
                                          const unsigned& q,
                                          const unsigned& i,
                                          const unsigned& k) const
    {
      // Determine number of nodes
      const unsigned n_node = nnode();
 
      // Allocate storage for local shape function and its derivatives
      // with respect to space
      Shape psif(n_node);
      DShape dpsifds(n_node, 2);
 
      // Find values of shape function (ignore jacobian)
      (void)this->dshape_local(s, psif, dpsifds);
 
      // Allocate memory for the jacobian and the inverse of the jacobian
      DenseMatrix<double> jacobian(2), inverse_jacobian(2);
 
      // Allocate memory for the derivative of the jacobian w.r.t. nodal coords
      DenseMatrix<double> djacobian_dX(2, n_node);
 
      // Allocate memory for the derivative w.r.t. nodal coords of the
      // spatial derivatives of the shape functions
      RankFourTensor<double> d_dpsifdx_dX(2, n_node, 3, 2);
 
      // Now calculate the inverse jacobian
      const double det =
        local_to_eulerian_mapping(dpsifds, jacobian, inverse_jacobian);
 
      // Calculate the derivative of the jacobian w.r.t. nodal coordinates
      // Note: must call this before "transform_derivatives(...)" since this
      // function requires dpsids rather than dpsidx
      dJ_eulerian_dnodal_coordinates(jacobian, dpsifds, djacobian_dX);
 
      // Calculate the derivative of dpsidx w.r.t. nodal coordinates
      // Note: this function also requires dpsids rather than dpsidx
      d_dshape_eulerian_dnodal_coordinates(
        det, jacobian, djacobian_dX, inverse_jacobian, dpsifds, d_dpsifdx_dX);
 
      // Get the index at which the velocity is stored
      const unsigned u_nodal_index = u_index_axi_nst(i);
 
      // Initialise value of dudx
      double interpolated_d_dudx_dX = 0.0;
 
      // Loop over the local nodes and sum
      for (unsigned l = 0; l < n_node; l++)
      {
        interpolated_d_dudx_dX +=
          nodal_value(l, u_nodal_index) * d_dpsifdx_dX(p, q, l, k);
      }
 
      return (interpolated_d_dudx_dX);
    }
 
    /// Compute the volume of the element
    double compute_physical_size() const
    {
      // Initialise result
      double result = 0.0;
 
      // Figure out the global (Eulerian) spatial dimension of the
      // element by checking the Eulerian dimension of the nodes
      const unsigned n_dim_eulerian = nodal_dimension();
 
      // Allocate Vector of global Eulerian coordinates
      Vector<double> x(n_dim_eulerian);
 
      // Set the value of n_intpt
      const unsigned n_intpt = integral_pt()->nweight();
 
      // Vector of local coordinates
      const unsigned n_dim = dim();
      Vector<double> s(n_dim);
 
      // Loop over the integration points
      for (unsigned ipt = 0; ipt < n_intpt; ipt++)
      {
        // Assign the values of s
        for (unsigned i = 0; i < n_dim; i++)
        {
          s[i] = integral_pt()->knot(ipt, i);
        }
 
        // Assign the values of the global Eulerian coordinates
        for (unsigned i = 0; i < n_dim_eulerian; i++)
        {
          x[i] = interpolated_x(s, i);
        }
 
        // Get the integral weight
        const double w = integral_pt()->weight(ipt);
 
        // Get Jacobian of mapping
        const double J = J_eulerian(s);
 
        // The integrand at the current integration point is r
        result += x[0] * w * J;
      }
 
      // Multiply by 2pi (integrating in azimuthal direction)
      return (2.0 * MathematicalConstants::Pi * result);
    }
  };
 
  //////////////////////////////////////////////////////////////////////////////
  //////////////////////////////////////////////////////////////////////////////
  //////////////////////////////////////////////////////////////////////////////
 
 
  //==========================================================================
  /// Crouzeix_Raviart elements are Navier--Stokes elements with quadratic
  /// interpolation for velocities and positions, but a discontinuous linear
  /// pressure interpolation
  //==========================================================================
  class GeneralisedNewtonianAxisymmetricQCrouzeixRaviartElement
    : public virtual QElement<2, 3>,
      public virtual GeneralisedNewtonianAxisymmetricNavierStokesEquations
  {
  private:
    /// Static array of ints to hold required number of variables at nodes
    static const unsigned Initial_Nvalue[];
 
  protected:
    /// Internal index that indicates at which internal data the pressure is
    /// stored
    unsigned P_axi_nst_internal_index;
 
    /// Velocity shape and test functions and their derivs
    /// w.r.t. to global coords  at local coordinate s (taken from geometry)
    /// Return Jacobian of mapping between local and global coordinates.
    inline double dshape_and_dtest_eulerian_axi_nst(const Vector<double>& s,
                                                    Shape& psi,
                                                    DShape& dpsidx,
                                                    Shape& test,
                                                    DShape& dtestdx) const;
 
    /// Velocity shape and test functions and their derivs
    /// w.r.t. to global coords at ipt-th integation point (taken from geometry)
    /// Return Jacobian of mapping between local and global coordinates.
    inline double dshape_and_dtest_eulerian_at_knot_axi_nst(
      const unsigned& ipt,
      Shape& psi,
      DShape& dpsidx,
      Shape& test,
      DShape& dtestdx) const;
 
    /// Shape/test functions and derivs w.r.t. to global coords at
    /// integration point ipt; return Jacobian of mapping (J). Also compute
    /// derivatives of dpsidx, dtestdx and J w.r.t. nodal coordinates.
    inline double dshape_and_dtest_eulerian_at_knot_axi_nst(
      const unsigned& ipt,
      Shape& psi,
      DShape& dpsidx,
      RankFourTensor<double>& d_dpsidx_dX,
      Shape& test,
      DShape& dtestdx,
      RankFourTensor<double>& d_dtestdx_dX,
      DenseMatrix<double>& djacobian_dX) const;
 
 
    /// Pressure shape functions at local coordinate s
    inline void pshape_axi_nst(const Vector<double>& s, Shape& psi) const;
 
    /// Pressure shape and test functions at local coordinte s
    inline void pshape_axi_nst(const Vector<double>& s,
                               Shape& psi,
                               Shape& test) const;
 
 
  public:
    /// Constructor, there are three internal values (for the pressure)
    GeneralisedNewtonianAxisymmetricQCrouzeixRaviartElement()
      : QElement<2, 3>(),
        GeneralisedNewtonianAxisymmetricNavierStokesEquations()
    {
      // Allocate and add one Internal data object that stores the three
      // pressure values
      P_axi_nst_internal_index = this->add_internal_data(new Data(3));
    }
 
    /// Number of values (pinned or dofs) required at local node n.
    virtual unsigned required_nvalue(const unsigned& n) const;
 
    /// Return the pressure values at internal dof i_internal
    /// (Discontinous pressure interpolation -- no need to cater for hanging
    /// nodes).
    double p_axi_nst(const unsigned& i) const
    {
      return internal_data_pt(P_axi_nst_internal_index)->value(i);
    }
 
    /// Return number of pressure values
    unsigned npres_axi_nst() const
    {
      return 3;
    }
 
    /// Function to fix the internal pressure dof idof_internal
    void fix_pressure(const unsigned& p_dof, const double& pvalue)
    {
      this->internal_data_pt(P_axi_nst_internal_index)->pin(p_dof);
      internal_data_pt(P_axi_nst_internal_index)->set_value(p_dof, pvalue);
    }
 
    /// Compute traction at local coordinate s for outer unit normal N
    void get_traction(const Vector<double>& s,
                      const Vector<double>& N,
                      Vector<double>& traction) const;
 
    /// Overload the access function for the pressure's local
    /// equation numbers
    inline int p_local_eqn(const unsigned& n) const
    {
      return internal_local_eqn(P_axi_nst_internal_index, n);
    }
 
    /// Redirect output to NavierStokesEquations output
    void output(std::ostream& outfile)
    {
      GeneralisedNewtonianAxisymmetricNavierStokesEquations::output(outfile);
    }
 
    /// Redirect output to NavierStokesEquations output
    void output(std::ostream& outfile, const unsigned& n_plot)
    {
      GeneralisedNewtonianAxisymmetricNavierStokesEquations::output(outfile,
                                                                    n_plot);
    }
 
 
    /// Redirect output to NavierStokesEquations output
    void output(FILE* file_pt)
    {
      GeneralisedNewtonianAxisymmetricNavierStokesEquations::output(file_pt);
    }
 
    /// Redirect output to NavierStokesEquations output
    void output(FILE* file_pt, const unsigned& n_plot)
    {
      GeneralisedNewtonianAxisymmetricNavierStokesEquations::output(file_pt,
                                                                    n_plot);
    }
 
    /// The number of "DOF types" that degrees of freedom in this element
    /// are sub-divided into: Velocity and pressure.
    unsigned ndof_types() const
    {
      return 4;
    }
 
    /// Create a list of pairs for all unknowns in this element,
    /// so that the first entry in each pair contains the global equation
    /// number of the unknown, while the second one contains the number
    /// of the "DOF type" that this unknown is associated with.
    /// (Function can obviously only be called if the equation numbering
    /// scheme has been set up.) Velocity=0; Pressure=1
    void get_dof_numbers_for_unknowns(
      std::list<std::pair<unsigned long, unsigned>>& dof_lookup_list) const;
  };
 
  // Inline functions
 
  //=======================================================================
  /// Derivatives of the shape functions and test functions w.r.t. to global
  /// (Eulerian) coordinates. Return Jacobian of mapping between
  /// local and global coordinates.
  //=======================================================================
  inline double GeneralisedNewtonianAxisymmetricQCrouzeixRaviartElement::
    dshape_and_dtest_eulerian_axi_nst(const Vector<double>& s,
                                      Shape& psi,
                                      DShape& dpsidx,
                                      Shape& test,
                                      DShape& dtestdx) const
  {
    // Call the geometrical shape functions and derivatives
    double J = this->dshape_eulerian(s, psi, dpsidx);
    // Loop over the test functions and derivatives and set them equal to the
    // shape functions
    for (unsigned i = 0; i < 9; i++)
    {
      test[i] = psi[i];
      dtestdx(i, 0) = dpsidx(i, 0);
      dtestdx(i, 1) = dpsidx(i, 1);
    }
    // Return the jacobian
    return J;
  }
 
  //=======================================================================
  /// Derivatives of the shape functions and test functions w.r.t. to global
  /// (Eulerian) coordinates. Return Jacobian of mapping between
  /// local and global coordinates.
  //=======================================================================
  inline double GeneralisedNewtonianAxisymmetricQCrouzeixRaviartElement::
    dshape_and_dtest_eulerian_at_knot_axi_nst(const unsigned& ipt,
                                              Shape& psi,
                                              DShape& dpsidx,
                                              Shape& test,
                                              DShape& dtestdx) const
  {
    // Call the geometrical shape functions and derivatives
    double J = this->dshape_eulerian_at_knot(ipt, psi, dpsidx);
    // Loop over the test functions and derivatives and set them equal to the
    // shape functions
    for (unsigned i = 0; i < 9; i++)
    {
      test[i] = psi[i];
      dtestdx(i, 0) = dpsidx(i, 0);
      dtestdx(i, 1) = dpsidx(i, 1);
    }
    // Return the jacobian
    return J;
  }
 
  //=======================================================================
  /// Define the shape functions (psi) and test functions (test) and
  /// their derivatives w.r.t. global coordinates (dpsidx and dtestdx)
  /// and return Jacobian of mapping (J). Additionally compute the
  /// derivatives of dpsidx, dtestdx and J w.r.t. nodal coordinates.
  ///
  /// Galerkin: Test functions = shape functions
  //=======================================================================
  inline double GeneralisedNewtonianAxisymmetricQCrouzeixRaviartElement::
    dshape_and_dtest_eulerian_at_knot_axi_nst(
      const unsigned& ipt,
      Shape& psi,
      DShape& dpsidx,
      RankFourTensor<double>& d_dpsidx_dX,
      Shape& test,
      DShape& dtestdx,
      RankFourTensor<double>& d_dtestdx_dX,
      DenseMatrix<double>& djacobian_dX) const
  {
    // Call the geometrical shape functions and derivatives
    const double J = this->dshape_eulerian_at_knot(
      ipt, psi, dpsidx, djacobian_dX, d_dpsidx_dX);
 
    // Loop over the test functions and derivatives and set them equal to the
    // shape functions
    for (unsigned i = 0; i < 9; i++)
    {
      test[i] = psi[i];
 
      for (unsigned k = 0; k < 2; k++)
      {
        dtestdx(i, k) = dpsidx(i, k);
 
        for (unsigned p = 0; p < 2; p++)
        {
          for (unsigned q = 0; q < 9; q++)
          {
            d_dtestdx_dX(p, q, i, k) = d_dpsidx_dX(p, q, i, k);
          }
        }
      }
    }
 
    // Return the jacobian
    return J;
  }
 
  //=======================================================================
  /// Pressure shape functions
  //=======================================================================
  inline void GeneralisedNewtonianAxisymmetricQCrouzeixRaviartElement::
    pshape_axi_nst(const Vector<double>& s, Shape& psi) const
  {
    psi[0] = 1.0;
    psi[1] = s[0];
    psi[2] = s[1];
  }
 
  /// Define the pressure shape and test functions
  inline void GeneralisedNewtonianAxisymmetricQCrouzeixRaviartElement::
    pshape_axi_nst(const Vector<double>& s, Shape& psi, Shape& test) const
  {
    // Call the pressure shape functions
    pshape_axi_nst(s, psi);
    // Loop over the test functions and set them equal to the shape functions
    for (unsigned i = 0; i < 3; i++) test[i] = psi[i];
  }
 
 
  //=======================================================================
  /// Face geometry of the GeneralisedNewtonianAxisymmetric
  /// Crouzeix_Raviart elements
  //=======================================================================
  template<>
  class FaceGeometry<GeneralisedNewtonianAxisymmetricQCrouzeixRaviartElement>
    : public virtual QElement<1, 3>
  {
  public:
    FaceGeometry() : QElement<1, 3>() {}
  };
 
  //=======================================================================
  /// Face geometry of face geometry of the
  /// GeneralisedNewtonianAxisymmetric Crouzeix_Raviart elements
  //=======================================================================
  template<>
  class FaceGeometry<
    FaceGeometry<GeneralisedNewtonianAxisymmetricQCrouzeixRaviartElement>>
    : public virtual PointElement
  {
  public:
    FaceGeometry() : PointElement() {}
  };
 
 
  ////////////////////////////////////////////////////////////////////////////
  ////////////////////////////////////////////////////////////////////////////
 
  //=======================================================================
  /// Taylor--Hood elements are Navier--Stokes elements
  /// with quadratic interpolation for velocities and positions and
  /// continous linear pressure interpolation
  //=======================================================================
  class GeneralisedNewtonianAxisymmetricQTaylorHoodElement
    : public virtual QElement<2, 3>,
      public virtual GeneralisedNewtonianAxisymmetricNavierStokesEquations
  {
  private:
    /// Static array of ints to hold number of variables at node
    static const unsigned Initial_Nvalue[];
 
  protected:
    /// Static array of ints to hold conversion from pressure
    /// node numbers to actual node numbers
    static const unsigned Pconv[];
 
    /// Velocity shape and test functions and their derivs
    /// w.r.t. to global coords  at local coordinate s (taken from geometry)
    /// Return Jacobian of mapping between local and global coordinates.
    inline double dshape_and_dtest_eulerian_axi_nst(const Vector<double>& s,
                                                    Shape& psi,
                                                    DShape& dpsidx,
                                                    Shape& test,
                                                    DShape& dtestdx) const;
 
    /// Velocity shape and test functions and their derivs
    /// w.r.t. to global coords  at local coordinate s (taken from geometry)
    /// Return Jacobian of mapping between local and global coordinates.
    inline double dshape_and_dtest_eulerian_at_knot_axi_nst(
      const unsigned& ipt,
      Shape& psi,
      DShape& dpsidx,
      Shape& test,
      DShape& dtestdx) const;
 
    /// Shape/test functions and derivs w.r.t. to global coords at
    /// integration point ipt; return Jacobian of mapping (J). Also compute
    /// derivatives of dpsidx, dtestdx and J w.r.t. nodal coordinates.
    inline double dshape_and_dtest_eulerian_at_knot_axi_nst(
      const unsigned& ipt,
      Shape& psi,
      DShape& dpsidx,
      RankFourTensor<double>& d_dpsidx_dX,
      Shape& test,
      DShape& dtestdx,
      RankFourTensor<double>& d_dtestdx_dX,
      DenseMatrix<double>& djacobian_dX) const;
 
    /// Pressure shape functions at local coordinate s
    inline void pshape_axi_nst(const Vector<double>& s, Shape& psi) const;
 
    /// Pressure shape and test functions at local coordinte s
    inline void pshape_axi_nst(const Vector<double>& s,
                               Shape& psi,
                               Shape& test) const;
 
  public:
    /// Constructor, no internal data points
    GeneralisedNewtonianAxisymmetricQTaylorHoodElement()
      : QElement<2, 3>(),
        GeneralisedNewtonianAxisymmetricNavierStokesEquations()
    {
    }
 
    /// Number of values (pinned or dofs) required at node n. Can
    /// be overwritten for hanging node version
    inline virtual unsigned required_nvalue(const unsigned& n) const
    {
      return Initial_Nvalue[n];
    }
 
    /// Which nodal value represents the pressure?
    virtual int p_nodal_index_axi_nst() const
    {
      return 3;
    }
 
    /// Access function for the pressure values at local pressure
    /// node n_p (const version)
    double p_axi_nst(const unsigned& n_p) const
    {
      return nodal_value(Pconv[n_p], p_nodal_index_axi_nst());
    }
 
    /// Return number of pressure values
    unsigned npres_axi_nst() const
    {
      return 4;
    }
 
    /// Fix the pressure at local pressure node n_p
    void fix_pressure(const unsigned& n_p, const double& pvalue)
    {
      this->node_pt(Pconv[n_p])->pin(p_nodal_index_axi_nst());
      this->node_pt(Pconv[n_p])->set_value(p_nodal_index_axi_nst(), pvalue);
    }
 
    /// Compute traction at local coordinate s for outer unit normal N
    void get_traction(const Vector<double>& s,
                      const Vector<double>& N,
                      Vector<double>& traction) const;
 
    /// Overload the access function for the pressure's local
    /// equation numbers
    inline int p_local_eqn(const unsigned& n) const
    {
      return nodal_local_eqn(Pconv[n], p_nodal_index_axi_nst());
    }
 
    /// Redirect output to NavierStokesEquations output
    void output(std::ostream& outfile)
    {
      GeneralisedNewtonianAxisymmetricNavierStokesEquations::output(outfile);
    }
 
    /// Redirect output to NavierStokesEquations output
    void output(std::ostream& outfile, const unsigned& n_plot)
    {
      GeneralisedNewtonianAxisymmetricNavierStokesEquations::output(outfile,
                                                                    n_plot);
    }
 
    /// Redirect output to NavierStokesEquations output
    void output(FILE* file_pt)
    {
      GeneralisedNewtonianAxisymmetricNavierStokesEquations::output(file_pt);
    }
 
    /// Redirect output to NavierStokesEquations output
    void output(FILE* file_pt, const unsigned& n_plot)
    {
      GeneralisedNewtonianAxisymmetricNavierStokesEquations::output(file_pt,
                                                                    n_plot);
    }
 
    /// Returns the number of "DOF types" that degrees of freedom
    /// in this element are sub-divided into: Velocity and pressure.
    unsigned ndof_types() const
    {
      return 4;
    }
 
    /// Create a list of pairs for all unknowns in this element,
    /// so that the first entry in each pair contains the global equation
    /// number of the unknown, while the second one contains the number
    /// of the "DOF type" that this unknown is associated with.
    /// (Function can obviously only be called if the equation numbering
    /// scheme has been set up.) Velocity=0; Pressure=1
    void get_dof_numbers_for_unknowns(
      std::list<std::pair<unsigned long, unsigned>>& dof_lookup_list) const;
  };
 
  // Inline functions
 
  //==========================================================================
  /// Derivatives of the shape functions and test functions w.r.t to
  /// global (Eulerian) coordinates. Return Jacobian of mapping between
  /// local and global coordinates.
  //==========================================================================
  inline double GeneralisedNewtonianAxisymmetricQTaylorHoodElement::
    dshape_and_dtest_eulerian_axi_nst(const Vector<double>& s,
                                      Shape& psi,
                                      DShape& dpsidx,
                                      Shape& test,
                                      DShape& dtestdx) const
  {
    // Call the geometrical shape functions and derivatives
    double J = this->dshape_eulerian(s, psi, dpsidx);
    // Loop over the test functions and derivatives and set them equal to the
    // shape functions
    for (unsigned i = 0; i < 9; i++)
    {
      test[i] = psi[i];
      dtestdx(i, 0) = dpsidx(i, 0);
      dtestdx(i, 1) = dpsidx(i, 1);
    }
    // Return the jacobian
    return J;
  }
 
  //==========================================================================
  /// Derivatives of the shape functions and test functions w.r.t to
  /// global (Eulerian) coordinates. Return Jacobian of mapping between
  /// local and global coordinates.
  //==========================================================================
  inline double GeneralisedNewtonianAxisymmetricQTaylorHoodElement::
    dshape_and_dtest_eulerian_at_knot_axi_nst(const unsigned& ipt,
                                              Shape& psi,
                                              DShape& dpsidx,
                                              Shape& test,
                                              DShape& dtestdx) const
  {
    // Call the geometrical shape functions and derivatives
    double J = this->dshape_eulerian_at_knot(ipt, psi, dpsidx);
    // Loop over the test functions and derivatives and set them equal to the
    // shape functions
    for (unsigned i = 0; i < 9; i++)
    {
      test[i] = psi[i];
      dtestdx(i, 0) = dpsidx(i, 0);
      dtestdx(i, 1) = dpsidx(i, 1);
    }
    // Return the jacobian
    return J;
  }
 
  //==========================================================================
  /// Define the shape functions (psi) and test functions (test) and
  /// their derivatives w.r.t. global coordinates (dpsidx and dtestdx)
  /// and return Jacobian of mapping (J). Additionally compute the
  /// derivatives of dpsidx, dtestdx and J w.r.t. nodal coordinates.
  ///
  /// Galerkin: Test functions = shape functions
  //==========================================================================
  inline double GeneralisedNewtonianAxisymmetricQTaylorHoodElement::
    dshape_and_dtest_eulerian_at_knot_axi_nst(
      const unsigned& ipt,
      Shape& psi,
      DShape& dpsidx,
      RankFourTensor<double>& d_dpsidx_dX,
      Shape& test,
      DShape& dtestdx,
      RankFourTensor<double>& d_dtestdx_dX,
      DenseMatrix<double>& djacobian_dX) const
  {
    // Call the geometrical shape functions and derivatives
    const double J = this->dshape_eulerian_at_knot(
      ipt, psi, dpsidx, djacobian_dX, d_dpsidx_dX);
 
    // Loop over the test functions and derivatives and set them equal to the
    // shape functions
    for (unsigned i = 0; i < 9; i++)
    {
      test[i] = psi[i];
 
      for (unsigned k = 0; k < 2; k++)
      {
        dtestdx(i, k) = dpsidx(i, k);
 
        for (unsigned p = 0; p < 2; p++)
        {
          for (unsigned q = 0; q < 9; q++)
          {
            d_dtestdx_dX(p, q, i, k) = d_dpsidx_dX(p, q, i, k);
          }
        }
      }
    }
 
    // Return the jacobian
    return J;
  }
 
  //==========================================================================
  /// Pressure shape functions
  //==========================================================================
  inline void GeneralisedNewtonianAxisymmetricQTaylorHoodElement::
    pshape_axi_nst(const Vector<double>& s, Shape& psi) const
  {
    // Local storage
    double psi1[2], psi2[2];
    // Call the OneDimensional Shape functions
    OneDimLagrange::shape<2>(s[0], psi1);
    OneDimLagrange::shape<2>(s[1], psi2);
 
    // Now let's loop over the nodal points in the element
    // s1 is the "x" coordinate, s2 the "y"
    for (unsigned i = 0; i < 2; i++)
    {
      for (unsigned j = 0; j < 2; j++)
      {
        /*Multiply the two 1D functions together to get the 2D function*/
        psi[2 * i + j] = psi2[i] * psi1[j];
      }
    }
  }
 
  //==========================================================================
  /// Pressure shape and test functions
  //==========================================================================
  inline void GeneralisedNewtonianAxisymmetricQTaylorHoodElement::
    pshape_axi_nst(const Vector<double>& s, Shape& psi, Shape& test) const
  {
    // Call the pressure shape functions
    pshape_axi_nst(s, psi);
    // Loop over the test functions and set them equal to the shape functions
    for (unsigned i = 0; i < 4; i++) test[i] = psi[i];
  }
 
  //=======================================================================
  /// Face geometry of the GeneralisedNewtonianAxisymmetric Taylor_Hood elements
  //=======================================================================
  template<>
  class FaceGeometry<GeneralisedNewtonianAxisymmetricQTaylorHoodElement>
    : public virtual QElement<1, 3>
  {
  public:
    FaceGeometry() : QElement<1, 3>() {}
  };
 
  //=======================================================================
  /// Face geometry of the face geometry of the
  /// GeneralisedNewtonianAxisymmetric Taylor_Hood elements
  //=======================================================================
  template<>
  class FaceGeometry<
    FaceGeometry<GeneralisedNewtonianAxisymmetricQTaylorHoodElement>>
    : public virtual PointElement
  {
  public:
    FaceGeometry() : PointElement() {}
  };
 
 
  //==========================================================
  /// GeneralisedNewtonianAxisymmetric Taylor Hood upgraded to become
  /// projectable
  //==========================================================
  template<class TAYLOR_HOOD_ELEMENT>
  class GeneralisedNewtonianProjectableAxisymmetricTaylorHoodElement
    : public virtual ProjectableElement<TAYLOR_HOOD_ELEMENT>
  {
  public:
    /// Specify the values associated with field fld.
    /// The information is returned in a vector of pairs which comprise
    /// the Data object and the value within it, that correspond to field fld.
    /// In the underlying Taylor Hood elements the fld-th velocities are stored
    /// at the fld-th value of the nodes; the pressures (the dim-th
    /// field) are the dim-th values at the vertex nodes etc.
    Vector<std::pair<Data*, unsigned>> data_values_of_field(const unsigned& fld)
    {
      // Create the vector
      Vector<std::pair<Data*, unsigned>> data_values;
 
      // Velocities dofs
      if (fld < 3)
      {
        // Loop over all nodes
        unsigned nnod = this->nnode();
        for (unsigned j = 0; j < nnod; j++)
        {
          // Add the data value associated with the velocity components
          data_values.push_back(std::make_pair(this->node_pt(j), fld));
        }
      }
      // Pressure
      else
      {
        // Loop over all vertex nodes
        unsigned Pconv_size = 3;
        for (unsigned j = 0; j < Pconv_size; j++)
        {
          // Add the data value associated with the pressure components
          unsigned vertex_index = this->Pconv[j];
          data_values.push_back(
            std::make_pair(this->node_pt(vertex_index), fld));
        }
      }
 
      // Return the vector
      return data_values;
    }
 
    /// Number of fields to be projected: dim+1, corresponding to
    /// velocity components and  pressure
    unsigned nfields_for_projection()
    {
      return 4;
    }
 
    /// Number of history values to be stored for fld-th field. Whatever
    /// the timestepper has set up for the velocity components and
    /// none for the pressure field (includes current value!)
    unsigned nhistory_values_for_projection(const unsigned& fld)
    {
      if (fld == 3)
      {
        // pressure doesn't have history values
        return 1;
      }
      else
      {
        return this->node_pt(0)->ntstorage();
      }
    }
 
    /// Number of positional history values (includes current value!)
    unsigned nhistory_values_for_coordinate_projection()
    {
      return this->node_pt(0)->position_time_stepper_pt()->ntstorage();
    }
 
    /// Return Jacobian of mapping and shape functions of field fld
    /// at local coordinate s
    double jacobian_and_shape_of_field(const unsigned& fld,
                                       const Vector<double>& s,
                                       Shape& psi)
    {
      unsigned n_dim = this->dim();
      unsigned n_node = this->nnode();
 
      if (fld == 3)
      {
        // We are dealing with the pressure
        this->pshape_axi_nst(s, psi);
 
        Shape psif(n_node), testf(n_node);
        DShape dpsifdx(n_node, n_dim), dtestfdx(n_node, n_dim);
 
        // Domain Shape
        double J = this->dshape_and_dtest_eulerian_axi_nst(
          s, psif, dpsifdx, testf, dtestfdx);
        return J;
      }
      else
      {
        Shape testf(n_node);
        DShape dpsifdx(n_node, n_dim), dtestfdx(n_node, n_dim);
 
        // Domain Shape
        double J = this->dshape_and_dtest_eulerian_axi_nst(
          s, psi, dpsifdx, testf, dtestfdx);
        return J;
      }
    }
 
 
    /// Return interpolated field fld at local coordinate s, at time
    /// level t (t=0: present; t>0: history values)
    double get_field(const unsigned& t,
                     const unsigned& fld,
                     const Vector<double>& s)
    {
      unsigned n_node = this->nnode();
 
      // If fld=3, we deal with the pressure
      if (fld == 3)
      {
        return this->interpolated_p_axi_nst(s);
      }
      // Velocity
      else
      {
        // Find the index at which the variable is stored
        unsigned u_nodal_index = this->u_index_axi_nst(fld);
 
        // Local shape function
        Shape psi(n_node);
 
        // Find values of shape function
        this->shape(s, psi);
 
        // Initialise value of u
        double interpolated_u = 0.0;
 
        // Sum over the local nodes at that time
        for (unsigned l = 0; l < n_node; l++)
        {
          interpolated_u += this->nodal_value(t, l, u_nodal_index) * psi[l];
        }
        return interpolated_u;
      }
    }
 
 
    /// Return number of values in field fld
    unsigned nvalue_of_field(const unsigned& fld)
    {
      if (fld == 3)
      {
        return this->npres_axi_nst();
      }
      else
      {
        return this->nnode();
      }
    }
 
 
    /// Return local equation number of value j in field fld.
    int local_equation(const unsigned& fld, const unsigned& j)
    {
      if (fld == 3)
      {
        return this->p_local_eqn(j);
      }
      else
      {
        const unsigned u_nodal_index = this->u_index_axi_nst(fld);
        return this->nodal_local_eqn(j, u_nodal_index);
      }
    }
  };
 
 
  //=======================================================================
  /// Face geometry for element is the same as that for the underlying
  /// wrapped element
  //=======================================================================
  template<class ELEMENT>
  class FaceGeometry<
    GeneralisedNewtonianProjectableAxisymmetricTaylorHoodElement<ELEMENT>>
    : public virtual FaceGeometry<ELEMENT>
  {
  public:
    FaceGeometry() : FaceGeometry<ELEMENT>() {}
  };
 
 
  //=======================================================================
  /// Face geometry of the Face Geometry for element is the same as
  /// that for the underlying wrapped element
  //=======================================================================
  template<class ELEMENT>
  class FaceGeometry<FaceGeometry<
    GeneralisedNewtonianProjectableAxisymmetricTaylorHoodElement<ELEMENT>>>
    : public virtual FaceGeometry<FaceGeometry<ELEMENT>>
  {
  public:
    FaceGeometry() : FaceGeometry<FaceGeometry<ELEMENT>>() {}
  };
 
 
  //==========================================================
  /// Crouzeix Raviart upgraded to become projectable
  //==========================================================
  template<class CROUZEIX_RAVIART_ELEMENT>
  class GeneralisedNewtonianProjectableAxisymmetricCrouzeixRaviartElement
    : public virtual ProjectableElement<CROUZEIX_RAVIART_ELEMENT>
  {
  public:
    /// Specify the values associated with field fld.
    /// The information is returned in a vector of pairs which comprise
    /// the Data object and the value within it, that correspond to field fld.
    /// In the underlying Crouzeix Raviart elements the
    /// fld-th velocities are stored
    /// at the fld-th value of the nodes; the pressures are stored internally
    Vector<std::pair<Data*, unsigned>> data_values_of_field(const unsigned& fld)
    {
      // Create the vector
      Vector<std::pair<Data*, unsigned>> data_values;
 
      // Velocities dofs
      if (fld < 3)
      {
        // Loop over all nodes
        const unsigned n_node = this->nnode();
        for (unsigned n = 0; n < n_node; n++)
        {
          // Add the data value associated with the velocity components
          data_values.push_back(std::make_pair(this->node_pt(n), fld));
        }
      }
      // Pressure
      else
      {
        // Need to push back the internal data
        const unsigned n_press = this->npres_axi_nst();
        // Loop over all pressure values
        for (unsigned j = 0; j < n_press; j++)
        {
          data_values.push_back(std::make_pair(
            this->internal_data_pt(this->P_axi_nst_internal_index), j));
        }
      }
 
      // Return the vector
      return data_values;
    }
 
    /// Number of fields to be projected: dim+1, corresponding to
    /// velocity components and  pressure
    unsigned nfields_for_projection()
    {
      return 4;
    }
 
    /// Number of history values to be stored for fld-th field. Whatever
    /// the timestepper has set up for the velocity components and
    /// none for the pressure field (includes current value!)
    unsigned nhistory_values_for_projection(const unsigned& fld)
    {
      if (fld == 3)
      {
        // pressure doesn't have history values
        return 1;
      }
      else
      {
        return this->node_pt(0)->ntstorage();
      }
    }
 
    /// Number of positional history values (includes current value!)
    unsigned nhistory_values_for_coordinate_projection()
    {
      return this->node_pt(0)->position_time_stepper_pt()->ntstorage();
    }
 
    /// Return Jacobian of mapping and shape functions of field fld
    /// at local coordinate s
    double jacobian_and_shape_of_field(const unsigned& fld,
                                       const Vector<double>& s,
                                       Shape& psi)
    {
      unsigned n_dim = this->dim();
      unsigned n_node = this->nnode();
 
      if (fld == 3)
      {
        // We are dealing with the pressure
        this->pshape_axi_nst(s, psi);
 
        Shape psif(n_node), testf(n_node);
        DShape dpsifdx(n_node, n_dim), dtestfdx(n_node, n_dim);
 
        // Domain Shape
        double J = this->dshape_and_dtest_eulerian_axi_nst(
          s, psif, dpsifdx, testf, dtestfdx);
        return J;
      }
      else
      {
        Shape testf(n_node);
        DShape dpsifdx(n_node, n_dim), dtestfdx(n_node, n_dim);
 
        // Domain Shape
        double J = this->dshape_and_dtest_eulerian_axi_nst(
          s, psi, dpsifdx, testf, dtestfdx);
        return J;
      }
    }
 
 
    /// Return interpolated field fld at local coordinate s, at time
    /// level t (t=0: present; t>0: history values)
    double get_field(const unsigned& t,
                     const unsigned& fld,
                     const Vector<double>& s)
    {
      // unsigned n_dim =this->dim();
      // unsigned n_node=this->nnode();
 
      // If fld=n_dim, we deal with the pressure
      if (fld == 3)
      {
        return this->interpolated_p_axi_nst(s);
      }
      // Velocity
      else
      {
        return this->interpolated_u_axi_nst(t, s, fld);
      }
    }
 
 
    /// Return number of values in field fld
    unsigned nvalue_of_field(const unsigned& fld)
    {
      if (fld == 3)
      {
        return this->npres_axi_nst();
      }
      else
      {
        return this->nnode();
      }
    }
 
 
    /// Return local equation number of value j in field fld.
    int local_equation(const unsigned& fld, const unsigned& j)
    {
      if (fld == 3)
      {
        return this->p_local_eqn(j);
      }
      else
      {
        const unsigned u_nodal_index = this->u_index_axi_nst(fld);
        return this->nodal_local_eqn(j, u_nodal_index);
      }
    }
  };
 
 
  //=======================================================================
  /// Face geometry for element is the same as that for the underlying
  /// wrapped element
  //=======================================================================
  template<class ELEMENT>
  class FaceGeometry<
    GeneralisedNewtonianProjectableAxisymmetricCrouzeixRaviartElement<ELEMENT>>
    : public virtual FaceGeometry<ELEMENT>
  {
  public:
    FaceGeometry() : FaceGeometry<ELEMENT>() {}
  };
 
 
  //=======================================================================
  /// Face geometry of the Face Geometry for element is the same as
  /// that for the underlying wrapped element
  //=======================================================================
  template<class ELEMENT>
  class FaceGeometry<FaceGeometry<
    GeneralisedNewtonianProjectableAxisymmetricCrouzeixRaviartElement<ELEMENT>>>
    : public virtual FaceGeometry<FaceGeometry<ELEMENT>>
  {
  public:
    FaceGeometry() : FaceGeometry<FaceGeometry<ELEMENT>>() {}
  };
 
 
} // namespace oomph
 
#endif