********************
Regularization Model
********************

::

   Regularization Model = <MODEL>

-------------------
Description / Usage
-------------------

This card is used to specify the model for the regularization used in the generalized 
Newtonian viscosity.

Currently used for *HERSCHEL_BULKLEY* models

Choices for <MODEL> in *HERSCHEL_BULKLEY* fluids:

EPSILON
   Epsilon regularization applied to both the power law term and the yield term (default for HERSCHEL_BULKLEY)

    .. math::

       \mu = \mu_0 (\dot{\gamma} + \epsilon)^{n-1} + \frac{\tau_y}{(\dot{\gamma} + \epsilon)}
                                                                                                   
PAPANASTASIOU
   Papanastasiou regularization applied to the yield parameter (default for HERSCHEL_BULKLEY_PAPANASTASIOU)

    .. math::
      
       \mu = \mu_0 \dot{\gamma}^{n-1} + (1-exp(-f \dot{\gamma})) \frac{\tau_y}{\dot{\gamma}}

EPSILON_YIELD_ONLY
   Epsilon regularization applied only to the yield term

    .. math::

       \mu = \mu_0 (\dot{\gamma})^{n-1} + \frac{\tau_y}{(\dot{\gamma} + \epsilon)}

PAPANASTASIOU_EPSILON
   Papanastasiou regularization applied to the yield term and an epsilon on the power law term

    .. math::
      
       \mu = \mu_0 (\dot{\gamma}+ \epsilon)^{n-1} + (1-exp(-f \dot{\gamma})) \frac{\tau_y}{\dot{\gamma}}




--------
Examples
--------

The following is a sample card that sets Papanastasiou regularization to a HERSCHEL_BULKLEY fluid
with an epsilon on the power law term

::

   Liquid Constitutive Equation = HERSCHEL_BULKLEY
   ...
   Regularization Model         = PAPANASTASIOU_EPSILON
   Yield Exponent               = CONSTANT 200
   Epsilon Regularization       = CONSTANT 1e-12

Without the epsilon term

::

   Liquid Constitutive Equation = HERSCHEL_BULKLEY
   ...
   Regularization Model         = PAPANASTASIOU
   Yield Exponent               = CONSTANT 200

Similarly applying only to the Yield term for epsilon regularization

::

   Liquid Constitutive Equation = HERSCHEL_BULKLEY
   ...
   Regularization Model         = EPSILON_YIELD_ONLY
   Epsilon Regularization       = CONSTANT 1e-5

Or to both terms

::

   Liquid Constitutive Equation = HERSCHEL_BULKLEY
   ...
   # optional this is the default
   Regularization Model         = EPSILON
   Epsilon Regularization       = CONSTANT 1e-5

--------------------
Technical Discussion
--------------------

See Description/Usage for *Liquid Constitutive Equation*