nanoFluidX now supports both temperature-viscosity and
non-Newtonian modeling behavior. For temperature-viscosity, three models were implemented:
polynomial, Sutherland, and power law. For non-Newtonian, the Cross model is available,
which can be used to approximate power law behavior without risking instabilities due to
viscosity unboundness of the power model.
Important: Temperature-viscosity and non-Newtonian modeling behavior
cannot be used simultaneously. For more information, refer to
Viscosity Models in the
Command Reference
Guide.
Non-Newtonian Viscosity Model (Cross model)
Newtonian fluids, where shear and strain rate have a linear relationship through a
single constant called viscosity, are one of the simplest models used to describe
fluid behavior. However, some common fluids exhibit more complex behavior where the
shearing resistance changes depending on the duration of the shear or the rate of
strain applied to the fluid. Some lubricants, many colloidal suspensions, yogurt,
ketchup and blood are a few examples of such fluids, collectively called
non-Newtonian fluids.
In their simplest form, non-Newtonian fluids may have an effective viscosity that
only depends on strain rate without being dependent on the history of the
deformation. Also referred to as generalized Newtonian fluids, these simpler
non-Newtonian fluids may exhibit shear thickening (dilatant) or shear thinning
(pseudoplastic) behaviors with a stress-strain rate relationship of the form for an
incompressible fluid.
(1)
Where,
-
- Stress tensor
-
- Strain (shear) rate tensor
-
- Effective viscosity as a function of strain rate
In Cross fluids, the effective viscosity
varies smoothly from a predefined zero shear rate
viscosity
in
Pa.s
to an infinite shear rate viscosity
in
Pa.s.
To control the transition between these two ends, a coefficient
[s] and a dimensionless power of
are used in the following form to formalize the relationship with shear rate
in 1/s.
(2)
In Cross fluids, shear thickening or shear thinning behavior is achieved by changing
and
while
and
control
the curve connecting these two values.
In the absence of an explicit definition for reference viscosity, the code will use
the maximum value specified for
and
. This may not be ideal for shear thickening flows as
shear rate levels producing the maximum viscosity may not appear in the simulation.
It is not recommended to set
and/or
to zero as this may result in inviscid fluid at
certain shear rates. In addition, while it may be possible to use negative
,
or
values without jeopardizing the simulation, these scenarios were not tested and are
not recommend as they may result in negative viscosity or viscosity values beyond
the interval defined by
and
. Setting
to zero recovers Newtonian behavior with a viscosity value equal to
.
Temperature-viscosity Models
Temperature-viscosity models are useful when temperature changes significantly
influence the viscosity of the fluid. The time-scale of heat transfer in the fluid
must be considered in order for these models to exhibit their function. For example,
attempting to capture very slow heat transfer dynamics in a complex multiphase flow
while having very high resolution and high reference velocities is likely
prohibitively, computationally expensive. This feature works best in cases where
heat transfer is happening on the same time-scale as the fluid dynamics (low
velocity flows) and high temperature gradients are present in the fluid.
Important: In order to enable temperature-viscosity coupling models, both
the
varvisc_temperature and
energy_transport switches in the
Simulation Parameters section must be
set to true.
There are three temperature-viscosity models available in
nanoFluidX:
- Polynomial
Specified by five coefficients (4th order polynomial), such
that the dynamic viscosity is given by:
(3)
Where,
-
- Temperature of the particle
-
- Coefficients
For air, the viscosity can be approximated by a linear
function with
.
- Sutherland law
Abides by the following expression:
(4)
Where,
-
- Reference viscosity
-
- Reference temperature
-
- Sutherland temperature (coefficient)
For air, these values are:
= 1.72 x 10-5Pas,
= 273.15 K and
S = 110.4.
- Power law
Defined by:
(5)
Where
is the exponent. For air, the power law
values are:
= 1.72 x 10-5 Pas,
= 273.15 K and
= 0.66.
Important:
- Since the viscosity field is updated after establishing the time step,
reference viscosity must be specified in the Domain Parameter section as ref_visc. The
reference viscosity should be the highest expected viscosity during the
simulation.
- After the reference viscosity has been set, the Viscosity Models section must be defined.
- If the varvisc_temperature or
varvisc_non_Newtonian options are turned on, it is
recommended that all of the fluid phase viscosities in the case be defined
in the viscosity models section of the configuration file.