# Viscosity Models

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)
$\underset{¯}{\tau }=2{\mu }_{e}\left(\left|\underset{¯}{\stackrel{˙}{\epsilon }}\right|\right)\underset{¯}{\stackrel{˙}{\epsilon }}$
Where,
$\tau$
Stress tensor
$\stackrel{˙}{\epsilon }$
Strain (shear) rate tensor
${\mu }_{e}\left(\stackrel{˙}{\epsilon }\right)$
Effective viscosity as a function of strain rate

In Cross fluids, the effective viscosity ${\mu }_{e}$ varies smoothly from a predefined zero shear rate viscosity ${\mu }_{0}$ in Pa ⋅ s to an infinite shear rate viscosity ${\mu }_{\infty }$ in Pa ⋅ s. To control the transition between these two ends, a coefficient 𝑘 [s] and a dimensionless power of 𝑛 are used in the following form to formalize the relationship with shear rate $\stackrel{˙}{\epsilon }$ in 1/s.

(2)
${\mu }_{e}={\mu }_{\infty }+\frac{{\mu }_{0}-{\mu }_{\infty }}{1+{\left(k\left|\underset{¯}{\stackrel{˙}{\epsilon }}\right|\right)}^{n}}$

In Cross fluids, shear thickening or shear thinning behavior is achieved by changing ${\mu }_{0}$ and ${\mu }_{\infty }$ while 𝑘 and 𝑛 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 ${\mu }_{0}$ and ${\mu }_{\infty }$ . 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 ${\mu }_{0}$ and/or ${\mu }_{\infty }$ to zero as this may result in inviscid fluid at certain shear rates. In addition, while it may be possible to use negative ${\mu }_{\infty }$ , 𝑘 or 𝑛 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 ${\mu }_{0}$ and ${\mu }_{\infty }$ . Setting 𝑘 to zero recovers Newtonian behavior with a viscosity value equal to ${\mu }_{0}$ .

## 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:
1. Polynomial
Specified by five coefficients (4th order polynomial), such that the dynamic viscosity is given by:(3)
$\mu ={C}_{1}+{C}_{2}T+{C}_{3}{T}^{2}+{C}_{4}{T}^{3}+{C}_{5}{T}^{4}$
Where,
$T$
Temperature of the particle
${C}_{n}$
Coefficients

For air, the viscosity can be approximated by a linear function with ${C}_{1}=5×{10}^{-8}$ .

2. Sutherland law
Abides by the following expression:(4)
$\mu =\frac{{\mu }_{0}{\left(\frac{T}{{T}_{0}}\right)}^{\frac{3}{2}}\left({T}_{0}+S\right)}{T+S}$
Where,
${\mu }_{0}$
Reference viscosity
${T}_{0}$
Reference temperature
$S$
Sutherland temperature (coefficient)

For air, these values are: ${\mu }_{0}$ = 1.72 x 10-5Pas, ${T}_{0}$ = 273.15 K and S = 110.4.

3. Power law
Defined by:(5)
$\mu ={\mu }_{0}{\left(\frac{T}{{T}_{0}}\right)}^{n}$

Where $n$ is the exponent. For air, the power law values are: ${\mu }_{0}$ = 1.72 x 10-5 Pas, ${T}_{0}$ = 273.15 K and $n$ = 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.