# Fluid Rheology¶

## Basic Concepts¶

Fluids are gases and liquids that flow when subject to an applied shear stress. For single-phase fluids, momentum transport is governed by the fluid density and viscosity. For multi-phase systems, the surface tension must also be defined to describe dynamics at the interface between two phases.

M-Star CFD can handle both Newtonian and non-Newtonian fluid rheology. The available fluid models, along with the relevant simulation parameters, are described in the sections that follow.

Turbulent fluid flows in M-Star CFD are typically filtered using large eddy simulation (LES). The effects of the filtering are controlled by the user-defined Smagorinsky Coefficient. Additional theoretical details related to the LES model are provided in the Theory and Implementation section of this manual.

## General Fluid Parameters¶

- Density
Density of fluid, [kg/m^3]

- Surface Tension, [N/m]
Surface tension of the fluid in air.

Only relevant for free surface or immiscible fluid simulations.

Turbulence Model, [Auto, DNS, ILES, LES]

DNS: Direct Numerical Simulation. DNS simulations attempt to capture all fluid motion across all eddy scales. DNS simulations will diverge if the eddy size approaches the lattice spacing. ILES: Implicit Large Eddy Simulations. These models use a larger (27-vector) lattice stencil and a cumulant-based momentum integrator to maintain stability at higher Reynolds numbers. LES: Large Eddy Simulations. These models compute a local eddy viscosity using the local shear rate to capture the effects of sub-grid turbulence. LES models tend to be stable at arbitrary Reynolds numbers. Auto: If maximum Reynolds number detected by the simulation is below 5000, the codes runs a DNS simulation. Above the Reynolds number, an LES model with static Smagorinsky coefficient of 0.10 is applied.

## Newtonian Fluid¶

A Newtonian fluid has a constant viscosity, such that the viscous stresses arising from flow are linearly proportional to the local strain rate.

- Kinematic Viscosity
Kinematic viscosity of fluid, [m^2/s]

- Max User Shear Rate
Max allowable shear rate, [1/s]

Shear rates above and below these will use a constant viscosity equal to that realized at these maximum and minimum rates

- Min User Shear Rate
Min allowable shear rate, [1/s]

Shear rates above and below these will use a constant viscosity equal to that realized at these maximum and minimum rates

- Viscosity At Max Shear
Viscosity that corresponds to the MaxUserShearRate, [m^2/s]

- Viscosity At Min Shear
Viscosity that corresponds to the MinUserShearRate, [m^2/s]

## Power Law Fluid¶

A power law fluid is generalized Newtonian fluid where the shear stress, \(\tau\), is related to the shear rate, \(\dot{\gamma}\) , such that:

where \(\rho\) is the fluid density, \(K\) is the flow consistency, and \(n\) is the fluid behavior index. The units on \(\rho\) are taken to be \(kg/m^3\) , the units on \(K\) are taken to be \(m^2/s^{2-n}\) and \(n\) is dimensionless.

From this constitutive relationship, the apparent viscosity \(\nu_a\) of a power-law fluid is then defined as:

where the units \(\nu_a\) are \(m^2/s\).

This definition of apparent viscosity is used to calculate the spatiotemporal variation in viscosity across the fluid volume due to spatiotemporal variations in strain rate.

- Power Law K
Flow consistency index, [\(m^2/s^{2-n}\)]

- Power Law N
Flow behavior index “n”, [-]

When a yield stress is added to a power law fluid, we have a Herschel-Bulkley fluid. The Herschel-Bulkley model describes the behavior of non-Newtonian yield stress fluids:

where \(\tau\) is the shear stress, \(\dot{\gamma}\) the shear rate, \(\tau_{0}\) the yield stress, \(n\) the consistency index, and \(k\) the flow index.

Like the power law expression, the effective viscosity is then defined as:

- Yield Stress
Yield shear stress, [N / m^2]

- Power Law K
Flow consistency index, [\(m^2/s^{2-n}\)]

- Power Law N
Flow behavior index “n”, [-]

Important

-Users should specify the fluid behavior index in anticipation that `1`

will be subtracted from the specified value when evaluating the
local viscosity.

## Carreau Fluid¶

A Carreau fluid is a generalized Newtonian fluid with an effetive viscosity, \(\mu_{\operatorname{eff}}\) , defined by:

- Carreau Vinf
Viscosity at infinite shear, [m^2/s]

- Carreau V0
Viscosity at zero shear, [m^2/s]

- Carreau Lambda
Relaxation time, [s]

- Carreau N
Power index, [-]

## Herschel-Bulkley Fluid¶

The Herschel-Bulkley model describes the behavior of non-Newtonian yield stress fluids:

where \(\tau\) is the shear stress, \(\dot{\gamma}\) the shear rate, \(\tau_{0}\) the yield stress, \(n\) the consistency index, and \(k\) the flow index.

The effective viscosity is then defined as:

- Yield Stress
Yield shear stress, [N / m^2]

- K
Flow consistency index, [m^2 / s]

- N
Flow behavior index ‘n’ [dimensionless]

## Custom Fluid¶

- Custom Expression
Analytic expression F(s) for the kinematic viscosity in units [m^2/s]. Can be a function of local shear rate ‘s’ with units [1/s], global time ‘t’ with units [s], local temperature ‘T’ with units [K], and the local concentration of any user-defined scalar field [mol].

Additional examples of entering formulas are presented in - User Defined Expression Syntax