# 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.

Turbulence Model

Closure model to be applied to the simulation

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. No modification to the viscosity.

LES (Large Eddy Simulation)

These models compute a local eddy viscosity using the local shear rate to capture the effects of sub-grid turbulence. This eddy viscosity is then superimposed on the molecular viscosity when solving the Navier-Stokes equations LES models tend to be stable at arbitrary Reynolds numbers. The effects of the filter on fluid flow decrease with increasing resolution.

Smagorinsky Coefficient

Only relevant for LES simulation. This value is set to 0.1, a value motivated by predictions from direct numerical solution.

## 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 non-Newtonian fluid where the shear stress, $$\tau$$, is related to the shear rate, $$\dot{\gamma}$$ , such that:

$\tau=\rho K \dot{\gamma}^n$

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:

$\nu_a=K\dot{\gamma}^{n-1}$

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:

$\tau = \tau_{0} + k \dot{\gamma} ^ {n}$

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:

$\begin{split}\nu_{\operatorname{eff}} = \begin{cases} \nu_0, & |\dot{\gamma}| \leq \dot{\gamma}_0 \\ k |\dot{\gamma}|^{n-1}+\tau_0 |\dot{\gamma}|^{-1} , & |\dot{\gamma}| \geq \dot{\gamma}_0 \end{cases}\end{split}$
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 non-Newtonian fluid with an effective viscosity, $$\mu_{\operatorname{eff}}$$ , defined by:

$\nu_{\operatorname{eff}}(\dot \gamma) = \nu_{\operatorname{\inf}} + (\nu_0 - \nu_{\operatorname{\inf}}) \left[1+\left(\lambda \dot \gamma\right) ^a \right] ^ {\frac {N-1} {a}}$
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, [-]

Carreau a

Transition index, [-]

## Herschel-Bulkley Fluid¶

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

$\tau = \tau_{0} + k \dot{\gamma} ^ {n}$

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:

$\begin{split}\nu_{\operatorname{eff}} = \begin{cases} \nu_0, & |\dot{\gamma}| \leq \dot{\gamma}_0 \\ k |\dot{\gamma}|^{n-1}+\tau_0 |\dot{\gamma}|^{-1} , & |\dot{\gamma}| \geq \dot{\gamma}_0 \end{cases}\end{split}$
Yield Stress

Yield shear stress, [N / m^2]

K

Flow consistency index, [m^2 / s]

N

Flow behavior index ‘n’ [dimensionless]

## Cross Fluid¶

A Cross fluid is a generalized non-Newtonian fluid with an effective viscosity, $$\mu_{\operatorname{eff}}$$ , defined by:

$\nu_{\operatorname{eff}}(\dot \gamma) = \nu_{\operatorname{\infty}} + \frac{\nu_0 - \nu_{\operatorname{\infty}}}{1+\left(k \dot{\gamma}\right)^n }$
Cross Vinf

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

Cross V0

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

Cross Gamma

Relaxation time, [s]

Cross n

Power index, [-]

## Custom Fluid¶

Custom Expression

Analytic expression F(s) for the kinematic viscosity in units [m^2/s].

Arbitrary C++ expression that may be an arbitrary function of strain, stress, age, species concentration, miscible fluid volume fraction, custom variables, particle concentration, and temperature. May include conditional statements. Evaluated at runtime at each lattice site at each timestep using current physical properties.

This function should return a physically realistic viscosity across all possible input parameters. Physically realistic viscosities typically range from $$10^{-7}-10^0$$ m^2/s. Higher viscosities may require smaller timesteps to remain stable.

Note

• In addition to returning physically realistic numbers, care must be taken to ensure that the values returned by this user-defined function are real. For example, a negative base raised to a non-integer exponent is not defined in real space and should not be used to define a transport coefficient.