Emulsion Rheology Model Reference

The Emulsion Rheology model is used for fluids that consist of liquid droplets that are suspended in another liquid. The mixture viscosity depends on the volume fraction of the dispersed phase, and the fluid exhibits non-Newtonian behavior.

At a critical volume fraction, the phases can invert, and the dynamic viscosities of the continuous and dispersed phases swap.

Table 1. Emulsion Rheology Model Reference
Theory	See Rheology of Emulsions and Suspensions.
Provided By	[phase interaction] > Models > Optional Models
Example Node Path	[phase interaction] > Models > Emulsion Rheology
Requires	A Eulerian Multiphase simulation with the following models activated: Material: Multiphase Multiphase Model: Eulerian Multiphase (EMP) Viscous Regime: Laminar or Turbulent For Turbulent: EMP Turbulence: Mixture Turbulence Turbulence: Reynolds-Averaged Navier-Stokes Reynolds-Averaged Turbulence: K-Epsilon Turbulence or Reynolds Stress Turbulence For K-Epsilon Turbulence K-Epsilon Turbulence Models: One of EB K-Epsilon, Lag EB K-Epsilon, Standard K-Epsilon Low-Re Wall Treatment: any For Reynolds Stress Turbulence: Reynolds Stress Turbulence Models: Elliptic Blending or Linear Pressure Strain Two-Layer Wall Treatment: All y+ Wall Treatment (for Elliptic Blending) or Two-Layer All y+ Wall Treatment (for Linear Pressure Strain Two-Layer) Two Liquid phases. A Continuous-Dispersed Topology phase interaction is required. The following phase interaction model activated: Optional Models: Emulsion Rheology
Properties	Key properties are: Maximum Packing. See Emulsion Rheology Properties.
Activates	Model Controls (child nodes)	Anisotropy Tensor Calibration Prefactor See Emulsion Rheology Child Nodes.
	Materials	Under the Multiphase Material node in the phase interaction: Normal Relative Viscosity Osmotic Pressure Relative Viscosity Relaxation Time Surface Tension Yield Relative Viscosity Yield Stress See Materials and Methods.
	Field Functions	See Field Functions.

Emulsion Rheology Properties

Maximum Packing

The maximum packing fraction is defined in a similar way as for suspensions of particles.

The specified value is multiplied by the calibration prefactor $C$ to produce $ϕ_{m}$ , which is used in Eqn. (2429) and Eqn. (2432) (for the Morris and Boulay model), and Eqn. (2428) (for the Krieger-Dougherty model).

A calibration prefactor of 1 reproduces the physics of hard particles, but for soft deformable particles the value is greater than 1.

Emulsion Rheology Child Nodes

Anisotropy Tensor

The Morris and Boulay model was derived from experiments in pipes and Couette devices where the flow becomes aligned in a particular direction that depends on the geometry. The normal stress was shown to be anisotropic and, at different strengths, lead to anisotropic normal stress differences.

Method	Corresponding Method Node
Isotropic Normal Stress Tensor	Sets the Anisotropy tensor equal to the identity tensor Eqn. (2431), and gives equal weighting to all directions. This method is the default and should be used for 3D flow.
X-Direction Aligned Anisotropic Normal Stress Tensor	Used for shear flow in the x-direction. The tensor is given in Eqn. (2430).

Calibration Prefactor

This factor

C

modifies the specified value of maximum packing fraction to account for changes that are due to shear-rate, temperature, or deformation of droplets. For example, in some emulsions, the deformation of droplets means that the maximum packing value can approach 1. You can use a constant or a field function to specify this factor.

See Maximum Packing.

Materials and Methods

Normal Relative Viscosity

Describes the normal relative viscosity of the mixture, which is similar to a particle pressure. If you set the Relative Viscosity property to use the Krieger-Dougherty model [491], this value defaults to zero.

Method

Corresponding Method Node

Morris and Boulay Model

The normal relative viscosity is defined using the Morris and Boulay model (see Eqn. (2432)).

Morris and Boulay Model

The Morris and Boulay Model has the following properties:

Maximum Viscosity
Specifies the maximum value of the relative viscosity that is allowed in the simulation.

In Eqn. (2432), the normal relative viscosity $η_{n}$ tends to infinity as the dispersed phase reaches the limit of maximum packing.
Contact Contribution
The contact contribution, $K_{n}$ in Eqn. (2432).

Shear Thinning Model

The normal relative viscosity depends on the shear strain-rate of the flow (see Eqn. (2435)).

Shear Thinning Model

The Shear Thinning Model has the following properties:

Maximum Viscosity
Sets the maximum viscosity cut-off for the zero strain rate Morris and Boulay normal viscosity. This value is $η_{n 0} (ϕ)$ in Eqn. (2433).
Maximum Viscosity Infinity
Sets the maximum viscosity cut-off for the infinite strain rate Morris and Boulay normal viscosity. This value is $η_{n \infty} (ϕ)$ in Eqn. (2434).
Maximum Packing at Zero Shear Rate
Sets the asymptote for the zero-shear normal viscosity curves. This value is $ϕ_{m 0}$ in Eqn. (2433).
Maximum Packing at Infinite Shear Rate
Sets the asymptote for the infinite-shear normal viscosity curves. This value is $ϕ_{m \infty}$ in Eqn. (2434).
Normal Zero-Strain Contact Parameter
Sets the strength of the particle interaction for the zero-shear normal viscosity curve. This value is $K_{n 0}$ in Eqn. (2433).
Normal Infinite-Strain Contact Parameter
Sets the strength of the particle interaction for the infinite-shear normal viscosity curve. This value is $K_{n \infty}$ in Eqn. (2434).
Use Normal Yield Stress
Activates the use of the Yield Stress material property.

Shear Thinning Model > Shear-thinning index

Sets the index

n

in Eqn. (2435) and Eqn. (2439).

To specify this index, you can use a constant, a field function, or the following method:


Method	Corresponding Properties
Linear Relation Describes $n$ as a function of the dispersed phase volume fraction (see Eqn. (2436)).	n1 Sets $n_{1}$ in Eqn. (2436). n2 Sets $n_{2}$ in Eqn. (2436).

Osmotic Pressure

The osmotic pressure is used to stop particles from exceeding their maximum packing volume fraction. This setting is useful for situations where the dispersed volume fraction could become large, such as in particle settling experiments.

The osmotic pressure is added to the particle momentum source in a similar way to the normal relative viscosity. The osmotic pressure is calculated using Eqn. (2442).

Method	Corresponding Method Node
Hard Sphere Osmotic Pressure Specifies $Π$ using the Hard Sphere Osmotic Pressure formulation.	Hard Sphere Osmotic Pressure The Hard Sphere Osmotic Pressure model has the following properties: Maximum Packing The maximum critical packing fraction (assuming random close-packing); $ϕ_{m}$ in Eqn. (2443). This value is 0.645 for hard spheres. This value is a theoretical limit and is lower in most practical cases. The value is lower again for non-spherical particles. Maximum Pressure Specifies the maximum value of the osmotic pressure $Π$ . When the particles start to overlap, the osmotic pressure takes this maximum value, representing the infinite hard sphere potential. When the dispersed phase volume fraction is above the specified maximum packing value, $Π$ is given this value to prevent the volume fraction from increasing much above the maximum packing.

Method

Corresponding Method Node

Hard Sphere Osmotic Pressure

Specifies $Π$ using the Hard Sphere Osmotic Pressure formulation.

Hard Sphere Osmotic Pressure

The Hard Sphere Osmotic Pressure model has the following properties:

Maximum Packing
The maximum critical packing fraction (assuming random close-packing); $ϕ_{m}$ in Eqn. (2443). This value is 0.645 for hard spheres.

This value is a theoretical limit and is lower in most practical cases. The value is lower again for non-spherical particles.
Maximum Pressure
Specifies the maximum value of the osmotic pressure $Π$ . When the particles start to overlap, the osmotic pressure takes this maximum value, representing the infinite hard sphere potential.

When the dispersed phase volume fraction is above the specified maximum packing value, $Π$ is given this value to prevent the volume fraction from increasing much above the maximum packing.

When the dispersed phase volume fraction is below the specified maximum packing value, the osmotic pressure is calculated using Eqn. (2442).

Relative Viscosity

This dimensionless value describes the viscosity of the multiphase mixture. The relative viscosity (see Eqn. (2422)) tends to infinity as the dispersed phase reaches the limit of maximum packing.

Method	Corresponding Method Node
Krieger-Dougherty Model Describes non-Newtonian flow behavior in rigid sphere suspensions (see Eqn. (2428)). This model takes into account the interactions between neighboring spherical particles. See Krieger-Dougherty Model.	Krieger-Dougherty Model The Krieger-Dougherty model has the following properties: Maximum Viscosity Specifies the maximum value of the relative viscosity that is allowed in the simulation. In Eqn. (2428), the relative viscosity $η_{r}$ tends to infinity as the dispersed phase reaches the limit of maximum packing. Intrinsic Viscosity Exponent The intrinsic viscosity; $[η]$ in Eqn. (2428). This value is 2.5 for spherical particles.
Morris and Boulay Model Describes the migration phenomenon that is observed in curvilinear flows of concentrated suspensions ([521]). This model uses shear-induced normal stresses to provide the driving force for migration (see Eqn. (2429)).	Morris and Boulay Model The Morris and Boulay model has the following properties: Maximum Viscosity Specifies the maximum value of the relative viscosity that is allowed in the simulation. In Eqn. (2429), the relative viscosity $η_{r}$ tends to infinity as the dispersed phase reaches the limit of maximum packing. Contact Contribution The contact contribution, $K_{s}$ in Eqn. (2429).
Shear Thinning Model Blends a zero-shear and infinite shear Morris and Boulay relative viscosity model [521] with a Carreau Generalized Newtonian model (see Eqn. (2435)).	Shear Thinning Model The Shear Thinning model has the following properties: Maximum Relative Viscosity 0 Sets the maximum viscosity cut-off for the zero strain rate Morris and Boulay viscosity. This value is $η_{r 0} (ϕ)$ in Eqn. (2433). Maximum Relative Viscosity Infinity Sets the maximum viscosity cut-off for the infinite strain rate Morris and Boulay viscosity. This value is $η_{r \infty} (ϕ)$ in Eqn. (2434). Maximum Packing at Zero Shear Rate Sets the asymptote for the zero-shear viscosity curves. This value is $ϕ_{m 0}$ in Eqn. (2433). Maximum Packing at Infinite Shear Rate Sets the asymptote for the infinite-shear viscosity curves. This value is $ϕ_{m \infty}$ in Eqn. (2434). Zero-Shear Contact Parameter Sets the strength of the particle interaction for the zero-shear curve. This value is $K_{s 0}$ in Eqn. (2433). Infinite-Shear Contact Parameter Sets the strength of the particle interaction for the infinite-shear curve. This value is $K_{s \infty}$ in Eqn. (2434). Use Yield Stress Activates the use of the Yield Stress material property.

Method

Corresponding Method Node

Krieger-Dougherty Model

Describes non-Newtonian flow behavior in rigid sphere suspensions (see Eqn. (2428)). This model takes into account the interactions between neighboring spherical particles.

See Krieger-Dougherty Model.

Krieger-Dougherty Model

The Krieger-Dougherty model has the following properties:

Maximum Viscosity
Specifies the maximum value of the relative viscosity that is allowed in the simulation.

In Eqn. (2428), the relative viscosity $η_{r}$ tends to infinity as the dispersed phase reaches the limit of maximum packing.
Intrinsic Viscosity Exponent
The intrinsic viscosity; $[η]$ in Eqn. (2428). This value is 2.5 for spherical particles.

Morris and Boulay Model

Describes the migration phenomenon that is observed in curvilinear flows of concentrated suspensions ([521]). This model uses shear-induced normal stresses to provide the driving force for migration (see Eqn. (2429)).

Morris and Boulay Model

The Morris and Boulay model has the following properties:

Maximum Viscosity
Specifies the maximum value of the relative viscosity that is allowed in the simulation.

In Eqn. (2429), the relative viscosity $η_{r}$ tends to infinity as the dispersed phase reaches the limit of maximum packing.
Contact Contribution
The contact contribution, $K_{s}$ in Eqn. (2429).

Shear Thinning Model

Blends a zero-shear and infinite shear Morris and Boulay relative viscosity model [521] with a Carreau Generalized Newtonian model (see Eqn. (2435)).

Shear Thinning Model

The Shear Thinning model has the following properties:

Maximum Relative Viscosity 0
Sets the maximum viscosity cut-off for the zero strain rate Morris and Boulay viscosity. This value is $η_{r 0} (ϕ)$ in Eqn. (2433).
Maximum Relative Viscosity Infinity
Sets the maximum viscosity cut-off for the infinite strain rate Morris and Boulay viscosity. This value is $η_{r \infty} (ϕ)$ in Eqn. (2434).
Maximum Packing at Zero Shear Rate
Sets the asymptote for the zero-shear viscosity curves. This value is $ϕ_{m 0}$ in Eqn. (2433).
Maximum Packing at Infinite Shear Rate
Sets the asymptote for the infinite-shear viscosity curves. This value is $ϕ_{m \infty}$ in Eqn. (2434).
Zero-Shear Contact Parameter
Sets the strength of the particle interaction for the zero-shear curve. This value is $K_{s 0}$ in Eqn. (2433).
Infinite-Shear Contact Parameter
Sets the strength of the particle interaction for the infinite-shear curve. This value is $K_{s \infty}$ in Eqn. (2434).
Use Yield Stress
Activates the use of the Yield Stress material property.

Relaxation Time

Sets the strain-rate at which shear-thinning occurs. This property is equivalent to the relaxation time $λ$ in the Carreau-Yasuda non-Newtonian model (see Eqn. (740)).

The relaxation time is typically a constant or a function of volume fraction. It is entered as a scalar profile.

Note	Available only when the Relative Viscosity material property uses the Shear Thinning Model.

Surface Tension

The surface tension coefficient, $σ$ , expresses the ease with which the two fluid phases can be mixed and is defined as the amount of work necessary to create a unit area of free surface. Its magnitude depends on the nature of the fluids in contact: for immiscible fluids, the value is always positive. The surface tension coefficient is entered as a scalar profile.

Yield Relative Viscosity

The yield relative viscosity is analogous to the yield viscosity $μ_{0}$ in the Herschel-Bulkley non-Newtonian model (see Eqn. (741)). This quantity is typically a large value that describes the material when it is solid-like (below the critical shear-strain rate).

The yield relative viscosity is typically a constant. It is entered as a scalar profile.

Note	Available only when the Relative Viscosity material property uses the Shear Thinning Model.

Yield Stress

The yield stress is analogous to the yield stress $τ_{0}$ in the Herschel-Bulkley non-Newtonian model (see Eqn. (741)).

Note	Available only when the Relative Viscosity material property uses the Shear Thinning Model.

Method	Corresponding Method Node
Emulsion Yield Stress The stress above which the emulsion behaves as a fluid. If the applied stress is less than the emulsion yield stress, the emulsion behaves as a solid. The emulsion yield stress $τ_{y}$ is calculated in Eqn. (2440).	Emulsion Yield Stress The Emulsion Yield Stress model has the following properties: Maximum Packing The maximum critical packing fraction; $ϕ_{m}$ in Eqn. (2440). Emulsions can have a much larger maximum packing fraction than particle suspensions due to the deformability of liquid droplets. Maximum Yield Stress Sets the maximum value of the Emulsion Yield Stress, $τ_{y}$ , that is used in Eqn. (2439). Prefactor A constant prefactor; $τ_{y 0}$ in Eqn. (2440). The default value is 0.5, as used in [516].

Method

Corresponding Method Node

Emulsion Yield Stress

The stress above which the emulsion behaves as a fluid. If the applied stress is less than the emulsion yield stress, the emulsion behaves as a solid.

The emulsion yield stress $τ_{y}$ is calculated in Eqn. (2440).

Emulsion Yield Stress

The Emulsion Yield Stress model has the following properties:

Maximum Packing
The maximum critical packing fraction; $ϕ_{m}$ in Eqn. (2440).

Emulsions can have a much larger maximum packing fraction than particle suspensions due to the deformability of liquid droplets.
Maximum Yield Stress
Sets the maximum value of the Emulsion Yield Stress, $τ_{y}$ , that is used in Eqn. (2439).
Prefactor
A constant prefactor; $τ_{y 0}$ in Eqn. (2440). The default value is 0.5, as used in [516].

Field Functions

The following field functions are available:

Mixture Viscosity of [phase interaction]
Normal Relative Viscosity of [phase interaction]
Osmotic Pressure of [phase interaction]
Relative Viscosity of [phase interaction]
Relaxation Time of [phase interaction]
Slip Viscosity of [phase interaction]
Yield Relative Viscosity of [phase interaction]
Yield Stress of [phase interaction]