Suspension Rheology Model Reference

The Suspension Rheology model describes the non-Newtonian viscosity of a fluid that consists of solid particles suspended in a liquid. The mixture viscosity depends on the volume fraction of the dispersed phase, and the fluid exhibits non-Newtonian behavior.

Table 1. Suspension Rheology Model Reference
Theory	See Rheology of Emulsions and Suspensions.
Provided By	[physics continuum] > Models > Optional Models
Example Node Path	[physics continuum] > Models > Suspension Rheology
Requires	An Eulerian Multiphase simulation with the following models activated: Material: Multiphase Multiphase Model: Eulerian Multiphase (EMP) Viscous Regime: Laminar Optional Models: Suspension Rheology One liquid phase and one or more solid particle phases: Phase 1, the continuous phase: Material: Liquid or Multi-Component Liquid Phase 2, the dispersed phase: Material: Particle or Multi-Component Particle A Continuous-Dispersed Topology phase interaction is required.
Activates	Models	Under the Eulerian Multiphase > Eulerian Phases > [phase] node: For Liquid phases: Continuous Liquid Rheology For Particle phases: Solid Particle Rheology
	Properties	Key property of Solid Particle Rheology is: Maximum Solid Fraction. See Solid Particle Rheology Properties.
	Materials	Under the Eulerian Multiphase > Mixture > Material Properties node: Normal Relative Viscosity Osmotic Pressure Relative Viscosity Suspension Anisotropy See Materials and Methods.
	Field Functions	See Field Functions.

Solid Particle Rheology Properties

Maximum Solid Fraction: The maximum solid fraction, assuming random close packing, is 0.645 for hard spheres. This value is a theoretical limit; the actual limit is lower for most practical cases. The value is lower again for non-spherical particles.

Materials and Methods

Normal Relative Viscosity

Describes the normal relative shear viscosity of the mixture. This value controls the shear-induced migration of the particles. If you set the Relative Viscosity property to use the Krieger-Dougherty model [491], this value should be set to constant 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 normal 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). This value controls the strength of the shear-induced migration of particles.

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 normal 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). This value controls the strength of the shear-induced migration of particles.

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. This option is temperature-dependent. It is available when an energy model is selected for the individual phases, or the Segregated Fluid Isothermal model is assigned to each phase.	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.

This option is temperature-dependent. It is available when an energy model is selected for the individual phases, or the Segregated Fluid Isothermal model is assigned to each phase.

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 shear contact contribution, $K_{s}$ in Eqn. (2429).

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 shear contact contribution, $K_{s}$ in Eqn. (2429).

Suspension Anisotropy

Specifies the material property to be anisotropic. This property is a tensor profile, although it has limited options to only diagonal form. This property is handled in a similar way to anisotropic conductivity, that is, the tensor appears in the region. The only available method is Anisotropic, although the tensor can be made isotropic by adjusting the Physics Values in the region to Isotropic Tensor.

Field Functions

The following field functions are available:

Mixture Viscosity
Normal Relative Viscosity
Osmotic Pressure
Phase Particle Pressure of [phase]
Phase-Pair Eotvos Number of [phase interaction]
Phase-Pair Reynolds Number of [phase interaction]
Phase-Pair Single-Particle Reynolds Number of [phase interaction]
Relative Viscosity of [phase interaction]
Relaxation Time of [phase interaction]
Slip Viscosity of [phase interaction]
Solid Viscosity of [solid phase]