FluidFlowPDEComponent

yields a flow PDE term with variables vars and parameters pars.

Details

FluidFlowPDEComponent models flow of viscous fluids subject to applied forces and constraints.
FluidFlowPDEComponent returns a sum of differential operators to be used as a part of partial differential equations:

FluidFlowPDEComponent creates PDE components for stationary, time-dependent, parametric analysis.
FluidFlowPDEComponent models fluid flow phenomena with velocities , and in units of [] as dependent variables, as independent variables in units of [] and time variable in units of [].
FluidFlowPDEComponent creates PDE components in two and three space dimensions.
Stationary variables vars are vars={{u[x₁,…,x_n],v[x₁,…,x_n],…,p[x₁,…,x_n]},{x₁,…,x_n}}.
Time-dependent or eigenmode variables vars are vars={{u[t,x₁,…,x_n],v[t,x₁,…,x_n],…,p[x₁,…,x_n]},t,{x₁,…,x_n}}.
The equations for different analysis types that FluidFlowPDEComponent generates depend on the form of vars.
FluidFlowPDEComponent creates a system of equations with the vector-valued Navier–Stokes equation combined with the continuity equation.
The time-dependent equilibrium equation of the fluid dynamics PDE FluidFlowPDEComponent with mass density [ $TemplateBox[{InterpretationBox[, 1], {"kg", , "/", , {"m", ^, 3}}, kilograms per meter cubed, {{(, "Kilograms", )}, /, {(, {"Meters", ^, 3}, )}}}, QuantityTF]$ ], fluid velocity [], time [], viscous stress tensor [], pressure [], identity matrix and body load vectors [ $TemplateBox[{InterpretationBox[, 1], {"N", , "/", , {"m", ^, 3}}, newtons per meter cubed, {{(, "Newtons", )}, /, {(, {"Meters", ^, 3}, )}}}, QuantityTF]$ ] is based on the Navier–Stokes equation and the continuity equation:

Here [] is the dynamic viscosity, and infinitesimal, small deformation strain rate measure [1/] is given as:

FluidFlowPDEComponent creates a PDE model for compressible or incompressible fluid flow, depending on the nature of the values of the mass density .
The compressible fluid dynamics model is given as:

For constant values of the mass density , the mass continuity equation simplifies to a volumetric continuity equation , and with that, the viscous stress tensor simplifies to .
The incompressible fluid dynamics model is given as:

The stationary equilibrium equations have .
The units of the Navier–Stokes model terms are a force density in [ $TemplateBox[{InterpretationBox[, 1], {"N", , "/", , {"m", ^, 3}}, newtons per meter cubed, {{(, "Newtons", )}, /, {(, {"Meters", ^, 3}, )}}}, QuantityTF]$ ].
The units of the mass continuity equation model terms are a force density in [ $TemplateBox[{InterpretationBox[, 1], {"kg", , "/(", , {"m", ^, 3}, , "s", , ")"}, kilograms per meter cubed second, {{(, "Kilograms", )}, /, {(, {{"Meters", ^, 3}, , "Seconds"}, )}}}, QuantityTF]$ ], and for the volumetric continuity [1/].
Laminar flow is typical for , where is the Reynolds number.
The Reynolds number is defined as , where [] is a characteristic length and the flow velocity.
The following parameters pars can be given:

parameter	default	symbol
"DynamicViscosity"	-	, dynamic viscosity []
"FluidLoad"	0	, body force density [ $TemplateBox[{InterpretationBox[, 1], {"N", , "/", , {"m", ^, 3}}, newtons per meter cubed, {{(, "Newtons", )}, /, {(, {"Meters", ^, 3}, )}}}, QuantityTF]$ ]
"FluidDynamicsMaterialModel"	"Newtonian"	none
"MassDensity"	-	, density in [ $TemplateBox[{InterpretationBox[, 1], {"kg", , "/", , {"m", ^, 3}}, kilograms per meter cubed, {{(, "Kilograms", )}, /, {(, {"Meters", ^, 3}, )}}}, QuantityTF]$ ]
"Material"	-	none
"ModelForm"	"Conservative"	none
"ReynoldsNumber"	-

If a "Material" is specified, the material constants are extracted from the material data; otherwise, relevant material parameters need to be specified.
Instead of material parameters, a Reynolds number can be specified:

The default material model is a Newtonian flow model.
Alternate material models can be specified by setting the "FluidDynamicsMaterialModel" key in parameters pars.
The following non-Newtonian material models are available:
material model name

"PowerLaw"

"Carreou"

"Bingham-Papanastasiou"

"Herschel-Bulkley-Papanastasiou"
For compressible non-Newtonian fluids, the viscous stress tensor is defined as:

The apparent viscosity is a function of the shear rate .
Additional material model-specific parameters for a model with "ModelName" can be specified through "FluidDynamicsMaterialModel"-><|"ModelName"-><|...|>|>.
The "PowerLaw" model, a general-purpose model, implements .
The following parameters can be given for the "PowerLaw" model:

parameter	default	symbol
"Exponent"		, exponent
"InfiniteShearRateViscosity"		, viscosity at inifinite shear rate
"Lambda"		, relaxation time []
"ZeroShearRateViscosity"		, viscosity at zero shear rate

The "Carreou" model can also be used for a Cross model where with .
The "Bingham-Papanastasiou" model, useful for viscoplastic material, implements .
The following parameters can be given for the "Bingham-Papanastasiou" model:

The "Herschel-Bulkley-Papanastasiou" model, a mixture of a "PowerLaw" and "Bingham-Papanastasiou" model, implements . The model uses a power law to compute the plastic viscosity of the Bingham–Papanastasiou model, and parameters for both models can be set.
A custom apparent viscosity function fun can be specified as "FluidDynamicsMaterialModel"-><|"Custom"-><|"ApparentViscosityFunction"->myApparentViscosity[_,vars_,pars_,data__]|>|>.
Custom viscous stress tensor functions fun can be specified as "FluidDynamicsMaterialModel"->myViscousStressTensor[vars_,pars_,data__].
Non-isothermal flow can be modeled through the Boussinesq approximation.
FluidFlowPDEComponent uses "SIBase" units. The geometry has to be in the same units as the PDE.
If the FluidFlowPDEComponent depends on parameters that are specified in the association pars as …,keyp_i…,p_iv_i,…], the parameters are replaced with .