# FluidFlowPDEComponent

FluidFlowPDEComponent[vars,pars]

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,,xn],v[x1,,xn],,p[x1,,xn]},{x1,,xn}}.
• Time-dependent or eigenmode variables vars are vars={{u[t,x1,,xn],v[t,x1,,xn],,p[x1,,xn]},t,{x1,,xn}}.
• 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 NavierStokes equation combined with the continuity equation.
• The time-dependent equilibrium equation of the fluid dynamics PDE FluidFlowPDEComponent with mass density [], fluid velocity [], time [], viscous stress tensor [], pressure [], identity matrix and body load vectors [] is based on the NavierStokes equation and the continuity equation:
• In compressible form, the viscous stress tensor is given as:
• 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 NavierStokes model terms are a force density in [].
• The units of the mass continuity equation model terms are a force density in [], 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 [] "FluidDynamicsMaterialModel" "Newtonian" none "MassDensity" - , density in [] "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"
• 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 "MinimalShearRate" , minimal shear rate "ReferenceShearRate" , reference shear rate "PowerLawViscosity" , power law viscosity
• The "Carreou" model, useful for polymer flow, implements .
• The following parameters can be given for the "Carreou" 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 model, useful for viscoplastic material, implements .
• The following parameters can be given for the model:
•  parameter default symbol "PlasticViscosity" , plastic viscosity "YieldStress" , yield stress "ShearRateFactor" , shear rate factor
• The model, a mixture of a "PowerLaw" and model, implements . The model uses a power law to compute the plastic viscosity of the BinghamPapanastasiou 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 ,keypi,pivi,], the parameters are replaced with .

# Examples

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## Basic Examples(4)

Define a flow PDE model:

Define a symbolic flow PDE:

Define a stationary flow PDE model with a Reynolds number of 100:

Solve for the flow velocity and pressure in a driven cavity with a Reynolds number of 1000:

Visualize the velocity of the fluid:

## Scope(8)

Define a flow PDE model for a specific material:

Specify a flow PDE with a dynamic viscosity of and a mass density of :

Activate a flow PDE model for a specific material:

Specify a symbolic stationary fluid dynamics PDE in two dimensions with a dynamic viscosity of and a mass density of :

Specify a symbolic stationary fluid dynamics PDE in three dimensions with a dynamic viscosity of and a mass density of :

Define a time-dependent flow PDE model:

Specify a symbolic time-dependent fluid dynamics PDE in two dimensions with a dynamic viscosity of and a mass density of :

A compressible fluid dynamics PDE is generated if the mass density is a function of space or time :

An incompressible fluid dynamics PDE is generated if the mass density is a constant:

## Applications(3)

### Stationary Analysis(1)

Solve for the velocity and pressure in a cavity, where the flow is driven at the top of a box:

Visualize the velocity of the fluid:

### Non-Newtonian Flow(1)

Compute the fluid flow of a non-Newtonian fluid in an opening channel by using the power law fluid flow model.

Set up the region:

Set up variables, flow parameters and the non-Newtonian power law for parameters for the exponent and power law viscosity:

Solve the PDE with an inflow profile given as {1/2,0} and with the outflow pressure set to 0. The walls are set up as no-slip walls:

Visualize the velocity in the region:

Plot the flow profile from the middle of the channel to the top scaled to 1:

### Time-Dependent Analysis(1)

Solve a time-dependent driven cavity problem.

Create and visualize an auxiliary function to ramp up the flow speed on top of the box:

Set up the PDE:

Set up the boundary conditions:

Set up the initial conditions:

Monitor the solution process and measure the time it takes to solve the PDE:

Visualize rasterized frames of the velocity field for various times:

Wolfram Research (2023), FluidFlowPDEComponent, Wolfram Language function, https://reference.wolfram.com/language/ref/FluidFlowPDEComponent.html.

#### Text

Wolfram Research (2023), FluidFlowPDEComponent, Wolfram Language function, https://reference.wolfram.com/language/ref/FluidFlowPDEComponent.html.

#### CMS

Wolfram Language. 2023. "FluidFlowPDEComponent." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/FluidFlowPDEComponent.html.

#### APA

Wolfram Language. (2023). FluidFlowPDEComponent. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FluidFlowPDEComponent.html

#### BibTeX

@misc{reference.wolfram_2024_fluidflowpdecomponent, author="Wolfram Research", title="{FluidFlowPDEComponent}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/FluidFlowPDEComponent.html}", note=[Accessed: 25-April-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_fluidflowpdecomponent, organization={Wolfram Research}, title={FluidFlowPDEComponent}, year={2023}, url={https://reference.wolfram.com/language/ref/FluidFlowPDEComponent.html}, note=[Accessed: 25-April-2024 ]}