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FluidFlowPDEComponent
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FluidFlowPDEComponent

Updated in 14.1[Experimental]

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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 [TemplateBox[{InterpretationBox[, 1], {"m", , "/", , "s"}, meters per second, {{(, "Meters", )}, /, {(, "Seconds", )}}}, QuantityTF]] as dependent variables, as independent variables in units of [TemplateBox[{InterpretationBox[, 1], "m", meters, "Meters"}, QuantityTF]] and time variable in units of [TemplateBox[{InterpretationBox[, 1], "s", seconds, "Seconds"}, QuantityTF]].
  • 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 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 [TemplateBox[{InterpretationBox[, 1], {"m", , "/", , "s"}, meters per second, {{(, "Meters", )}, /, {(, "Seconds", )}}}, QuantityTF]], time [TemplateBox[{InterpretationBox[, 1], "s", seconds, "Seconds"}, QuantityTF]], viscous stress tensor [TemplateBox[{InterpretationBox[, 1], "Pa", pascals, "Pascals"}, QuantityTF]], pressure [TemplateBox[{InterpretationBox[, 1], "Pa", pascals, "Pascals"}, QuantityTF]], 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:
  • In compressible form, the viscous stress tensor is given as:
  • Here [TemplateBox[{InterpretationBox[, 1], {"s", , "Pa"}, second pascals, {"Pascals", , "Seconds"}}, QuantityTF]] is the dynamic viscosity, and infinitesimal, small deformation strain rate measure [1/TemplateBox[{InterpretationBox[, 1], "s", seconds, "Seconds"}, QuantityTF]] 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/TemplateBox[{InterpretationBox[, 1], "s", seconds, "Seconds"}, QuantityTF]].
  • Laminar flow is typical for , where is the Reynolds number.
  • The Reynolds number is defined as , where [TemplateBox[{InterpretationBox[, 1], "m", meters, "Meters"}, QuantityTF]] is a characteristic length and the flow velocity.
  • The following parameters pars can be given:
  • parameterdefaultsymbol
    "DynamicViscosity"-
  • , dynamic viscosity [TemplateBox[{InterpretationBox[, 1], {"s", , "Pa"}, second pascals, {"Pascals", , "Seconds"}}, QuantityTF]]
    • "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"
    "Carreau"
    "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:
  • parameterdefaultsymbol
    "PowerLawExponent"
  • , exponent
    • "MinimalShearRate", minimal shear rate
    "ReferenceShearRate", reference shear rate
    "PowerLawViscosity", power law viscosity
  • The generalized "Carreau" model, useful for polymer or blood flow, implements .
  • The following parameters can be given for the "Carreau" model:
  • parameterdefaultsymbol
    "PowerLawExponent"
  • , exponent
    • "TransitionExponent"2, exponent
    "InfiniteShearRateViscosity", viscosity at inifinite shear rate
    "Lambda", relaxation time [TemplateBox[{InterpretationBox[, 1], "s", seconds, "Seconds"}, QuantityTF]]
    "ZeroShearRateViscosity", viscosity at zero shear rate
  • The "Carreau" 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:
  • parameterdefaultsymbol
    "PlasticViscosity"
  • , plastic viscosity
    • "YieldStress", yield stress
    "ShearRateFactor", shear rate factor
  • 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"->fun|>|>.
  • A custom apparent viscosity function fun has a function signature fun[name_,vars_,pars_,data__].
  • Custom viscous stress tensor functions fun can be specified as "FluidDynamicsMaterialModel"->fun.
  • A custom viscous stress tensor function fun has a function signature fun[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

open allclose all

Basic Examples  (4)Summary of the most common use cases

Define a flow PDE model:

In[1]:=1
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https://wolfram.com/xid/08dv9hamowb75lzjutugi-id0ily

Define a symbolic flow PDE:

In[1]:=1
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https://wolfram.com/xid/08dv9hamowb75lzjutugi-566p88

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

In[1]:=1
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https://wolfram.com/xid/08dv9hamowb75lzjutugi-oo9jax

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

In[1]:=1
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https://wolfram.com/xid/08dv9hamowb75lzjutugi-et0gl1

Visualize the velocity of the fluid:

In[2]:=2
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https://wolfram.com/xid/08dv9hamowb75lzjutugi-dev2d8
Out[2]=2

Scope  (8)Survey of the scope of standard use cases

Define a flow PDE model for a specific material:

In[1]:=1
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https://wolfram.com/xid/08dv9hamowb75lzjutugi-uyclgw

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

In[1]:=1
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https://wolfram.com/xid/08dv9hamowb75lzjutugi-fct83p

Activate a flow PDE model for a specific material:

In[1]:=1
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https://wolfram.com/xid/08dv9hamowb75lzjutugi-mq3ooq
Out[1]=1

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

In[1]:=1
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https://wolfram.com/xid/08dv9hamowb75lzjutugi-c8xbsh
Out[1]=1

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

In[1]:=1
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https://wolfram.com/xid/08dv9hamowb75lzjutugi-m3sk29
Out[1]=1

Define a time-dependent flow PDE model:

In[1]:=1
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https://wolfram.com/xid/08dv9hamowb75lzjutugi-ih1psi

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

In[1]:=1
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https://wolfram.com/xid/08dv9hamowb75lzjutugi-c9vur1
Out[1]=1

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

In[1]:=1
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https://wolfram.com/xid/08dv9hamowb75lzjutugi-g8bv8u
Out[1]=1

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

In[2]:=2
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https://wolfram.com/xid/08dv9hamowb75lzjutugi-y9p97w
Out[2]=2

Applications  (3)Sample problems that can be solved with this function

Stationary Analysis  (1)

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

In[1]:=1
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https://wolfram.com/xid/08dv9hamowb75lzjutugi-xgs3g5

Visualize the velocity of the fluid:

In[2]:=2
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https://wolfram.com/xid/08dv9hamowb75lzjutugi-ucl7di
Out[2]=2

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:

In[1]:=1
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https://wolfram.com/xid/08dv9hamowb75lzjutugi-ih67wc

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

In[2]:=2
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https://wolfram.com/xid/08dv9hamowb75lzjutugi-pdaygi

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:

In[4]:=4
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https://wolfram.com/xid/08dv9hamowb75lzjutugi-0r61v6
Out[4]=4

Visualize the velocity in the region:

In[5]:=5
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https://wolfram.com/xid/08dv9hamowb75lzjutugi-oyf1ue
Out[5]=5

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

In[6]:=6
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https://wolfram.com/xid/08dv9hamowb75lzjutugi-vx754r
Out[6]=6

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:

In[1]:=1
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https://wolfram.com/xid/08dv9hamowb75lzjutugi-js9898
Out[3]=3

Set up the PDE:

In[4]:=4
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https://wolfram.com/xid/08dv9hamowb75lzjutugi-dqq6qy

Set up the boundary conditions:

In[5]:=5
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https://wolfram.com/xid/08dv9hamowb75lzjutugi-y8ssd8

Set up the initial conditions:

In[6]:=6
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https://wolfram.com/xid/08dv9hamowb75lzjutugi-sf03em

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

In[7]:=7
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https://wolfram.com/xid/08dv9hamowb75lzjutugi-tdbfgy
Out[7]=7

Visualize rasterized frames of the velocity field for various times:

In[8]:=8
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https://wolfram.com/xid/08dv9hamowb75lzjutugi-fn27wk
Out[10]=10
Wolfram Research (2024), FluidFlowPDEComponent, Wolfram Language function, https://reference.wolfram.com/language/ref/FluidFlowPDEComponent.html (updated 2024).
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Wolfram Research (2024), FluidFlowPDEComponent, Wolfram Language function, https://reference.wolfram.com/language/ref/FluidFlowPDEComponent.html (updated 2024).

Text

Wolfram Research (2024), FluidFlowPDEComponent, Wolfram Language function, https://reference.wolfram.com/language/ref/FluidFlowPDEComponent.html (updated 2024).

✖
Wolfram Research (2024), FluidFlowPDEComponent, Wolfram Language function, https://reference.wolfram.com/language/ref/FluidFlowPDEComponent.html (updated 2024).

CMS

Wolfram Language. 2024. "FluidFlowPDEComponent." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2024. https://reference.wolfram.com/language/ref/FluidFlowPDEComponent.html.

✖
Wolfram Language. 2024. "FluidFlowPDEComponent." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2024. https://reference.wolfram.com/language/ref/FluidFlowPDEComponent.html.

APA

Wolfram Language. (2024). FluidFlowPDEComponent. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FluidFlowPDEComponent.html

✖
Wolfram Language. (2024). FluidFlowPDEComponent. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FluidFlowPDEComponent.html

BibTeX

@misc{reference.wolfram_2025_fluidflowpdecomponent, author="Wolfram Research", title="{FluidFlowPDEComponent}", year="2024", howpublished="\url{https://reference.wolfram.com/language/ref/FluidFlowPDEComponent.html}", note=[Accessed: 26-March-2025 ]}

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@misc{reference.wolfram_2025_fluidflowpdecomponent, author="Wolfram Research", title="{FluidFlowPDEComponent}", year="2024", howpublished="\url{https://reference.wolfram.com/language/ref/FluidFlowPDEComponent.html}", note=[Accessed: 26-March-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_fluidflowpdecomponent, organization={Wolfram Research}, title={FluidFlowPDEComponent}, year={2024}, url={https://reference.wolfram.com/language/ref/FluidFlowPDEComponent.html}, note=[Accessed: 26-March-2025 ]}

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@online{reference.wolfram_2025_fluidflowpdecomponent, organization={Wolfram Research}, title={FluidFlowPDEComponent}, year={2024}, url={https://reference.wolfram.com/language/ref/FluidFlowPDEComponent.html}, note=[Accessed: 26-March-2025 ]}