ConservativeConvectionPDETerm

ConservativeConvectionPDETerm[vars,α]

represents a conservative convection term with conservative convection coefficient and model variables vars.

ConservativeConvectionPDETerm[vars,α,pars]

uses model parameters pars.

Details

  • Conservative convection is typically used to model transport due to a bulk movement and should be used when the divergence of convection velocity is nonzero.
  • Convection with a conservative convection coefficient is the process of transport of the dependent variable :

    • .

  • ConservativeConvectionPDETerm returns a differential operators term to be used as a part of partial differential equations:
  • ConservativeConvectionPDETerm can be used to model conservative convection equations with dependent variable , independent variables and time variable .
  • Stationary model variables vars are vars={u[x1,…,xn],{x1,…,xn}}.
  • Time-dependent model variables vars are vars={u[t,x1,…,xn],{x1,…,xn}} or vars={u[t,x1,…,xn],t,{x1,…,xn}}.
  • The conservative convection term in context with other PDE terms is given by:
  • ConservativeConvectionPDETerm is similar to ConvectionPDETerm but affects the meaning of NeumannValue and has a coefficient that is part of the divergence .
  • The conservative convection coefficient has the following form:
  • {α1,…,αn}(alpha_(1),...,alpha_(n))vector
  • For a system of PDEs with dependent variables {u1,…,um}, the conservative convection represents:
  • The conservative convection term in context systems of PDE terms:
  • The conservative convection coefficient is a tensor of rank 3 of the form , where each submatrix is a vector of length that is specified in the same way as for a single dependent variable.
  • The conservative convection coefficient can depend on time, space, parameters and the dependent variables.
  • The following parameters pars can be given:
  • parameterdefaultsymbol
    "RegionSymmetry"None
  • A possible choice for the parameter "RegionSymmetry" is "Axisymmetric".
  • "Axisymmetric" region symmetry represents a truncated cylindrical coordinate system where the cylindrical coordinates are reduced by removing the angle variable as follows:
  • dimensionreductionequation
    1D
    2D
  • All quantities that do not explicitly depend on the independent variables given are taken to have zero partial derivative.

Examples

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Basic Examples  (3)

Define a time-independent conservative convection term:

Define a time-dependent conservative convection term:

Define a symbolic conservative convection term:

Scope  (5)

Define a 1D axisymmetric time-independent conservative convection term:

Apply Activate to the term:

Verify that the axisymmetric case is a consequence of using a truncated cylindrical coordinate system using the operators that compose the conservative convection term:

Define a 2D stationary conservative convection term:

Define a 2D axisymmetric time-independent conservative convection term:

Apply Activate to the term:

Verify that the axisymmetric case is a consequence of using a truncated cylindrical coordinate system using the operators that compose the conservative convection term:

Define a conservative convection term with multiple dependent variables:

Define an axisymmetric conservative convection term with multiple dependent variables:

Possible Issues  (2)

A conservative convection term with a 0-flow velocity field evaluates to 0:

A symbolic conservative convection coefficient is interpreted as a vector convection coefficient:

A subsequent substitution must account for that:

An alternative is to specify the symbolic convection coefficient as a vector:

Wolfram Research (2020), ConservativeConvectionPDETerm, Wolfram Language function, https://reference.wolfram.com/language/ref/ConservativeConvectionPDETerm.html.

Text

Wolfram Research (2020), ConservativeConvectionPDETerm, Wolfram Language function, https://reference.wolfram.com/language/ref/ConservativeConvectionPDETerm.html.

CMS

Wolfram Language. 2020. "ConservativeConvectionPDETerm." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/ConservativeConvectionPDETerm.html.

APA

Wolfram Language. (2020). ConservativeConvectionPDETerm. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ConservativeConvectionPDETerm.html

BibTeX

@misc{reference.wolfram_2024_conservativeconvectionpdeterm, author="Wolfram Research", title="{ConservativeConvectionPDETerm}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/ConservativeConvectionPDETerm.html}", note=[Accessed: 04-November-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_conservativeconvectionpdeterm, organization={Wolfram Research}, title={ConservativeConvectionPDETerm}, year={2020}, url={https://reference.wolfram.com/language/ref/ConservativeConvectionPDETerm.html}, note=[Accessed: 04-November-2024 ]}