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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

open allclose all

Basic Examples  (3)Summary of the most common use cases

Define a time-independent conservative convection term:

Define a time-dependent conservative convection term:

Define a symbolic conservative convection term:

Scope  (5)Survey of the scope of standard use cases

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)Common pitfalls and unexpected behavior

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_2025_conservativeconvectionpdeterm, author="Wolfram Research", title="{ConservativeConvectionPDETerm}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/ConservativeConvectionPDETerm.html}", note=[Accessed: 07-June-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_conservativeconvectionpdeterm, organization={Wolfram Research}, title={ConservativeConvectionPDETerm}, year={2020}, url={https://reference.wolfram.com/language/ref/ConservativeConvectionPDETerm.html}, note=[Accessed: 07-June-2025 ]}