ConvectionPDETerm

ConvectionPDETerm[vars,β]

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

ConvectionPDETerm[vars,β,pars]

uses model parameters pars.

Details

  • Convection terms are used in a number of domains such as thermodynamics, acoustics, structural mechanics and fluid dynamics.
  • Convection is also known as advection.
  • Convection with a convection coefficient is the process of transport of the dependent variable due to a bulk movement:
  • ConvectionPDETerm returns a differential operators term to be used as a part of partial differential equations:
  • ConvectionPDETerm can be used to model 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 convection term in context with other PDE terms is given by:
  • During convection, the medium in which the convection happens is the transport mechanism, in contrast to diffusion where the medium remains stationary.
  • The convection coefficient has the following form:
  • {β1,…,βn}(beta_(1),...,beta_(n))vector
  • For a system of PDEs with dependent variables {u1,…,um}, the convection represents:
  • The convection term in context systems of PDE terms:
  • The 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 coefficient does not affect the meaning of NeumannValue.
  • All quantities that do not explicitly depend on the independent variables given are taken to have zero partial derivative.
  • The ConservativeConvectionPDETerm is closely related.

Examples

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Basic Examples  (4)

Define a time-independent convection term:

Define a time-dependent convection term:

Define a 2D stationary convection term:

Solve a convection diffusion equation build with basic terms:

Visualize the result:

Scope  (4)

Define a symbolic convection term:

Define a stationary convection term with a parametric convection coefficient replaced:

Define a convection term with multiple dependent variables:

Define a Stokes-flow model:

Set up a symbolic equation:

Applications  (3)

Use DiffusionPDETerm to model species diffusion under a dam. Set up the region:

Set up the model:

Solve the equation:

Compute the flux:

Visualize the result:

Find the concentration of species under the dam. Construct the model:

Solve the equation:

Visualize the species concentration:

Define a Stokes-flow model:

Set up the equation:

Define a narrowing region:

Set up a boundary condition:

Solve the equation:

Visualize the solution:

Extend a Stokes-flow model to a Navier–Stokes flow model. Define a Stokes-flow model:

Define a Navier–Stokes flow model:

Set up the equation:

Define a region:

Set up a boundary condition:

Solve the equation:

Visualize the solution:

Possible Issues  (2)

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

A symbolic 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), ConvectionPDETerm, Wolfram Language function, https://reference.wolfram.com/language/ref/ConvectionPDETerm.html.

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2023_convectionpdeterm, author="Wolfram Research", title="{ConvectionPDETerm}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/ConvectionPDETerm.html}", note=[Accessed: 18-April-2024 ]}

BibLaTeX

@online{reference.wolfram_2023_convectionpdeterm, organization={Wolfram Research}, title={ConvectionPDETerm}, year={2020}, url={https://reference.wolfram.com/language/ref/ConvectionPDETerm.html}, note=[Accessed: 18-April-2024 ]}