# 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} 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 NavierStokes flow model. Define a Stokes-flow model:

Define a NavierStokes 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_2024_convectionpdeterm, author="Wolfram Research", title="{ConvectionPDETerm}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/ConvectionPDETerm.html}", note=[Accessed: 20-July-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_convectionpdeterm, organization={Wolfram Research}, title={ConvectionPDETerm}, year={2020}, url={https://reference.wolfram.com/language/ref/ConvectionPDETerm.html}, note=[Accessed: 20-July-2024 ]}