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ConvectionPDETerm

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,…,x_n],{x₁,…,x_n}}.
Time-dependent model variables vars are vars={u[t,x₁,…,x_n],{x₁,…,x_n}} or vars={u[t,x₁,…,x_n],t,{x₁,…,x_n}}.
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,…,β_n} vector
For a system of PDEs with dependent variables {u₁,…,u_m}, 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.
A symbolic convection coefficient can be specified through a VectorSymbol. »
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 (10)

Define a symbolic convection term:

Define a stationary dependent convection term with a symbolic vector convection coefficient:

Activate the term:

Define a stationary dependent convection term with a symbolic vector convection coefficient replaced:

Define a time-dependent convection term with a symbolic vector convection coefficient replaced:

Define a 1D axisymmetric time-independent convection term:

Apply Activate to the term:

Define a time-independent 2D convection term:

Define a nonlinear time-independent 2D convection term:

Define a nonlinear time-dependent 2D convection term:

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 needs to be of sufficient length:

The way to specify the symbolic vector convection coefficient is through a VectorSymbol:

Activate the term:

Substitute a VectorSymbol with an actual value:

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

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