# 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} 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:
•  parameter default symbol "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:
•  dimension reduction equation 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: