ConservativeConvectionPDETerm

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`ConservativeConvectionPDETerm`

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ConservativeConvectionPDETerm[vars,α]

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

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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,…,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 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,…,α_n} vector
For a system of PDEs with dependent variables {u₁,…,u_m}, 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

open allclose all

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

Define a time-independent conservative convection term:

In[1]:=1

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https://wolfram.com/xid/0b6g6nrf9pi3g2wga779sq-0tawq3

Out[1]=1

Define a time-dependent conservative convection term:

In[1]:=1

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https://wolfram.com/xid/0b6g6nrf9pi3g2wga779sq-l1q57y

Out[1]=1

Define a symbolic conservative convection term:

In[1]:=1

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https://wolfram.com/xid/0b6g6nrf9pi3g2wga779sq-mq8yxf

Out[1]=1

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

Define a 1D axisymmetric time-independent conservative convection term:

In[1]:=1

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https://wolfram.com/xid/0b6g6nrf9pi3g2wga779sq-gwt41u

Out[1]=1

Apply Activate to the term:

In[4]:=4

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https://wolfram.com/xid/0b6g6nrf9pi3g2wga779sq-bc9drq

Out[4]=4

Verify that the axisymmetric case is a consequence of using a truncated cylindrical coordinate system using the operators that compose the conservative convection term:

In[5]:=5

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https://wolfram.com/xid/0b6g6nrf9pi3g2wga779sq-fi9rc8

Out[5]=5

Define a 2D stationary conservative convection term:

In[1]:=1

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https://wolfram.com/xid/0b6g6nrf9pi3g2wga779sq-gmmpen

Out[1]=1

Define a 2D axisymmetric time-independent conservative convection term:

In[1]:=1

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https://wolfram.com/xid/0b6g6nrf9pi3g2wga779sq-f41qy6

Out[1]=1

Apply Activate to the term:

In[2]:=2

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https://wolfram.com/xid/0b6g6nrf9pi3g2wga779sq-if9vt

Out[2]=2

Verify that the axisymmetric case is a consequence of using a truncated cylindrical coordinate system using the operators that compose the conservative convection term:

In[3]:=3