- Of particular importance to our simulation is the Darcy Velocity:

$$\vec{u} = -\frac{\mathbf{K}}{\mu} \nabla p$$$- The Darcy velocity appears in the heat convection-conduction equation: $$C\left( \frac{\partial T}{\partial t} + \epsilon \vec{u}\cdot\nabla T \right) - \nabla \cdot k \nabla T = 0$$$

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- The primary unknown (aka nonlinear variable) is the pressure.
- Once the pressure is computed, the AuxiliarySystem can compute the velocity field and write it to the output file.
- We must couple in the primary nonlinear variable's gradient and make use of Material properties in the auxiliary computation.
- Auxiliary variables come in two flavors: Nodal and Elemental
- Nodal Auxiliary variables cannot couple to gradients of nonlinear variables.
* Gradients of $$C^0$$ continuous nonlinear variables are not well-defined at the nodes.
- Elemental Auxiliary variables (i.e. family = MONOMIAL) can couple to gradients of nonlinear variables -- they are only evaluated in element interiors.

[Auxiliary Variables](/wiki/MooseSystems/AuxVariables)

[Auxiliary Kernels](/wiki/MooseSystems/AuxKernels)

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# Source Code

[DarcyVelocity.h](https://github.com/idaholab/moose/blob/devel/tutorials/darcy_thermo_mech/step04_velocity_aux/include/auxkernels/DarcyVelocity.h)

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[DarcyVelocity.C](https://github.com/idaholab/moose/blob/devel/tutorials/darcy_thermo_mech/step04_velocity_aux/src/auxkernels/DarcyVelocity.C)

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[step4.i](https://github.com/idaholab/moose/blob/devel/tutorials/darcy_thermo_mech/step04_velocity_aux/problems/step4.i)

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# Create a Test

- Build a test for DarcyVelocity.
- To simplify the test, we:
* Remove Kernels and disable the solve.
* Create an artificial pressure field using FunctionIC.

[tutorials/darcy_thermo_mech/step04_velocity_aux/tests/auxkernels/velocity_aux/tests](https://github.com/idaholab/moose/blob/devel/tutorials/darcy_thermo_mech/step04_velocity_aux/tests/auxkernels/velocity_aux/tests)

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[tutorials/darcy_thermo_mech/step04_velocity_aux/tests/auxkernels/velocity_aux/velocity_aux.i](https://github.com/idaholab/moose/blob/devel/tutorials/darcy_thermo_mech/step04_velocity_aux/tests/auxkernels/velocity_aux/velocity_aux.i)