.. _navier_stokes-stokes_slip_bc:

navier_stokes/stokes_slip_bc.py
===============================

**Description**


Incompressible Stokes flow with Navier (slip) boundary conditions, flow driven
by a moving wall and a small diffusion for stabilization.

This example demonstrates the use of `no-penetration` boundary conditions as
well as `edge direction` boundary conditions together with Navier or slip
boundary conditions.

Find :math:`\ul{u}`, :math:`p` such that:

.. math::
    \int_{\Omega} \nu\ \nabla \ul{v} : \nabla \ul{u}
    - \int_{\Omega} p\ \nabla \cdot \ul{v}
    + \int_{\Gamma_1} \beta \ul{v} \cdot (\ul{u} - \ul{u}_d)
    + \int_{\Gamma_2} \beta \ul{v} \cdot \ul{u}
    = 0
    \;, \quad \forall \ul{v} \;,

    \int_{\Omega} \mu \nabla q \cdot \nabla p
    + \int_{\Omega} q\ \nabla \cdot \ul{u}
    = 0
    \;, \quad \forall q \;,

where :math:`\nu` is the fluid viscosity, :math:`\beta` is the slip
coefficient, :math:`\mu` is the (small) numerical diffusion coefficient,
:math:`\Gamma_1` is the top wall that moves with the given driving velocity
:math:`\ul{u}_d` and :math:`\Gamma_2` are the remaining walls. The Navier
conditions are in effect on both :math:`\Gamma_1`, :math:`\Gamma_2` and are
expressed by the corresponding integrals in the equations above.

The `no-penetration` boundary conditions are applied on :math:`\Gamma_1`,
:math:`\Gamma_2`, except the vertices of the block edges, where the `edge
direction` boundary conditions are applied. Optionally, Dirichlet boundary
conditions can be applied on the inlet, see the code below.

The mesh is created by ``gen_block_mesh()`` function - try different mesh
dimensions and resolutions below. For large meshes use the ``'ls_i'`` linear
solver - PETSc + petsc4py is needed in that case.

See also :ref:`navier_stokes-stokes_slip_bc_penalty`.


.. image:: /../gallery/images/navier_stokes-stokes_slip_bc.png


:download:`source code </../examples/navier_stokes/stokes_slip_bc.py>`

.. literalinclude:: /../examples/navier_stokes/stokes_slip_bc.py