:orphan:

Analytic solutions for the morphology of the final state of the front.
======================

This section demonstrates how to use the Shape_result class which allows to compute the shape of the flow front at the final step of the flow.

Coussot's model
---------------

Coussot et al proposed a formula in 1996 to approximate the geometry of the frontal lobe of a flow at a final instant for rheological test on an inclined surface:

.. math::
	D = - H - \ln{(1 - H)}
	
with :math:`D` and :math:`H` being the normalized version of the distance of the front :math:`d` and the fluid depth :math:`h`, obtained by computing:

.. math::
	D = \frac{\rho g d (\sin(\theta))^2}{\tau_c \cos(\theta)} \text{  and  } H = \frac{\rho g h \sin(\theta)}{\tau_c}
	
where

 - :math:`h`: fluid depth.
 - :math:`x`: spatial dimension.
 - :math:`g`: gravitational acceleration.
 - :math:`\rho`: fluid density.
 - :math:`\tau_c`: threshold constraint.
 - :math:`\theta`: slope of the surface.

In a case where :math:`\theta = 0`, the equations are slightly different:

.. math::
	D^* = \frac{{H^*}^2}{2}

with:

.. math::
	D^* = \frac{\rho g d}{\tau_c} \text{  and  } H^* = \frac{\rho g h}{\tau_c}


References
----------
Coussot, P., Proust, S., & Ancey, C., 1996, Rheological interpretation of deposits of yield stress fluids, Journal of Non-Newtonian Fluid Mechanics, v. 66(1), p. 55–70, doi:10.1016/0377-0257(96)01474-7.


Examples
--------


.. raw:: html

    <div class="sphx-glr-thumbnails">

.. thumbnail-parent-div-open

.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="This example demonstrates the final shape of simulated flow using Coussot&#x27;s equation.">

.. only:: html

  .. image:: /auto_examples/Analytic_Solutions_Shape/images/thumb/sphx_glr_plot_as_sh_coussot_thumb.png
    :alt:

  :doc:`/auto_examples/Analytic_Solutions_Shape/plot_as_sh_coussot`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Coussot's model example</div>
    </div>


.. thumbnail-parent-div-close

.. raw:: html

    </div>


.. toctree::
   :hidden:

   /auto_examples/Analytic_Solutions_Shape/plot_as_sh_coussot


.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-gallery

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download all examples in Python source code: Analytic_Solutions_Shape_python.zip </auto_examples/Analytic_Solutions_Shape/Analytic_Solutions_Shape_python.zip>`

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download all examples in Jupyter notebooks: Analytic_Solutions_Shape_jupyter.zip </auto_examples/Analytic_Solutions_Shape/Analytic_Solutions_Shape_jupyter.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_