.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_examples/Analytic_Solutions_Shape/plot_as_sh_coussot.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_auto_examples_Analytic_Solutions_Shape_plot_as_sh_coussot.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_auto_examples_Analytic_Solutions_Shape_plot_as_sh_coussot.py:


Coussot's model example
===========================

This example demonstrates the final shape of simulated flow using Coussot's equation.

The frontal lobe shape of the simulated flow at the final step is given by

    .. math::
            D = - H - \ln(1 - H)

where
 - :math:`H`: normalized fluid depth.
 - :math:`D`: normalized distance of the front from the origin.

:math:`H` and :math:`D` are obtained with these expressions:

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

with:
 - :math:`h`: fluid depth.
 - :math:`d`: distance of the front from the origin.
 - :math:`g`: gravitational acceleration.
 - :math:`\rho`: fluid density.
 - :math:`\tau_c`: threshold constraint.
 - :math:`\theta`: slope of the surface.

Implementation
---------------

.. GENERATED FROM PYTHON SOURCE LINES 34-36

First import required packages and define the context. For this example we will use a fluid with a density of :math:`\rho = 1000 kg/m^3`:
and :math:`\tau_c = 50 Pa`, with a slope of :math:`\theta = 10°`:

.. GENERATED FROM PYTHON SOURCE LINES 36-44

.. code-block:: Python

    from tilupy.analytic_sol import Coussot_shape
    import matplotlib.pyplot as plt

    case_1 = Coussot_shape(rho=1000, tau=50, theta=10)
    case_1.compute_rheological_test_front_morpho()
    plt.plot(case_1.d, case_1.h, color="black")
    plt.show()




.. image-sg:: /auto_examples/Analytic_Solutions_Shape/images/sphx_glr_plot_as_sh_coussot_001.png
   :alt: plot as sh coussot
   :srcset: /auto_examples/Analytic_Solutions_Shape/images/sphx_glr_plot_as_sh_coussot_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 45-54

If :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}

.. GENERATED FROM PYTHON SOURCE LINES 54-60

.. code-block:: Python


    case_2 = Coussot_shape(rho=1000, tau=50, theta=0, h_final=1)
    case_2.compute_rheological_test_front_morpho()
    plt.plot(case_2.d, case_2.h, color="black")
    plt.show()




.. image-sg:: /auto_examples/Analytic_Solutions_Shape/images/sphx_glr_plot_as_sh_coussot_002.png
   :alt: plot as sh coussot
   :srcset: /auto_examples/Analytic_Solutions_Shape/images/sphx_glr_plot_as_sh_coussot_002.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 61-65

Original reference:

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.


.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 0.069 seconds)


.. _sphx_glr_download_auto_examples_Analytic_Solutions_Shape_plot_as_sh_coussot.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

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

      :download:`Download Jupyter notebook: plot_as_sh_coussot.ipynb <plot_as_sh_coussot.ipynb>`

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

      :download:`Download Python source code: plot_as_sh_coussot.py <plot_as_sh_coussot.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_as_sh_coussot.zip <plot_as_sh_coussot.zip>`


.. only:: html

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

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