.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/fluid_engineering/viscous/plot_01_couette_viscometer.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

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

        :ref:`Go to the end <sphx_glr_download_api_gallery_fluid_engineering_viscous_plot_01_couette_viscometer.py>`
        to download the full example code or to run this example in your browser via JupyterLite.

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

.. _sphx_glr_api_gallery_fluid_engineering_viscous_plot_01_couette_viscometer.py:


Couette's rotating-cylinder viscometer (1890)
=============================================

Maurice Couette, in his 1890 thesis at Paris, measured the viscosity of
water in the narrow gap between a fixed inner cylinder and a rotating outer
one, from the torque on the inner cylinder. Rotating the outer cylinder
keeps the flow laminar to high speeds. For a narrow gap the flow is the
plane shear flow between a fixed and a sliding plate, with a straight
velocity profile and uniform stress :math:`\mu U/h`
(:func:`~engineeringkit.fluid_engineering.couette_velocity`,
:func:`~engineeringkit.fluid_engineering.couette_shear_stress`); the exact
torque for any gap is
:func:`~engineeringkit.fluid_engineering.couette_viscometer_torque`. This
example reads a viscosity back from a torque and adds a pressure gradient
to the plane flow.

.. GENERATED FROM PYTHON SOURCE LINES 20-25

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from engineeringkit.fluid_engineering import couette_flow_rate, couette_shear_stress, couette_velocity, couette_viscometer_torque








.. GENERATED FROM PYTHON SOURCE LINES 26-32

Reading a viscosity
-------------------

A bob of radius 20 mm and wetted length 60 mm in a cup of radius 21 mm,
the cup turning at 60 rpm, reads a torque of 4.2 mN m on a glycerine
solution. The torque is linear in :math:`\mu`, so one reading gives it.

.. GENERATED FROM PYTHON SOURCE LINES 32-39

.. code-block:: Python


    R1, R2, L, omega = 0.020, 0.021, 0.060, 2 * np.pi
    torque = 4.2e-3
    mu = torque / couette_viscometer_torque(1.0, omega, R1, R2, L)
    narrow = couette_shear_stress(mu, omega * R2, R2 - R1) * 2 * np.pi * R1 * L * R1
    print(f"viscosity {mu:.4f} Pa s; the narrow-gap estimate of the torque is {narrow * 1e3:.2f} mN m")





.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    viscosity 0.2061 Pa s; the narrow-gap estimate of the torque is 4.10 mN m




.. GENERATED FROM PYTHON SOURCE LINES 40-46

Couette-Poiseuille flow
-----------------------

Between flat plates 1 mm apart, the top one sliding at 1 m/s, a pressure
gradient bends the line into a parabola. An adverse gradient beyond
:math:`2\mu U/h^2` makes the fluid run backward near the fixed wall.

.. GENERATED FROM PYTHON SOURCE LINES 46-65

.. code-block:: Python


    h, U = 1e-3, 1.0
    y = np.linspace(0, h, 100)
    critical = 2 * mu * U / h**2
    fig, ax = plt.subplots(figsize=(5, 4))
    for k in [-1.0, 0.0, 1.0, 3.0]:
        ax.plot(couette_velocity(y, h, U, mu, k * critical), y * 1000, label=f"dp/dx = {k:g} × 2μU/h²")
    ax.axvline(0, color="k", lw=0.5)
    ax.set(xlabel="velocity [m/s]", ylabel="height above fixed plate [mm]", title="Couette flow")
    ax.legend(fontsize="small")
    fig.tight_layout()
    print(f"flow per metre width with no gradient: {couette_flow_rate(h, U, mu) * 1000:.2f} L/s")

    assert np.isclose(couette_viscometer_torque(mu, omega, R1, R2, L), torque)
    assert abs(narrow / torque - 1) < 0.05
    assert np.isclose(couette_flow_rate(h, U, mu, critical * 3), U * h / 2 - h**3 * 3 * critical / (12 * mu))
    assert couette_velocity(h / 10, h, U, mu, 3 * critical) < 0

    plt.show()



.. image-sg:: /api/gallery/fluid_engineering/viscous/images/sphx_glr_plot_01_couette_viscometer_001.png
   :alt: Couette flow
   :srcset: /api/gallery/fluid_engineering/viscous/images/sphx_glr_plot_01_couette_viscometer_001.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    flow per metre width with no gradient: 0.50 L/s





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

   **Total running time of the script:** (0 minutes 0.082 seconds)


.. _sphx_glr_download_api_gallery_fluid_engineering_viscous_plot_01_couette_viscometer.py:

.. only:: html

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

    .. container:: lite-badge

      .. image:: images/jupyterlite_badge_logo.svg
        :target: ../../../../lite/lab/index.html?path=api/gallery/fluid_engineering/viscous/plot_01_couette_viscometer.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_couette_viscometer.ipynb <plot_01_couette_viscometer.ipynb>`

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

      :download:`Download Python source code: plot_01_couette_viscometer.py <plot_01_couette_viscometer.py>`

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

      :download:`Download zipped: plot_01_couette_viscometer.zip <plot_01_couette_viscometer.zip>`


.. only:: html

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

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