.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/fluid_engineering/drag/plot_01_stokes_drag.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_drag_plot_01_stokes_drag.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_drag_plot_01_stokes_drag.py:


Stokes's drag on a sphere (1851)
================================

George Gabriel Stokes, correcting pendulum experiments for the air dragged
along with the bob, solved the slow viscous flow past a sphere in a paper
read to the Cambridge Philosophical Society in December 1850 and printed in
1851. The drag is :math:`3\pi\mu dV`
(:func:`~engineeringkit.fluid_engineering.stokes_drag`), so a small sphere
settles at :math:`(\rho_p - \rho)gd^2/(18\mu)`
(:func:`~engineeringkit.fluid_engineering.stokes_settling_velocity`). The
law fails once inertia matters, near :math:`Re = 1`; the measured drag
curve (:func:`~engineeringkit.fluid_engineering.sphere_drag_coefficient`)
takes over. This example draws the curve and finds how fast sand grains of
different sizes fall through water
(:func:`~engineeringkit.fluid_engineering.terminal_velocity`).

.. GENERATED FROM PYTHON SOURCE LINES 20-31

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from engineeringkit.fluid_engineering import (
        reynolds_number,
        sphere_drag_coefficient,
        stokes_drag,
        stokes_settling_velocity,
        terminal_velocity,
    )








.. GENERATED FROM PYTHON SOURCE LINES 32-37

The sphere drag curve
---------------------

Stokes's :math:`C_D = 24/Re` is the straight line at the left; the curve
levels off near 0.4 in Newton's range before the drag crisis.

.. GENERATED FROM PYTHON SOURCE LINES 37-47

.. code-block:: Python


    Re = np.geomspace(0.01, 2e5, 200)
    fig, ax = plt.subplots(figsize=(6, 4))
    ax.loglog(Re, sphere_drag_coefficient(Re), label="drag curve (White's fit)")
    ax.loglog(Re[Re < 10], 24 / Re[Re < 10], "--", label="Stokes, 24/Re")
    ax.set(xlabel="Reynolds number", ylabel="drag coefficient $C_D$", title="Drag on a smooth sphere")
    ax.legend()
    ax.grid(True, which="both", lw=0.3)
    fig.tight_layout()




.. image-sg:: /api/gallery/fluid_engineering/drag/images/sphx_glr_plot_01_stokes_drag_001.png
   :alt: Drag on a smooth sphere
   :srcset: /api/gallery/fluid_engineering/drag/images/sphx_glr_plot_01_stokes_drag_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 48-53

Settling sand
-------------

Quartz sand (2650 kg/m³) falling through water at 20 °C. Stokes's
velocity is accurate for silt but overestimates coarse sand badly.

.. GENERATED FROM PYTHON SOURCE LINES 53-60

.. code-block:: Python


    rho, mu, rho_p = 998.0, 1.0e-3, 2650.0
    for d in [20e-6, 100e-6, 500e-6, 2e-3]:
        V = terminal_velocity(d, rho_p, rho, mu)
        V_stokes = stokes_settling_velocity(d, rho_p, rho, mu)
        print(f"d = {d * 1e6:6.0f} µm: V = {V * 1000:7.3f} mm/s, Stokes {V_stokes * 1000:9.3f} mm/s, Re = {reynolds_number(rho, V, d, mu):.3g}")





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

 .. code-block:: none

    d =     20 µm: V =   0.359 mm/s, Stokes     0.360 mm/s, Re = 0.00717
    d =    100 µm: V =   8.042 mm/s, Stokes     9.000 mm/s, Re = 0.803
    d =    500 µm: V =  76.071 mm/s, Stokes   225.008 mm/s, Re = 38
    d =   2000 µm: V = 247.453 mm/s, Stokes  3600.130 mm/s, Re = 494




.. GENERATED FROM PYTHON SOURCE LINES 61-63

The drag on a 20 µm silt grain at its settling speed equals its weight
less its buoyancy.

.. GENERATED FROM PYTHON SOURCE LINES 63-73

.. code-block:: Python


    d = 20e-6
    V = stokes_settling_velocity(d, rho_p, rho, mu)
    weight = (rho_p - rho) * 9.80665 * np.pi * d**3 / 6

    assert np.isclose(stokes_drag(mu, V, d), weight)
    assert np.isclose(terminal_velocity(d, rho_p, rho, mu), V, rtol=1e-2)
    assert terminal_velocity(2e-3, rho_p, rho, mu) < 0.2 * stokes_settling_velocity(2e-3, rho_p, rho, mu)

    plt.show()








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

   **Total running time of the script:** (0 minutes 0.188 seconds)


.. _sphx_glr_download_api_gallery_fluid_engineering_drag_plot_01_stokes_drag.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/drag/plot_01_stokes_drag.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_stokes_drag.ipynb <plot_01_stokes_drag.ipynb>`

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

      :download:`Download Python source code: plot_01_stokes_drag.py <plot_01_stokes_drag.py>`

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

      :download:`Download zipped: plot_01_stokes_drag.zip <plot_01_stokes_drag.zip>`


.. only:: html

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

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