.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/fluid_engineering/pipe_flow/plot_01_hagen_poiseuille.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_pipe_flow_plot_01_hagen_poiseuille.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_pipe_flow_plot_01_hagen_poiseuille.py:


Hagen and Poiseuille: laminar pipe flow (1839-1840)
===================================================

Gotthilf Hagen in 1839, timing water through brass tubes, and Jean Léonard
Marie Poiseuille in 1840, studying blood flow in glass capillaries, found
independently that slow flow through a tube grows with the pressure drop
and with the fourth power of the radius. The law
:math:`Q = \pi R^4\Delta p/(8\mu L)`
(:func:`~engineeringkit.fluid_engineering.poiseuille_flow`) follows from a
parabolic velocity profile
(:func:`~engineeringkit.fluid_engineering.poiseuille_velocity`), and in
Darcy's form it is the laminar friction factor :math:`f = 64/Re`
(:func:`~engineeringkit.fluid_engineering.laminar_friction_factor`).

.. GENERATED FROM PYTHON SOURCE LINES 18-31

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np
    from scipy import integrate

    from engineeringkit.fluid_engineering import (
        G,
        darcy_head_loss,
        laminar_friction_factor,
        poiseuille_flow,
        poiseuille_velocity,
        reynolds_number,
    )








.. GENERATED FROM PYTHON SOURCE LINES 32-37

The parabolic profile
---------------------

Water (:math:`\mu = 1.0\times10^{-3}` Pa s) in a 2 mm capillary, 0.5 m
long, with 20 Pa across it.

.. GENERATED FROM PYTHON SOURCE LINES 37-51

.. code-block:: Python


    rho, mu, R, L, dp = 998.0, 1.0e-3, 1.0e-3, 0.5, 20.0
    r = np.linspace(-R, R, 101)
    u = poiseuille_velocity(np.abs(r), dp, R, L, mu)
    Q = poiseuille_flow(dp, R, L, mu)
    v_mean = Q / (np.pi * R**2)
    print(f"Q = {Q * 1e9:.2f} mm³/s, mean speed {1e3 * v_mean:.2f} mm/s, centreline {1e3 * u.max():.2f} mm/s")

    fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(9, 3.8))
    ax1.plot(1e3 * u, 1e3 * r)
    ax1.axvline(1e3 * v_mean, color="k", ls=":", label="mean")
    ax1.set(xlabel="velocity [mm/s]", ylabel="radius [mm]", title="Hagen-Poiseuille profile")
    ax1.legend()




.. image-sg:: /api/gallery/fluid_engineering/pipe_flow/images/sphx_glr_plot_01_hagen_poiseuille_001.png
   :alt: Hagen-Poiseuille profile
   :srcset: /api/gallery/fluid_engineering/pipe_flow/images/sphx_glr_plot_01_hagen_poiseuille_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    Q = 15.71 mm³/s, mean speed 5.00 mm/s, centreline 10.00 mm/s

    <matplotlib.legend.Legend object at 0x110fa3750>



.. GENERATED FROM PYTHON SOURCE LINES 52-57

The fourth power of the radius
------------------------------

Halving the bore cuts the flow sixteenfold, which is why narrowed arteries
matter so much.

.. GENERATED FROM PYTHON SOURCE LINES 57-65

.. code-block:: Python


    radii = np.geomspace(0.25e-3, 2e-3, 20)
    ax2.loglog(1e3 * radii, poiseuille_flow(dp, radii, L, mu) * 1e9, marker="o")
    ax2.set(xlabel="radius [mm]", ylabel="flow [mm³/s]", title="Q grows as R⁴")
    fig.tight_layout()
    slope = np.polyfit(np.log(radii), np.log(poiseuille_flow(dp, radii, L, mu)), 1)[0]
    print(f"log-log slope of Q against R: {slope:.3f}")





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

 .. code-block:: none

    log-log slope of Q against R: 4.000




.. GENERATED FROM PYTHON SOURCE LINES 66-71

The same law in Darcy's form
----------------------------

The profile integrates to the flow, and the Darcy head loss with
:math:`f = 64/Re` reproduces the applied pressure drop.

.. GENERATED FROM PYTHON SOURCE LINES 71-81

.. code-block:: Python


    Re = reynolds_number(rho, v_mean, 2 * R, mu)
    head = darcy_head_loss(laminar_friction_factor(Re), L, 2 * R, v_mean)
    print(f"Re = {Re:.1f}; Darcy loss {rho * G * head:.2f} Pa against {dp:.2f} Pa applied")

    assert np.isclose(integrate.quad(lambda s: 2 * np.pi * s * poiseuille_velocity(s, dp, R, L, mu), 0, R)[0], Q)
    assert np.isclose(u.max(), 2 * v_mean) and np.isclose(slope, 4.0)
    assert np.isclose(rho * G * head, dp)

    plt.show()




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

 .. code-block:: none

    Re = 10.0; Darcy loss 20.00 Pa against 20.00 Pa applied





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

   **Total running time of the script:** (0 minutes 0.062 seconds)


.. _sphx_glr_download_api_gallery_fluid_engineering_pipe_flow_plot_01_hagen_poiseuille.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/pipe_flow/plot_01_hagen_poiseuille.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_hagen_poiseuille.ipynb <plot_01_hagen_poiseuille.ipynb>`

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

      :download:`Download Python source code: plot_01_hagen_poiseuille.py <plot_01_hagen_poiseuille.py>`

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

      :download:`Download zipped: plot_01_hagen_poiseuille.zip <plot_01_hagen_poiseuille.zip>`


.. only:: html

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

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