.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/fluids/viscous_flow/plot_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_fluids_viscous_flow_plot_stokes_drag.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_fluids_viscous_flow_plot_stokes_drag.py:


Stokes drag and the terminal velocity of a settling sphere
=============================================================

In the creeping-flow limit :math:`Re = 2R|v|/\nu \ll 1`, the nonlinear
advection term in the Navier-Stokes equations is negligible next to viscous
diffusion, leaving the linear Stokes equations
:math:`\mu\nabla^2\mathbf{u}=\nabla p` -- George Stokes' 1851 exact solution
for uniform flow past a sphere of radius :math:`R`, giving a drag force

.. math::

    F_D = 6\pi\mu R v.

A sphere of density :math:`\rho_s` settling under gravity through a fluid of
density :math:`\rho_f` and viscosity :math:`\mu` reaches a terminal velocity
:math:`v` where this drag exactly balances its net (gravity minus buoyancy)
weight,

.. math::

    6\pi\mu R v = \tfrac{4}{3}\pi R^3 (\rho_s - \rho_f)\,g.

This example solves that balance for the terminal velocity across a range of
sphere radii and checks, via
:func:`~physicskit.fluids.utils.dimensionless.reynolds_number`, that Stokes'
law remains a good approximation only while the resulting Reynolds number
stays small -- exactly the regime the law assumes.

.. GENERATED FROM PYTHON SOURCE LINES 30-38

.. code-block:: Python


    import matplotlib.pyplot as plt
    import numpy as np

    from physicskit.fluids.systems.viscous_flow import stokes_drag
    from physicskit.fluids.utils.dimensionless import reynolds_number
    from physicskit.fluids.visualizers import theme








.. GENERATED FROM PYTHON SOURCE LINES 39-43

Solve for the terminal velocity at each radius
-------------------------------------------------
At terminal velocity, Stokes drag balances net weight:
:math:`6\pi\mu R v = \tfrac{4}{3}\pi R^3 (\rho_s - \rho_f) g`.

.. GENERATED FROM PYTHON SOURCE LINES 43-52

.. code-block:: Python


    mu, rho_fluid, rho_sphere, g = 1.0e-3, 1000.0, 2500.0, 9.81  # water, glass bead
    radii = np.geomspace(1e-5, 1e-3, 40)
    net_weight = (4.0 / 3.0) * np.pi * radii**3 * (rho_sphere - rho_fluid) * g
    v_terminal = net_weight / np.array([stokes_drag(mu, R, 1.0) for R in radii])

    nu = mu / rho_fluid
    Re = np.array([reynolds_number(velocity=v, length=2 * R, nu=nu) for v, R in zip(v_terminal, radii)])








.. GENERATED FROM PYTHON SOURCE LINES 53-57

Plot terminal velocity and its Reynolds number
-------------------------------------------------
The Reynolds number rises sharply with radius; Stokes' law is only
self-consistent while it stays well below 1.

.. GENERATED FROM PYTHON SOURCE LINES 57-76

.. code-block:: Python


    fig, axes = plt.subplots(1, 2, figsize=(10, 4))
    axes[0].loglog(radii, v_terminal, color=theme.PRIMARY, marker="o", markersize=3)
    axes[0].set_xlabel("sphere radius (m)")
    axes[0].set_ylabel("terminal velocity (m/s)")
    axes[0].set_title("Stokes terminal velocity")

    axes[1].loglog(radii, Re, color=theme.ACCENT, marker="o", markersize=3)
    axes[1].axhline(1.0, color=theme.MUTED, ls="--", lw=1.0, label="Re = 1 (Stokes' law breaks down)")
    axes[1].set_xlabel("sphere radius (m)")
    axes[1].set_ylabel("Reynolds number")
    axes[1].legend()
    axes[1].set_title("Validity of Stokes' law")
    fig.tight_layout()

    valid = Re < 1.0
    print(f"Stokes' law is self-consistent (Re < 1) for radii up to {radii[valid][-1] * 1e6:.1f} microns")

    plt.show()



.. image-sg:: /api/gallery/fluids/viscous_flow/images/sphx_glr_plot_stokes_drag_001.png
   :alt: Stokes terminal velocity, Validity of Stokes' law
   :srcset: /api/gallery/fluids/viscous_flow/images/sphx_glr_plot_stokes_drag_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    Stokes' law is self-consistent (Re < 1) for radii up to 52.2 microns





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

   **Total running time of the script:** (0 minutes 0.138 seconds)


.. _sphx_glr_download_api_gallery_fluids_viscous_flow_plot_stokes_drag.py:

.. only:: html

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

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

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

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

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

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

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


.. only:: html

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

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