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

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

.. _sphx_glr_api_gallery_fluids_vortex_dynamics_plot_vortex_pair.py:


Co-rotating pair versus translating dipole
=============================================

An isolated point vortex of circulation :math:`\Gamma` induces a purely
azimuthal velocity of magnitude :math:`\Gamma/(2\pi r)` at distance `r` (the
2D Biot-Savart law); for two vortices separated by distance :math:`d`, this
is exactly the speed each one advects with, carried by the field of the
*other*. What the pair then does depends only on the relative sign of their
circulation. Equal circulations :math:`\Gamma` each orbit the common
midpoint at radius :math:`d/2`, so their induced speed
:math:`\Gamma/(2\pi d)` translates into a rigid-body angular velocity

.. math::

    \Omega = \frac{\Gamma/(2\pi d)}{d/2} = \frac{\Gamma}{\pi d^2}.

Opposite circulations :math:`\pm\Gamma` instead induce velocity in the
*same* direction on each other (a "vortex dipole"), so the pair
self-propels together in a straight line at constant speed

.. math::

    v = \frac{\Gamma}{2\pi d},

perpendicular to the line joining them, advecting each other sideways
forever rather than orbiting. Both trajectories come from exactly the same
:class:`~physicskit.fluids.systems.vortex_dynamics.PointVortexSystem` --
only the sign of one circulation changes between them.

.. GENERATED FROM PYTHON SOURCE LINES 31-38

.. code-block:: Python


    import matplotlib.pyplot as plt
    import numpy as np

    from physicskit.fluids.systems.vortex_dynamics import PointVortexSystem
    from physicskit.fluids.visualizers import theme








.. GENERATED FROM PYTHON SOURCE LINES 39-43

A co-rotating pair
--------------------
Equal circulations orbit their midpoint at angular velocity
:math:`\Omega = \Gamma / (\pi d^2)`.

.. GENERATED FROM PYTHON SOURCE LINES 43-49

.. code-block:: Python


    Gamma, d = 1.0, 1.0
    pair = PointVortexSystem(positions=[[d / 2, 0.0], [-d / 2, 0.0]], circulations=[Gamma, Gamma])
    period = 2 * np.pi / (Gamma / (np.pi * d**2))
    _, trajectory_pair = pair.trajectory(dt=period / 2000, n_steps=2000)








.. GENERATED FROM PYTHON SOURCE LINES 50-54

A translating dipole
----------------------
Equal and opposite circulations instead translate together at constant
speed :math:`\Gamma / (2\pi d)`, perpendicular to the line joining them.

.. GENERATED FROM PYTHON SOURCE LINES 54-59

.. code-block:: Python


    dipole = PointVortexSystem(positions=[[0.0, d / 2], [0.0, -d / 2]], circulations=[Gamma, -Gamma])
    travel_time = 4.0 * d / (Gamma / (2 * np.pi * d))
    _, trajectory_dipole = dipole.trajectory(dt=travel_time / 2000, n_steps=2000)








.. GENERATED FROM PYTHON SOURCE LINES 60-62

Plot both trajectories
-------------------------

.. GENERATED FROM PYTHON SOURCE LINES 62-83

.. code-block:: Python


    fig, axes = plt.subplots(1, 2, figsize=(11, 5))
    axes[0].plot(trajectory_pair[:, 0, 0], trajectory_pair[:, 0, 1], color=theme.PRIMARY, label="vortex 1")
    axes[0].plot(trajectory_pair[:, 1, 0], trajectory_pair[:, 1, 1], color=theme.ACCENT, label="vortex 2")
    axes[0].set_aspect("equal")
    axes[0].set_title("Co-rotating pair (same sign): rigid orbit")
    axes[0].legend()

    axes[1].plot(trajectory_dipole[:, 0, 0], trajectory_dipole[:, 0, 1], color=theme.PRIMARY, label="vortex 1")
    axes[1].plot(trajectory_dipole[:, 1, 0], trajectory_dipole[:, 1, 1], color=theme.ACCENT, label="vortex 2")
    axes[1].set_aspect("equal")
    axes[1].set_title("Vortex dipole (opposite sign): self-propulsion")
    axes[1].legend()
    fig.tight_layout()

    separation_pair = np.hypot(*(trajectory_pair[-1, 0] - trajectory_pair[-1, 1]).T)
    separation_dipole = np.hypot(*(trajectory_dipole[-1, 0] - trajectory_dipole[-1, 1]).T)
    print(f"pair separation: started at {d:.3f}, ended at {separation_pair:.3f} (conserved by rigid rotation)")
    print(f"dipole separation: started at {d:.3f}, ended at {separation_dipole:.3f} (conserved by rigid translation)")

    plt.show()



.. image-sg:: /api/gallery/fluids/vortex_dynamics/images/sphx_glr_plot_vortex_pair_001.png
   :alt: Co-rotating pair (same sign): rigid orbit, Vortex dipole (opposite sign): self-propulsion
   :srcset: /api/gallery/fluids/vortex_dynamics/images/sphx_glr_plot_vortex_pair_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    pair separation: started at 1.000, ended at 1.000 (conserved by rigid rotation)
    dipole separation: started at 1.000, ended at 1.000 (conserved by rigid translation)





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

   **Total running time of the script:** (0 minutes 0.060 seconds)


.. _sphx_glr_download_api_gallery_fluids_vortex_dynamics_plot_vortex_pair.py:

.. only:: html

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

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

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

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

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

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

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


.. only:: html

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

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