Arnold–Beltrami–Childress flow
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The Arnold–Beltrami–Childress (ABC) flow or Gromeka–Arnold–Beltrami–Childress (GABC) flow is a three-dimensional
incompressible In fluid mechanics or more generally continuum mechanics, incompressible flow ( isochoric flow) refers to a flow in which the material density is constant within a fluid parcel—an infinitesimal volume that moves with the flow velocity. An eq ...
velocity field In continuum mechanics the flow velocity in fluid dynamics, also macroscopic velocity in statistical mechanics, or drift velocity in electromagnetism, is a vector field used to mathematically describe the motion of a continuum. The length of the f ...
which is an exact solution of
Euler's equation 200px, Leonhard Euler (1707–1783) In mathematics and physics, many topics are named in honor of Swiss mathematician Leonhard Euler (1707–1783), who made many important discoveries and innovations. Many of these items named after Euler include ...
. Its representation in Cartesian coordinates is the following: : \dot = A \sin z + C \cos y, : \dot = B \sin x + A \cos z, : \dot = C \sin y + B \cos x, where (\dot,\dot,\dot) is the
material derivative In continuum mechanics, the material derivative describes the time rate of change of some physical quantity (like heat or momentum) of a material element that is subjected to a space-and-time-dependent macroscopic velocity field. The material der ...
of the Lagrangian motion of a fluid parcel located at (x(t),y(t),z(t)). It is notable as a simple example of a fluid flow that can have chaotic trajectories. It is named after
Vladimir Arnold Vladimir Igorevich Arnold (alternative spelling Arnol'd, russian: link=no, Влади́мир И́горевич Арно́льд, 12 June 1937 – 3 June 2010) was a Soviet and Russian mathematician. While he is best known for the Kolmogorov– ...
, Eugenio Beltrami, and Stephen Childress. Ippolit S. Gromeka's (1881) name has been historically neglected, though much of the discussion has been done by him first.Zermelo, Ernst. Ernst Zermelo-Collected Works/Gesammelte Werke: Volume I/Band I-Set Theory, Miscellanea/Mengenlehre, Varia. Vol. 21. Springer Science & Business Media, 2010.


See also

* Beltrami flow


References

* V. I. Arnold. "Sur la topologie des ecoulements stationnaires des fluides parfaits". '' C. R. Acad. Sci. Paris'', 261:17–20, 1965. * Chaos theory Fluid dynamics Differential equations {{fluiddynamics-stub