The Arnold–Beltrami–Childress (ABC) flow or Gromeka–Arnold–Beltrami–Childress (GABC) flow is a three-dimensional incompressible velocity field which is an exact solution of Euler's equation. Its representation in Cartesian coordinates is the following: ^{[1] [2]}

${\displaystyle {\dot {x}}=A\sin z+C\cos y,}$

${\displaystyle {\dot {y}}=B\sin x+A\cos z,}$

${\displaystyle {\dot {z}}=C\sin y+B\cos x,}$

where ${\displaystyle ({\dot {x}},{\dot {y}},{\dot {z}})}$ is the material derivative of the Lagrangian motion of a fluid parcel located at ${\displaystyle (x(t),y(t),z(t)).}$

It is notable as a simple example of a fluid flow that can have chaotic trajectories.

It is named after Vladimir Arnold, Eugenio Beltrami, and Stephen Childress. Ippolit S. Gromeka's (1881) ^[3] name has been historically neglected, though much of the discussion has been done by him first. ^[4]

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.
• Bouya, Ismaël; Dormy, Emmanuel (March 2013). "Revisiting the ABC flow dynamo". Physics of Fluids. 25 (3): 037103–037103–10. arXiv: 1206.5186. Bibcode: 2013PhFl...25c7103B. CiteSeerX  10.1.1.759.9218. doi: 10.1063/1.4795546. ISSN  1070-6631. S2CID  118722952.

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