Arnold–Beltrami–Childress flow

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

1. Xiao-Hua Zhao, Keng-Huat Kwek, Ji-Bin Li and Ke-Lei Huang. "Chaotic and Resonant Streamlines in the ABC Flow". SIAM Journal on Applied Mathematics. Vol. 53, No. 1 (Feb., 1993), pp. 71–77. Published by: Society for Industrial and Applied Mathematics.
2. T. Dombre, U. Frisch, J. M. Greene, M. Hénon, A. Mehr, and A. M. Soward (1986). "Chaotic streamlines in the ABC flows". Journal of Fluid Mechanics, 167, pp. 353–391 doi:10.1017/S0022112086002859
3. Gromeka, I. "Some cases of incompressible fluid motion." Scientific notes of the Kazan University (1881): 76-148.
4. Zermelo, Ernst. Ernst Zermelo-Collected Works/Gesammelte Werke: Volume I/Band I-Set Theory, Miscellanea/Mengenlehre, Varia. Vol. 21. Springer Science & Business Media, 2010.

• 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. arXiv:1206.5186. Bibcode:2013PhFl...25c7103B. CiteSeerX 10.1.1.759.9218. doi:10.1063/1.4795546. ISSN 1070-6631.