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]
where is the material derivative of the Lagrangian motion of a fluid parcel located at
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
References
- ↑ 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.
- ↑ 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
- ↑ Gromeka, I. "Some cases of incompressible fluid motion." Scientific notes of the Kazan University (1881): 76-148.
- ↑ 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): 037103–037103–10. 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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