View source for Arnold–Beltrami–Childress flow

The '''Arnold–Beltrami–Childress''' ('''ABC''') '''flow''' or '''Gromeka–Arnold–Beltrami–Childress''' ('''GABC''') '''flow''' is a three-dimensional [[incompressible flow|incompressible]] [[velocity field]] which is an exact solution of [[Euler equations (fluid dynamics)|Euler's equation]]. Its representation in [[Cartesian coordinates]] is the following:<ref>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.</ref><ref>T. Dombre, [[Uriel Frisch|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</ref>

: <math>\dot{x} = A \sin z + C \cos y,</math>
: <math>\dot{y} = B \sin x + A \cos z,</math>
: <math>\dot{z} = C \sin y + B \cos x,</math>

where <math>(\dot{x},\dot{y},\dot{z})</math> is the [[material derivative]] of the [[Lagrangian and Eulerian specification of the flow field|Lagrangian motion]] of a [[fluid parcel]] located at <math>(x(t),y(t),z(t)).</math>

This ABС flow was analyzed by Dombre et al. 1986 who gave it the name A-B-C because this example was independently introduced by Arnold (1965) and Childress (1970) as an interesting class of Beltrami flows. For some values of the parameters, e.g., A=B=0, this flow is very simple because particle trajectories are helical screw lines. For some other values of the parameters, however, these flows are ergodic and particle trajectories are everywhere dense. The last result is a counterexample to some statements in traditional textbooks on fluid mechanics that vortex lines are either closed or they can not end in the fluid. That is, because for the ABC flows we have <math>\omega=\lambda v</math>, vortex lines coincide with the particle trajectories and they are also everywhere dense for some values of the parameters A, B,  and C.<ref>{{cite book|author=Andrew J. Majda, Andrea L. Bertozzi|title=Vorticity and Incompressible Flow|url=https://books.google.com/books?id=Lmcz0uwHj_8C&dq=Arnold-Beltrami-Childress&pg=PA60|year=2002|pages=60|publisher=Cambridge University Press |isbn=978-0-521-63948-4 }}</ref>

It is notable as a simple example of a fluid flow that can have chaotic trajectories.<ref>{{cite journal|author=Y. Yamakoshi, K. Muto, and Z. Yoshida|title=Numerical analysis of quasiperiodic perturbations for the Alfvén wave|journal=Physical Review E|year=1994|url=https://journals.aps.org/pre/abstract/10.1103/PhysRevE.50.1437|volume=50|issue=2 |pages=1437–1443|lang=en|doi=10.1103/PhysRevE.50.1437|pmid=9962112 }}</ref>

It is named after [[Vladimir Arnold]], [[Eugenio Beltrami]], and [[Stephen_Childress_(mathematician) |Stephen Childress]]. [[Ippolit S. Gromeka]]'s (1881)<ref>Gromeka, I. "Some cases of incompressible fluid motion." Scientific notes of the Kazan University (1881): 76-148.</ref> name has been historically neglected, though much of the discussion has been done by him first.<ref>Zermelo, Ernst. Ernst Zermelo-Collected Works/Gesammelte Werke: Volume I/Band I-Set Theory, Miscellanea/Mengenlehre, Varia. Vol. 21. Springer Science & Business Media, 2010.</ref>

==See also==
*[[Beltrami flow]]

==References==
{{reflist}}
* [[Vladimir Arnold|V. I. Arnold]]. "Sur la topologie des ecoulements stationnaires des fluides parfaits". ''[[Comptes rendus de l'Académie des sciences|C. R. Acad. Sci. Paris]]'', '''261''':17–20, 1965.
*{{Cite journal|date=March 2013|title=Revisiting the ABC flow dynamo|journal=Physics of Fluids|volume=25|issue=3|pages=037103–037103–10|doi=10.1063/1.4795546|issn=1070-6631|author1-first=Ismaël|author1-last=Bouya|author2-first=Emmanuel|author2-last=Dormy|arxiv = 1206.5186 |bibcode = 2013PhFl...25c7103B |citeseerx=10.1.1.759.9218|s2cid=118722952 }}


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[[Category:Chaos theory]]
[[Category:Fluid dynamics]]
[[Category:Differential equations]]


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