Arnold–Beltrami–Childress flow

The Arnold–Beltrami–Childress (ABC) flow or Gromeka–Arnold–Beltrami–Childress (GABC) flow is a three-dimensional

Cartesian coordinates is the following:^[1]^[2]

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

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

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

where $({\dot {x}},{\dot {y}},{\dot {z}})$ is the material derivative 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

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]

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.

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.
S2CID 118722952
.

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[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.

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See also

References