Arnold–Beltrami–Childress flow

The Arnold–Beltrami–Childress (ABC) 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]

{\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 Vladimir Arnold, Eugenio Beltrami, and Steven Childress.

Notes and 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

V. I. Arnold. "Sur la topologie des ecoulements stationnaires des fluides parfaits". C. R. Acad. Sci. Paris, 261:17–20, 1965.

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