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's notable as a simple example of a fluid flow that can have chaotic trajectories.

Notes and references

Inline

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

Other

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