Arnold–Beltrami–Childress flow

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]

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

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

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

where ${\displaystyle ({\dot {x}},{\dot {y}},{\dot {z}})}$ is the material derivative of the Lagrangian motion of a fluid parcel located at ${\displaystyle (x(t),y(t),z(t)).}$

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 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]