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

incompressible Incompressible may refer to: * Incompressible flow, in fluid mechanics * incompressible vector field, in mathematics * Incompressible surface, in mathematics * Incompressible string, in computing {{Disambig ...

velocity field In continuum mechanics the flow velocity in fluid dynamics, also macroscopic velocity in statistical mechanics, or drift velocity in electromagnetism, is a vector field used to mathematically describe the motion of a continuum. The length of the f ...

which is an exact solution of Euler's equation. It is named after

Vladimir Arnold Vladimir Igorevich Arnold (or Arnol'd; , ; 12 June 1937 – 3 June 2010) was a Soviet and Russian mathematician. He is best known for the Kolmogorov–Arnold–Moser theorem regarding the stability of integrable systems, and contributed to s ...

, Eugenio Beltrami, and Stephen Childress. Ippolit S. Gromeka's (1881) name has been historically neglected, though much of the discussion has been done by him first. It is notable as a simple example of a fluid flow that can have chaotic trajectories. Its representation in

Cartesian coordinates In geometry, a Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called ''coordinates'', which are the signed distances to the point from two fixed perpendicular o ...

is the following: :

\dot = A \sin z + C \cos y,

\dot = B \sin x + A \cos z,

\dot = C \sin y + B \cos x,

where

(\dot,\dot,\dot)

is the

material derivative In continuum mechanics, the material derivative describes the time rate of change of some physical quantity (like heat or momentum) of a material element that is subjected to a space-and-time-dependent macroscopic velocity field. The material de ...

of the Lagrangian motion of a

fluid parcel In fluid dynamics, a fluid parcel, also known as a fluid element or material element, is an infinitesimal volume of fluid, identifiable throughout its dynamic history while moving with the fluid flow. As it moves, the mass of a fluid parcel rema ...

located at

(x(t),y(t),z(t)).

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

\omega=\lambda v

, vortex lines coincide with the particle trajectories and they are also everywhere dense for some values of the parameters ''A'', ''B'' and ''C''.

References

* V. I. Arnold. "Sur la topologie des ecoulements stationnaires des fluides parfaits". '' C. R. Acad. Sci. Paris'', 261:17–20, 1965. * Chaos theory Fluid dynamics Differential equations {{fluiddynamics-stub

See also

References