Compressible Helical Beltrami Flow

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Consider the case of a compressible fluid undergoing helical (Beltrami) flow with nonuniform vorticity and velocity. This is a generalization of the ABC (Arnold–Beltrami–Childress) flow for the compressible case [1] in which chaotic dynamics appear [2] that involve a "Q-flow" (flows with quasisymmetry).

Contributed by: Enrique Zeleny (June 2014)
Open content licensed under CC BY-NC-SA


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Details

The Q-flows have a Hamiltonian ; then

,

,

,

where is a perturbation parameter (when the advection equations are integrable) and

with and .

Now, introducing a soliton-like function related to the Beltrami flow condition for a scalar function (condition for compressibility), in this case of the form

where the variable is used instead of and are related according to , is a constant, and is a scale parameter along . Consider a generating function of the type

;

then, rewriting the equations, we have

,

,

.

References

[1] A. Morgulis, V. I. Yudovich, and G. M. Zaslavsky, "Compressible Helical Flows," Communications on Pure and Applied Mathematics, 48(5), 2006 pp. 571–582. doi:10.1002/cpa.3160480505

[2] G. M. Zaslavsky, Hamiltonian Chaos and Fractional Dynamics, New York: Oxford University Press, 2005.



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