Abstract
We study the integrability of an eight-parameter family of three-dimensional spherically confined steady Stokes flows introduced by Bajer and Moffatt. This volume-preserving flow was constructed to model the stretch–twist–fold mechanism of the fast dynamo magnetohydrodynamical model. In particular we obtain a complete classification of cases when the system admits an additional Darboux polynomial of degree one. All but one such case are integrable, and first integrals are presented in the paper. The case when the system admits an additional Darboux polynomial of degree one but is not evidently integrable is investigated by methods of differential Galois theory. It is proved that the four-parameter family contained in this case is not integrable in the Jacobi sense, i.e. it does not admit a meromorphic first integral. Moreover, we investigate the integrability of other four-parameter {textit{STF}} systems using the same methods. We distinguish all the cases when the system satisfies necessary conditions for integrability obtained from an analysis of the differential Galois group of variational equations.
Highlights
1.1 Origin of STF SystemsOne of the important problems of geo- and astrophysics is an explanation of the origin of magnetic fields of stars and planets
The first involves the determination of cases when the general STF system (1.9) admits a linear Darboux polynomial
The second part of our results contains theorems which give necessary or necessary and sufficient conditions for the integrability of distinguished families of the STF system obtained by an application of the differential Galois methods
Summary
One of the important problems of geo- and astrophysics is an explanation of the origin of magnetic fields of stars and planets. The authors extended the velocity field considered in Moffatt and Proctor (1985), adding to it the appropriate additional potential field such that the two required conditions were satisfied. As a result, they obtained the following differential system. Let u0(x) denote the vector field given by the right-hand sides of (1.1) It has zero divergence and can be considered as a velocity field of an incompressible fluid. It is a divergence-free vector field, and the unit ball B3 and the unit sphere S2 are invariant with respect to its flow This is the most general polynomial vector field of degree two having these two properties. It is worth mentioning that some experimental realizations of the STF flows have been conducted, see e.g. Fountain et al (1998, 2000)
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