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Abstract
A class of exact solutions to the Navier-Stokes equations for axisymmetric vortex flows of incompressible fluids is obtained. Invariant manifolds of flows with rotational symmetry relative to a given axis in three-dimensional coordinate space are identified, and the structure of the solutions is described. It is established that typical invariant regions of such flows are rotational figures homeomorphic to a torus, forming a structure of topological fibration, such as in a sphere, cylinder, and more complex configurations composed of such figures. The results are extended to similar solutions of the magnetohydrodynamics (MHD) system and Maxwell’s electrodynamics equations, which possess ℝ3 analogous properties. Examples of axisymmetric vortex vector fields and the topological fibrations they generate on manifolds invariant ℝ3 under the dynamical systems defined by these fields are provided.
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