  Non-ideal fluids are generally subject to the occurrence of non-isotropic
pressure tensors, whose determination is fundamental in order to characterize
their dynamical and thermodynamical properties. This requires the
implementation of theoretical frameworks provided by appropriate microscopic
and statistical kinetic approaches in terms of which continuum fluid fields are
obtained. In this paper the case of non-relativistic magnetized fluids forming
equilibrium toroidal structures in external gravitational fields is considered.
Analytical solutions for the kinetic distribution function are explicitly
constructed, to be represented by a Chapman-Enskog expansion around a
Maxwellian equilibrium. In this way, different physical mechanisms responsible
for the generation of non-isotropic pressures are identified and proved to be
associated with the kinetic constraints imposed on single and collective
particle dynamics by phase-space symmetries and magnetic field. As a major
outcome, the validity of a polytropic representation for the kinetic pressure
tensors corresponding to each source of anisotropy is established, whereby
directional pressures exhibit a specific power-law functional dependence on
fluid density. The astrophysical relevance of the solution for the
understanding of fluid plasma properties in accretion-disc environments is
discussed.
