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Abstract

This review summarizes recent advances in the theoretical analysis of the stability of magnetized flows in a non-uniformly rotating layer of electrically conductive nanofluid, incorporating the effects of Brownian diffusion and thermophoresis. In the absence of temperature gradients, different forms of magnetorotational instability (MRI): standard (SMRI), azimuthal (AMRI), and helical (HMRI) are investigated for nanofluid layers subjected to axial, azimuthal, and helical magnetic fields. The corresponding growth rates and instability regions are analyzed in relation to the rotation profile (quantified by the Rossby number $\textrm{Ro}$) and the radial wave number $k$. When temperature gradients and nanoparticle concentration effects are present, stationary convective modes in both axial and helical magnetic fields are examined under conditions of non-uniform rotation. Analytical expressions for the critical Rayleigh number $\textrm{Ra}_{st}$ are derived, and neutral stability curves are constructed as functions of the angular velocity profile, the azimuthal magnetic field inhomogeneity (magnetic Rossby number $\textrm{Rb}$), and the wave number $k$. The study identifies and discusses key mechanisms responsible for the stabilization or destabilization of stationary convection in axial and spiral magnetic field configurations, highlighting the role of nanoparticle-driven effects in modifying classical magnetoconvective behavior.