Unification theory of instabilities of visco-diffusive swirling flows

By employing local geometrical optics stability analysis adapted to visco-diffusive flows, we derive novel explicit instability criteria for isothermal and non-isothermal swirling flows, induced by the combination of rotation and shear in orthogonal directions and ubiquitous in various natural phenomena, such as tornadoes and tropical cyclones. Our advance stems from an observation overlooked in previous research: the neutral stability curves in these problems possess an envelope, which we have analytically determined using the connection between envelopes and polynomial discriminants. Our analytical results offer a general theory of instabilities across a wide range of swirling flows.