  A universal theory of instabilities in swirling flows, occurring in both
natural settings and industrial applications, is formulated. The theory
encompasses a wide range of open and confined flows, including spiral
isothermal flows and baroclinic flows driven by radial temperature gradients
and natural gravity in rotating fluids. By employing short-wavelength local
analysis, the theory generalizes previous findings from numerical simulations
and linear stability analyses of specific swirling flows, such as spiral
Couette flow, spiral Poiseuille flow, and baroclinic Couette flow. A general
criterion, extending and unifying existing criteria for instability to both
centrifugal and shear-driven perturbations in swirling flows is derived, taking
into account viscosity and thermal diffusion and guiding experimental and
numerical investigations in the otherwise inaccessible parameter regimes.
