  We investigate the effect of inertial particles dispersed in a circular patch
of finite radius on the stability of a two-dimensional Rankine vortex in
semi-dilute dusty flows. Unlike the particle-free case where no unstable modes
exist, we show that the feedback force from the particles triggers a novel
instability. The mechanisms driving the instability are characterized using
linear stability analysis for weakly inertial particles and further validated
against Eulerian-Lagrangian simulations. We show that the particle-laden vortex
is always unstable if the mass loading $M>0$. Surprisingly, even non-inertial
particles destabilize the vortex by a mechanism analogous to the centrifugal
Rayleigh-Taylor instability in radially stratified vortex with density jump. We
identify a critical mass loading above which an eigenmode $m$ becomes unstable.
This critical mass loading drops to zero as m increases. When particles are
inertial, modes that fall below the critical mass loading become unstable,
whereas, modes above it remain unstable but with lower growth rates compared to
the non-inertial case. Comparison with Eulerian-Lagrangian simulations shows
that growth rates computed from simulations match well the theoretical
predictions. Past the linear stage, we observe the emergence of high-wavenumber
modes that turn into spiraling arms of concentrated particles emanating out of
the core, while regions of particle-free flow are sucked inward. The vorticity
field displays similar pattern which lead to the breakdown of the initial
Rankine structure. This novel instability for a dusty vortex highlights how the
feedback force from the disperse phase can induce the breakdown of an otherwise
resilient vortical structure.
