An axisymmetric instability of a dusty vortex

In this work, we study the three-dimensional stability of a particle-laden vortex - an isolated columnar vortex with non-uniformly distributed inertial particles. We perform a temporal linear stability analysis by considering an Eulerian-Eulerian framework of a two-fluid model for the particle and fluid phases, further reduced to a single-fluid model for weakly inertial particles. We obtain a 'particle modified Rayleigh criterion', identifying the possibility of an axisymmetric instability of a monotonically decaying vorticity profile due to particle feedback force. For the specific scenario of a vortex column, our calculations reveal that the axisymmetric perturbations render the system unstable for non-zero mass loading, even in the limit of vanishing particle inertia. With increasing particle inertia, the growth rates of the unstable modes are reduced; however, for large wavenumbers, the stationary unstable modes morph into a pair of oscillatory unstable modes. Thus, a novel axisymmetric instability is observed in a columnar vortex due to the presence of inertial particles.