Stability characteristics for flows of the vortex sheet type

The interfacial conditions for a cylindrical vortex sheet or a cylindrical fluid layer obtained earlier are examined further for the case of incompressible fluids. Only temporal perturbations are considered. Unlike the single role of destabilization played by the velocity in two dimensional stratified flows or axisymmetric jet flows, the rotating velocity in vortex motions plays a dual role in flow stability. The radial velocity gradient generates tangential shear and the fluid rotation creates a centrifugal force field. While the former always destabilizes the flow, the latter can either stabilize or destabilize the flow depending upon whether the resultant force field is centrifugally stable or unstable. These characteristics are demonstrated by examining three general types of perturbations. We further show that deformations of the vortex sheet or the fluid layer affect the flow field in two ways: disturbing the total pressure field and perturbing the centrifugal force field created by the azimuthal components of the velocity and magnetic fields. The latter, even though it seems to be straightforward, has often been overlooked in previous analyses. It is shown that failure to consider such a perturbation to a stable centrifugal force field will lead to the improper destabilization of modes with smaller axial and azimuthal wave numbers.