  We theoretically study the instability of helical shear flows, in which one
fluid component flows along the vortex core of the other, in phase-separated
two-component Bose-Einstein condensates at zero temperature. The helical shear
flows are hydrodynamically classified into two regimes: (1) a helical vortex
sheet, where the vorticity is localized on the cylindrical interface and the
stability is described by an effective theory for ripple modes, and (2) a
core-flow vortex with the vorticity distributed in the vicinity of the vortex
core, where the instability phenomena are dominated only by the
vortex-characteristic modes: Kelvin and varicose modes. The helical shear-flow
instability shows remarkable competition among different types of instabilities
in the crossover regime between the two regimes.
