A linear model describing the instability behavior of annular, swirling, inviscid sheets subject to inner and outer gas flows of differing velocities is presented. The model considers three‐dimensional disturbances and contains previous flat sheet, cylindrical jet, and annular jet analyses as limiting cases. Model predictions show that, in the absence of swirl, (i) an increase in axial Weber number causes the range of unstable axial disturbance modes to increase, (ii) when the axial Weber numbers are small (<8), inner gas flows lead to slightly faster growing axial instability modes than outer gas flows at equivalent inner and outer Weber numbers, but inner and outer gas flows have the same effect when Weber numbers are high (≳10), (iii) the wavenumber for the axial mode having the highest growth rate decreases with a decrease in axial Weber number, (iv) an increase in the density of the atomizing gas results in a slight increase in the wavenumber of the axial disturbance mode having the highest growth rate. When swirl is present, model predictions demonstrate that (v) swirl reduces the wavenumber for the axial disturbance mode having the highest growth rate and reduces growth rates as well, (vi) an increase in the swirl Weber number beyond the stabilizing region increases the range of unstable axial and circumferential modes and increases growth rates as well for nonzero axial Weber numbers, (vii) increasing the swirl Weber number increases the axial wavenumber for the disturbance mode having the highest growth rate, but a circumferential mode number of zero is retained until the swirl Weber number exceeds about 8, at which point the axial wavenumber for the disturbance having the highest growth rate falls to zero and the circumferential wavenumber jumps to a finite value of n at which time further increases in swirl Weber number serve to increase n, (viii) up to two local nondimensional growth rate maxima can exist, and the instability domain can be simply connected or can consist of two separate regions separated by an area where disturbances are stable. The topology of the growth rate surface depends on the ratio of the annulus inner to outer radii. These findings are used to explain some observations of practical atomizer performance. © 1996 American Institute of Physics.