Weakly nonlinear instability of annular viscous sheets

A second-order perturbation analysis has been performed on the nonlinear temporal instability of para-sinuous disturbances on annular viscous sheets moving in an inviscid stationary gas medium. The mathematical expressions of second-order interface disturbances, velocity, and pressure have been derived. The nonlinear instability of annular viscous sheets has several characteristics which differ from that of planar viscous sheets: (1) both the first-order interface disturbances and the second-order interface disturbances contribute to breakup; (2) the zero-wavenumber component of interface disturbances in the second-order solution is nonzero; (3) the second-order interface disturbance is para-varicose in most cases, but para-sinuous for some cases. As with planar viscous sheets, it was found that viscosity plays a dual role in the nonlinear instability of annular viscous sheets. However, with the decrease in the ratio of inner radius to sheet thickness, the interval between the upper and the lower critical Reynolds numbers shrinks, and when the ratio of inner radius to sheet thickness is less than a certain value, the dual effect of viscosity vanishes.