Nonlinear dual-mode instability of planar liquid sheets

The nonlinear temporal instability of gas-surrounded planar liquid sheets, whose linear instability contains both sinuous and varicose modes, is studied. Both the weakly nonlinear analysis using a second-order perturbation expansion and the numerical simulation using a boundary integral method have been applied. Their comparison shows that the weakly nonlinear analysis can precisely predict the shapes of sheets for most of the time of disturbance evolution and qualitatively explain the instability mechanism when sheets break up. Both the first harmonics of the linear sinuous mode and linear varicose mode are varicose; they contribute to the breakup of sheets, but the first harmonic generated by the coupling between the linear sinuous and varicose modes is sinuous; it plays an important role in modulating the wave profile. The instability with various initial phase differences between the upper and lower interfaces is examined. Except for the varicose initial disturbance, the linear sinuous mode dominates in the shapes of sheets when their amplitudes grow large. Within the second-order analysis, the major modes that can cause the breakup include the linear varicose mode, the first harmonic of the linear sinuous mode and the first harmonic of the linear varicose mode. The effects of various flow parameters have been investigated. At relatively large wavenumbers where approximate analytical and numerical results agree well when sheets break up, increasing the wavenumber reduces the wave amplitude. Reducing the initial disturbance amplitude makes the first harmonic of the linear sinuous mode the dominant mode in causing the breakup. Increasing the Weber number or gas-to-liquid density ratio significantly reduces breakup time and enhances instability.