Effect of gas velocity on the weakly nonlinear instability of a planar viscous sheet

Abstract

A weakly nonlinear spatial instability of a two-dimensional planar viscous sheet for sinuous disturbances in a co-flowing inviscid gas stream is investigated theoretically, with an emphasis on the effect of the surrounding gas velocity. The solutions of the second-order interface disturbances are derived and the wave deformation has been computed. The results indicate that the second-order surface disturbance of the fundamental sinuous mode is varicose, which causes the thinning and the subsequent breakup of the liquid sheet. The nonlinear behaviors of the planar sheet are quite sensitive to variations in gas-to-liquid velocity ratio. The deviation of the velocity ratio from the value of unity leads to a larger growth rate, a larger second-order initial amplitude, and a shorter breakup length, and therefore enhances the instability. The growth rates predicted by the present nonlinear analysis according to the shortest breakup length are generally smaller than the linear predictions and can better conform to the experimental measures of Barreras et al. [“Linear instability analysis of the viscous longitudinal perturbation on an air-blasted liquid sheets,” Atomization Sprays 11, 139 (2001)]. Furthermore, the wave deformations of the most unstable disturbances are presented. The nonlinear instability of the planar sheet for a fixed velocity difference is performed. An equal increase of the gas and liquid velocity reduces the spatial growth rate and increases the breakup length, but generally has no influences on the second-order initial amplitude and the wavelength of the disturbance.