Nonlinear capillary wave distortion and disintegration of thin planar liquid sheets

Abstract

Linear and nonlinear dilational and sinuous capillary waves on thin inviscid infinite and semi-infinite planar liquid sheets in a void are analysed in a unified manner by means of a method that reduces the two-dimensional unsteady problem to a one-dimensional unsteady problem. For nonlinear dilational waves on infinite sheets, the accuracy of the numerical solutions is verified by comparing with an analytical solution. The nonlinear dilational wave maintains a reciprocal relationship between wavelength and wave speed modified from the linear theory prediction by a dependence of the product of wavelength and wave speed on the wave amplitude. For the general dilational case, nonlinear numerical simulations show that the sheet is unstable to superimposed subharmonic disturbances on the infinite sheet. Agreement for both sinuous and dilational waves is demonstrated for the infinite case between nonlinear simulations using the reduced one-dimensional approach, and nonlinear two-dimensional simulations using a discrete-vortex method. For semi-infinite dilational and sinuous distorting sheets that are periodically forced at the nozzle exit, linear and nonlinear analyses predict the appearance of two constant-amplitude waves of nearly equal wavelengths, resulting in a sheet disturbance characterized by a long-wavelength envelope of a short-wavelength oscillation. For semi-infinite sheets with sinuous waves, qualitative agreement between the dimensionally reduced analysis and experimental results is found. For example, a half-wave thinning and a sawtooth wave shape is found for the nonlinear sinuous mode. For the semi-infinite dilational case, a critical frequency-dependent Weber number is found below which one component of the disturbances decays with downstream distance. For the semi-infinite sinuous case, a critical Weber number equal to 2 is found; below this value, only one characteristic is emitted in the positive time direction from the nozzle exit.