Nonlinear capillary wave distortion and disintegration of thin planar liquid sheets

Linear and nonlinear dilational and sinuous capillary waves on thin invisci d 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 uns teady problem to a one-dimensional unsteady problem. For nonlinear dilation al waves on infinite sheets, the accuracy of the numerical solutions is ver ified by comparing with an analytical solution. The nonlinear dilational wa ve maintains a reciprocal relationship between wavelength and wave speed mo dified 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 s uperimposed subharmonic disturbances on the infinite sheet. Agreement for b oth sinuous and dilational waves is demonstrated for the infinite case betw een nonlinear simulations using the reduced one-dimensional approach, and n onlinear two-dimensional simulations using a discrete-vortex method. For se mi-infinite dilational and sinuous distorting sheets that are periodically forced at the nozzle exit, linear and nonlinear analyses predict the appear ance of two constant-amplitude waves of nearly equal wavelengths, resulting in a sheet disturbance characterized by a long-wavelength envelope of a sh ort-wavelength oscillation. For semi-infinite sheets with sinuous waves, qu alitative agreement between the dimensionally reduced analysis and experime ntal results is found. For example, a half-wave thinning and a sawtooth wav e shape is found for the nonlinear sinuous mode. For the semi-infinite dila tional case, a critical frequency-dependent Weber number is found below whi ch one component of the disturbances decays with downstream distance. For t he 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.