Breakup of a falling liquid sheet caused by amplified modulational waves

We investigate the breakup process of a falling inviscid liquid sheet in the absence of surrounding fluids. Analysis is based on the unsteady nonlinear equations derived under the membrane approximation. Particular boundary conditions upstream are numerically found for non-oscillatory steady and stable solutions. When unsteady modulational waves propagate on such a steady flow, weakly nonlinear analysis shows that only the antisymmetric mode is amplified propagating downstream; from it the symmetric mode is additionally induced. On the other hand, numerical analysis shows that amplification of the antisymmetric mode agrees well with the weakly nonlinear theory, whereas the magnitude of the induced symmetric mode is much larger than that. It is expected that this induced mode together with thinning of the sheet leads to the sheet breakup downstream.