Instability of a planar liquid sheet with surrounding fluids between two parallel walls

We investigate the linear and nonlinear instability of a planar liquid sheet with surrounding fluids between two parallel plane solid walls. Linear analysis shows that the maximum temporal growth rate and unstable wave number region of disturbances increase for the dilational and sinuous modes when the gap between the sheet and the wall decreases. The walls have more influence on the instability when the density ratio of the surrounding fluid to the sheet and/or the Weber number decrease. On the other hand, nonlinear analysis is performed by means of the discrete vortex method, where double vortex rows and their mirror images are placed so as to satisfy the boundary condition on the walls. Numerical results show that the walls enhance nonlinearity, which causes deformation and distortion of the sheet, whereas the nonlinearity diminishes linear growth rates except for long dilational disturbances. In particular, as the walls are placed more closely to the sheet, local sheet thinning becomes more pronounced in the long dilational mode, while the dilational mode is more strongly induced from the sinuous mode through monotonic or periodic energy exchanges between the two modes.