The analysis of Lin is extended to investigate the nonlinear stability of a liquid film with respect to three-dimensional side-band disturbances. Near the upper branch of the linear-stability curve where the amplification ci is O (ε2), ε being proportional to the ratio of the amplitude to the film thickness, the nonlinear evolution of initially infinitestimal three-dimensional disturbances of a finite band width is shown to obey the nonlinear Schrödinger equation. Near the lower branch of the neutral curve, the nonlinear evolution is stronger. An equation is derived which describes this strong nonlinear development of relatively long three-dimensional waves. It is shown that the supercritically stable, finite amplitude, long monochromatic wave is stable to three-dimensional side-band disturbances under modal interaction if the bandwidth is less in magnitude than ε.
The nonlinear stability of a viscous liquid film flowing steadily down an inclined plane is studied in the phase plane. Explicit expressions of the critical points of the governing differential system are obtained. The nature of the critical points and the integral curves corresponding to various flow parameters, initial conditions, and disturbance characteristics are determined numerically. Based on the phase plane analysis and the numerical results, the following conclusions are reached: The film which is unstable according to linear theory may be stable with respect to a finite three-dimensional disturbance if the initial amplitude and the side-band width are sufficiently small. The particular values of the initial amplitude and the side-band width beyond which the film becomes unstable depend on the relevant flow parameters. The film which is stable according to linear theory is also shown to be stable with respect to three-dimensional small finite amplitude disturbances with finite bandwidths.
SUMMARYWe present in this paper a nine-point, fourth-order difference scheme for the numerical solution of the quasilinear Poisson equation in polar co-ordinates,
1Aurr + -ur + -U e e = f (r, 0, U , U r , Ue) r r2 with appropriate boundary conditions. A separate difference scheme of order four valid at r = 0 has also been obtained. The method is based on five evaluations of the function f. The numerical results of two problems obtained using this scheme are listed. The numerical results demonstrate the fourth-order accuracy of the scheme.
Surface tension and shear stress are shown to cause a viscous liquid film falling from the edge of an inclined plane to deflect toward the underside of the plane.
SUMMARYWe propose in this paper a nine-point, fourth-order difference method for the numerical solution of the quasilinear Poisson equation with appropriate boundary conditions. The method is based on five evaluations of f. The numerical results of four problems obtained using this method are listed. The results demonstrate the fourth-order accuracy of the method.
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