This paper validates the theory of linear stability as applied to a fluid flowing down a vertically inclined plane using experimental comparison. The theoretically predicted mode 1 instability, corresponding to a surface wave oscillating at half the forcing frequency, is found to exist alongside the mode 2 instability, corresponding to a surface wave oscillating at the forcing frequency. Instability onset amplitude and frequency were compared, and an investigation of wavenumber with forcing amplitude at several distinct frequencies further confirmed the validity of the theory. A study of the wavenumber trend revealed distinct differences in the stability and development of the surface wave with increasing forcing amplitude. An analysis of the Womersley number, which expresses the ratio of the oscillatory inertia force to the viscous shear force, provides a physical indication of the differences in the wavenumber trend observed.

The problem of the vibration induced instability of a liquid film flow is formulated, thanks to a linear approach. For the solution of the problem, the disturbances are expanded using a spectral method based on Chebyshev polynomials leading to a system of ordinary differential equations. This system is solved by using Floquet theory. The natural and vibration induced instabilities are studied in terms of frequency, temporal growth rate, and eigenfunction. Numerical results show that the mean flow velocity alters the surface wave frequency, which is not predictable by Mathieu's equation model (which assumes that the fluid is at rest). Above a threshold amplitude, multiple vibration induced instability modes are triggered. At specific amplitudes above the latter, two wavenumbers coexist at the free surface. A relation between the eigenfunction shape and the instability mode is observed.

This paper constitutes the final part of an investigation into the vibration-induced instability of a fluid flowing down an inclined plane. The first part ("Vibration-induced instability of a fluid film flowing down a vertically inclined plane: Experimental and theoretical comparison." Physics of Fluids 29.10 (2017): 104103.) involved validating the theory of linear stability as applied to a fluid flowing down a vertically inclined plane via experimental comparison. In this second part, a two-dimensional direct numerical simulation is carried out, and is compared to experiment. The numerical solver was compared to experiment by evaluating the frequency spectra at specific forcing amplitudes where mode 2 & 3 instabilities become dominant. The simulation reproduced the frequency spectra in all cases to a high degree of accuracy and demonstrates that a consideration of a two-dimensional case is su cient to accurately resolve the onset and growth of higher mode non-linear instabilities without the need to account for three-dimensional e↵ects.

This study pertains to the three-dimensional direct numerical simulation (DNS) of a vertically oscillating vessel containing an incompressible Newtonian liquid, surrounded by air at rest and ambient conditions. Squire's theorem was extended and shown to apply in this case, allowing for the theory of linear stability to be implemented and a comparison to be made with the DNS results. It was further discovered that the method by which a fluid instability is initiated in the numerical simulation affects the initial development of the instability. This phenomenon was confirmed through an optimal perturbations analysis. A possible physical explanation of this effect is also presented.