Bi-global instability analysis of sidewall effects on an oscillating liquid film

The stability of a horizontally oscillating liquid film under sidewall confinement is investigated through a bi-global Floquet analysis. The classical two-dimensional oscillating liquid film is extended to a three-dimensional configuration with oscillating sidewalls. This framework resolves the high-dimensional eigenvalue problem arising from the spatio-temporal coupling in this three-dimensional system. For long-wave instabilities, a semi-analytical asymptotic approach is adapted to capture the growth rates efficiently, while the arbitrary-wavelength instabilities are solved via Chebyshev spectral collocation methods. The asymptotic and numerical results reveal that the sidewalls suppress both long-wave and finite-wavelength instabilities by contracting the unstable frequency range and elevating the critical Reynolds numbers. Besides, two distinct instability modes are identified: a long-wave mode dominated by surface shear stress, and a finite-wavelength mode driven by Reynolds stress. Energy budget analysis confirms that sidewalls stabilise the flow by dampening energy production. This work provides fundamental insights into controlling oscillating film instabilities through geometric confinement.