Linear stability of a viscoelastic film with insoluble surfactants under quasi-periodic oscillation

The linear stability of a viscoelastic liquid film containing insoluble surfactant on a horizontal quasi-periodic oscillating plane for disturbances with arbitrary wavenumbers is investigated. The combined effects of viscoelasticity and quasi-periodic oscillation on the instability mechanism in the presence of surfactant are described with the Floquet theory. For long-wavelength instability, the solution in the limit of long-wave perturbations is obtained by the asymptotic expansion method. The results indicate that, in the absence of gravity, the stability of the oscillating film is determined by the competition between viscoelastic and surfactant effects. In the presence of gravity, the U-shaped neutral curves with separation bandwidth appear. Due to the existence of viscoelasticity, the effects of different frequency ratios and amplitude ratios on the U-shaped neutral curve in the high-frequency region are significantly different from the results under Newtonian fluids. Furthermore, whether gravity exists or not, the viscoelastic effect exhibits a dual effect of initially destabilising, subsequently stabilising and finally destabilising again in the high-frequency oscillating liquid film. The finite-wavelength instability is solved numerically based on the Chebyshev spectral collocation method. The study demonstrates that viscoelasticity results in a non-monotonic trend in the critical Reynolds number of finite-wavelength modes. In the high-frequency region, a bifurcation phenomenon appears due to competition between different standing-wave modes. Increasing the viscoelastic parameter, frequency ratio or amplitude ratio significantly destabilises the flow under high-frequency oscillations.