Effect of Horizontal Quasi-Periodic Oscillation on the Interfacial Instability of Two Superimposed Viscous Fluid Layers in a Vertical Hele-Shaw Cell

We investigate the effect of horizontal quasi-periodic oscillation on stability two superimposed immiscible fluid layers confined in a Hele-Shaw cell. To approximate real oscillations, with incommensurate frequencies is considered. Thus, linear analysis leads to oscillator, damping, which describes evolution amplitude interface. Two types instabilities occur: low-wavenumber Kelvin-Helmholtz instability and large-wavenumber resonances. mainly show that, for equal amplitudes accelerations, low irrational frequency ratio, there competition between several resonance modes allowing very large selection wavenumber from lower higher values. This way control sizes waves. Furthermore, increasing ratio has stabilizing both whose thresholds are found correspond solutions using spectrum. For displacement unity less than unity, number resonances their also become significant intermediate values frequencies. The effects other physical geometrical parameters, such as damping coefficient, density heights layers, examined.