Nonlinear Stability of a Thin Liquid Layer Flowing Down an Oscillating Plane

Abstract

The nonlinear stability of a thin liquid layer flowing down an inclined plane which is oscillating in a direction perpendicular to its surface is analyzed using the power series of amplitude in the form of the Fourier series with the coefficients depending on the time. Although the growth rate of the disturbances is influenced by the oscillation of the plane, its influence vanishes on an average. In the flow down an oscillating plane, the oscillation is synchronous and there is an equilibrium state with time-periodic amplitude and phase velocity. The magnitude of the time variation of the amplitude and the phase velocity increases as the amplitude of the oscillation of the plane increases and as the frequency decreases. However, taking the average, the equilibrium amplitude in the oscillating plane is smaller than that in the fixed plane. The flow is stabilized by the oscillating plane.