Stability and bifurcations of parametrically excited thin liquid films

We investigate the stability and bifurcations of parametrically excited thin liquid films. A recently derived nonlinear evolution equation for the two-dimensional spatio-temporal dynamics of falling liquid films on an oscillating vertical wall is expanded to low order Fourier modes. A fourth-order modal dynamical system is validated to yield the primary bifurcation structure of the fundamental falling film dynamics described by the Benney equation, and accurately predicts the quasi-periodic structure of the temporally modulated Benney equation (TMBE). The stability of fundamental steady and periodic solutions is analytically and numerically investigated so as to reveal the threshold for nonstationary and chaotic solutions corresponding to aperiodic modulated traveling waves. The reduced modal dynamical system enables construction of a comprehensive bifurcation structure, which is verified by numerical simulation of the evolution equation.