Nonlinear buckling of a floating fluid film

Using a long-wave analysis, buckling under compression is explored for a thin film of viscous fluid floating above a deep, less viscous bath. Buckling competes against buoyancy and bending stress in this problem, leading to a preferred wavelength in linear theory. Nonlinearity arrests the growth of such unstable modes, slowing amplification from the initially exponential growth to a long-term algebraic growth. When multiple buckling modes are present, the nonlinearity of the thin-film model leads to an exact closure in which only those wavelengths remain excited. Moreover, the suppression of the longest waves by buoyancy triggers a direct cascade in which mode coupling gradually transfers excess length from the gravest modes to higher wavenumber. Buckling patterns then inexorably decrease in wavelength until the mode with the shortest wavelength is reached. The dynamics of this nonlinear cascade is interrogated in detail and its temporal control is deciphered. The analogous dynamics for a thin film of power-law fluid is discussed. In this case, nonlinear effects are more complicated owing to the nonlinearity present in the viscosity law.