A model of chaotic evolution of an ultrathin liquid film flowing down an inclined plane

Chemical and colloidal transport processes in partially saturated porous and fractured media are dependent on liquid flow along a solid surface, which may be chaotic even for small Reynolds numbers. This paper presents the derivation of a one-dimensional evolution equation describing the slow motion (small Reynolds numbers, R<<1) of a very thin liquid film flowing down an inclined impermeable plane. In this equation, gravitational, capillary, and molecular forces are taken into account. The addition of the molecular force term leads to a highly nonlinear equation governing the spatial and temporal evolution of film thickness. In a weakly nonlinear limit, this evolution equation is rescaled to a canonical form. The latter predicts a chaotic hydrodynamic instability for the film surface. This chaotic behavior is illustrated using the 3D projections of pseudo-phase space attractors for the spatial and temporal variations of the dimensionless film thickness.