Ooshida, Takeshi

The long- wave expansion method initiated by Benney reduces the dynamics of falling film flows into a surface equation (an equation which contains only one dependent variable h but the equation thus obtained, as well as its higher order version, is known to be valid only slightly far beyond the criticality (up to R=1 for vertical films). This means that Benney's method is not applicable to the typical experimental conditions, R=10 and W=100, where R is as large as µ-1 ~ and which could be referred to as "far beyond the criticality." This lecture introduces the regularized long-wave expansion method, which consists of regarding Benney's long-wave expansion as a power series expansion by ¶x and then replacing it by the Pad\'e approximation. After deriving a "regularized equation" by this method, its numerical solutions are compared with the calculations of the full Navier=Stokes equation as well as with experimental measurements, resulting in a good agreement even at R=10.

Unlike the "boundary-layer" equation which is also valid far beyond the criticality, the regularized equation is a surface equation, i.e. it governs the surface evolution alone without postulating to resolve the velocity field. The structure of the regularized equation, however, has a correspondence to the depth-averaged "boundary-layer" equations, which facilitates to discuss the physical mechanism of the wave dynamics. In particular, it helps to clarify how the inertia effect de stabilizes the flat film and how the wave growth is saturated subsequently. By taking notice of solitary waves appearing in the saturated waves, their tail length L and its R-dependence is calculated, which reveals two distinct regimes in the L - R diagram. The second regime, which is not predicted by Benney's method, arises when the inertia effect becomes significant.