Waves on a Falling Liquid Film

FIGURE 5.11 Flow of a liquid film down an inclined plane. 

The velocity distribution in a film of uniform thickness h, where the flow is laminar, is well known (Bird et al., 2002)  

We seek information about characteristics of the traveling waves that develop when the interface is deformed. Because of the initial velocity, a direct stability analysis based on the differential equations of change is more complex mathematically than the analyses presented above, although it has been carried out (Benjamin, 1957 Krantz and Goren, 1970 Yih, 1963). We present here a simpler but shghtly less accurate analysis based on the integral momentum equation. It follows for the most part the procedure of Kapitsa (1948) which was described by Levich (1962). 

We can simplify this equation for wavelengths much longer than film thickness by using the usual boimdary layer approximations  

With this equation and condition (a). Equation 5.142 becomes 

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