Stability of Liquid Film Flow Down an Oscillating Wall

The stability of a liquid film flowing down an inclined oscillating wall is analyzed. First, the linear theory growth rates of disturbances are calculated to second order in a disturbance wave number. It is shown that this growth rate is simply the sum of the same growth rate expansions for a nonoscillating film on an inclined plate and an oscillating film on a horizontal plate. These growth rates were originally calculated by Yih (1963, 1968). The growth rate formula derived here shows that long wavelength disturbances to a vertical falling film, which are unstable at all nonzero values of the Reynolds number when the wall is stationary, can be stabilized by sufficiently large values of wall oscillation in certain frequency ranges. Second, the full time-dependent stability equations are solved in terms of a wall oscillation amplitude expansion carried to about 20 terms. This expansion shows that for values of mean flow Reynolds number less than about ten, the wall oscillations completely stabilize the film against all the unstable disturbances of the steady film.