II-2

by S. P. Lin

Clarkson Univerity

Abstract

A uniform layer of Newtonian liquid flowing down a vertical wall is known to be inherently unstable. It has been found recently that the onset of instability can be suppressed by imparting an inplane oscillation to the vertical wall. The relevant flow parameters are the Reynolds number, the Weber number, the Strouhal number which is the ratio of the wall oscillation velocity to the film flow velocity, the oscillation amplitude in the unit of the film thickness, and the wave number of disturbances. For certain ranges of the Strouhal number and the dimensionless wall oscillation amplitude, with the rest of the flow parameters fixed, more than one branch of neutral satiability curve have been found in the wave number-Reynolds number space. Between two neighboring branches of neutral curves originating from the zero wave-number line there exists a stable region where it would be unstable if there were no wall oscillation. thus, suppression of the film flow instability may be achieved by appropriate wall oscillation.

Whether the window of stability which is opened for infinitesimal amplitude disturbances may stay open for finite amplitude disturbances remains a question. For various sets of parameters the narrowest part of the window of stability occurs at zero wave number. Thus successfully using nonlinear stability analysis for answering the above raised questions may be achieved with a long wave expansion solution of the Navier-Stokes equations subject to the boundary conditions. The difficulty of the analysis lies in the fact that the time and space variables appear explicitly in the governing nonlinear partial differential systems. The time scales of the externally imparted oscillation and the disturbance growth are very far apart as suggested by the linear theory. this allows us to use the method of multiple-time scale. However the quasi-steady approach used successfully for the case of steady basic flow cannot be applied. A system of nonlinear partial differential equations which describes the evolution of finite amplitude two dimensional disturbances is obtained by extracting the slow time part of the solution systematically. The independent variables are the slow time and the space distance in the basic flow direction. The mechanism of the stabilization will be elucidated with the evaluation of the sign of the coefficients of various orders of spatial derivatives, with the aid of numerical results.