DERIVATION OF A THIN-FILM EQUATION BY A DIRECT APPROACH

A new model of the thin viscous fluid film, constrained between two tr anslating, flexible surfaces, is presented in this paper The unsteady inertia of the film is included in the model. The derivation starts wi th the reduced three-dimensional Navier-Stokes equations for an incomp ressible viscous fluid with a small Reynolds number. By introduction o f an approximate velocity field which satisfies the continuity equatio n and the no-slip boundary conditions exactly, into weighted integrals of the three-dimensional equations over the film thickness, a two-dim ensional thin film equation is obtained explicitly in a closed form. T he lth thin film equation is obtained when the velocity field is appro ximated by 2lth order polynominals, and the three-dimensional viscous film is described with increasing accuracy by thin film equations of i ncreasing order. Two cases are used to illustrate the coupling of the film to the vibration of the structure and to show that the second thi n film equation can be applied successfully to the prediction of a cou pled film-structure response in the range of most applications. A redu ced thin film equation is derived through approximation of the second thin film equation that relates the film pressure to transverse accele rations and velocities, and to slopes and slope rates of the two trans lating surfaces.