Asymptotic solution of the Orr-Sommerfeld equation for surface waves on an inclined plane.

Bruno Pelisson Chimetta¹; Erick de Moraes Franklin¹

¹ University of Campinas - UNICAMP

doi:10.20906/CPS/COB-2015-1657

Resumo

Asymptotic solution of the Orr-Sommerfeld equation for surface waves on an inclined plane. Bruno Pelisson Chimetta*, Erick de Moraes Franklin + University of Campinas - UNICAMP Faculty of Mechanical Engineering Rua Mendeleyev, 200 - CEP 13083-860 - Campinas - SP * bruno_chimetta@yahoo.com.br + franklin@fem.unicamp.br ABSTRACT This work presents an analysis of the initial behavior of the flow on inclined planes with free surface where, under some conditions, long-wave instability may appear. These instabilities may evolve to surface-wave, that often appear on thin liquid films. Such knowledge is useful in industry, once liquid films help to remove the heat from solid surfaces, and also reduces the friction between high viscosity fluids and pipe walls, by injecting close to the wall a less viscous fluid. The surface-wave instability is governed by the Orr-Sommerfeld equation. In this work, we present a long-wave solution for the Orr-Sommerfeld equation based on asymptotic expansions. For the long-wave perturbations, the wave number can be treated as a small parameter, and the form of the equations suggests that the speed and the amplitude of the eigenfunction can be sought as a power series of the wave number. The asymptotic solution gives the growth rate, the wave speed, the wavelength and the critical Froude number of the liquid film at the instability threshold. The results are compared with previously published data. Keywords: Surface-wave instability, Orr-Sommerfeld equation, Asymptotic Solution. REFERENCES [1] C. S. Yih, "Stability of Liquid Flow dow

Palavras-chave: Surface-wave instability; Orr-Sommerfeld equation; Asymptotic Solution