An elliptic velocity profile-based two-equation model in viscous film

Abstract

An elliptic velocity profile-based depth-averaged two-equation model is derived for a viscous falling film in terms of the film thickness h(x, t) and the flow rate q(x, t), which is consistent up to first order in inertia terms and consistent up to second order in viscous diffusion terms. It is observed that the proposed depth-averaged two-equation model recovers the available analytical, numerical, and experimental findings of the literature very well as the free parameter involving eccentricity of the ellipse increases. In fact, the present depth-averaged two-equation model converges to the existing parabolic velocity profile-based depth-averaged two-equation model as the eccentricity approaches one because the elliptic velocity profile becomes a parabolic velocity profile. Furthermore, we see that [20, 100] is the suitable range of the free parameter for capturing the existing findings of the literature. In addition, the full second-order depth-averaged model, which is consistent up to second-order in inertia and viscous terms, is also developed in Appendix A.