Combined parabolic and elliptic velocity profile-based low-dimensional model in falling film

Based on the assumption of a combined parabolic and elliptic velocity profile, the simplified second-order depth-averaged equations are derived. As the parameter A relating to the eccentricity of the ellipse increases, new results adequately capture available findings. However, A = 2.23219 provides a relatively more accurate result. Maximum amplitude and speed of the steady-state traveling wave increase with rising values of A. The backflow phenomenon occurs in the capillary regime. Interestingly, the combined velocity profile detects the point of inflection in the capillary region at A = 2.23219, but it disappears at higher values of A, signaling a strong influence of the elliptic part.