STUDY ON THE EVOLUTION CHARACTERISTICS OF SOLITARY WAVES AND INTERNAL VORTICES IN LIQUID FILM FLOW ON INCLINED WALLS

This paper investigates the evolution of solitary waves and internal vortices in a two-dimensional incompressible viscous liquid film flowing down an inclined wall under the influence of gravity. Using the method of small parameter perturbation and the traveling wave transformation, we first derive the general zeroth-order and first-order evolution equations for the film thickness on a non-smooth inclined substrate. These equations are then simplified and numerically solved using Mathematica. The analysis results show that in the waveform of the solitary wave, a capillary wave appears at the wavefront, and a fully open vortex forms at the trough of the capillary wave. Flow rate analysis indicates that it follows the same variation trend as the solitary wave profile and is positively correlated with the wave speed. For double-peak and triple-peak solitary waves, the flow rate of the preceding peak is greater than that of the subsequent peaks. When the wave speed increases beyond a certain critical value, vortices appear within the solitary wave peaks. Calculations show that this critical wave speed is directly proportional to the inclination angle. For double-peak and triple-peak solitary waves, further increase in wave speed results in vortices forming within the subsequent peaks as well. By analyzing the velocity distribution on the free surface, it is concluded that the generation of vortices is due to the combined effect of the velocity gradient of the vertical velocity at the wavefront and wave tail and the horizontal velocity exceeding the wave speed, with more wave peaks facilitating vortex formation. Analysis of the streaklines obtained in the moving coordinate system reveals that the vortex area is also proportional to the wave speed and rotates clockwise. Overall, it is inferred that the vortex starts to form at the surface of the solitary wave, with the fluid within the wave peaks rolling down along the wall.