Simulation of the long-term nonlinear evolution and interaction of disturbances in a liquid film

L. A. Prokudina

South Ural State University (National Research University), Chelyabinsk, Russian Federation

Abstract: We present a nonlinear mathematical model describing the evolution of disturbances on a liquid film surface. The model is formulated as a nonlinear parabolic equation for the envelope amplitude of a narrow wave packet. The equation coefficients are expressed in terms of wave characteristics, including growth rate, frequency, and their first- and second-order derivatives.
Using calculations in the liquid film instability region, we identified inflection points of the growth rate and the wave numbers corresponding to the maximum growth rate for moderate Reynolds numbers. We show that the nonlinear evolution of disturbances changes abruptly at the wave numbers where the growth rate is maximal, while the equation coefficients remain nonzero and vary only slightly. We also demonstrate that the phase velocity reaches a minimum at the point of maximum growth rate. Wave packet amplitudes perturbed near the neutral stability point experience damping.

Keywords: liquid film, instability, increment, phase velocity, wave package, nonlinear parabolic equation.