Numerical And Theoretical Investigation On Faraday Waves In Hele-Shaw Cells

Parametric excitation,an issue both ancient and modern,simple and profound.It has been almost two hundred years since Faraday waves were discovered in the 1830 s.The scientific topic returned to the sight of researchers of hydrodynamics in the middle of the last century after silence for a long time because of a theoretical breakthrough.By using the Mathieu equation,the stability regions held by Faraday waves can be conveniently displayed,which in turn benefits people to take the opportunity to appreciate the physical uncertainty.This pioneering linear theory has a great practical significance,no analytical solution though.However,the weakly nonlinear theory revitalized this fluid mechanical problem.A wide variety of Faraday wave patterns have been well explained by the nonlinear interaction among different wave vectors.Weakly nonlinear studies also shed light on stability analysis.The rise of Faraday waves in Hele-Shaw cells expanded people's vision again.The strong nonlinearity and the localized wave profile can both threaten the current mechanical theory.How to face the challenge.The problem itself gives meaning to the study of Faraday waves in Hele-Shaw cells.First,based on Kelvin-Helmholtz-Darcy theory,a gap-averaged Navier–Stokes equation is developed to fit the parametric vibration of the liquids in Hele-Shaw cells.By using an open source code Gerris Faraday waves in this particular container are accurately simulated for the first time.Its validity is also examined by experimental data.The characteristics of the flow field are analyzed after the data of numerical computation is acquired,especially for the mechanism of stationary wave motions.Besides,the major features of the coupled wave profiles in the two-layer condition are also captured by this method,which deepens our understanding of the horizontal oscillation of the lower waves.Actually,the viscous shear friction induced by the upper layer fluid motion leads to the onset of the lower free surface.Second,by embracing a dynamic contact line model from molecular kinetic theory,the gap-averaged equation is improved to make some theoretical analysis.A novel amplitude equation of Faraday waves in Hele-Shaw cells is obtained by multiscale analysis.The stability analysis based on this autonomous equation by Lyapunov's first method gives a better prediction than the previous theory compared with our experimental observation.This theory can also explain the hysteresis found in experiments.Since the liquid film yielded on the cell wall by Faraday waves reduces the contact line friction coefficient due to wetting,when driving force lowers down the damping declines as well.Accordingly,the supercritical value is naturally smaller than its counterpart.Hence,the conclusion of detuning resulting in hysteresis is questionable.Third,A logarithmic scaling law is obtained for Faraday waves in Hele-Shaw cells by dimensional analysis.Different types of dispersion relations are also compared with each other.A conclusion is made combined with experimental support that the traditional dispersion relation of Faraday waves is so far the best fit for this problem.Trying to find a “theoretically correct” dispersion relation for liquid with parametric forcing issues is not recommended until a much better theory with an accurate estimate of damping and powerful capacity of dealing with nonlinearity can be found.