Study On Nonlinear Stability Of Plane Liquid Jet Multi-level Surface Wave

In this paper,a nonlinear physical model of the non-viscous planar liquid film jet into the incompressible axial flow in the fragmentation process is established.The nonlinear governing equations of continuity,kinematics and dynamics are deduced to derive the explicit expression of the second harmonic complex circular frequency and surface wave amplitude.First,the initial function of the surface wave amplitude of Jazayeri?paper published in Journal of Fluid Mechanics,International Journal of Fluid Mechanics,Cambridge,England?was solved.Then the first-order wave differential equation of Jazayeri antisymmetric waveform was deduced and The expression of the solution is exactly the same as Jazayeri's derivation and extends the derivation to a positive symmetry waveform.After that,we set up our own initial function of the space-time model and derive the first-order wave differential equations and the expressions of the antisymmetric and symmetrical waveforms.Further,the expressions of the second-order wave differential equations and their solutions of antisymmetric and symmetric waveforms are deduced.Draw the waveforms of the first and second wave,as well as the superimposed waveforms of the first and second wave.The results show that:?1?Due to the ordinate symmetry of cosh of hyperbolic cosine function,the positive and negative of Jazayeri's antisymmetric waveform's first-order surface wave growth rate ??? have no effect on the numerical results.Our antisocial space-time model,when ??? is positive,the surface wave continues to grow,when ??? is negative,the surface wave becomes more and more stable.?2?Under the condition of the same input parameters and low liquid viscosity,the nonlinear explicit growth parameters-the dominantsurface wave growth rate ???_i1-dom,the fluctuation term parameters-the dominant ???_i1-dom)and dominant wave numbers k_1-dom.The numerical results are based on the Gaster transformed and linear Implicit Temporal Dispersion Criterion Relationships The results of numerical calculations using the Muller method are almost identical and can be substituted to simplify the calculation.?3?The amplitude of the second-order wave significantly increases with time.?4?The antisymmetric wavefronts are mainly affected by the second-order wavefronts and the fragmentation length.The influence of the disintegration time and the fragmentation length of the antisymmetric wavefronts are greatly shortened.?5?The Jazayeri model,our ???_i1-dom)are positive and negative spatio-temporal models,all have exactly the same results for the crack length of the liquid film.Note that although the initial function of surface wave is different,but the calculation results are exactly the same.Since the initial function expression of the first-order wave amplitude in our spatio-temporal pattern is consistent with the linear stability,the theoretical system is more complete and logistic.?6?The first wave of the symmetrical waveform will cause the liquid film to crack.The second-wave superposition will affect the waveform of the first-order wave,but it will not shorten the fragmentation time and fragmentation length.