Uniform droplets generated by Rayleigh breakup of liquid jet are widely applied in science and engineering. The droplets are produced by imposing a periodic velocity perturbation on a micro-sized liquid jet. In practical situations, the frequency of droplet generation is not perfectly steady like the preset perturbation frequency. This unwanted timing jitter poses kinds of problems. We studied the fluid mechanism of the jitter at short working distance and its dependence on various parameters. We found that at short distance, the jitter is mainly affected by the reduction rather than the dispersion in the droplet velocity. The magnitude of the jitter is related to the velocity reduction and the unsteadiness of the perturbation. The velocity difference between the droplet and the jet is analytically obtained based on one-dimensional linear analysis of drop formation in liquid jet, and numerical simulations validate the results. The influence of the unsteady perturbation is explained by the evolution of control volumes with different initial amplitudes. The degree of jitter is finally deduced, and its relationships with each variable are compared with experiments. Optimization methods are given to mitigate jitter by adjusting the perturbation parameters and jet properties.
Study on propagation of cylindrical electromagnetic waves in various inhomogeneous and nonlinear media is of fundamental importance, which can be described by the cylindrical nonlinear Maxwell's equations. In recent years, finding exact solutions for these equations has emerged as a popular research topic. The exact solutions play an irreplaceable role in understanding and predicting physical phenomena, and developing numerical calculation methods, and so on. However, due to the extreme complexity of nonlinear partial differential equations, exact solutions of the cylindrical Maxwell's equations were only able to be obtained in a nonlinear and nondispersive medium whose dielectric function is an exponential function in previous researches. Actually, there is no general method at present which can exactly solve arbitrary cylindrical nonlinear Maxwell's equations. Therefore, finding physically admissible solutions meeting certain particular condition for the cylindrical nonlinear Maxwell's equations might be feasible. In this paper, we discuss the traveling wave solutions which are very important in electromagnetic theory, especially in solitary wave theory. To our knowledge, research on obtaining traveling wave solutions of the cylindrical nonlinear Maxwell's equations is still lacking. The main conclusions in this paper are listed as follows. Firstly, we introduce the cylindrical nonlinear Maxwell's equations mentioned in some previous publications, which can describe cylindrical electromagnetic waves propagation in inhomogeneous nonlinear and nondispersive media. In this paper, we focus on the nondispersive media with arbitrary nonlinearity and power-law inhomogeneity. Secondly, we point out that the electric field component E of the model has no plane traveling wave solutions E=g(r-kt), after theoretical analysis and study. Then generalized traveling wave solutions in form of E=g(lnr-kt) for the electric field component are obtained by finding correct variable substitution and solving second-order nonlinear ordinary differential equation.Finally, we provide two examples to show the physical meanings of our generalized traveling wave solutions. We find that the transmitting speeds of vibrations vary with different points of the electric field. Actually, the transmitting speed of the vibration of a certain point closer to the cylinder center is lower. As a result, we observed a physical phenomenon similar to that of self-steepening. Our work can be used to analyze the electromagnetic properties of ferroelectric materials and new materials. Theoretically, it can also provide an approach to studying the cylindrical nonlinear Maxwell's equations.
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