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竖直振动无黏液滴的法拉第不稳定性分析

INSTABILITY ANALYSIS OF INVISCID DROPLET SUBJECT TO VERTICAL VIBRATION

  • 摘要: 由于外部周期性的振动而在液滴表面产生的Faraday不稳定效应广泛存在于超声雾化、喷涂加工等应用中, 对Faraday不稳定性进行分析对研究振动液滴的表面动力学有着重要意义. 本文将Faraday不稳定性问题从径向振动拓展到竖直振动, 研究了竖直振动无黏液滴表面波的不稳定性. 竖直方向的振动使得液滴动量方程为含有空间相关项和时间周期系数的Mathieu方程. 采用Floquet理论进行求解, 得到了竖直振动液滴表面波线性增长率与模态数以及流动参数之间的色散关系. 通过求解一个关于表面变形模态的特征值问题, 得到了竖直振动无黏液滴在Faraday不稳定性下的中性稳定边界, 并比较了竖直振动与径向振动的液滴中性稳定边界的差异. 通过大模态数假设下的近似计算, 得到了仰角θ对中性不稳定边界的影响规律. 结果表明竖直振动的液滴与径向振动相比, Faraday不稳定区域更小, 激发的模态范围更窄, 并且不会出现亚简谐的不稳定波. 另外, 对于竖直振动的液滴, 仰角θ越大的位置, 中性不稳定区域越小, 在受到外部激励时液滴表面越容易保持稳定.

     

    Abstract: Faraday instability on the droplet surface due to external periodic oscillation is widely used in ultrasonic atomization, spraying processing and other applications. The analysis of Faraday instability is of great significance to the study of the surface dynamics of vibrating droplet. In this paper, the Faraday instability problem is extended from radial vibration to vertical vibration, and the instability of inviscid droplet surface wave in vertical vibration is studied. The vertical vibration makes the droplet momentum equation a Mathieu equation with spatial correlation term and time periodic coefficient. The dispersion relations between the growth rate, the mode number and flow parameters of vertically vibrating droplet surface waves are obtained by using Floquet theory. The neutral stable boundary of vertically vibrating inviscid droplet under Faraday instability is obtained by solving an eigenvalue problem of surface deformation modes. The difference of droplet neutral stability boundary between vertical vibration and radial vibration is compared. The influence of elevation angle θ on the neutral instability boundary is obtained by the approximate calculation under the assumption of large mode number. The results show that the difference between vertically vibrating droplets and radially vibrating droplets is obvious. The differences are as follows: in the case of harmonic, the unstable region of droplet surface wave becomes smaller, and the droplet will be more difficult to destabilize under external excitation; In the case of subharmonic, the neutral stable boundaries of the droplet surface wave coincide, and the droplet unstable wave will not appear subharmonic mode. Besides, for vertically vibrating droplets, the larger the elevation angle θ, the smaller the neutral instability region, and the easier it is for the droplet surface to remain stable under external excitation.

     

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