Mhatre S E, Deshmukh S D, Thaokar R M
Department of Chemical Engineering, Indian Institute of Technology Bombay, 400 076 Mumbai, India.
Eur Phys J E Soft Matter. 2012 May;35(5):39. doi: 10.1140/epje/i2012-12039-4. Epub 2012 May 29.
The effect of conductor boundaries on the deformation and stability of a charged drop is presented. The motivation for such a study is the occurrence of a charged conductor drop near a conductor wall in experiments (Millikan-like set-up in studies on Rayleigh break-up) and applications (such as electrospraying, ink-jet printing and ion mass spectroscopy). In the present work, analytical (linear stability analysis (LSA)) and numerical methods (boundary element method (BEM)) are used to understand the instability. Two kinds of boundaries are studied: a spherical, conducting, grounded enclosure (similar to a spherical capacitor) and a planar conducting wall. The LSA of a charged drop placed at the center of a spherical cavity shows that the Rayleigh critical charge (corresponding to the most unstable l = 2 Legendre mode) is reduced as the non-dimensional distance ̂d = (b - a)/a decreases, where a and b are the radii of the drop and spherical cavity, respectively. The critical charge is independent of the assumptions of constant charge or constant potential conditions. The trans-critical bifurcation diagram, constructed using BEM, shows that the prolate shapes are subcritically unstable over a much wider range of charge as [Formula: see text] decreases. The study is then extended to the stability of a charged conductor drop near a flat conductor wall. Analytical theory for this case is difficult and the stability as well as the bifurcation diagram are constructed using BEM. Moreover, the induced charges in the conductor wall lead to attraction of the drop to the wall, thereby making it difficult to conduct a systematic analysis. The drop is therefore assumed to be held at its position by an external force such as the electric field. The case when the applied field is much smaller than the field due to inherent charge on the drop ((a(3)ρg)/(3ε(0)Ψ(2)) ≪ 1 is considered. The wall breaks the fore-aft symmetry in the problem, and equilibrium, predominantly prolate shapes corresponding to the legendre mode, l = 2 , are observed. The deformation increases with increasing charge on the drop. The breakup of the prolate equilibrium shapes is independent of the legendre modes of the initial perturbations. The prolate perturbations are subcritically unstable. Since the equilibrium prolate shapes cannot continuously exchange instability with equilibrium oblate shapes, an imperfect transcritical bifurcation is observed. A variety of highly deformed equilibrium oblate shapes are predicted by the BEM calculations.
本文介绍了导体边界对带电液滴变形和稳定性的影响。开展此类研究的动机源于实验（如瑞利破裂研究中的类似密立根装置）和应用（如电喷雾、喷墨打印和离子质谱分析）中导体壁附近出现带电导体液滴的情况。在本工作中，采用解析方法（线性稳定性分析（LSA））和数值方法（边界元法（BEM））来理解不稳定性。研究了两种边界情况：一个球形的、导电的、接地的封闭腔体（类似于球形电容器）和一个平面导电壁。对置于球形腔中心的带电液滴进行的线性稳定性分析表明，随着无量纲距离 ̂d = (b - a)/a减小，瑞利临界电荷（对应于最不稳定的l = 2勒让德模式）减小，其中a和b分别为液滴和球形腔的半径。临界电荷与恒定电荷或恒定电势条件的假设无关。使用边界元法构建的跨临界分岔图表明，随着[公式：见原文]减小，长椭球形在更宽的电荷范围内是亚临界不稳定的。然后将研究扩展到平面导体壁附近带电导体液滴的稳定性。这种情况下的解析理论较为困难，因此使用边界元法构建稳定性和分岔图。此外，导体壁中的感应电荷会导致液滴被吸引到壁上，从而难以进行系统分析。因此假设液滴通过诸如电场等外力保持在其位置。考虑外加场远小于液滴固有电荷产生的场的情况（(a(3)ρg)/(3ε(0)Ψ(2)) ≪ 1）。壁打破了问题中的前后对称性，观察到对应于勒让德模式l = 2的平衡态，主要是长椭球形。变形随着液滴上电荷的增加而增大。长椭球形平衡态的破裂与初始扰动的勒让德模式无关。长椭球形扰动是亚临界不稳定的。由于平衡长椭球形不能与平衡扁球形连续地交换不稳定性，因此观察到一个不完全的跨临界分岔。边界元法计算预测了各种高度变形的平衡扁球形。
