帶電環型液體噴流在軸向磁場作用下之穩定性分析

Abstract

充電的環型噴流在受到靜電力的強力撕裂作用下，通常會觸發更短波長的擾動波，在各種不穩定機制的非線性交互作用下，最後導致噴流連續破裂成微米等級以下的霧滴。此相關現象在引擎燃燒室中之燃油霧化技術有相當重要的應用。本論文主要在以線性穩定性分析探討環型噴流在受到徑向電場作用下的初始不穩定行為，並藉以了解各種物理機制對噴流不穩定變形的影響。另外，在一些應用中(例如中空奈米纖維之製造)，噴流的不穩定現象必須儘量被避免，而加入軸向磁場被認為是穩定噴流的方式之一。因此本文也同時對噴流在軸向磁場下的穩定性做分析，試圖了解磁場的穩定效果與其機制特性。 本理論模型乃耦合電磁場與流場等現象之複雜系統，方程式包括描述電磁場的Maxwell方程與統御流場之Navier-stokes方程式。其數學模型處理起來極端困難(尤其具有內外兩自由液面)，因此必須透過合理的假設以簡化分析。文中指示當流體的導電度在某一範圍內，數學模型可以被近似成準靜電的方程式。在線性微擾化以及正規模態展開後，我們獲得一組24階的常微分方程式，最後使用契比雪夫配置法(Chebyshev collocation method)來求解特徵值問題。研究結果顯示，由於內層表面的出現，環型噴流通常會存在兩種變形模態：同相(in phase)與異相(out of phase)。其中，同相的模態永遠比異相更不穩定，且當環型噴流的厚度越得越薄時，同相的成長率將會增加，但異相的模態將被抑制。此外，我們也發現在某些特殊情況下，m = 2的模態可以是最不穩定的，這暗示環型噴流有連續分裂成多條噴流的傾向，此現象是一般圓柱噴流所沒有的。最後，當磁場的強度增加時，原本同相的模態會轉變成異向的模態，反之異相的模態則會變成同向的。此現象初步推斷可能是由於羅倫茲力的扭轉機制使得內外表面的擾動波成長不同步所造成。 本論文藉由詳盡的參數分析試圖揭露各種導致環型電噴流不穩定變形背後的物理機制，並繪製了參數地圖以劃分各種擾動型態的參數操作範圍，豐富的結果將提供實驗設計者一個定性的參考依據。

Subject to the electrically repulsive force, charged annular jets are more unstable than uncharged jets. The short-wavelength disturbances often nonlinearly grow due to the interaction between various instability mechanisms, and eventually leads to the breakup into micron-scale drops. The involving phenomena have important applications in various engineering and technologies over the past 20 years. For example, the performance of engine combustors strongly depends on the atomization process of fuel oils. A complete understanding of the mechanisms beyond the instability phenomena is essential to the practical applications. This study focuses on the effects of various physical mechanisms on the onset of instability of a charged annular jet. In addition, we also examine the stabilizing ability of the magnetic field to benefit the fabrication of hollow nanofibers, in which the instability is undesirable. The presented theoretical model is a complex system coupled with electromagnetic field and flow motion, and the governing equations, including the Maxwell equations and the Navier-stokes equations with moving internal and external interface, make the mathematical procedure extremely difficult to handle with. Thus, simplicity with some reasonable assumption is necessary. In the context, we clearly reveal that when the fluid conductivity is limited, the mathematical model can be approximately simplified as a quasi-electrostatic form. After linearization and normal mode expansion, we obtain an eigenvalue problem in terms of ordinary differential equations, which are then solved by using Chebyshev collocation method. The results show that an annular jet usually possesses two unstable modes (i.e. para-sinuous and para-varicose modes) due to the presence of the inner surface, and the para-sinuous mode is found to be always more unstable than the para-varicose mode. A decreasing thickness of the annular jet enhances the growth rate of the para-sinuous mode while suppressing that of the para-varicose modes. In addition, we also found that the asymmetric mode with m=2 may dominate the instability in a small parameter range. This implies that the annular jet under some special conditions tends to split into multiple jets. Such a phenomenon has not been found in the system of round jets. For the cases with the magnetic effect, the para-sinuous and the para-varicose modes can transform with one another at a sufficiently high magnetic field strength. This phenomenon is possibly resulted from the twisted effect caused by the Lorenz force, which makes non-synchronized growth of the inner and outer surfaces. This research devotes to the understanding of the physical mechanisms leading to unstable deformation of charged annular jets by carrying out a detailed parameters analysis. The resultant parameter maps, in which the boundaries between various instability types are clearly depicted, should be able to provide engineers a useful guideline for novel designs in future.