Nair Ashwin, Sharma Gaurav
Department of Chemical Engineering, Indian Institute of Technology Roorkee, Roorkee 247667, India.
Phys Rev E. 2020 Aug;102(2-1):023111. doi: 10.1103/PhysRevE.102.023111.
The stability of surfactant-laden liquid film flow over a cylindrical rod is examined in creeping flow limit using standard temporal linear stability analysis. The clean film flow configuration (i.e., in absence of surfactant) is well-known to become unstable owing to Rayleigh-Plateau instability of cylindrical liquid interfaces. Previous studies demonstrated that for a static liquid film (i.e., zero basic flow) coating a rod, the presence of interfacial surfactant decrease the growth of Rayleigh-Plateau instability, but is unable to suppress it completely. Further, the presence of interfacial surfactant is known to introduce an additional mode, referred to as surfactant mode in the present work. To the best of our knowledge, the stability of surfactant mode has not been analyzed in the context of cylindrical film flows. Thus, we reexamined the stability of surfactant-laden cylindrical liquid film flow to analyze the stability behavior of the above said two modes when the basic flow is turned on. The present study reveals that the incorporation of basic flow in stability analysis leads to the complete suppression of Rayleigh-Plateau instability due to the presence of interfacial surfactants as compared to the partial suppression obtained for a stationary liquid film. Three nondimensional parameters appear for this problem: Bond number (denoted as Bo) which characterizes the strength of basic flow, Marangoni number (denoted as Ma) which signifies the presence of surfactant, and ratio of rod radius to film thickness denoted as S. In creeping flow limit, the characteristic equation is quadratic with one root belonging to Rayleigh-Plateau mode and the other to surfactant mode. We first carried out an asymptotic analysis to independently capture the eigenvalues corresponding to both the modes in limit of long-wave disturbances. The long-wave results show that the Rayleigh-Plateau instability is completely suppressed on increasing the Marangoni number above a critical value while the surfactant mode always remains stable in low wave-number limit. The continuation of long-wave results to arbitrary wavelength disturbances show that the suppression of Rayleigh-Plateau instability mode still holds, however, the surfactant mode becomes unstable at sufficiently high values of Marangoni number. Further, this surfactant mode instability shifts toward low wave numbers with critical Marangoni number for instability scaling with wave number in a particular fashion. We used this scaling and carried out an asymptotic analysis to capture this instability in low wave-number limit. Depending on S and Bo, we observed the existence of a stable gap in terms of Ma where both the eigen-modes remain stable. Our results indicate that for a given Bond number, the width of stable gap in terms of Ma decreases with decrease in S and the stable gap vanishes when S is sufficiently small. The effect of increasing Bond number (or equivalently, the strength of basic flow) is found to be stabilizing for the film flow configuration.
在蠕动流极限情况下，使用标准的时间线性稳定性分析方法，研究了含表面活性剂的液膜在圆柱棒上的流动稳定性。众所周知，由于圆柱液体界面的瑞利 - 普拉托不稳定性，清洁液膜流动构型（即不存在表面活性剂时）会变得不稳定。先前的研究表明，对于涂覆在棒上的静态液膜（即基本流为零），界面表面活性剂的存在会降低瑞利 - 普拉托不稳定性的增长，但不能完全抑制它。此外，已知界面表面活性剂的存在会引入一种额外的模式，在本工作中称为表面活性剂模式。据我们所知，尚未在圆柱膜流动的背景下分析表面活性剂模式的稳定性。因此，我们重新研究了含表面活性剂的圆柱液膜流动的稳定性，以分析当开启基本流时上述两种模式的稳定性行为。本研究表明，与静止液膜获得的部分抑制相比，在稳定性分析中引入基本流会由于界面表面活性剂的存在而导致瑞利 - 普拉托不稳定性被完全抑制。这个问题出现了三个无量纲参数：表征基本流强度的邦德数（记为Bo）、表示表面活性剂存在的马兰戈尼数（记为Ma）以及棒半径与膜厚度的比值记为S。在蠕动流极限情况下，特征方程是二次的，一个根属于瑞利 - 普拉托模式，另一个属于表面活性剂模式。我们首先进行了渐近分析，以独立捕获长波扰动极限下对应于两种模式的特征值。长波结果表明，当马兰戈尼数增加到临界值以上时，瑞利 - 普拉托不稳定性被完全抑制，而表面活性剂模式在低波数极限下始终保持稳定。将长波结果扩展到任意波长扰动表明，瑞利 - 普拉托不稳定性模式的抑制仍然成立，然而，表面活性剂模式在马兰戈尼数足够高时变得不稳定。此外，这种表面活性剂模式不稳定性以特定方式随波数的临界马兰戈尼数标度向低波数移动。我们使用这种标度并进行了渐近分析，以在低波数极限下捕获这种不稳定性。根据S和Bo，我们观察到在Ma方面存在一个稳定间隙，其中两种本征模式都保持稳定。我们的结果表明，对于给定的值，Ma方面稳定间隙的宽度随着S的减小而减小，并且当S足够小时稳定间隙消失。发现增加邦德数（或等效地，基本流的强度）的效果对液膜流动构型具有稳定作用。
