Effect of imposed shear stress on the stability of surfactant-laden liquid film flow over a rod

Abstract

The linear stability of gravity-driven flow of surfactant-laden liquid film over a rod is examined in creeping flow limit in presence of an applied shear stress at gas–liquid (GL) interface. This flow system admits two instability modes: (i) a surface-tension driven Rayleigh-Plateau (RP) mode, and (ii) a surface-tension gradient driven surfactant mode. In absence of imposed shear stress, the surfactant completely suppresses the RP instability when Marangoni number (Ma), increases above a critical value. On further increase of Ma to high enough values, the surfactant mode becomes unstable, and as a result, a gap in terms of Ma exists where the film flow remains stable. The present work shows that these stability characteristics are dramatically modified in presence of imposed shear stress. When shear stress acts in a direction to assist gravity-driven flow (i.e. positive stress), it has a stabilizing effect on RP mode in addition to the stabilizing effect of surfactant. In contrast, the positive applied shear destabilizes the surfactant mode when shear stress exceeds above a critical value. Below this critical stress value, it is still possible to obtain stable film flow for a range of Marangoni number values. However, above this critical value, the shear stress induced surfactant mode instability engulf whole region from low wave number to finite wave number perturbations for any value of Ma. For negative values of imposed shear (i.e. when shear stress acts opposite to gravity-driven flow direction), the effect of shear stress on RP mode is destabilizing (stabilizing) when the magnitude of applied stress is lower (higher) than a threshold value. On the other hand, the effect of applied negative shear is found to be exactly opposite for surfactant mode. The overall analysis of results for negative shear shows that it is not possible to obtain stable flows when magnitude of shear stress is above a certain value in a similar manner as shown for positive applied stress. It was also observed that the finite wave number perturbations become important for a wide range of parameters for shear stress induced destabilization of film flow configuration. This was not the case in absence of applied stress in which case the dominant perturbations were always long wavelength perturbations.