FILM FLOWS INSTABILITIES: EFFECT OF SURFACTANT, IMPOSED SHEAR AND POROUS SUBSTRATE

Abstract

Cylindrical liquid film flows present inside or outside of a vertical tube or wire is ubiquitous and garnered a lot of interest in literature due to its applications in industrial, technological and biological processes. These film flows are prone to the disturbances where the amplification of the disturbances over time may distort the interfaces and render it unstable. These instabilities may be desirable for enhanc ing the heat and mass transfer phenomena in the distillation unit, heat exchangers, falling film reactors and also for the formation of liquid drops in microfluidic de vices. There are situations where the perfectly cylindrical interfaces are required such as during the coating of wire. The distortions at the interface creates an un even film thickness which affects the final quality of the product. Therefore, many strategies have been developed to control or manipulate these instabilities such as the use of surface-active agents at the gas-liquid or liquid-liquid interfaces, applica tion of an externally imposed shear stress at the free surface, by replacing the rigid solid wall with the soft or deformable solid substrate, etc. Consequently, the stabil ity of these film flows has been an important area of research both experimentally and theoretically. Cylindrical liquid films present inside or outside of a tube always remain unsta ble due to the surface tension driven capillary instability which is termed as the Rayleigh-Plateau (RP) instability. This RP instability exists even in the absence of inertia. However, the presence of inertia excites an another mode of instability at the free surface that is known as free surface instability, which exists for both planar and cylindrical film flows. Furthermore, for the stationary cylindrical film, the effect of the presence of an insoluble surfactant was found to be stabilising. It decreases the growth rate of the Rayleigh-Plateau instability but could not eliminate it completely (Camapana et al. 2004; Carroll & Lucassen 1974; Cassidy et al. 1999; Halpern & Grotberg 1993). However, in the presence of the basic flow, the RP instability was completely eliminated above a critical value of the Marangoni number (Nair & Sharma 2020). They also investigated that the surfactant mode remains stable in the low wavenumber limit and become unstable for a range of arbitrary wavenumber ii for high Marangoni number. Similar observation of complete suppression of RP instability was made by (Ogrosky 2021) for the liquid film present inside of a tube but did not mention anything about finite wavenumber instability as investigated by (Nair & Sharma 2020). Apart from the effect of presence of surfactant at the free surface (passive gas), number of studies are available where the surfactant loaded gas-liquid interface is subjected to an externally applied shear stress for planar and cylindrical film flows. The presence of an imposed shear was found to be destabilis ing or stabilising (downstream or upstream direction) for the gas-liquid mode. The surfactant mode was found to be unstable solely due to the presence of basic shear (Bhat & Samanta 2019; Wei 2005b; Zhou et al. 2014). The existing literature has explored the effect of the surfactant for the liquid film present inside of a tube using lubrication or the long wave approximation. There fore, in one of the objective of this thesis: (1) We investigated the effect of the surfactant on the liquid film present inside of a vertical tube in presence of the base f low using the long-wave and numerical shooting method. Furthermore, to the best of our knowledge no study has examined the effect of the imposed shear stress at the gas-liquid for the liquid film flowing outside of a vertical fibre. Hence, the second objective is: (2) To examine the effect of an imposed shear stress on the stability of gravity-driven surfactant doped Newtonian liquid film flowing outside of a verti cal rod or fibre. Both the studies were conducted in the limit of vanishingly small Reynolds number (Re “ 0). We carried out the linear stability analysis using the long-wave asymptotic analytical method and further extended our results to finite or arbitrary wavenumbers using the numerical shooting method. In Chapter 2 of this thesis, the stability of the gravity-driven and surfactant-doped Newtonian liquid film present inside of a vertical tube was examined using the lin ear stability analysis in the creeping flow limit. This flow configuration exhibits two normal modes of instability. (i) Surface tension-driven Rayleigh-Plateau mode. (ii) Surfactant mode, which arises due to the gradients of surface tension in presence of surfactant. It has been observed that in the presence of the base flow (character ized by Bond number, Bo), the Rayleigh-Plateau instability can be eliminated com pletely, when the Marangoni number increases above a critical value. In contrast, the surfactant mode which was stable for stationary film now remains unstable for any non-zero value of the Marangoni number, when the base flow was included into iii the stability analysis. This surfactant mode instability was observed to endure for f inite or high wavenumber depending on the liquid film thickness and Bond number. It was noted that the maximum growth rate for the surfactant mode become higher than that of the RP mode for a clean film (Ma=0). Therefore, for moderate and high Bond number, the long-wave nature of instability observed for static film shifts to f inite or short wave surfactant mode instability present into the system. Therefore, the presence of surfactant has an overall destabilizing effect on film flows and the long-wave model is not sufficient to determine the overall stability characteristics of such film flows. In chapter 3 of this thesis, the linear stability of gravity-driven flow of a surfactant laden Newtonian liquid film over a rod is examined in creeping flow limit, in the presence of an externally applied shear stress at gas-liquid (GL) interface. The im posed shear can be applied either in a direction assisting the flow (positive shear stress) or opposite to the direction of the gravity-driven flow (negative shear stress). The two instability modes exist for this flow configuration: (i) Rayleigh-Plateau (RP) mode, and (ii) surfactant mode. Earlier studies revealed that, in absence of im posed shear stress, the surfactant can completely suppress the RP instability above a critical value of Marangoni number. With further increase in Marangoni number to sufficiently high values, the surfactant mode becomes unstable. Hence, there ex ists a window in terms of Marangoni number, where the liquid film remains stable. However, in the presence of imposed shear stress at the gas-liquid interface, the dy namics of the liquid film changes significantly and alters the stability characteristics of the film flow. We first examined the problem using the long-wave asymptotic analysis, it was demonstrated that the positive stress has a stabilizing effect on the RP mode in addition to the stabilizing contribution due to the presence of the sur factant and destabilizing impact on the surfactant mode. Therefore, for any physical situation where the surfactant is present in the sufficiently low concentration (small values of Ma), the application of the suitable magnitude of the positive stress can be used to completely suppress the RP instability. The surfactant mode which remains stable in the absence of an external shear, may become unstable, when the magni tude of the positive stress increases above a threshold value (τ ą τc) and remains stable for τ ă τc. Thus, the effect of increasing the positive shear was found to be destabilizing for the surfactant mode, in contrast to the stabilizing effect for the RP iv mode. It is important to note the the critical value of stress above which surfactant mode destabilises is lower in magnitude to the critical stress required for the sta bilisation of the RP mode. Hence, it is not possible to eliminate the RP instability without triggering the surfactant mode instability. Below this critical value of the stress, long-wave RP instability will be present in to the system whereas above τc, the surfactant mode instability occupies the whole region from low to finite wave perturbations for any value of Marangoni number. Therefore, the positive stress is found to have an overall destabilising effect on the system. However, the extension of the results from low to finite or high wavenumber shows the existence of a stable gap in terms of Marangoni number for small values of the imposed shear stress. This stable gap vanishes, when the shear stress increases beyond the critical value required for the destabilisation of the surfactant mode. For the negative imposed shear stress, the low wavenumber results predicts that for the effect of negative shear stress is destabilizing (stabilizing) for RP (surfac tant) mode, when the magnitude of applied stress is lower than a threshold value (τ ă τc “ 4 d). However, the effect of the imposed shear reverses for both modes when τ ą τc. Note that the critical value of the imposed shear remains same for both the RP or surfactant mode unlike in case of positive shear, where critical stress (τc) was different for both the modes. The analysis concerned with the results of negative shear stress shows that with increase in the magnitude of imposed shear, the finite wavenumber perturbations are destabilised and occupies the whole region in Marangoni versus wavenumber plane. Therefore, it results into the change in stability behavior from long-wave (for stress free interface) to finite wavelength dominated instability. Hence, it is not possible to obtain stable flows when magni tude of shear stress is above a certain value in a similar manner as shown for the positive applied stress. In Chapter 4 of this thesis, we conducted the linear stability analysis of gravity driven Newtonian liquid film flow over a porous inclined plane. The flow in the f luid andporouslayer is modelled with Navier Stokes and volume-averaged Navier Stokes (VANS) equation respectively. In this study, three different modes of insta bility was found: surface mode, shear mode and the porous mode. The surface mode is the long-wave instability mode which triggers due to the deformation of the free surface above a critical value of Reynolds number. In regard of the rigid v plane, it has been observed that for the low angle of inclination, the mode which be come unstable at finite wavenumber was termed as shear or hard mode. However, the porous mode is observed only in the presence of permeable porous substrate instead of rigid plane. The porous mode instability occurs as a results of perme ation of the fluid flow perturbations inside the porous region. We re-examined the research conducted by Camporeale et al. (2013) and made corrections in the pertur bation equations that was initially done by Camporeale et al. (2013). They missed many terms in the boundary condition at the fluid-solid interface (normal stress bal ance) and added an extra term in fourth order ODE for porous layer. It has been observed that the corrected equations significantly change the stability features for the range of parameters typically for large δ. Camporeale et al. (2013) observed that the high wave number porous mode exists with surface and shear for all chosen value of angle of inclination θ. However, such porous mode instability remains ab sent in our analysis. We observed that porous mode do exists but in the range low to finite wavenumber for very small angle 0.03˝ at δ “ 0.02. We also studied the effect of different parameter (d,δ,α,Re,θ) on the flow stability behaviour of the system.