MANIPULATION AND CONTROL OF FILM FLOW INSTABILITIES USING A SOFT, DEFORMABLE SOLID COATING

Abstract

Film flows occur in a variety of natural, engineering and physical settings, and are vulnerable to interfacial instabilities. Engineering examples where film flow is important include coating and coextrusion process, distillation units, condensers and heat exchangers, microfluidics, pulmonary flows, etc. It is often required to control and manipulate (suppress or enhance) the interfacial instabilities in a process or as per the desired product specifications. Shankar and coworkers (summarized by Gaurav & Shankar (2015)) have extensively examined a number of different configurations and proposed the idea of using a soft solid layer as the passive and efficient method for the purpose of instability control and manipulation. In reference to film flows past flexible surfaces, Shankar and coworkers examined only clean film flows, i.e. devoid of any surfactant/impurity present at gas-liquid interface. Further, the effect of presence of an active gas layer which could exert a shear stress at the free surface remains unexplored. In this thesis, we examine the above two aspects for the gravity-driven flow of a (i) viscoelastic liquid film with the free surface covered with an insoluble surfactant, and (ii) Newtonian liquid film with the gasliquid interface exposed to a constant shear stress. Surfactant-contaminated film flows are frequently encountered in biological phenomena and various technological applications. The pulmonary surfactant helps to prevent lung airway closure by suppressing surface-tension-driven instabilities (Halpern & Grotberg 1993). Surfactants are widely used in industrial coating operations in order to produce coatings of specific (or desired) properties, and to make the process operable over a wider range of operating conditions (Quere et al. 1997). Gravity-driven flow of a Newtonian film with a surfactant monolayer covering its free surface exhibits two normal modes in low Reynolds number regime: (i) the usual Yih (1963) type fluid-fluid interfacial mode, also known as gas-liquid (GL) or free-surface mode, and (ii) the Marangoni mode originating due to surface tension gradients developed by the alteration in surfactant concentration at gas-liquid interface. For the case of falling film down an inclined plane, some finite inertia is always required to destabilize the Yih’s mode (Yih 1963). It is well known that the surfactant has a stabilizing effect on this GL mode. On the other hand, Marangoni mode remains stable both at zero and finite inertia as long as the free surface is exposed to a passive atmosphere and the GL interface remains stress-free (Blyth & Pozrikidis 2004a). Therefore, both the normal modes remain suppressed for the creeping gravity-driven flow of surfactant-doped Newtonian film down an inclined plane. The possibility of using a soft solid coating to destabilize this stable film flow configuration past a rigid incline has been explored recently by Tomar et al. (2017). They demonstrated that the Marangoni mode can be excited even in the absence of inertia when the inclined plane is coated with a soft solid layer. With the help of low wavenumber asymptotic analysis, it was shown that for a given solid thickness, the Marangoni mode becomes unstable as soon as the solid layer becomes sufficiently soft such that the wall deformability parameter, defined as G = μV /EsR, exceeds a critical value. Here V, Es, R and μ are the free surface velocity, shear modulus of solid layer, film thickness and viscosity, respectively. It was reported to be the first ever case when the free surface flow exhibits a Marangoni instability in the absence of an interfacial shear at the gas-liquid interface (or fluid-fluid interface in general). It is worth finding out if this soft-wall-driven destabilization of Marangoni mode is limited to Newtonian film or is a more general phenomenon which remains independent of fluid rheological characteristics. In Chapter 2 of this thesis, we have carried out the linear stability analysis of a freely falling surfactant-covered viscoelastic liquid film past an inclined plane lined with a soft/deformable solid layer in creeping flow limit. Unlike Newtonian fluid, the viscoelastic film flow down a rigid inclined plane undergoes an elasticity-driven instability even at zero Reynolds number (Shaqfeh et al. 1989). The free surface in such a film flow becomes unstable as soon as the Weissenberg number (W ) exceeds a threshold value (Wgl). Weissenberg number is a non-dimensional parameter characterizing fluid relaxation time or elasticity and is defined as W = τRV /R, where τR,V and R are the relaxation time, free surface velocity and film thickness, respectively. Hence, the dynamics of undertaken system is qualitatively different from the Newtonian case as the Yih type fluid-fluid interfacial mode remains suppressed in the latter in an inertialess approximation and the stability is primarily dominated by the Marangoni mode. Our long-wave analysis indicates that the presence of surfactant is stabilizing for the free-surface mode and this contribution remains identical to that for a Newtonian film (Blyth & Pozrikidis 2004a; Tomar et al. 2017). Hence, the presence of surfactant increases the critical Weissenberg number required to destabilize the gas-liquid mode (Wgl) as compared to an uncontaminated viscoelastic liquid film flow down a rigid/flexible inclined plane. More importantly, our low wavenumber analysis shows that the wall deformability has opposite effects on the free surface and Marangoni modes: it is stabilizing to the free-surface mode but has destabilizing contribution for Marangoni mode. This destabilizing role of soft wall to the Marangoni mode is identical to Tomar et al. (2017), and hence, is independent of fluid rheology. Therefore, the stability of configuration considered in Chapter 2 is driven by the competition between GL mode stabilization and Marangoni mode destabilization. It is important to point out that both the surfactant and soft solid contributions are stabilizing for free-surface mode, or in other words, the soft solid can be used to provide additional stabilization when the surfactant contribution is insufficient to suppress the GL mode instability. Our numerical analysis for arbitrary wavenumbers reveals that both the long-wave GL and Marangoni mode instabilities extend up to finite wavenumbers but vanish for short wave perturbations. For the parametric space where the free surface remains stable in rigid limit (W < Wgl), the possibility of Marangoni mode destabilization has been demonstrated on replacing the rigid plane with a soft wall. For the parameter range where the rigid configuration already exhibits a gas-liquid interfacial instability (W > Wgl), the solid elasticity reduces the GL mode growth-rates and suppresses the free-surface instability as soon as the wall deformability parameter G exceeds a threshold value Ggl. But at the same time, the soft solid may excite the Marangoni mode which is otherwise stable for the flow down a rigid incline. Let GM be the critical value of deformability parameter above which the Marangoni mode becomes unstable for a given solid thickness. We show that there exists a sufficiently wide range of deformability parameter G (or equivalently, the shear modulus Es) such that the free-surface mode is suppressed without exciting Marangoni or any other mode of instability (Ggl < G < GM). For the parameter sets where this stabilization is not possible, we show that the Marangoni mode can be selectively destabilized. The wall thickness acts as a deciding parameter to obtain either a stability window (where all the modes remain stable for the complete band of perturbation wavelengths) or the selective destabilization of Marangoni mode. This surfactant-laden film flow also exhibits a short-wave liquid-solid (LS) interfacial mode instability at higher values of deformability parameter G ≥ O(1). This corresponds to very low values of shear moduli (Es ≤ 10 Pa) which are unachievable, and therefore, this LS instability remains practically irrelevant for the configuration undertaken in Chapter 2. Liquid films sheared by a gas stream can be witnessed in a number of engineering applications covering absorption, evaporation, condensation, cooling towers, desalination plants, distillation columns, and in cooling of miniaturized electronic equipment, etc. The hydrodynamic stability of system involving liquid-gas interactions plays an important role in predicting the flow behavior in both the phases, as well as the phenomena taking place at the interface. For the gravity-driven flow of a Newtonian liquid film down an inclined plane when the liquid free surface is exposed to a constant shear stress (exerted by co- or counter-current gas flow), the low wavenumber asymptotic analysis carried out by Smith (1990) shows that the applied shear stress has destabilizing contribution on the GL mode when it acts in the downstream direction (i.e. along the gravity component parallel to plane). Hence, a shear stress assisting gravity reduces the critical Reynolds number above which the Yih mode becomes unstable and prepones the onset of long-wave instability. A soft solid layer is known to have stabilizing role on the Yih mode under the assumption of passive gas atmosphere, and the possibility of complete instability suppression has been shown for the planar falling film flow (Gaurav & Shankar 2015). It is pertinent to ask whether this stabilization is valid when the liquid film is sheared by a gas stream or not? In order to explore this aspect, and in general, the effect of presence of both imposed shear and deformable wall on the stability of film flow, we undertake (in Chapter 3) the gravity-driven flow of Newtonian film down an inclined plane coated with a soft solid layer when the free surface is exposed to a constant shear stress. Our long-wave asymptotic analysis reveals that the Yih mode, which becomes unstable only at finite inertia for the case of passive gas environment, can be destabilized even in the absence of inertia when the applied shear stress acts in the upstream direction (opposite to the gravity component parallel to plane) and exceeds a threshold value. This low wavenumber instability in creeping flow limit is attributed to the coupling between imposed shear stress and deformable wall, and it vanishes if either of the contributions is absent. Our numerical results for arbitrary wavelength perturbations show that this long-wave GL instability continues to finite wavenumbers and dominates the stability of system at zero Reynolds number (Re = 0). To the best of our knowledge, this is the first instance when the soft solid layer has a destabilizing contribution on the gas-liquid interfacial mode in an inertialess flow. However, in the presence of even a very small inertia (Re 􀀋 1), we observed that the fluid-solid composite system considered in Chapter 3 exhibits a class of unstable modes. These modes are found to be similar to the unstable modes reported by Gaurav & Shankar (2009, 2010b) for fluid flow in deformable tube and channel, and therefore, these are identified as inertial liquid-solid (ILS) modes. These modes belong to a category of shear waves which exist for the film flow past flexible surfaces, but remain suppressed in the absence of inertia. Out of all the ILS modes appearing for a particular set of parameters, we have restricted our attention only to the most unstable (or critical) ILS mode. The ILS modes are found to become unstable at much lower values of wall deformability parameter G as compared to the long-wave GL mode. Hence, though we report the first case of free-surface mode destabilization due to the presence of soft wall, this instability becomes insignificant as ILS mode becomes the critical mode of instability. Also, it becomes essential to point out that the approximation of zero Reynolds number (Re = 0) cannot be used to recover creeping flow limit (Re → 0) for this fluid-solid coupled flow, i.e. results obtained by setting Re identically equal to zero could be misleading for this particular configuration. ILS modes continue to remain critical modes at non-zero Reynolds number (Re 􀀎= 0) as well and dominate the stability of sheared film flow undertaken in Chapter 3 for moderate angles of inclination. The most unstable ILS mode appears at G ∼ O(10−3), and this corresponds to flexible solid with shear modulus Es ∼ 104 Pa. Soft walls with shear moduli of this order can be fabricated and used to test the predictions reported in this thesis. In contrast to the moderate inclination angles, we have shown for a vertically falling film that the low wavenumber GL instability could indeed become critical when the inclined plane is coated with thin layer of deformable wall. We have also explored the possibility of instability suppression due to wall elasticity for the parametric space when the sheared film flow down a rigid inclined plane is unstable.