STABILITY OF SURFACTANT-LADEN LIQUID FILM FLOW DOWN A FLEXIBLE INCLINED PLANE

Abstract

Liquid film flows with free-surface occurs in a wide variety of technological appli- cations such as coating flows, falling film reactors, absorption column etc. These film flows fall prey to free-surface instability due to jump in fluid properties (like viscosity, elasticity etc.) across the gas-liquid interface. Such interfacial instabil- ities are desirable in some applications while undesirable in others. For example, instabilities are detrimental to product quality in coating flows as they result in non-uniform film thickness. On the other hand these instabilities are useful for heat and mass transfer applications, or for patterning applications (Bandyopadhyay et al. 2008; Mukherjee & Sharma 2015). Thus, it is frequently required to control and manipulate instabilities observed in liquid film flows with free surface. In the present thesis, we explore ways to manipulate and control the interfacial instabili- ties observed for the single liquid film with free surface flowing down an inclined plane. For single liquid film flow down an inclined plane, only a gas-liquid interface is present. Further, these film flows are often accompanied by surface active agents or surfactants in several technological and physiological flow systems. It is well known that the presence of surfactant have significant effect on the stability of dif- ferent interfaces present in a particular flow configuration. Thus, we also consider the presence of surfactant at gas-liquid interface in the present thesis. In view of the above discussion, we explore the use of a passive deformable solid coating as a means to manipulate and control the interfacial instabilities for single liquid film flow down an inclined plane when the gas-liquid interface is contaminated with a mono-layer of insoluble surfactant. We investigate the linear stability of a surfactant-laden single liquid film with free surface flowing down an inclined plane under the action of gravity when the inclined plane is coated with a deformable solid layer. We first examine the stabil- ity of flow configuration in creeping flow (or Re = 0) limit. In this zero Reynolds number limit, the surfactant covered liquid film flowing down a rigid inclined wall admits two normal eigenmodes: (i) a gas-liquid (GL) interfacial or free surface mode, and (ii) a surfactant-induced Marangoni mode. The GL free surface mode ii is the usual Yih (1963, 1967) type mode which is present because of jump in vis- cosity across the gas-liquid interface. This GL interfacial mode remains stable in creeping flow limit. The Marangoni mode arise because of the convective motion of surfactant along the gas-liquid interface and this mode also remains stable for liquid film flow down a rigid inclined wall. We examine how the stability characteristics change when the surfactant-laden liquid film flow occurs down a flexible inclined wall instead of a rigid inclined wall. The effect of presence of deformable wall or soft solid coating on GL mode has already been discussed by Shankar and cowork- ers (Gaurav & Shankar 2007; Sahu & Shankar 2016; Shankar & Sahu 2006). The primary aim of this study is to explore the role of wall deformability on the sta- bility characteristics of Marangoni mode. Two parameters, namely, shear modulus and thickness of deformable solid layer appear in presence of a deformable solid coating in addition to the parameters which were present for flow down a rigid in- clined wall. We performed a long-wave asymptotic analysis and observed that the Marangoni mode becomes unstable in presence of deformable solid coating. This long wave instability was further continued to finite wavelength perturbations us- ing a numerical shooting procedure. Specifically, we have shown that for a given solid thickness, the Marangoni mode becomes unstable when the shear modulus of solid layer decreases below a critical value (i.e. the solid layer becomes suffi- ciently soft). The effect of increasing solid thickness is found to be destabilizing. The liquid-solid (LS) interfacial mode also becomes unstable at high wave num- bers below a threshold value of shear modulus, however, this value is much smaller than that required to trigger Marangoni mode instability. This implies that as the solid coating becomes more and more deformable, the Marangoni mode becomes unstable first followed by the LS interfacial mode. The GL mode was always found to be stable in creeping flow limit. Further, our long-wave analysis shows that the solid deformability has an additional stabilizing effect on GL mode. We defined a non-dimensional solid deformability parameter as G=mV=EsR, where m is the vis- cosity of the liquid, V is the characteristic velocity, R is the liquid layer thickness, and Es is the shear modulus of solid layer. Note that higher value of G implies lower Es value, and hence, more soft (deformable) solid layer. We plotted neutral stability diagram in terms of this non-dimensional parameter G (or equivalently shear mod- ulus) vs. wavenumber for all the unstable modes, and these diagrams clearly show iii that the Marangoni mode is the dominant mode of instability in creeping flow limit. This observation related to destabilization of Marangoni mode due to the presence of deformable wall for free surface liquid film flow is important because we believe that this is the first example of the case where the instability of the Marangoni mode is observed when the fluid-fluid interface (here, GL interface) remains stress-free in the basic state. We further investigated the linear stability of the surfactant-loaded liquid film flowing down a flexible inclined plane in presence of inertia (Re 6= 0). For non-zero Reynolds number, it is well known that the GL interfacial mode becomes unsta- ble for a clean liquid film flowing down a rigid inclined plane when Re increases above a critical value. The presence of surfactant at GL interface is known to sup- press this GL mode instability when the Marangoni number (Ma) increases above a threshold value (Blyth & Pozrikidis 2004a). The Marangoni number is defined as Ma = EG0=s0, where E refers to surface elasticity, G0 denotes the surfactant con- centration at GL interface in steady base state configuration, and s0 represents the corresponding unperturbed GL interfacial tension. Recall that we have shown that when the rigid wall is replaced by a deformable wall, the Marangoni mode becomes unstable in creeping flow limit. We first continue this Marangoni mode instability to finite Reynolds number, and observed that this Marangoni instability persists at non-zero values of Reynolds number. The GL interfacial mode can also become un- stable as Reynolds number increases above the critical value for (clean or surfactant- covered) film flow down an inclined (flexible or rigid) plane. We observed that as Reynolds number is increased above zero, the Marangoni mode remains the domi- nant mode of instability for small Reynolds number until the GL mode also becomes unstable with increase in Reynolds number. Once, the GL mode becomes unstable, it dominates the stability of falling film configuration. Thus, there is an exchange in the dominant mode of instability with the increase in Reynolds number. Previous works have also demonstrated the potential of using a deformable solid coating in suppressing the interfacial instabilities for a wide variety of configurations (Gau- rav & Shankar 2015; Shankar 2015). The presence of surfactant also suppresses GL interfacial instability when the Ma increases above the critical value. We ex- plore whether it is possible to use a deformable solid coating to achieve stable flow configuration for surfactant-loaded film when the stabilizing contribution from sur- iv factant is not sufficient to suppress the GL instability (i.e. when Ma remains below the threshold value to stabilize the GL mode for a given Reynolds number). This question becomes important in view of the recently observed Marangoni mode in- stability solely induced due to the presence of the deformable wall (Tomar et al: 2017). We show in the present thesis that for such cases as well, a deformable solid coating could be employed to suppress free surface instability without triggering Marangoni or liquid-solid interfacial modes. Specifically, we have shown that for a given solid thickness, as the shear modulus of the solid layer decreases (i.e. the solid becomes more deformable) the GL mode instability is suppressed. With further de- crease in shear modulus, the Marangoni and liquid-solid interfacial modes become unstable. Thus, there exists a stability window in terms of shear modulus where the surfactant-laden film flow remains stable even when the Marangoni number is below the critical value required for free surface instability suppression. Based on our numerical results (primarily G vs. wavenumber neutral stability curves), we estimated that typical values of shear modulus of elasticity for the deformable solid layer is of the order of 104 Pa to obtain stable flow of surfactant-laden liquid film down a flexible inclined plane. Further, when the Marangoni number is greater than the critical value so that the GL mode remains stable in the rigid limit or with the deformable wall, the increase in wall deformability or solid thickness triggers Marangoni mode instability and thus, renders a stable flow configuration into an unstable one. Thus, we show that the soft solid layer can be used to manipulate and control the stability of surfactant-laden film flows.