Surfactant-laden Film Research Articles

Linear stability of a contaminated two-layer films flowing down a slippery inclined plane: Parametric study and physical mechanism

A linear stability analysis of isothermal two-layer surfactant-laden films flowing down a slippery inclined plane is carried out, with a focus on the shear mode instabilities. The intention is to extend our previous study [Bhat and Samanta, “Linear stability for surfactant-laden two-layer film flows down a slippery inclined plane,” Chem. Eng. Sci. 220, 115611 (2020)] in the high Reynolds number regime. The stability analysis is performed under the umbrella of the Orr–Sommerfeld-type boundary value problem. The method of energy budget is employed to figure out the proper physical mechanisms responsible for the growth of the shear modes under the influence of various flow parameters. Numerical results reveal that the flow configuration under consideration is susceptible to two distinct shear modes in the high Reynolds number regime. The shear mode associated with the lower fluid layer is stabilized if the viscosity ratio, wall slip, and interfacial surfactant concentration are increased. However, increasing density ratio has a destabilizing impact on the lower-layer shear mode. On the other hand, the shear mode associated with the upper fluid layer is destabilized if the viscosity ratio and interfacial surfactant concentration are increased. However, the density ratio exhibits a non-monotonic impact, but the wall slip exhibits a stabilizing influence on the upper-layer shear mode. The energy budget analysis predicts that the primary energy source terms for the growth of the shear mode instabilities are base shear stresses, which transfer energy to the disturbance through the Reynolds stress term.

Linear instability of a surfactant-laden shear imposed falling film over an inclined porous bed

The influence of externally imposed shear on a surfactant-laden gravity-driven fluid flow over an inclined porous substrate is studied using the linear perturbation theory. The hydrodynamic instability of the flow system corresponding to infinitesimal disturbances is examined in the framework of the Orr–Sommerfeld (OS) boundary value problem. Furthermore, the generalized OS model is obtained by including the Marangoni stress and external shear on the flow dynamics. The formulated stability problem is solved as an eigenvalue problem by the Chebyshev spectral collocation technique. The analysis encounters the existence of different classes of unstable modes, namely, the surface, surfactant, and shear modes. The surface mode instability occurs in the low range of Reynolds number and is the dominant mode of instability in particular parameter ranges. The imposed shear at the top surface along and opposite to the flow direction induces possible destabilization and stabilization of the flow, respectively. The permeability and porosity of the porous medium have a mixed impact on the surface mode instability. The temporal growth rate of the surface mode enhances for a thicker porous medium. The surface mode of the flow contaminated by an insoluble surfactant is less unstable than that of the clean free surface flow. This is due to the co-existence of the damped surfactant mode together with the unstable surface mode. On the other hand, the shear mode instability is identified at higher Reynolds numbers for a very small inclination angle, and the shear mode propagates faster for stronger imposed shear in the downstream direction. This trend is reversed for the upstream imposed shear. Moreover, the Marangoni effects exhibit the stabilizing influence on the shear mode. Conclusively, the external shear force would be helpful in regulating the instability of the surfactant-laden film flow down a porous medium.

Surfactant-laden film lining an oscillating cap: problem formulation and weakly nonlinear analysis

A surfactant-laden liquid film that lines the inside of an oscillating spherical cap is considered as a model of lung alveoli. Pulmonary surfactant solubility is described by Langmuir adsorption kinetics, modified by incorporating the intrinsic compressibility of the adsorbed monolayer. A novel boundary condition, supported by experimental data and scaling arguments, is applied at the rim. The condition enforces mass conservation of water and surfactant by matching the ‘large-scale’ dynamics of the alveolus to ‘small-scale’ equilibrium over mid-alveolar septa of small but finite thickness. Linear and weakly nonlinear analysis around the conditions in a non-oscillating cap indicates that the occurrence of shearing motion in the liquid is related to the non-zero film thickness over the rim, and shearing velocity at the interface is predicted an order-of-magnitude lower than the velocity of radial oscillation. Marangoni stresses dominate the interfacial dynamics, but capillary stresses affect significantly the interior flow field. In particular, they produce spatial modulations in flow rate, surface concentration of surfactant and wall shear stress, whose length scale varies with $Ca^{-1/3}$ , i.e. is determined by a balance between capillary and viscous forces. Non-zero adsorption kinetics modifies at first order only the amplitude and phase of surface concentration, but affects all other variables at second order. In particular, it sets a steady drift of surfactant away from the alveolus and towards the rim. Finally, an attempt is made to relate the present predictions to physiological findings about air flow and particle deposition inside alveoli, and about shear stress-inflicted damage in diseased lungs.

Manipulation and control of instabilities for surfactant-laden liquid film flowing down an inclined plane using a deformable solid layer

We analyzed the linear stability of surfactant-laden liquid film with a free surface flowing down an inclined plane under the action of gravity when the inclined plane is coated with a deformable solid layer. For a flow past a rigid incline and in the presence of inertia, the gas-liquid (GL) interface is prone to the free surface instability and the presence of surfactant is known to stabilize the free surface mode when the Marangoni number increases above a critical value. The rigid surface configuration also admits a surfactant induced Marangoni mode which remains stable for film flows with a free surface. This Marangoni mode was observed to become unstable for a surfactant covered film flow past a flexible inclined plane in a creeping flow limit when the wall is made sufficiently deformable. In view of these observations, we investigate the following two aspects. First, what is the effect of inertia on Marangoni mode instability induced by wall deformability? Second, and more importantly, whether it is possible to use a deformable solid coating to obtain stable flow for the surfactant covered film for cases when the Marangoni number is below the critical value required for stabilization of free surface instability. In order to explore the first question, we continued the growth rates for the Marangoni mode from the creeping flow limit to finite Reynolds numbers (Re) and observed that while the increase in Reynolds number has a small stabilizing effect on growth rates, the Marangoni mode still remains unstable for finite Reynolds numbers as long as the wall is sufficiently deformable. The Marangoni mode remains the dominant mode for zero and small Reynolds numbers until the GL mode also becomes unstable with the increase in Re. Thus, for a given set of parameters and beyond a critical Re, there is an exchange of dominant mode of instability from the Marangoni to free surface GL mode. With respect to the second important aspect, our results clearly demonstrate that for cases when the stabilizing contribution of surfactant is not sufficient for suppressing GL mode instability, 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 decrease 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. 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.

Interfacial mechanisms for stability of surfactant-laden films.

Thin liquid films are central to everyday life. They are ubiquitous in modern technology (pharmaceuticals, coatings), consumer products (foams, emulsions) and also serve vital biological functions (tear film of the eye, pulmonary surfactants in the lung). A common feature in all these examples is the presence of surface-active molecules at the air-liquid interface. Though they form only molecular-thin layers, these surfactants produce complex surface stresses on the free surface, which have important consequences for the dynamics and stability of the underlying thin liquid film. Here we conduct simple thinning experiments to explore the fundamental mechanisms that allow the surfactant molecules to slow the gravity-driven drainage of the underlying film. We present a simple model that works for both soluble and insoluble surfactant systems in the limit of negligible adsorption-desorption dynamics. We show that surfactants with finite surface rheology influence bulk flow through viscoelastic interfacial stresses, while surfactants with inviscid surfaces achieve stability through opposing surface-tension induced Marangoni flows.

Effect of periodic grooving topography on dynamics of Insoluble surfactant-laden thin film flow

Targeting the flowing of insoluble surfactant-laden film over topography substrate, the lubrication theory is adopted to derive the evolution equations of thin liquid film thickness and interfacial surfactant concentration. The flowing characteristics of the film on topography surfaces, and the influence of topography structure are examined based on the numerical simulation with PDECOL code. The results show that when the thin film of insoluble surfactant flows over periodic grooving topography, the depression appears at the negative step, while the ridge is shaped at the positive step, both of which increase gradually with time going by. Compared with the case of the flat base, the surfactant-laden film spreading speed is enhanced. Increasing the groove depth or reducing the groove steepness leads to the increase of the rupture possibility of the film. Improving the groove width promotes the film flowing. The decrease of the steepness can cause the film to form a ridge feature before entering into the first groove. Gravity has the opposite effects on the up-hilling and down-hilling processes of liquid film flow, which causes the flow stability to deteriorate. Conjoining pressure induced by intermolecular forces can accelerate the film, giving rise to a notable dewetting phenomenon, while disjoining pressure has an adverse effect.

Dynamics of a climbing surfactant-laden film – I: Base-state flow

The dynamics of a surfactant-laden film climbing up an inclined plane is investigated through a two-dimensional (2-D), nonlinear evolution equation for the interface coupled to convective-diffusion equations for the surfactant, derived using lubrication theory. One-dimensional (1-D) solutions, representing the base-state flow, are investigated for constant flux and constant volume configurations; these flows are parameterised by capillarity, gravity, convection–diffusion ratios (represented by Péclét numbers at the surface and bulk), a solubility parameter, sorption kinetics constants, the number of surfactant monomers in a micelle, and the nonlinearity of the surfactant equation of state. In both configurations studied, a front develops spreading up the substrate against the direction of gravity whereby the leading edge of the front follows a power–law as a function of time. The effect of system parameters on the base-state flow is explored through an extensive parametric study, while the stability of the above-mentioned system to spanwise perturbations is the focus of Part II [1].

Dynamics of a climbing surfactant-laden film II: Stability

The linear and nonlinear stability of a spreading film of constant flux and a drop of constant volume, discussed in [1], are examined here. A linear stability analysis (LSA) is carried out to investigate the stability to spanwise perturbations, by linearisation of the two-dimensional (2-D) evolution equations derived in [1] for the film thickness and surfactant concentration fields. The latter correspond to convective–diffusion equations for the surfactant, existing in the form of monomers (present at the free surface and in the bulk) and micelles (present in the bulk). The results of the LSA indicate that the thinning region, present upstream of the leading front in the constant flux case, and the leading ridge in the constant volume case, are unstable to spanwise perturbations. Numerical simulations of the 2-D system of equations demonstrate that the above-mentioned regions exhibit finger formation; the effect of selected system parameters on the fingering patterns is discussed.